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UNIVERSAL:AGUIDETOTHECOSMOS

COVERTITLEPAGECOPYRIGHTDEDICATIONACKNOWLEDGEMENTS

1.THESTORYOFTHEUNIVERSE2.HOWOLDARETHINGS3.WEIGHINGTHEEARTH4.THEDISTANCETOTHESTARS5.EINSTEIN’STHEORYOFGRAVITY6.THEBIGBANG7.WEIGHINGTHEUNIVERSE8.WHATHAPPENEDBEFORETHEBIGBANG?9.OURPLACE

APPENDIXEVOLUTIONOFTHEUNIVERSEILLUSTRATIONCREDITSINDEX

ForBrian’sdad,David

ACKNOWLEDGEMENTS

Fortheirspecifichelpwithvariouspartsofthebook,we’dliketothankRichardBattye,SarahBridle,MikeBowman,Bill&PaulineChamberlain,EdCopeland,Mrinal Dasgupta, Neal Jackson, Scott Kay, Kevin Kilburn, Peter Millington,Tim O’Brien, Michael Oates, Subir Sarkar, Bob Seymour and Martin Yates.ParticularthanksgotoMikeSeymour,withwhomwehavehadmanyenjoyableandhelpfuldiscussions.

SpecialthanksarealsoduetotheteamatPenguinandespeciallyTomPenn,oureditor,andTomEtherington,whoproducedthefigures.

ThanksalsotoDianeandSuefortheircontinuedguidanceandsupport.

Finally, we would like to thank Peter Saville for his influence on the book,whichextendsbeyondthebeautifulcoverdesign.

Thisbookhasbeenalongtimeinthemakingandwearedeeplygratefulforthesupportandencouragementofourfamilies.

1.THESTORYOFTHEUNIVERSE

WedaretoimagineatimewhentheentireobservableUniversewascompressedinto a region of space smaller than an atom. And we can do more than justimagine.Wecancompute.Wecancomputehowhundredsofbillionsofgalaxiesemerged from a single subatomic-sized patch of space dwarfed by a mote ofdust, and there is precise agreement between those computations and ourobservations of the cosmos. It seems that human beings can know about theoriginsoftheUniverse.

Cosmologyissurelythemostaudaciousbranchofscience.TheideathattheMilkyWay,ourhomegalaxyof400billionstars,wasoncecompressed intoaregion so vanishingly small is outlandish enough. That the entire visiblecongregationofbillionsofgalaxiesonceoccupiedsuchasubatomic-sizedpatchsounds like insanity. But to many cosmologists this claim isn’t even mildlycontroversial.

This isnotabookaboutknowledgehandeddownfromonhigh.Morethananything,itisabouthowwe–allofus–cangainanunderstandingoftheUniversebydoingscience.Youmightthinkthatit’simpossiblefortheaveragepersontoexplore the Universe in much detail: don’t we need access to Hubble SpaceTelescopes andLargeHadronColliders?The answer is no, not always. SomefundamentalquestionsaboutourEarth,ourSun,oursolarsystem,andeventheUniverse beyond, are answerable from your back garden. How old are they?Howbigarethey?Howmuchdotheyweigh?Wewillanswerthesequestionsbydoing science.We will observe, measure and think. One of the great joys ofscienceistounderstandsomethingforthefirsttime–toreallyunderstand,whichisverydifferentfrom,andfarmoresatisfyingthan,knowingthefacts.WewillmakeourownmeasurementsofthemotionofNeptune,followinthefootstepsofthepioneeringcosmologistEdwinHubbleindiscoveringthatourUniverseisexpanding, andmake an apparently trivial observation standing on a beach insouthWales.

Asthebookunfolds,ourgazewillinevitablyturnoutwardstowardsthestar-filled galaxies. To understand them, we will rely on observations andmeasurementsthatwecannotmakeourselves.Butwecanimaginebeingapart

oftheteamsofastronomerswhocan.Howfarawayarethestarsandgalaxies?Howbig is theUniverse?What is itmade of?Whatwas it like in the distantpast?Theanswerstothesequestionswillgenerateacascadeofnewideas,and,beforethebookisfinished,wewillbeequippedtoenquireabouttheoriginsoftheUniverse.Scienceisanenchantingjourneyofexploration.Itisanexciting,rewarding process and one that leaves scientistswith a feeling of being betterconnectedtotheworldaroundthem.Itleavesasenseofaweandhumilitytoo;afeeling that the world is beautiful beyond imagination and that we are veryprivilegedtobeheretowitnessit.

Beforewebeginourjourney,however,wewillallowourselvesaglimpseofthedestination.WhatfollowsnextisthestoryofhowourUniverseevolvedfromasubatomicpatchofspaceintotheoceansofgalaxiesweseetoday.Perhaps,bytheendofthebook,youwilljudgethatitmightjustbetrue.

ConsidertheUniversebeforetheBigBang.By‘BigBang’wemeanatime13.8billionyears agowhenall thematerial thatmakesup theobservableUniversecame into being in the form of a hot, dense plasma of elementary particles.Before this time, the Universe was very different. It was relatively cold anddevoidofparticles, and space itselfwas expandingvery rapidly,whichmeansthatanyparticles itmayhavecontainedweremovingawayfromeachotherathigh speeds.Theaveragedistancebetweenparticleswasdoublingevery10−37seconds.This isa staggering,almost incomprehensible, rateofexpansion: twoparticlesonecentimetreapartatoneinstantwereseparatedby10billionmetresonly4×10−36secondslater;morethantwentytimesthedistancefromtheEarthtotheMoon.WedonotknowforhowlongtheUniverseexpandedlikethis,butit continued for at least 10−35 seconds. This pre-Big Bang phase of rapidexpansionisknowntocosmologistsastheepochofinflation.

Letusfocusonatinyspeckofspaceabilliontimessmallerthanaproton,theatomicnucleusofahydrogenatom.Atfirstglance,thereisnothingparticularlyspecial about this tiny patch. It is one small part of a much larger, inflatingUniverse,and it looksmuch thesameasall theotherpatches that surround it.Theonlyreasonthisparticularpatchdeservesourattentionisthatitisdestined,over 13.8 billion years, to grow into our observable Universe: the region ofspace containing all the galaxies and quasars and black holes and stars andplanetsandnebulaevisiblefromEarthtoday.TheUniverseisfarbiggerthantheobservableUniverse,butwecan’tseeitallbecauselightcanonlytravelafinitedistancein13.8billionyears.

Before the Big Bang, the Universe was filled with something called the‘inflaton’field;amaterialthing,likeastilloceanfillingspace.ThegravitationaleffectoftheenergystoredintheinflatonfieldcausedtheUniverse’sexponentialexpansion, and this is the origin of its name: it is the field responsible forinflatingtheUniverse.Onthewhole,theinflatonfieldremainedundisturbedastheUniverseexpanded,butitwasnotperfectlyuniform.Ithadtinyripplesinit,asrequiredbythelawsofquantumphysics.

BythetimeourobservableUniversewasthesizeofamelon, theperiodofinflation was drawing to a close as the energy driving it drained away. Thisenergywasnotlost,however;itwasconvertedintoaseaofelementaryparticles.In an instant, a cold, empty Universe became a hot, dense one. This is howinflation ended and theBigBang began, delivering aUniverse filledwith theparticlesthatweredestinedtoevolveintogalaxies,stars,planetsandpeople.

WedonotcurrentlyknowwhichparticleswerepresentatthemomentoftheBigBang,butwedoknowthat theheaviestparticlessoondecayedtoproducethe lighter ones we know today: electrons, quarks, gluons, photons, neutrinosanddarkmatter.1WecanalsobeconfidentabouttheparticlesthatpopulatedtheUniversewhenitwasaroundatrillionthofasecondold,becauseweareabletore-create these conditions onEarth, at theLargeHadronCollider.2This is thetimewhenemptyspacebecamefilledwiththeHiggsfield,whichcausedsomeoftheelementaryparticlestoacquiremass.3Theweaknuclearforce,responsiblefor the reactions that allow the stars to shine, became distinct from theelectromagneticforceatthistime.

AmillionthofasecondaftertheBigBang,whenthehotplasmahadcooledto 10 trillion degrees celsius, the quarks and gluons formed into protons andneutrons, the building blocks of atomic nuclei. Although this primordialUniverse consisted of an almost uniform soup of particles, there were slightvariationsinthedensityofthesoup–animprintofthequantum-inducedripplesin the inflaton field. These variationswere the seeds fromwhich the galaxieswouldlatergrow.

OneminuteaftertheBigBang,ataroundabilliondegrees,theUniversewascoolenoughforsomeoftheprotonsandneutronstoclustertogetherinpairstoform deuterium nuclei.Most of these thenwent on to partnerwith additionalprotonsandneutrons to formheliumand, in tinyamounts, lithium.This is theepochofnucleosynthesis.

Forthenext100,000yearsorso,littlehappenedastheUniversecontinuedtoexpand and cool. Towards the end of this time, however, the dark matter

graduallybegan toclumparound the seeds sownby the ripples in the inflatonfield.Regions of theUniversewhere therewas a slight excess of darkmattergrewdenser, as theirgravitypulled inyetmorematter from the surroundings.Thisisthestartofthegravitationalclumpingofmatterthatwilleventuallyleadto the formation of galaxies. Meanwhile, photons, electrons and the atomicnucleibouncedandzig-zaggedaround,hittingeachothersofrequentlythattheyformedsomethingresemblingafluid.After380,000years,whentheobservableUniversewasathousandtimessmallerthanitistoday,temperaturesdroppedtothosefoundonthesurfaceofanaveragesun-likestar,coolenoughforelectronsto be captured in orbit around the electrically charged hydrogen and heliumnuclei.Suddenly,across theUniverse, thefirstatomsformedand theUniverseunderwentarapidtransitionfromahotplasmaofelectricallychargedparticlesto a hot gas of electrically neutral particles. This had dramatic consequences,because photons interact far lesswith electrically neutral atoms.TheUniversebecame transparent,whichmeans thephotons stoppedzig-zaggingaroundandstarted to head off in straight lines. The majority of these photons continuedonwards,travellinginstraightlinesforthenext13.8billionyears.Someofthemare just arriving at our Earth today in the form ofmicrowaves. These ancientphotonsaremessengersfromtheearliesttimes,andtheycarryatreasuretroveofinformationthatcosmologistshavelearnttodecode.

AstheUniversecontinuedtoexpand,itsdenserregions,composedmainlyofdark matter, became ever denser under the action of gravity. Hydrogen andheliumatomsclusteredaroundthedarkmatter,andswirlingatomiccloudsgrewuntil the densest regions collapsed inwards, increasing the pressure andtemperatureattheircoretosuchanextentthattheybecamenuclearfurnaces;thefusion of hydrogen into helium was initiated, and stars formed across theUniverse. A hundred million years after the Big Bang, the cosmic dark agescametoanendandtheUniversewasfloodedwithstarlight.Themostmassivestars had brief lives and, as they ran out of hydrogen fuel, they began to fuseheavier elements in an ultimately futile battle with gravity: carbon, oxygen,nitrogen, iron–the elements of life–weremade thisway.When the fuel finallyranout, these stars scattered thenewlymintedheavyelementsacross spaceastheyendedtheirlivesasbrightplanetarynebulaeorexplodingsupernovae.Inafinal flourish, the violent shock of each exploding supernova synthesized theheaviestelements, includinggoldandsilver.Newstarsformedfromthedebrisoftheold,andcongregatedintheirhundredsofbillionsinthefirstgalaxies.Thegalaxies, numbered in hundreds of billions, were moulded into the giant

filamentarywebs that criss-cross theUniverse by the gravitational pull of thedominantdarkmatter.

4.6billionyearsagointheMilkyWaygalaxy,agascloudenrichedinstellardebriscollapsedtoformourSun.Shortlyafterwards,theEarthformedfromtheremainsof the cloud.Then, 4billionyears ago, in a great ocean created fromhydrogenformedinthefirstminuteoftheUniverse’slifeandoxygenforgedinlong-deadstars,thegeochemistryoftheyoungEarthbecamebiochemistry:lifebegan. In 1687 Isaac Newton published the Principia Mathematica. We’veobviouslyskippedabitofbiology.

This is the broad outline of the story of the evolution of the Universe, frombefore theBigBang to IsaacNewton. It seems that collections of atoms on acooling cinder, in possession of a precious thing called science for barely aninstant,havefoundawaytoglimpsethefiresofcreation.Therestofthisbookisthestoryofhowwedidit.

2.HOWOLDARETHINGS

TheEarthis4.55billionyearsold,giveortake50millionyears.Thisisafigureconsistent with independent measurements of the age of the Universe, whichplace the Big Bang 13.8 billion years ago. It is also consistent with physicalbiological evidence and our understanding of evolution by natural selection,which suggest that the first living things appeared onEarth around3.8 billionyearsago.Thelifecyclesofstarsfitintothistimelinetoo.TheageofourSunisestimatedat4.6billionyears,andsimilarstarsarepredictedtoliveforaround10billion years before they die.Moremassive stars havemuch shorter lifetimes.TheremusthavebeentimeforatleastsomestarstoliveanddiebeforetheEarthformed, because the Earth is made out of heavy chemical elements like iron,carbon and oxygen: elements that are made inside stars. Leaping forward intime, the basalt columns of the Giant’s Causeway in Ireland were formed 60million years ago, around the time the dinosaurs became extinct. The oldestlivingtreeisabristleconepinethatlivesintheWhiteMountainsinCalifornia.Itis–asof2016–5066yearsold.

All these dates are determined using very different kinds of science, but,remarkably–impressively–they fit together without contradiction. There isnothing special about this particular list; we chose this eclectic bunch simplybecausetheyreflectavarietyofdifferent‘old’things,andwecouldhavechosenadifferentlist.Thisraisesthequestion:howdoweknowhowoldsomethingis?Ageisnotatrivialthingtodetermine,especiallyforveryoldthings,becauseitmustbe inferred indirectly.Wecan’t sit aroundandwatchwhile theUniverseevolvesfromthehotplasmaofitsbirth.Wecan’tevenpointtodirectevidencefortheageoftheoldesttree;nobodywasaroundtowriteaboutitandrecordthedate when it was a tiny sapling. But we don’t need to have been present:knowledgecanbeacquiredindirectlyifwedoalittledetectiveworktocollectevidenceandthenapplysimplelogictodrawconclusions.Thisbookisallabouttakingascientificapproachtosecuringknowledgeoftheworldaroundus.Thisapproach is incremental–a framework of knowledge grows over time as weunderstandmoreabouttheUniverse–anditsitsinstarkoppositiontohaphazardthinking: you don’t build a computer by trial and error and you are prone to

mistakesifyoudon’tentertainthelikelihoodthatyoumaybewrong.Wetrustour lives to scientific knowledge, in hospitals and aeroplanes, and exactly thesametypeofthinkingcanbeusedtogreateffectelsewhereinourlives.Inthisbook,wewillshowhowfaritispossibletotravelinunderstandingtheUniverseby taking simple, reasoned steps coupled with careful observations. In thischapter,wearegoingtobeginbyexploringthesciencethatallowsustomeasuretheageofthingswithsuchconfidenceandprecision.

Let’sbeginwiththeageoftheEarth.Averyobviouswaytostartistolookatwhatwecansee:toaskwhetherthereareanyfeaturesontheEarth’ssurfacethatmightgiveusacluetoitsage.TotakeacarefullookatNature,inotherwords,andseewhatwecanworkoutfromsimpleobservation.Forexample,weknowthat river valleys are cut by flowing water, and that coastlines are subject toerosion. These are features that change with time; therefore, observing themcarefully and understanding the physical processes that formed them shouldallowustoestimatetheirages.Onlargerscalesstill,couldthefamiliarshapesofthe continents and oceans also tell us something about the way they haveevolved,andhowlongithastakenthemtodoso?

Figure2.1TheMid-AtlanticRidge.

Figure2.1 isamapof theAtlanticOceanand the landmasses thatsurround it.SouthAmericaandAfricainparticularlookasiftheyfittogether.Let’ssupposethis fit is no accident and make a proposal: the continents were snuggledtogether at some time in the past, and have been graduallymoving apart eversince.Ifthistheoryiscorrect,thenwecanmakearoughestimateoftheageoftheAtlanticOcean.Ofcourse,thisisn’tanewidea–AlfredWegener’sideaofaglobal-supercontinent thatbrokeupover timeasa resultofcontinentaldrift isover 100 years old. The point here, and throughout this book, is that we canuncoverthescienceforourselves–wewanttofollowinthefootstepsofthegreatscientists, to appreciate how irresistible progress comes from simple thoughts.Asafirststep,weneedtoconfirmthatthebroadoutlineofourhypothesis(thatSouthAmerica andAfricawereonce joinedandhavebeenmovingapart eversince)isplausiblebycheckingwhethertheAtlanticisstillgrowingtoday.If itis,we canmeasure the current rate of separation of the continents, and–ifwemakethefurtherassumptionthatthisratehasstayedconstantsincethetimethatthecontinentsbegantoseparate–wewillbeabletomakeanestimateoftheageoftheAtlantic.Therearealotofassumptionshere,butlet’sgetonwithitandseewhatwefind.

If we were very committed experimentalists, we could measure themovements of the continents ourselves. We could pack a couple of GPSreceivers into a rucksack, fly to the eastern coast of Brazil, fix one of thereceivers to the ground, fly back across the Atlantic to northwest Africa–adistanceofaround4000km–andsetupthesecondGPSreceiver.Overthenextfewyears,wecouldmonitorhowthereceiversmoverelativetoeachother.Wedon’t need to do this, because geologists have already been making suchmeasurements for many years. Quite wonderfully, apart from using GPSreceivers, the distance between North America and Europe has also beenmeasuredusingapairof radio telescopes(one inEuropeandone in theUSA)each focused on a distant quasar.Quasars are active galactic nuclei thatmostprobably originate as matter accretes onto super-massive black holes in thecentresofgalaxies, and theyareamong thebrightestobjects and therefore themostdistantwecansee.Because theyaresofaraway, theyserveasexcellentfixedpointsonthesky,whichisimportantfortriangulatingthedistancebetweenEuropeandtheUSA.WedescribethemeasurementinalittlemoredetailinBox1.Doyourememberthoseschoolscienceexperimentswhereyouhadtobeginwith the heading ‘Apparatus: two large radio telescopes and a grid systemcomprisingactivegalacticnucleioverabillionlightyearsfromEarth’?

Figure2.3showsasummaryoftheresultsmeasuringthepresent-dayratesatwhichthevarioustectonicplatesaremoving.ItshowsthattheAtlantic,betweennorthernBrazilandnorthwestAfrica,iscurrentlyexpandingatarateof2.5cmperyear,whichisthespeedatwhichfingernailsgrow.

BOX1.MEASURINGCONTINENTALDRIFT

Thedistancebetween two radio telescopeson theEarth’s surfacecanbedeterminedusingatechniqueknownasVeryLongBaselineInterferometry.Thetwotelescopeslookatthesamedistantobjectinthe sky, and from the difference in arrival time of light signals–determined using very precise clocks accurate to 1 second in 1million years–the distance between the telescopes can bedetermined tomillimetreaccuracy.Quasarsare sobright that theyarevisibleatdistancesofmanybillionsoflightyears,andbeingsofar away guarantees that they appear still during the time of themeasurement. Over twenty years, telescopes in Westford,Massachusetts, and Wettzell, Germany, have been used todeterminetherateatwhichtheAtlanticisopeningbetweenEuropeand the United States. The data are shown in Figure 2.2, whichshows a rate of spreading in this region of 1.7 cm/year. Satellitelaser ranging,which involvesbouncing laser lightoffsatellites,andGPSmeasurementsarealsousedalongthelengthoftheNorthandSouthAtlantic,andgiveconsistentresults.

Figure2.2ThesteadyrateatwhichGermanyandtheUSAhavebeen

Figure2.2ThesteadyrateatwhichGermanyandtheUSAhavebeen

recedingfromeachotherintherecentpast,asmeasuredbyapairof

radiotelescopestrainedondistantastronomicalobjects.

Figure2.3Howthecontinentsaremovingaround.Thenumbersandarrows

indicatetherateanddirectionofmovement,incentimetresperyear.

Working on the assumption that the continents have always been movingapartatthisrate,wecannowestimatetheageoftheAtlanticOcean:4000km×40 years/metre = 160million years. If this figure is a good estimate, thenwenow also have a minimum age for the Earth–because obviously it can’t beyoungerthantheAtlanticOcean.

We’ve just done what could be described as a ‘back of the envelope’calculation. Obviously, we’d like to know if our number is anywhere nearcorrect;afterall,wedidmakeaboldassertionandaveryboldassumption.Weassertedthatthecontinentswereoncepartofasinglelandmassandassumedthatthey have beenmoving apart at a steady rate ever since. Let’s examine theseassumptionsmorecloselyandtrytojudgehowreasonabletheyare.

LookbackatthemapinFigure2.1.ItalsoshowsthetopologyoftheAtlanticOcean’sfloor.Thegreatrangeofunderwatermountainsrunningdownthecentreis called the Mid-Atlantic Ridge. This ridge clearly mirrors the shape of thecontinents on either side; it’s also bang in between the two continents, in themiddle of theAtlantic, and is currently spewingoutmaterial from theEarth’sinterior: lava that solidifies and formsacrust.This suggests amechanism thatcouldexplainwhythecontinentsarecontinuingtomoveaparttoday:newoceancrustisbeingformedalongtheMid-AtlanticRidge.

Allofwhichseemstoindicatethatourassertionisingoodshape.Wecould,of course, havebeen fooledby a series of coincidences: (i) that the coastlinesappeartofittogetherandmatchtheshapeoftheMid-AtlanticRidge;(ii)thattheMid-Atlantic Ridge lies midway between the continents; (iii) that the lavaerupting from the Mid-Atlantic Ridge has nothing to do with the currentlyobservedwideningof theocean.But althoughwe canbepretty confident thatthese are not simply coincidences, nothingwe have established so far impliesthattheseparationofthesetwocontinentshasbeenproceedingatthesameratefor over a hundred million years, and we must admit that, at this stage, thisassumptionisablindguess.

Figure2.4Theagesofthesea-floorrocks.

BOX2.SEAFLOORSPREADING

Theageofsea-floorrocksintheAtlanticisdeterminedbyexploitingthe fact that sea-floor basalt is magnetized in a stripy pattern, asshown inFigure2.5.Thestripes, typicallysome tensof kilometreswide,areformedasnewrockspewsoutoftheridgeandbecomesmagnetized by the Earth’smagnetic field.When the rock freezes,the magnetic orientation gets frozen within it. The stripes appearbecause the Earth’s magnetic field flips its direction from time totime,andthesechangesofdirectionareencoded in therocks.Wecan therefore map the temporal evolution of the sea floor bymeasuring the barcode-like pattern of stripes, known as ‘polaritychrons’,so longaswehavesomemethodofsetting the timescalefor the flips in the Earth’s magnetic field. And we do: radiometricdating methods have been used to date rocks in other locations,suchason-landlavaflows.Thebarcodepatternsmatch.

Figure2.5Theagesofrocksformingalongariftvalley,suchastheone

runningalongtheMid-AtlanticRidge.Thebarcodestripesarevery

distinctiveandareduetothefactthattheEarthflipsitsmagneticfield

aroundeverysooften.

In December 1968 and January 1969, the Glomar Challenger, ascientific research drill-ship, acquired a very important dataset by

drilling a series of seventeen holes in the equatorial and SouthAtlantic, many of them traversing the Mid-Atlantic Ridge. Thesamples the Glomar Challenger collected were dated mainly bypaleontologicalmethods,whichinvolvedlookingfortinyfossilsinthesamplecoresandmatchingthemwithknownstagesintheevolutionin the flora and fauna of the oceans (whose ages themselves arefixedusingradiometricmethods).Theshipboardscientists involvedanalysedthecoresandfoundanage–distancerelationshipfromtheMid-AtlanticRidgethatisremarkablyconsistentwiththeassumptionthattheseafloorhasbeenspreadingataconstantrate.Theyfoundsedimentssittingdirectlyabovethesea-floorwithagesrangingfrom10millionyearsforsamples200kmfromtheridge,allthewayto70million years for samples taken 1300 km from the ridge,correspondingtoasea-floorspreadingrateofcloseto2cm/year.

So let’s bring in some serious science. For decades, geoscientists havemeticulouslyexaminedoceanfloorsacrosstheglobeanddeterminedtheageofthe rocks on the seabed. This is a difficult task, and requires some beautifulsciencethatwewilldiscussinamoment(seealsoBox2).Fornow,letusjustpresent thedata,whichisshowninFigure2.4.There isaveryclearpattern intheAtlantic:theyoungestrocksliealongtheMid-AtlanticRidge;theoldestaretobefoundborderingthecontinents.ThisfitsverynicelywithourproposalthattheAtlanticwasformedbysea-floorspreadingfromtheMid-AtlanticRidge;ifweareright, therocksontheseabedshouldindeedgetprogressivelyolder thefurtherwe travel fromtheridge.Noticealso that therearenosharp transitionswhere the rocks suddenly getmuch older, nor are there any extended regionswheretherocksareallthesameage.Thisiswhatwewouldexpectiftherateatwhich new rock is being formed along theMid-Atlantic Ridge has remainedroughly constant during the time that the continents have beenmoving apart.Thefinalobservationwecanmakeis tolookat theageoftherocksliningtheoceanfloorsalongtheedgesofthecontinents.Thesearedatedtobearound180millionyearsold–inbroadagreementwithourback-of-the-envelopecalculation.

Wehaven’tyetdescribedhowwegoaboutdatingrocksdirectly.Butwecansay that our suggestion that theAtlanticOceanwas created by the continentsdrifting apart due to geological activity along the Mid-Atlantic Ridge isconsistentwiththemeasuredageoftherocksontheoceanfloor.

Logical consistency and the accumulation of evidence are very importantfeaturesof thewaymodernscienceworks.Consider, forexample,whatwouldhappentoourpreviouslogiciftheAtlanticweresignificantlyyoungerthan160millionyears.Forthesakeofargument,let’sgowithBishopJamesUssher,andsayit isaround10,000yearsold.Thisrathercasual levelofprecisionisdoingthegoodbishopadisservice,becausehewasveryspecific.Heassertedthattheworldwascreatedontheeveningof22October4004BC.Thebishopperformedhiscalculationsinthelateseventeenthcentury,usinghistoricalrecordsandtheBible.We,ontheotherhand,areoperatingonthebackofanenvelope,whichmeanswearecontenttoworkwithroundnumbers.

If we want to accommodate an Atlantic that is 10,000 years old, but stillaccept that the twocontinentswerebothclose togetheratsomepoint, then therate atwhich the continentsmoved apartwould have had to have beenmuchfasterthanthecurrentlyobserved2.5cmperyear.Instead,wewouldrequireanexpansionrateof theorderof400metresperyearformostof the10,000-yearperiod.

Theproblemwithanexpansionrateof400metresperyearisthattherocksalongtheAtlanticshoresaremeasuredtobe180,000yearsold,adatethatisingood agreement with the 2.5 cm per year spreading rate. If we insisted on a10,000-year-old Earth, then it must follow that the rock ages are wrong bypreciselythesamefactorasthespreadingrateestimateiswrong.Thiswouldbequiteacoincidence.

WithBishopUssher’sdatingstillinmind,asecondpossibilitymightbethatthecontinentswereneverinfactclosetogether,butinsteadtheywereoriginallycreated 4000 km apart, 10,000 years ago. In that scenario, the fact that theobserveddriftrateof2.5cmperyearjusthappenstobeconsistentwiththeageinferredfromdating the rocksmustbe regardedasameaninglesscoincidence,notleastbecausewewouldalsoneedtorejectaswrongthemethodsusedtodatetherocks.Inaddition,we’dalsohavetosupposethatthetwocontinentsandtheMid-AtlanticRidgeallfittogetherquitebyaccident.ItisclearthenthatthecaseforayoungEarthrequireswerejectthemostobviousinterpretationofthefactsand appeal instead to coincidence and error.We have only been studying thecaseoftheAtlanticOceansofarandwewillmeetsomemoreexamplesofveryoldthingsinduecourse.Itisuptoyoutojudgetheextenttowhichtheevidenceisconvincing.

The reason that it is so difficult tomake an argument against theAtlanticOcean being around 160million years old is that independent measurements,

relyingoncompletelydifferentscience,combinetoprovideaconsistentpictureofwhathappened.Itiseasytocookupascenario,howeverfanciful,thatcastsdoubt on some measurement or other. But it is usually extremely difficult toargueforaradicalchangeinoneareawithoutmakinglargepartsof thewholeinterlinkededificeinconsistent.Giventhatthescientificedificeisalsothethingthat keeps your lights on, keeps aircraft in the sky andmakes your computerwork, this is not usually a sustainable position to take. Ourmodern scientificworld-viewisauniversalone,and this isakeyreasonwhy it issorobustandsuccessful.

One of the most precise ways of dating old rocks is through radiometrictechniques. The key idea is that certain types of atoms are radioactive,whichmeansthattheycanspontaneouslytransformintoatomsofadifferenttype.(InBox3(pp.41–6),weprovideaprimeronthebasicsonatomsandradioactivity.)Thistransformationprocessisknownasradioactivedecay.Ifweknowtherateatwhichaparticulartypeofatomdecays,thenbycountingthenumberofthoseatomspresentinarocksamplewecanobtainameasureofhowmuchtimehaspassedsinceitwasformed.Wedonotneedtoknowwhatcausesatomstodecay(for thatwehave tounderstandsomequantumphysics);we justneed toknowthe rate at which atoms decay, which is called the half-life. The half-lifeexpresseshowlong,onaverage,ittakesforhalftheatomsinasampletodecay.For example, ifwe know that a rock sample initially containedN radioactiveatoms, and we measure that it currently contains N/4 atoms–that’s to say, aquarterof theradioactiveatomsoriginallypresent–wecandeduce immediatelythattwohalf-liveshaveelapsedsincetherockwasformed.

Someatomshaveshorthalf-livesofmuchlessthanonesecond;othershavelonghalf-lives, reaching into thebillionsofyears. Ifwewant todetermine theageofarock, thebestwaywouldbetocountatomswhosehalf-life isnot toodifferentfromtheageofthatrock.Ifthehalf-lifeismuchlessthantheage,mostoftheradioactiveatomswillhavedecayedaway,andwewillhaveadifficultjobcountingthesmallnumberthatremain.Ifthehalf-lifeismuchgreaterthantheage,veryfewatomswillhavedecayedandwemightstruggletodetermineanysignificant deviation from the initial number. None of this would be of anypracticaluse if radioactiveatomswere rarely found in rocks.Fortunately, theyarerelativelycommon.

Youmayhavealreadynoticed that therecouldbea flaw inourplan.Howcouldwepossibly knowhowmany radioactive atomswere present in a given

rock when it first formed? This might seem to scupper the half-life datingprocedure.However,thereisabeautifulwaytosidesteptheproblem,knownastheisochronmethod.

In order to understand the isochron method, let’s look at a specific atom,rubidium-87,whichwewill write as 87Rb. Rubidium is chemically similar topotassium,aboutasabundantaszinc, andoften found incommon,potassium-richminerals in rocks. It is radioactive,withavery longhalf-lifeof48billionyears.Beingbarely radioactive is abonus fordating theoldest rocksonEarthbecause, as we will see, they have ages of several billion years. When arubidiumatomdecays, itconverts intoanatomofstrontium-87(87Sr).Wecancountthenumbersof87Rband87Sratomsinasampleofrock.Thecleverpartofthe isochronmethod is to exploit the existenceof adifferent sort of strontiumatom, strontium-86. 87Sr and 86Sr are different isotopes of strontium; the onlydifferenceisthat86Srhas1lessneutroninitsnucleus.Thismeansthattheyarechemicallyidentical–andthis iscruciallyimportant.Alsocrucially,86Srcannotbeproducedthroughradioactivedecay,whichmeansthatany86Srnowpresentintherocksamplewastherewhenitwasoriginallyformed.

Figure2.6Twoisochronplotsusedtodatetheagesofrocks.Theupperone

isforsamplestakenfromthechondritemeteoriteTieschitzthatfellinwhatis

nowtheCzechRepublicin1878.Theloweroneisforsamplestakenfrom

IsuainGreenland.

In a sample from the rockwewant to date,we count the number of 87Rbatomsanddividebythenumberof86Sratoms.Wealsocountthenumberof87Sratomsanddividethatbythenumberof86Sratoms.Wemarkthesetworatiosasapointonagraph,asshowninFigure2.6.Werepeatthiscountingprocessforseveral different samples, each taken from the rock we want to date. Thedifferentsamplesmightbechunksofrocktakenfromalargebed,ortheymightbe samples of different minerals taken from the same piece of rock. If thesamples are all absolutely identical to each other, thenwewill find the same87Rbto86Srratiosinallofthem,andthiswillsimplygeneratelotsofpointsontopofeachotheronthegraph.Butifthedifferentsampleshaddifferinginitialamountsof87Rb,thentheratioswillbedifferentandwewillgetseveralpointsonourgraph.

Thestrikingthingabout thegraphsshowninFigure2.6–whichcontainrealdata–isthatthedifferentpoints,correspondingtodifferentinitialconcentrationsof rubidium atoms in each sample, lie on a straight line. This is not acoincidence;infact,it’sthecleverbit.

To understandwhat is happening here, imagine that wemake an isochronplot immediatelyafter the rockhas formed fromamolten state.Since the twoisotopesofstrontiumhaveidenticalchemicalproperties,theratio87Sr/86Srwillbethesameinitiallyineverysample.Forexample,ifthereare787Sratomsforevery 10 86Sr atoms throughout the initialmoltenmix, then this ratiowill bepreserved in everymineralwithin every newly solidified rock sample. This isbecausethereisnoreasonforanyparticularmineraltoformusingonestrontiumisotopeovertheother,sincethetwostrontiumisotopesarechemicallyidentical.Thismeansthatattime-zero,justaftertherockforms,theisochronplotwillbeahorizontalstraightline.ThisisillustratedinFigure2.7.TheRb/Srratiowillvarybecauserubidiumandstrontiumarechemicallydifferentfromeachother.Ifonesampleistakenfromapotassium-richmineral,forexample,itislikelytohavemorerubidiumthanasampletakenfromamineralwithlesspotassium,becauserubidium behaves like a potassium substitute. We therefore expect the initialRb/Srratiotovarywiththepotassiumcontentofthedifferentmineralsamples.

As time advances, rubidium atoms decay; as a result, the amount of 87Srinsidetherockstartstorise.Thismeansthatapointontheoriginalplotmovestotheleftbyanamountproportionaltothenumberofrubidiumatomsthathavedecayed.Thepointalsomovesupwardsbyexactlythesameamount,becauseforeverydecayingRbatomanewatomof87Sriscreated.Thishappensforalltheoriginalpoints,andresultsinthelinetiltingfromthehorizontal,asillustratedin

Figure2.7.Asmoretimepassessincetherockwasformed,thelinetiltsfurther.Iftherewasnoinitialrubidiumpresentinthesamplethentherecanneverbeanynewstrontiumgenerated, so the left-handpointon the linemust stay fixed,asindicatedinthefigure.Crucially,thepointsshouldremainonastraightline.Thelinesimplytiltswithtime,andthetiltfromthehorizontaliswhattellsustheageofthesample.

Figure2.7Illustratinghowthelineonanisochronplottiltsawayfromthe

horizontalastimepasses.PointAmovestoA’andBmovestoB’.

Thisishowtheingeniousisochronmethodworks.Weneverneededtoknowtheinitialconcentrationsofanyoftheatoms;wesimplyneedasetofdifferentsamples,withdifferentinitialconcentrationsofrubidiumandstrontium.

We’ll showwhatwemean. Supposewe happen tomeasure the rock’s ageafter1half-lifeofrubidium–whichisadmittedlydifficult,becausethat’slongerthan theageof theUniverse,butwearegoingforsimplemathematics. In thiscase,apointontheinitialisochronmoveshalfwaytotheleft,becausetherearehalfasmanyRbatomsastherewereinitially.Thepointalsomovesupwardsbyexactly the same amount (because an equal number of new 87Sr atoms are

created).Thiscorrespondstoatiltof45degreestothehorizontal.Anisochrontiltedat45degreesthereforemeansthatonehalf-lifehaselapsed.

For those who want a bit more detail: if g is the fraction of the originalrubidiumthathasdecayedsincetherockwasformed,thenapointontheinitialhorizontal isochron moves to the left a distance g times its initial value andmovesupbythesamedistance.Thismeansthetangentofthetiltrelativetothehorizontalisg/(1–g).Thisisverynice,becausegdependsonlyuponthehalf-lifeand the time since the samplewas formed.Measuringg from the slopeof thegraph tells us the age of the sample without us needing to know how muchrubidiumor strontiumwas initially present in it. For the case of themeteoriteillustrated in Figure 2.6, the tangent of the tilt is approximately (0.7325–0.699)/0.5=0.067.Thisimpliesgis0.063,whichmeanstherehavebeenmhalf-liveswhere(1/2)m=1–0.063=0.937,whichtellsusthatm=0.094.Sincethehalf-life of rubidium is 48 billion years, this particular rock (multiplying 48billionby0.094)isdatedat4.5billionyearsold.

Theisochronmethodrepresentsabeautifullysimplepieceofscience.Ittellsus howmuch time has passed since the isochronwas horizontal,which is thetimetherockcooledandturnedsolid.InorderforthepointsontheisochrontofallonastraightlineitmustbethecasethatforeveryRbatomthatdecaysintherockanew87Sratomappears.Moreover,therockhastohaveactedlikeasealedcapsule, so that no Rb or Sr atoms entered or left it following its originalformation.Iftherockhasnotbehavedlikeasealedcapsulethenthepointswillnotlineupinastraightline.Thisisnice,becauseitallowsustocheckwhethertherockinteractedwithitsenvironmentinsomeway,whichisanindicationthatthedatingwillbeunreliable.Theflipsideisthatifthedata-pointsdoalllineupwe can be supremely confident thatwe aremeasuring the time since the rockwas last in amolten state. The isochronmethod is self-checking, which addsgreatlytotheconfidencewecanhaveinitsresults.

TheNorthAtlanticcratonisanancientpartoftheEarth’scrustthatisexposedinGreenland, thecoastofLabrador innorthernCanadaandpartsofnorthwestScotland. It is primarily composed of granitoid gneiss, a type ofmetamorphicrock,whichmeansitwasformedfromotherrocksunderveryhightemperaturesand pressures. The lower plot in Figure 2.6 shows a rubidium–strontiumisochron for samples taken from the North Atlantic craton in Isua, on thesouthwestcoastofGreenland.Thedatapointslieonastraightline,whichtellsus that the rock has not been altered significantly since its formation. The

gradient of the line dates the rock samples to a common age of 3.66 billionyears, with an uncertainty of 0.06 billion years. If you followed through thecalculationaboveyouwillbeabletogetthisageforyourself.

Many other samples have been dated across the North Atlantic craton,sometimesusingdifferentradioactiveatomsandtheir isochrons.Themeasuredages vary mainly between 2.6 and 3 billion years, but the most ancient arearound 3.8 billion years old. These ages are consistent with the most ancientrocksfoundatsiteselsewhereintheworld.Theoldestknownrocksarefoundinthe Slave craton, in northwest Canada, and are close to 4 billion years old.Zircon, a mineral that contains small amounts of radioactive uranium andthorium, canbedatedby severalmethods, including auranium–lead isochron.ZircongrainshavebeenfoundinJackHillsinWesternAustraliathatare4.404billionyearsold,withanuncertaintyof0.008billionyears.ThesearetheoldestmaterialsthathavebeenfoundonEarth.

Nointactcrusthasbeenfoundthatisolderthan4billionyears.Thismaybeduetothefactthatthehot,youngEarthvigorouslyreprocesseditscrust,therebyresettingtheradiometricclocksintherocks.Perhapsthisreprocessingwasaidedbyaheavybombardmentofmeteorites–or,possibly,olderrocksdoexistandwehaven’t found them yet. Zircon grains can remain intact in more extremeconditions thanmost rocks,which isprobablywhy theyhavebeendiscoveredwith significantly older isochron-derived ages. In any case, the radiometricdatingofrocksandzircongrainsallowsustoclaimthattheEarthisatleast4.4billionyearsold.

Butwhatexactlydowemeanby‘theageof theEarth’?Presumablyitdidnotforminaninstant.WecanavoidthedifficultquestionoftheEarth’sbirthdayifthe planetary formation process was not too lengthy compared to its age.Theoreticalmodellingofplanetaryformationsuggests that this is thecase,andthatamere0.1billionyearsistheabsoluteupperlimitonthetimeitwouldhavetaken for the Earth to form from the solar nebula. Going further, we mightsuppose that all of the planets in the solar systemare approximately the sameage,because theyall formed from the samenebula.This is somethingwecantest,becausewehavemanyrocksamplesfromspace,intheformofmeteorites,andfourtonnesofMoonrocksreturnedbytheApolloastronauts.Manyofthesesampleshavebeendatedusingisochronmethods.

TheyoungestMoonrocksarefoundtobe3.2billionyearsold;theoldestare4.5billionyearsold.Twelvesampleshaveagesolderthan4.2billionyears,with

an uncertainty of around 0.1 billion years. The geology of these rocks isconsistentwiththemhavingformedasthelunarcrustcooled,andindicatethattheMoonis4.53billionyearsold.Theradiometricdatingofalmostahundredmeteorites, suchas theTieschitzmeteoritewemetabove, reveals that thevastmajorityformedbetween4.4and4.6billionyearsago,whiletheyoungeronesshow clear evidence of severe shock-heating and metamorphism, which willhaveresettheradiometricclocks.Bringingallthedatatogether,includingdatesfrom another precision technique called lead-isotopic dating, the current bestmeasurementoftheageoftheEarthis4.55billionyears,withanuncertaintyof0.02billionyears.

Figure2.8TheBarringerCraterinArizona.Itis1.2kmindiameterandwas

madebytheimpactofameteorite,knownasCanyonDiablo;over10tonnes

ofmeteoritehavebeenrecoveredfromtheneighbourhood.Typically,around

onemeteoriteofsize1metrehitstheEartheveryyearand,fortunately,

impactsfrommeteoritesbiggerthan1kminsizeareexpectedtooccuronce

impactsfrommeteoritesbiggerthan1kminsizeareexpectedtooccuronce

everymillionyearsorso.

Wecouldstophere,satisfiedthatwehaveaprecisionmeasurementoftheageofthe Earth gleaned frommaterial taken from different sites, and cross-checkedusingrocksfromtheMoonandmeteoritesthathavebeenwanderingthroughthesolar system untouched and uncontaminated since their formation, using avariety of radiometric datingmethods. In the spirit of this book, however,weshould ask if there is some other, completely independent, measurement thatdoes not rely on radioactive decay.And,wonderfully, there is such amethod.WecanmeasuretheageoftheSun.

TheSunshinesbyreleasingenergythroughthenuclearfusionofhydrogenintohelium. Fusion also took place in the first few minutes in the life of theUniverse,creatingtheprimordialheliumthatconstitutesaquarterofthevisiblematter present in the Universe today (the remaining three quarters beinghydrogen).When theSun formed, its initialcomposition roughly reflected thisuniversal ratioofhydrogen tohelium,but,over time, thefractionofheliumintheSunincreasedastheSunburned.IfwecanmeasuretheamountofheliumintheSunthathasbeenproducedbyfusionreactionsinitscore,andifweknowtherateofthosereactions,thenwecanestimatetheageoftheSunbycalculatinghowlongitwouldtaketomakethatamountofhelium.

Thissoundslikeatallorder.WecanhardlygototheSun,digintoitscoreandtakesamples.(Aswe’llseeinChapter3,itischallengeenoughtoworkoutthecompositionoftheEarth.)Remarkably,however,studiesoftheSunoverthepast fifty years have allowed scientists to investigate its interior, using atechniqueknownashelioseismology.PressurewavesintheSuncauseittoringlikeabell,andanalysingthewaythatitvibratesallowsforitscompositiontobededuced.1 The method is not unlike the way you might ascertain whethersomethingishollowbyknockingit.Thesestudiesindicatethataround4.2%ofthe Sun’s total mass is helium that has been produced as a result of nuclearfusion.Figure2.9showshowtheheliumabundancechangeswithdistancefromthecentreoftheSun.Youcanseethatitincreasestowardsthecoreandthatitlevelsoffataround27%,whichisclosetothefractionweexpecttobepresentasa resultofheliumformationshortlyafter theBigBang(we’llhavemore tosayaboutBigBangNucleosynthesis inChapter6).Thedark shaded region inthefigure,atradiismallerthanabout0.2,istheheliumproducedthroughfusion.

Weneednowtoworkouthowlongitwouldtakefor theSuntomakethatmuch helium.We can estimate this because we know the Sun’s total energyoutput, and we know how much energy is released by the fusion of fourhydrogenatomsintooneheliumatom.Let’sdealwitheachoftheseinturn.

TheSunradiatesenergyatarateof3.9×1026watts,anumberfirstmeasuredto a reasonable accuracy in 1838 by theFrench physicistClaudePouillet. It’srelativelysimpletodo:youcouldmakeanestimateyourselfusingsomewater,athermometer, awatch and a bucket. The basic idea is tomeasure how long ittakes direct sunlight falling on the Earth to raise the temperature of a knownvolumeofwater,withaknownsurfacearea,by1degree.Thisenablesyou toestimate how much solar energy per square metre per second arrives at theEarth’spositioninspace,93millionmilesawayfromtheSun.Thisquantityisknown as the solar constant. Modern measurements of the solar constant aremadebysatellitesabovetheEarth’satmosphere,butyou’llgetittowithin10%orsoifyouperformthemeasurementsatthetopofamountainandarecarefulwith yourmathematics andyour bucket–asPouilletwas.The solar constant isapproximately1.36kilowattsper squaremetre, andvariesbyabout0.1%withchangesinsolaractivityandbyabout7%overthecourseofayear,duetotheeccentricity of the Earth’s orbit. The total solar energy output can then becalculated,ifyouknowthedistancefromtheEarthtotheSun.(We’llseehowthiscanbemeasuredinChapter3.)

Figure2.9TheheliumfractionintheSun(bymass).Thereisaclearexcessin

thecore,i.e.forradiilessthan20%oftheSun’sradius.Thisistheresultof

hydrogenfusingintohelium.Theresiduallevelat27%istheheliumthatwas

presentintheswirlinggasoutofwhichtheSunformed.

Figure2.10Thereactionsthatdominatehelium-4productionintheSun.The

fusionoftwoprotonstoproduceadeuteronyields0.42MeV.Theemitted

positronannihilateswithanearbyelectrontoliberateafurther1.02MeV.And

thefusionofthedeuteronwithanotherprotontoproducehelium-3yields5.49

MeV.Twoofthesechainsyieldsatotalof2×(0.42+1.02+5.49)=13.86

MeV.Finally,thetwohelium-3nucleifusetoproducethehelium-4nucleus,

withafurther12.86MeVofenergyreleased.Theneteffectofallofthese

nuclearfusionreactionsisforfourprotonstoconvertintooneheliumnucleus

withthereleaseof26.72MeVinenergy.

ThedetailsofthefusionchainthatleadstoheliumproductionintheSunareillustratedinFigure2.10.It’squitecomplicated,butthefinalanswerissimple.Four hydrogen nuclei are consumed to make one helium nucleus, and in theprocess26.7MeVofenergyisreleased,2ofwhich98%isradiatedawayaslight.Themissing2%oftheenergyiscarriedawaybyneutrinos.

Because the Sun is releasing energy at a rate of 3.9 × 1026 watts, we candeduce that it ismaking3.9×1026/(26.72×1.6×10−13)=9.1×1037heliumnucleipersecond.Ifwesuppose that theSunhasbeendoingthiseversince itwasborn,thenwecandeducethattheageoftheSunisequaltothetotalmassofhelium produced by fusion divided by the mass of helium produced everysecond,whichis4.2%×1.99×1030kg/(6.64×10−27kg×9.1×1037)seconds.Wehavetaken1.99×1030kgforthemassoftheSunand6.64×10−27kgforthemass of one helium nucleus. This gives us an age of 1.4 × 1017 seconds,whichisequalto4.4billionyears.3Amazing!

We are surely allowed an exclamation mark. We have set great store onachieving consistency between scientific results. Here we have presented acalculationfortheageoftheSunthatusesourunderstandingofnuclearfusionreactions, a measurement of the amount of solar energy falling on the Earthevery second, the distance of the Earth to the Sun and measurements of theamountofheliumin theSun.Thiscalculationgivesanageconsistentwith theageoftheoldestrocksonEarthandontheMoon,andtheagesofmeteoritesthathavefallenontheEarth.Thesewerecalculatedusingourunderstandingof theradioactive decay of heavy atoms, which is a completely different physicalprocess.Remarkably,wehavefoundthattheagesoftheSunandEarthareverynearlythesame,whichfitsperfectlywiththeideathattheywereformedoutofthesamecloudofcollapsinggasanddustaround4.6billionyearsago.

BOX3.EVERYTHINGISMADEOFATOMS

If,insomecataclysm,allscientificknowledgeweretobedestroyed,and only one sentence passed on to the next generations ofcreatures,whatstatementwouldcontainthemostinformationinthefewest words? I believe it is the atomic hypothesis (or the atomicfact, or whatever you wish to call it) that all things are made ofatoms–little particles that move around in perpetual motion,attracting each other when they are a little distance apart, butrepelling upon being squeezed into one another. In that onesentence,youwillsee,thereisanenormousamountofinformationabouttheworld,ifjustalittleimaginationandthinkingareapplied.(RichardFeynman,SixEasyPieces,p.4)

Thechild-likesimplicityofthequestion‘WhatifIdividethisthinginhalf… and in half again.… and again…?’ belies its sophistication.The answer to this question is not fully known, and it leads usdownwards into theworld of atoms: aworld inwhich tinyparticlesdancearoundaccordingtothecrazyrulesofquantumphysics.Wedonotneedtodelveintothoseruleshere,insteadwe’dliketogetarough-and-readyappreciationofsomebasicpropertiesofeveryday‘stuff’.Let’sstartoutwithasummaryofwhatweknow.

Weknowthatallordinarymatter ismadeupofatoms, fromtheSun toa lumpof rock to theairwebreathe,and that thereare98different types of atom that occur naturally on the Earth (humanshavemanaged to build a further 20 different types). These are alllisted in the periodic table, shown in Figure 2.11. The mostcommonlyoccurringtypeofatomintheUniverseishydrogen;over90% of the atoms in the Sun are hydrogen (the remainder beingalmost entirely helium). On Earth, the atomic composition ismorediverse. The oceans are mainly hydrogen and oxygen, while themostabundantatomsintheEarth’scrustareoxygenandsilicon.Inthe atmosphere they are nitrogen and oxygen, while the core ispredominantlymadeofiron.

Weknowthateachatomismadeupofacentralnucleus,whichcontainsalmostallofthemassoftheatom,andsurroundingit isa‘swarm’ of tiny electrons (tiny in comparison to the nucleus). Thenucleusistypicallyjustafewfemtometresacross,andtheelectronsdancearoundit:infinitesimalspecksatdistancesofafewtenthsofa

nanometreacross.Togiveanideaofwhatthismeans,ifweweretoscale up an atom so that the nucleus was the size of a pea, theelectrons would be like tiny grains of sand hopping around akilometre away. In other words, an atom is almost entirely emptyspace.InBox4(pp.47–9)weshowhowyoucanmakeanestimateofthesizeofanatomyourself.

But thesetiny,point-likeelectrons,dancingatgreatdistances inrelation to the nucleus, are anything but ephemeral. For it is theirdancing motion, described by the rules of quantum theory, thatdeterminesthewaythatcollectionsofatomscommunicatewitheachother;inotherwords,theelectronsarewhatfixesthechemistry.

As far as the chemistry is concerned, the nucleus is an inertobjectandactsonlyasaheavysourceofpositiveelectriccharge,whichisresponsibleforholdingthenegativelychargedelectronsinakindoforbitaround it. Ineveryday life, theelectronsdo thehardworkofnegotiatinghowtheatomsandmoleculesbehave,whilethenucleisitinertattheheartofatoms,shieldedfromthemaelstromofelectronactivity.

Figure2.11Theperiodictableoftheelements.

The nuclei are far from boring, though. Breaking apart heavynuclei (fission) or fusing light nuclei together can be used togenerate vast amounts of energy. Fission is what underpins thereactors that are used throughout the world today, and fusionreactors promise to deliver a cleanand virtually limitless supply ofenergy. Looking inside nuclei is also mandated by our naturalhumancuriosity–wewanttoknowwhattheyaremadeof.Sofar,weknowthatnucleiaremadeofprotonsandneutronsandthatthey,inturn,arebuilt fromquarksandgluons.Thestoryseems tostopatquarks, gluons and electrons because the Standard Model ofparticle physics describes these objects without any need forsubstructure. In other words, it might not make any sense to ask‘WhathappensifIchopanelectroninhalf?’or‘Whatisanelectronmadefrom?’Thefact that the ‘Whathappens if Ichopthis thing inhalf’sequencemayeventuallycometoanendissomethingthat isreasonable inquantumphysics,not leastbecause themorewe trytopindown the locationof very tinyobjects themoreelusive theybecome.Sowhilequantumphysicsdoesnotexcludethepossibilitythatparticleslikeelectronsandquarkshavesubstructures,nordoesitdemandit.

Figure2.12Aringof48ironatomsadsorbedontoacoppersurface.The

picturesaretakenwithascanningtunnelingmicroscopeandthecircular

wavestrappedinsidethe‘corral’correspondtowavesofelectrondensity.

Itshowshowelectronsbehavelikewaves,whichisafeatureofquantum

theory.

Nuclei are also interesting because they can perform acts of

alchemy. By that we mean that a nucleus of one type canspontaneously change into a nucleus of a different type. Forexample, in nuclear alpha decay a nucleus can eject an ‘alphaparticle’,whichisinfactthenucleusofaheliumatom.Certainnucleihave a propensity to eject alpha particles: uranium and thoriumproduce most of the helium on Earth this way. The alpha emitterplutonium-238 isusedtopowerheartpacemakers,andamericium-241 (which ismade inside nuclear reactors and is a by-product oftheManhattanproject) isused insmokedetectors.Theamericiumproduces alpha particles that collide with air molecules, knockingelectronsoffthemtoproduceanelectriccurrent.Thecurrentfallsifsmoke particles enter the detector and prevent the alpha particlesfromionizingtheair,whichinturntriggersthealarm.Alphaemissionwasa totalmysteryprior to thearrivalofquantumtheory–not leastbecauseitisthesubatomicequivalentofthrowingatennisballatabrick wall and occasionally seeing it pass through. By which wemean that the alpha particle manages to escape from inside thenucleuseventhoughitoughttobetrappedwithinit,justasatennisballoughttobetrappedononesideofabrickwall.Thisweirdeffectiscalledquantumtunnelling.

Alsoextremelypuzzlingisthesimplefactthatnobodycanpredictexactly when a particular atom will undergo a radioactive decay,althoughwecansayhowlongitwilltakeonaverage.Forexample,wemightsaythatthereisa50%chanceofanatomtransmutinginacertainintervaloftime,calledthehalf-life.ThisisillustratedinFigure2.13, which shows how the number of radioactive caesium-137atomsmightchangeovertime.Caesium-137decaystobarium-137when one of its neutrons turns into a proton, with the concurrentemissionof anelectronandanelectron-antineutrino.1 This typeofdecayisknownasbetadecayanditisaconsequenceoftheweaknuclearforce.Figure2.13showsthat,ofahypotheticalinitialsampleof 1000 atoms, 474 remained after 30 years; 60 years later thenumberroughlyhalvedagain,to250atoms,andafter90yearsthenumberofatomsleftwas108.Thisatomthereforehasahalf-lifeof30 years. The curve shows the expected number of atomsremaining,basedupon the idealizedcase inwhichexactlyhalf theatomsdecayin30years(itis,inotherwords,anexponentialdecay).

Figure2.13Thenumberofatomsinahypotheticalsampleofatoms

whosehalf-lifeis30years.Thedotsareforaspecificsamplewhilethe

curveiswhatyouwouldgetifyouaveragedovermanysamples.

The randomness of radioactive decay is curious. For example,wemightsupposethatafreshlycreatednucleuswouldtendtolastlonger than an older one. But this is not the case–the decay of anucleus is totally randomandwithoutanydependenceonhow thenucleus was created or its history. We now understand that thisrandomnessisafundamentalfeatureoftheUniverse:itisadefiningcharacteristicofquantumphysics.

Likealphadecay,betadecayisalsoexploitedineverydaylife,forinstance inPET scanners,where anti-matter is exploited to imagethe human body. Fluorine-18 is unstable to beta decay, whichmeansthatitisliabletoconvertintooxygen-18withahalf-lifeofjustunder 2 hours. In this case, the decay process involves theconversion of a proton inside the fluorine nucleus into a neutron,with the concurrent emission of an antielectron (also known as apositron) and an electron neutrino. Positrons are identical toelectrons, with the sole exception that they have positive electriccharge.CruciallyforPETscanning,whenthepositronbumpsintoan

electron the two annihilate each other with the production of twophotons(particlesoflight).Thephotonshavealotmoremomentumthantheoriginalelectronandpositron,andsotheytravelawayfromeachother inoppositedirections.Bysurroundingthepatientwithaphoton detector, it is possible to detect the individual photons andascertainwhereaboutsinthebodytheywereproduced.Fluorine-18is particularly useful for mapping out brain function or locatingglucose-hungry cancer cells, because it can be incorporated intoglucosemolecules.Otherpositronemitterscanbeincorporatedintoa variety of molecules to trace out different biologically activeregionsofthebody.Thewaytheseradioactivesubstances,suchasfluorine-18, are produced is an interesting example of how blue-skiesresearchoftenbecomesrelevanttoeverydaylife.Specifically,fluorine-18 is manufactured using room-sized particle physicsaccelerators,bybombardingoxygen-18withprotonsthathavebeenaccelerated throughavoltageofa fewmillionvolts.PETscannersarealsoagoodillustrationofEinstein’sE=mc2 inaction,becausethemassassociatedwiththeinitialelectronandpositronisentirelyconverted into the energy of the photons: each and every photonthat is detected has energy equal to the mass of an electronmultiplied by the speed of light squared (511 keV). This energy islargeenoughtoguaranteethatthetwophotonswilltravelawayfromeachother inoppositedirections.This,combinedwith the fact thatallofthePETphotonscarrythesameenergy,helpsinthedetectionprocess. PET scanners are beautiful examples of highly esotericfundamentalphysicsineverydaylife.

Theperiodic tableorders theatomsaccording to thenumberofprotons in theirnucleus,which isequal to thenumberofelectronssurroundingit:thisiscalledthe‘atomicnumber’.Themassofeachatomisalsolistedinthetable,describedinunitswherethemassofa proton and a neutron are approximately equal to 1. All ofwhichmeanswecanusuallyfigureouthowmanyneutronsareinanatom.The number of neutrons, which should be an integer, should beapproximately equal to the atomic mass minus the number ofprotons. Gold has an atomic mass of 196.97 and contains 79protons:anatomofgoldshouldbyimplicationcontain118neutrons.Therearesomepeculiaritiesthough.Takealookatchlorine(atomic

number 17). It has an atomic mass of 35.453, which is midwaybetween35and36,soitseemsthatchlorineshouldcontainaround18.5neutrons,whichdoesnotmakesensebecausetherecanonlybeanintegernumberofneutrons.Thereasonforthisoddityisthatchlorine atoms come mostly in two types: one type contains 18neutrons and the other contains 20 neutrons. The lighter variationaccounts for 76% of the mass in naturally occurring chlorine; theheavier one accounts for the remaining 24%. Themass quoted intheperiodictableistheaverageofthesetwo,i.e.76%×35+24%×37=35.5.

We have already been making reference to the atomic massnumber, for example americium-241 is built from a total of 241protonsandneutrons.Varioustypesofthesameatomwithdifferentnumbers of neutrons are called isotopes. We say that naturallyoccurringchlorine iscomposedmainly from two isotopes:chlorine-35(oftenwritten35Cl)andchlorine-37(37Cl).Asfarasthechemistryisconcerned,isotopesofthesameelementbehaveidentically,sincethechemistryonlycaresabouttheelectrons.Incontrast,thenuclearpropertiesofdifferentisotopescanbeverydifferent:fluorine-18isapositron emitter, while fluorine-19 nuclei are stable, and thereforeuselessinPETscanners.Theperiodictableisofprimaryinteresttochemists,whichiswhytheinformationonisotopesisnotexplicit.

BOX4.HOWBIGISANATOM?

We can estimate the size of an atom by carefully dropping a tinyamount of oil onto the surface of somewater and thenmeasuringhowmuch it spreads out. The oilwill spread out because there isnothingtostopitfromspreading,anditwillformalayerafewatomsthick. This idea is attributed to the prolific physicist Lord Rayleighwho,onbeingawardedthe1902OrderofMeritatthecoronationofKing Edward VII, said: ‘The only merit of which I personally amconsciouswasthatofhavingpleasedmyselfbymystudies,andanyresultsthatmaybeduetomyresearcheswereowingtothefactthat

it has been a pleasure for me to become a physicist.’ Withoutknowingmore about howoil ismade fromatomswedo not knowhowmanyatoms thick the layer is, butwe certainly know that thelayer cannot be smaller than one atom thick. This means that bydetermining the thickness of the layer we can make a statementaboutthelargestpossiblesizeofanatom.

So ifweknowhowmuchoilweplacedonto thewater thenweare in business.For example, a 0.5mmdiameter dropof olive oilspreadsoutoverthewatertoadiameterof25cm.Youcantrythisexperimentathomeandshouldmeasurethemaximumextentoftheoildrop,becausewater-softeningagentsmightcause it tocontractafteraperiodoftimeastheyattacktheoil(soitwouldbebettertousedistilledwater).Thevolumeofoliveoilisequalto4πr2/3wheretheradiusr=0.25mm.Thisvolumeisalsoequaltothevolumeofthediskofoilonthewater,whichinturnisequaltoπR2d,whereR=12.5 cm and d is the thickness of the disk. Equating these twovolumesallowsustodeterminethethicknessoftheoillayer:d=1.3× 10-9 metres. So we know that an atom cannot be bigger thanabout1nanometreinsize.

Wecanbea littlemoredaringthanthisandmakeastabat thesizeofanatomifwearepreparedtoaccept that the layerofoil isonemolecule thickand thateachoilmolecule isachainofcarbonand hydrogen atoms with one end attached to the water. Thesechainsaretypicallyaround10atomslong(itdependsonthetypeofoil).Withthisextrainformation,wecanestimatethatthesizeofoneatomisroughlyequalto0.1nanometres.

Knowingthesizeofanatom,wecanaskhowmanyareinaglassofwater. If we assume that the atoms in a drop of olive oil areapproximatelythesamedistanceapartasthoseinadropofwater,wecanestimatethenumberofatomsinaglassfullofwatersimplyby dividing the volumeof the glass by the volumeof oneatom.A500mlglasshasavolumeofhalfalitre,whichwecandividebythevolumeofoneatomtofigureouthowmanyatomsareinthewater.500 ml is 500 cubic centimetres, and we shall assume that eachatomoccupiesavolumeofapproximately1cubicångstrom(i.e.oneatom fits inside a cube of side 1 ångstrom), which is 10-24 cubic

centimetres.Takingtheratioinformsusthattherearesomethinglike500×1024=5×1026atoms inahalf litreglassofwater.Wecannowusethisinformationtofigureoutthemassofawatermolecule.Ourglassofwaterhasamassof500gramsandwehaveestimatedthat it contains around 5 × 1026 atoms, so we can conclude thateach atom weighs around 10-27 kilograms. All these numbers arenottoofaroffthemark,andtheyarecertainlywithinafactorof10ofthe true values–which is a terrific achievement given how manyfactorsof10areinvolved.

Thereisasecondwaythatwecanestimatethesizeofanatom.We can use a more theoretical approach to figure out what weexpect theanswer tobe. Let’s focuson the simplest atom: that ofhydrogen,with itsoneprotonandoneelectron.Theproton isverymuchheavier thantheelectronandcanbethoughtofasprovidingan anchor about which the tethered electron dances.We want toknowhowfar theelectron is fromtheprotononaverage.Thenextparagraphisalittlemoremathematicalthanthenormforthisbook;skipitifyouneedto.

The distance that the electron is from the proton, d, can onlydepend on the size of the electrical charge of the proton andelectron(Q),theelectronmass(m)andanumberthatunderpinsthewhole of the quantum world: the quantum of action (ħ). ThedependenceonQandm is fairlyobvious:weexpect thatdshouldreduceasQincreases(theelectronwouldbemoretightlyboundifithadmoreelectriccharge)orasmincreases(theelectronwouldbelessinclinedtoflyawayifitisheavy).Thedependenceonħislessobvious–butthisistheonekeyparameterinquantumtheory,andanelectron dancing arounda proton certainly is sensitive to quantumeffects,soweshouldcontemplatethefactthatdmightdependuponit. This is not theplace for us todelve into thedetails of quantummechanics;sufficetosaythatthequantumofactioniswhatcontrolsthewavelengthofthoseelectronwavesinFigure2.12.Ifweassumethat these are the only quantities thatd depends on, thenwe cansay that d must be proportional to ħ2/(mQ2). We can be thisconfidentbecausethereisnootherwaytocombinethesequantitiesto give a quantity that can bemeasured inmetres. The chargeQcanbedeterminedby ingeniouslyexploitingCoulomb’sLaw,which

saysthattheforcebetweentwoelectricchargesofsizeQseparatedbyadistanceRisequaltoQ2/R2.TheresultisthatQ=4.8×10-10g1/2cm3/2/s.Themassofanelectronism=9.11×10-28gandħ=1.1 × 10-27 g cm2/s. Putting these numbers in gives d = 0.6ångstroms.Wecannotclaimanaccuracytobetterthanafactorofafew using thismethod–but it should give us a rough answer. Thisway of figuring things out by appealing to the units carried by thesalientquantitiesinaproblemisusedalotbyphysicists,becauseitprovides a quickway to estimate things thatmight bemuchmoredifficulttocomputewithprecision.Thereis,ofcourse,aproper(andmuch lengthier)waytodo thiscalculation.Aseveryundergraduatephysicsstudentknows,itinvolvessolvingtheSchrödingerequation.

The number we just obtained is a factor of 6 larger than thenumberobtainedbydroppingoilonwater.This really isverygoodagreement.Somanypowersof10are involvedinthenumbersweareworkingwiththatitwouldbeveryeasytogettotaldisagreementifwedidnotunderstand thingscorrectly.Withnopriorknowledge,exceptthatprovideddirectlybyoursenses,atomscouldconceivablybe any size at all smaller than something like a hundredth of amillimetre.Thatisarangespanninganinfinitenumberofpowersof10, so the fact that the theoretical estimate and the oil-dropmeasurementgiveanswers thatagree tobetter thanonepowerof10isveryimpressive.

3.WEIGHINGTHEEARTH

OurcolleagueMikeSeymourmadeaninterestingobservationwhileonholidayatOgmore-by-Sea.Standingon themudflatsby thewater’sedge,enjoying thecool of the salty waves over his mildly sunburnt feet, Mike noticed a buoyfloating in the Bristol Channel that appeared to be perched precisely on thehorizon:anobservationthatisinitselfenoughtomakearoughestimateofthesizeoftheEarth.Beingaphysicistatrest,1Mikedecidedtogatherthenecessaryinformation. Releasing his heels with a squelch he turned and walked withcarefulsharp-shellcadencetoashop,andboughtamap.Thisinformedhimthatthe buoy, known as the Fairy Buoy, was approximately 4 km away from hisvantage point on the beach,which ismarked by a red cross in Figure 3.1. Aquick sketch on the back of a seaside serviette ( Figure 3.2) allowed him todeduce that the Earth has a radius of roughly 5000 km. The actual value isaround6400km. Itmay impressyou thatMikemadea reasonableestimateofthe size of our planet simply by observing the region aroundOgmore-by-Sea.Equally,youmightbeunimpressedthathisansweris20%out.

Figure3.1MikestandsonTheFlatsatOgmore-by-Sea,attheposition

markedbyaredcross.Heestimatesthedistancetothehorizonbynoticing

thattheFairyBuoyisapproximately4kmaway.

Figure3.2Mike’seyesareh=1.6mabovethesurfaceoftheEarthatpointA.HeestimatesthatthehorizonvanishesatpointBbyobservingthebuoy

perchedperfectlyinfullviewonthehorizon.Themapinformshimthatthisis

adistanceD=4kmfromthebeach.UsingPythagoras’stheorem,theradiusoftheEarthRisgivenbyD2/(2h),whereDisthedistancebetweenMikeand

thebuoy.PuttingthenumbersingivesR=5000km.

Thecalculationworkson theassumption that thedistance to thebuoy is infactthedistancetothehorizon.ThequalityofMike’seyesightgovernshowwellheisabletodeterminewhetherornotthebuoyiscoincidentwiththehorizon.Apersonwithaverageeyesightcanjustaboutresolveasmallcoinatadistanceof40metres, corresponding to an angular resolution of about 0.03 degrees. ThismeansthatMikecouldperceivethebuoyasbeingcoincidentwiththehorizon,eventhoughthehorizonisslightlyinfrontoforslightlybehindit.Allhecansaywith certainty, therefore, is that the distance to the horizon lies somewherebetween two extremes, definedby the resolutionof his eyes,whichhe should

quote as an uncertainty on themeasurement.A little calculation2 reveals that,given the limits of his eyesight,Mike really ought to have concluded that theradiusoftheEarthcouldquiteeasilybeanywherebetween2000kmand36,000km.Thefactthathisserviettecalculationgotsoclosetothetruevalueislargelyacoincidence.

Estimatingtheuncertaintyonaresultisoftenasimportantastheresultitself.Itisonlywhenweareawareofourignorancethatwecanjudgebesthowtouseknowledge. In engineering or medical science, a deep understanding ofuncertaintycanbeamatteroflifeanddeath.Inpolitics,over-confidenceisoftenthe norm; uncertainty is seen as weakness when really it is a vital part ofdecision making. In this respect, science delivers an important lesson inhumility.

InMike’s case, hismeasurement,while inaccurate, does still give us someideaaboutthesizeoftheEarth.Toachieveabetterresult,Mikewouldneedtoimprove on the limiting resolution of his eyes,which can be done by using acamerawithalonglens.Fortunately,Mike’sdad,Bob,isakeenphotographer,and lives in Ogmore-by-Sea.We couldn’tmake this up.We asked Bob if hemightgodowntothebeachandtakesomephotographsoftheFairyBuoyforus.Hekindlyobliged:aselectionofhisphotosareshowninFigure3.3.

Figure3.3BobSeymour’sphotographsoftheFairyBuoy.Theyarealltaken

fromthesameplaceonthebeachandwiththesametripodheight.Thewaves

arecausingthebuoytobobupanddown.

Bob’sphotosweretakenwhentheseawasquitechoppy–whichisperhapsabonus,aswedonotneedtoworryaboutthebendingoflightduetoatmosphericeffects,somethingwhichismorepronetohappeningoncalmdays:wecanseethat atmospheric effects are not an issue here because the images are pretty

sharp. Bob adjusted the height of his camera such that the pictures show thebuoy perched directly on the horizon. (Lowering the camera would push thebuoy behind the horizon; raising it brings it in front of the horizon.) Thephotographs in the figure were taken at a height of 1.3 metres. The camerapositionwasdeterminedusingGPSandthepositionoftheFairyBuoywastakenfromtheTrinityHouse3officialrecords.Thedistancebetweenbuoyandcamerawas4.15km.Thesemore refinednumbersgive a radiusof theEarth equal to6600 km.Using a camera has significantly reduced the uncertainty caused bylimited resolution; the chief source of uncertainty that remains is now thedifficultyinascertainingthepreciseheightofthecameraabovetheaveragelevelof the waves. A 10 cm change in height leads to a 500 km change in thecalculatedradiusoftheEarth,whichwemightquoteasaconservativeestimateoftheuncertaintyonBob’smeasurement.

Thereare,ofcourse,farbetterwaysofmeasuringtheradiusoftheEarththanthis–but that isn’t the point.This is a good example of how simple, curiosity-driven observations, together with a little bit of careful thought, can lead tointeresting conclusions. Inwhatwe suspect is aworld first,Mike and his dadhave measured the size of our planet from Ogmore-by-Sea. We have alsolearnedavaluablelessoninquantifyinguncertainty:itiseasytobemisledintodrawingthewrongconclusionunlessweunderstandthedegreeofourignorance.

InhismeasurementoftheEarth,Mikefollowedinthefootstepsofthegreats(althoughifhe’dactuallystoodontheshouldersofgiantshewouldn’thavebeenable to make his measurement). One of the earliest documented attempts toestimatethesizeoftheEarthwasmadebyAristotlein350BCE.Aristotlenotedin his book On the Heavens that ‘there are stars seen in Egypt and in theneighbourhoodofCypruswhicharenotseeninthenortherlyregions,’andthatthesphereoftheEarthistherefore‘ofnogreatsize,forotherwisetheeffectofsoslightachangeofplacewouldnotbequicklyapparent’.Usingaverysimpleobservation,Aristotleruledoutthepossibilitythat theEarthhasaradiusmuchbigger than the distance betweenEgypt and the northern extent of the ancientworld–that’s tosay,a few thousandkilometres.And,unsurprisingly foroneofthemostinfluentialscientistsever,hewasright.Thisisaterrificillustrationofan ‘order-of-magnitude’ estimate. Order-of-magnitude estimates are quickcalculationsthatarenotsupposedtobeveryaccurate,andtheyareimportantinsciencebecausetheycanprovideagooddealofinsightwithverylittlework.

Thetitleofthischapteris‘WeighingtheEarth’,andthatseemslikeamuchtallerorderthanestimatingitsradius.Itis,butwecanalreadymakeanorder-of-

magnitudeestimateusingtheSeymourfamilymeasurements.Let’sassumethattheEarthisaperfectsphere.Itsvolumeis4/3πR3,whichis1.2×1021m3.Intheabsence of any other information, it isn’t too crazy to suppose that thewholeEarth might be a uniform sphere made of granite–a dense rock commonlyoccurringintheEarth’scrust–orsomethingwhoseaveragedensityisthesameasthatofgranite.Thedensityofgraniteis2.8grams/cubiccentimetre,whichgivesusaveryroughestimateforthemassoftheEarthas3.4×1024kg.Wehave,ofcourse, absolutely no way of knowing whether this is anywhere near rightwithout finding some other way of weighing the Earth–and it is not at allobvioushowthismightbedone.Let’sworkouthowtodoit.

We’llstartbyexaminingthemotionsoftheplanetsacrossthenightsky.Atfirstglancethismightseemtobeastrangepointofdeparture,butitwillservetoillustrateanimportantpoint.Veryofteninscience,workinoneareacanimpacton superficially unrelated areas. This is one of many reasons why scientistsshould be allowed and encouraged to roam around researching anything thatappears interesting.Nature is tremendously interconnected. Some time around1510, the Polish astronomer Nicolaus Copernicus wrote a manuscript, theCommentariolus, in which he expressed his dissatisfaction with the classicalEarth-centredcosmologyofPtolemy,formulatedsome1400yearspreviously.‘Ioftenconsider,’Copernicuspondered,‘whethertherecouldperhapsbefoundamorereasonablearrangementofcircles,fromwhicheveryapparentirregularitywouldbederivedwhileeverythinginitselfwouldmoveuniformly,asrequiredby the rule of perfect motion.’ The irregularities he was referring to are theoccasionalloopstheplanetsperform,asviewedfromEarth,astheymaketheirwayacrossthestarrybackgroundoverthecourseofweeksandmonths.BackinthesecondcenturyCE,Ptolemyhaddevisedacomplicatedsystemtopredictthemotionoftheplanets,whichworkedverywellbutwas,atleasttoCopernicus’smind, ugly. In the Commentariolus, Copernicus asserts that the Moon goesaroundtheEarth,theEarthandplanetsgoaroundtheSun,thatthedailymotionof theSunandstars isdue to therotationof theEarthonitsaxis,andthat thedistancefromtheEarthtotheSunisfarsmallerthanthedistancestothestars.HealsosuggeststhattheplanetaryloopsweobservearearesultoftheEarth’smotionrelativetotheplanets.Allofthesestatementsarecorrect.

Copernicus published his complete works in 1543 in the six-volume DeRevolutionibusorbiumcoelestium(OntheRevolutionsoftheHeavenlySpheres),which is rightly regarded as one of the foundational early works in modernscience.Heshowedthatthecomplexmotionsoftheplanetscanbeunderstoodif

itisassumedthattheyall,Earthincluded,moveinorbitsaroundtheSun,eachtakingadifferentlengthoftimetocompleteonecircuit.Copernicusthoughttheorbits were circles but, as we now know, this is only an approximation. Theplanetsfollowslightlyellipticaltrajectories.TheEarthtakes1yeartocircletheSun,Mercury takes88days,whileSaturnmakes the journey in29years.TheCopernicanSun-centredmodelgotaroughrideformanyyears,partlybecausethe idea was seen to run counter to scripture by demoting the Earth from itspreviously imaginedpositionat thecentreof theUniverse,andalsobecause itdoesn’tfeelasiftheEarthishurtlingthroughspace.Thisisn’tasillyobjection,andthedeeperramificationsofthefactthatwecan’ttellwhetherornotwearemoving were only truly appreciated by Einstein in his special and generaltheoriesofrelativity,publishedin1905and1915.WewillgettoEinsteinlater.

IfweacceptCopernicus’swisdom,andarrange theplanets innear circularorbitsaroundtheSun,wecanworkouttheirdistancestotheSun,intermsofthedistance from theEarth to theSun.For the innerplanets,MercuryandVenus,theorbital radius canbecalculated fromameasurementof the largest angularseparationbetweentheplanetandtheSun.(YoucanseethisinFigure3.4.)Wedefer to thenextchapter thetaskofmeasuringthedistancetoNeptune,oneoftheouterplanets,whichwedoexplicitlyusingadigitalcamera,agood tripodandsomephoto-editingsoftware.Theorbitalperiodsof theplanets–the time ittakes them toorbit once around theSun–are also easily determined. InFigure3.5weshowhowtodeterminethelengthoftimeittakesforJupitertoorbittheSunbymeasuring the timebetweensuccessiveoccasionswhen theSun,Earthandtheplanetlineupinthesky.

Figure3.4WecandeterminethedistancefromEarthtoVenusorMercuryby

measuringtheangle,θ,whichisthelargestangulardistanceonthesky

betweentheplanetandtheSun.Thedistancedisthenobtainedbybasictrigonometry,i.e.d=Dsinθ.ThesemethodsallowalldistancestobecomputedrelativetothedistancebetweentheEarthandtheSun.The

distancebetweenEarthandVenuscanalsobemeasuredbytiminghowlong

ittakesforaradarsignaltotraveltoVenusandback.Onceweknowthat

distance,wecandeterminetheorbitaldistancesofalloftheplanetsinthe

solarsysteminmetres.Thisradarmeasurementwasfirstachievedin1961,

usingthetwin26-metre-diameterradiotelescopesattheGoldstone

observatoryintheMojaveDesert,California.Theresultgaveavalueforthe

meanEarth–Sundistanceof149,599,000km.It’squitehardtobaseahigh-

precisionmeasurementsystemonsomethingdefinedsimplyas‘themean

distancefromtheEarthtotheSun’.So,inAugust2012,theInternational

AstronomicalUniondefinedtheAstronomicalUnit(AU)tobeprecisely

149,597,870,700metres.ThismeansthatthemeandistancefromtheEarthto

theSunisnotprecisely1AUanymore,butitneverwaspreciselyanything

anyway.Theprecisionofmodernmeasurementhasmeantthatastronomers

haveoutgrowntheold,intuitivedefinition.

Figure3.5MeasuringthetimeittakesforanouterplanettoorbittheSun.Sis

thetimebetweensuccessiveoccasionswhentheSun,Earthandtheplanet

alllineup(astronomerssaythattheplanetsare‘inopposition’whentheyline

uplikethis).Simplegeometrygives(S-E)/E=S/P,wherePisthetimeittakes

fortheplanettoorbittheSunandEisthetimeittakestheEarthtoorbitthe

Sun,i.e.1year.Thisimpliesthat1/E–1/S=1/P,aformulaCopernicususedfivecenturiesagoinDeRevolutionibus.

Just over half a century after Copernicus, the German astronomer JohannesKeplerspottedthatthereisasimplerelationshipbetweenthesizeofaplanet’sorbitandthetimetakentotraveloncearoundtheSun.UsingdatacollectedbyTychoBrahe,aDanishnoblemanandfellowastronomer,Keplernoticedthatthesquareof theorbital period (orT2 for short) isproportional to the cubeof theradiusoftheorbit(R3–moreprecisely,itisthecubeofthesemi-majoraxisofanellipticalorbit).Thismeansthat theratioofT2/R3shouldbethesameforeachplanetand,asthedatainTable3.1illustrate,Keplerwasontosomething.

In1687,thefirsteditionofIsaacNewton’sPhilosophiæNaturalisPrincipiaMathematica was published. In it, Newton demonstrated that the empiricalpatternKeplerhaddetectedisaconsequenceofadeeperphysicallaw.Theideathat regularities and patterns in Nature are often the sign of an underlyingsimplicity that canbe capturedbymathematical equations is a familiar one toscientists today–but in the late seventeenth century, Newton’s discovery wasrevolutionary.

The mathematical, law-based approach that Newton introduced in hisPrincipia is thefoundationforvirtuallyallofmodernphysics.HeshowedthatKepler’sT2/R3patternisaconsequenceoftheexistenceofaUniversalLawofGravitation,whichstatesthatallmassivebodiesattracteachotherwithaforcethatisproportionaltotheproductoftheirmasses,andinverselyproportionaltothesquareofthedistancebetweenthem.ForaplanetorbitingtheSun,thelawstatesthattheSunexertsaforceFontheplanet,andthatF=GMm/R2,whereM

andmarethemassesoftheSunandtheplanetrespectively,andRisthedistancebetween their centres. The quantity G is now known as the GravitationalConstant, and it is a number whose value describes the strength of thegravitational force. Newton also introduced what is now referred to as hisSecondLawofMotion,whichdescribeshowanobject–suchasaplanet–moveswhenaforceactsuponit.ThisSecondLawofMotionstatesthatforcesinduceaccelerationsaccordingtotheequationF=ma,wherem,inourcase,wouldbethemassoftheplanetandaistheacceleration.Withjustthesetwoequations,itispossibletounderstandtheoriginoftheellipticalplanetaryorbitsandKepler’sT2/R3pattern.Box5showsdetailsforthesimplestcaseofacircularorbit.

ItishardtooverstatetheradicalleapforwarddeliveredbyNewton,andwecouldspendtherestofthisbookexploringtheconsequencesofhislaws.Butinthischapterwe’refocusedononespecificgoal.WewanttoweightheEarth4–and in this context,Newton’s lawsoffer somethingextremely important.Theyrelatethemotionofsomething,suchastheorbitofamoonaroundaplanetoraplanetaroundtheSun,tothemassofthethingthatinducesthatmotion,throughtheUniversalLawofGravitation. In the languageofaphysicist, this ishighlynon-trivial.

BeforeNewton,therewasnoknownconnectionbetweenthesethings.AllofNewton’slawsareuniversal,whichmeansthattheydon’tonlyapplytomoonsandplanetsandstars.Theyaresupposedtoapplytoanyobjects,anywhere.Thisis also a highly non-trivial statement, because it means there is a commonframework that can describe themotion of the planets in the heavens and themotion of objects like cannonballs and swinging pendulums here on Earth.Perhapsyoucanseewhereweareheading.

Let’s consider a ball falling to the ground. How is this described usingNewton’slaws?Theforceactingontheball,acceleratingittowardstheground,isgivenbyF=GMm/r2,whereMisthemassoftheEarth,misthemassoftheball,andr is thedistancebetweenthecentreoftheEarthandthecentreoftheball.ThewaytheballrespondstothisforceisgivenbyNewton’sSecondLawofMotion,F=ma.AverysimplebitofalgebragivesustheaccelerationoftheballinducedbythegravitationalpulloftheEarth;a=GM/r2.Usingarulerandawatch,wecanmeasure theaccelerationofaballwhenwedrop it (we’llgetclose to9.8m/s2) and,with r=6370km,wecancalculate theproductof themassandtheGravitationalConstant:GM=4.0×1014m3/s2.

BOX5.KEPLER’SLAW

Newton’s lawofgravity, togetherwithhisequationF=ma,explainKepler’slaw.Thisiseasytounderstandinacasewheretheplanet’sorbitisaperfectcirclebut,withalittlemoremaths,italsoworksforelliptical orbits. Here we show how Newton’s law works out for acircular orbit.SinceF =ma =GMm/R2, it immediately follows thatthe planet accelerates towards the Sun with an acceleration ofGM/R2. But for things that go in circles at constant speed, thisaccelerationmustalsobeequaltov2/R,wherevisthespeedoftheplanet.EquatingthesetwoaccelerationsgivesGM/R=v2.Butv isrelated to the timeT it takes for theplanet toorbit theSunbyv=2πR/T,whichmeansthatGM/R=4π2R2/T2.ThistellsusthatT2/R3

=4π2/(GM),which is constant. In fact,whatwehavedone isonlyapproximatelycorrect.ThemassMappearinginKepler’slawreallyshouldbeM+m.This isbecause the forceFalsoactson theSun,causingittoacceleratetoo.ThisisasmalleffectifMismuchbiggerthanm,which isthecaseforallof theplanets inthesolarsystem.Generally speaking, two objects will orbit about a point a fractionm/(M+m)alongthelinejoiningthem.Forexample,twoequalmassobjectswillorbitaroundapointmidwaybetweenthetwo.

Or consider the orbit of theMoon around the Earth.We canmeasure theaveragedistancefromtheEarthtotheMoon,5R(385,000km),andtheperiodoftheMoon’sorbit,T(27.3days).Newton’sUniversalLawofGravitationrelatesthesequantitiestothemassoftheEarth,M,throughtheequationwederivedinBox5:T2/R3=4π2/(GM).As in thecaseofadroppedball,wehaveawayofdetermining the productGM, only this time from astronomical observations.PuttingthenumbersingivesGM=4.0×1014m3/s2,asbefore.WegetthesameanswerforadroppedballandfortheorbitingMoonbecauseNewton’slawsareuniversal; they encode information about the deeper physical structure of ouruniverse.

ArmedwiththevalueofGM,wecangoaheadanddetermineeitherthemassoftheEarth(M)orthegravitationalconstant(G),butonlyifwealreadyknowthevalueofoneofthem.SoweneedsomenewmethodtomeasureGorM,butthisisnotaneasythingtodo,becausegravityisacolossallyweakforce.

Intheory,asimplewaytodeterminethemassoftheEarthwouldbetomeasurethe amount by which a hanging plumb-line is attracted towards a large massplacednext to it,as inFigure3.6. Ifwepositionaballof leadwithamassof1000 tonnes anda radius justunder3metres, so that its centre ishorizontallyaligned with a hanging mass placed 3 metres away, the plumb-line will beattractedtotheballbyaminusculeangleof0.16arcsecondsfromthevertical6(1arcsecond is an angle equal to1/3600thof adegree).But although the idea istheoretically simple, it is clearly not an easy thing to do in practice. It isn’tcheap,either:aleadballthisbigwouldcostaroundamillionpoundsintoday’smoney.

Two classic experiments, performed by Nevil Maskelyne and HenryCavendishin1774and1798respectively,werethefirsttoallowforanaccuratedeterminationofG. It is testament to thedifficultyof themeasurements thatacentury passed after Newton published his theory of gravity before anyonemanaged to perform experiments capable of determining its strength. We’vebeen careful with our wording here because, for historical reasons, neitherMaskelynenorCavendishactuallyquoted thevalueofG; theywerebothonlyinterested in weighing the Earth. From a physicist’s perspective this doesn’tmatter at all, because once you have one quantity you can get the other in asinglelineofmathematics.

In 1774, theReverendNevilMaskelyne,AstronomerRoyal, led a team toPerthshire,intheScottishHighlands,tomeasurethedeflectionofaplumb-lineinthevicinityofamountaincalledSchiehallion.There,hecarriedoutwhatisinessence the experiment illustrated in Figure 3.6–except that he replaced the1000-tonne ball by an entire mountain. This magnified the deflection of theplumb-line, making it (just about) measureable. Using the stars as referencepoints,hesucceededinmeasuringaverysmalldeflectionof11.6arcseconds(infact,thiswasthesumoftwodeflectionscorrespondingtotheplumb-linebeinglocatedonthenorthandsouthsidesofthemountain).Fouryearslater,followinga detailed survey of the mountain, mathematician Charles Hutton usedMaskelyne’smeasureddeflectiontoestimatethedensityoftheEarth.Itwas,hecalculated,4.5timesthatofwater,whichisquiteabithigherthanthedensityofrock. Assuming that the Earth is a perfect sphere, and using the modernmeasurementof itsaverage radius,6370km, thisgivesusamassof theEarthequalto4.9×1024kg.

Figure3.6Ahangingplumb-linenexttoalargesphere.Thetangentofthe

angleδistheratioofthehorizontalpulloftheballtotheverticalpulloftheEarth.AccordingtoNewton’sUniversalLawofGravitation,thisisjust

(Gm/r2)/(GM/R2)whereRistheradiusoftheEarth.ThedependenceonGcancels,givingtanδ=(m/M)×(R/r)2.

ThenextmajorstepinweighingtheEarthwastakensometwentyyearslaterby the brilliant Henry Cavendish, using a remarkable experiment in his verylargehouseoverlookingClaphamCommon,insouthLondon.

Cavendish was an eccentric man of independent means; he wore old-fashionedclothes andwas famouslyvery shy.Andhewasoneof thegreatestscientists of all time–apart from the work we describe here, he made hugelyimportant contributions in the fields of chemistry and electricity. Today,

Cambridge University’s Department of Physics, the Cavendish Laboratory, isnamed after him. In 1798, Cavendish detailed the results of his experiment–including a drawing of the apparatus he devised for it–in his ‘Experiments toDetermine theDensityof theEarth’.Hebeginsbycrediting theex-CambridgeprofessorofgeologyReverend JohnMichell fordesigning the experiment andbuilding the first versionof it;Michell, though, died in 1793, before he couldmake any measurements, and the apparatus found its way to Cavendish viaanother Cambridge philosopher-priest, the Reverend Francis John HydeWollaston.

Figure3.7ThelayoutofHenryCavendish’singeniousapparatustoweighthe

Earth.

The ideabehind theexperiment isvery simple, although theprecisionwithwhich Cavendish made his measurements required the touch of a brilliantexperimenter.Twosmallballsarehungfromeitherendofalongbeam,whichisitselfsuspendedbyathinwire.Twomuchbiggerballsarethenmovedclosetothesmallballs,oneoneachsideofthebeam.AccordingtoNewton’stheory,thelargeballswillexertagravitationalforceonthesmallballs,therebycausingthe

beamtotwistslightly.Cavendishmeasuredthetwist,andusedthemeasurementtodeterminetheforcethatthelargeballsexertonthesmallones.7Toavoidanyexternaldisturbances,heputtheapparatusinaclosedroom.The6-footwoodenbeamwassuspendedfromthepointmarkedFinFigure3.7,andthetwosmalllead balls of 2-inch diameter were hung by threads inside the boxes markedDCB.CavendishwasabletomeasurethetwistofthebeamusingtwotelescopesT trained on ivory scalesmarked in gradations of hundredths of an inch. Thelarger 12-inch diameter lead balls, labelled W, were hung by copper rods(labelled r at the top andR at the bottom) andmoved into place by a pulleysystem operated from outside the room. Bymaking it possible to operate theapparatusremotely,CavendishimprovedMichell’soriginaldesignsubstantially,isolatinghisapparatusasmuchaspossiblefromunwanteddisturbances.

We’vegoneintosomedetailbecausewelovethissortofthing.Cavendish’sexperiment is very modern in many ways. It is a high-precision apparatusdesigned to measure the tiny gravitational force exerted by the heavy balls–aforce that amounts to no more than the weight of a grain of sand. Successdepended on Cavendish exercising extreme care, observing with extremeaccuracy, and developing a solid understanding of the principal sources ofuncertainty.He considered the effects ofmagnetism, temperature variations intheroom,variabilityinthestiffnessofthesuspensionwire,thegravitationalpullof the rods from which the heavy balls were suspended, air currents, thegravitationalattractionofthewoodencaseontheballsandthebeams,andmore.He operated according to the same scientific principles as the Seymours atOgmore-by-Sea,butwithaquitestunningattentiontodetailthatwasmandatedbythedelicacyandsubtletyofthepropertyofNaturehewishedtomeasure.

Inthefinalanalysis,hedeterminedthemeandensityoftheEarthtobe5.45times that of water, i.e. 5.45 grams/cm3. As Cavendish noted, in respectfullycircumspecttones:

AccordingtotheexperimentsmadebyDr.Maskelyne,ontheattractionofthehillSchehallien[sic],thedensityoftheearthis4½timesthatofwater;which differs rather more from [my measurement] than I should haveexpected. But I forbear entering into any consideration of whichdetermination is most to be depended on, till I have examined morecarefullyhowmuch[my]determinationisaffectedbyirregularitieswhosequantityIcannotmeasure.

Cavendish was taking care not to rush to conclusions but, as futuremeasurementsrevealed,hewasbangonthemoney.

Charles Hutton, the mathematician who had weighed the Earth using themeasurements from theSchiehallion experiment, remained extremely scepticalof Cavendish’s more delicate measurement to the day he died. In 1801,encouraged by Hutton, the Scottish philosopher-mathematician John Playfairconducted a new lithological survey of Schiehallion and, in 1811, revisedHutton’soriginalmeasurementupwards to4.71.Twoyearsbeforehisdeath in1823,and11yearsafterCavendishhaddied,Huttonmadehisviewsveryclearin his paper ‘On the mean density of the Earth’. ‘From the closest andmostscrupulousattention,’hewrote,‘thepreference,inpointofaccuracy,appearstobedecidedly infavourof the largemountainexperimentover thatof thesmallballs.’ He spoke condescendingly of Cavendish’s ‘pretty and amusing littleexperiment’ and sniffed at the veracity of results ‘produced by machinery socomplex’ and ‘calculated by theorems derived from intricate mathematicalinvestigations’.

Cavendish, though, was right. Today, the best measurement of the meanspecific density of the Earth is 5.515, which is only 1.2% different fromCavendish’s result. Cavendish’s ballsmay have been small, but hemore thanmadeupforitwithcarefulattentiontodetailandasteadyhand.

Thereisaninterestingpostscripttothisstory.In2007,anexpertinoilandgasexploration called John Smallwood revisited the Schiehallion measurement,using modern methods to determine the geometry and composition of themountain. He was provoked by a challenge issued in Charles Hutton’s 1821paper, inwhichHutton stated categorically that the Earthwas ‘very near fivetimesthedensityofwater;butnothigher’,before throwingdownthegauntlet:‘Letanyperson,whodoubts,lookoverandrepeatthecalculations…andtryifhecantofindaninaccuracyinthem.’Smallwooddidjustthat.Hisre-analysis,published in the Scottish Journal of Geology, concluded that Maskelyne’soriginal measurement allows one to conclude that the specific density of theEarthis5.48withanuncertaintyof0.25.

WepresentedCavendish’snumberashepublishedit,intermsoftheaveragedensity of theEarth.Hismeasurement corresponds to amass for theEarth of5.90×1024kg.Themodernvalueis5.97×1024kg.WhataringingendorsementofthebrillianceofCavendish.Ifyourecall, theSeymourfamilymeasurement,togetherwithourguessthattheEarthisauniformsphereofgranite,gaveusa

massof3.4×1024kg.Notbadforadayonthebeach.Actually, things get even better than ‘merely’ measuring the mass of the

Earth.Becausenow thatwehave themass of theEarthwe candetermine thegravitational constant, G = 4.0 × 1014 m3/s2/5.97 × 1024 kg = 6.7 × 10−11m3/s2/kg. And withG we hit the jackpot, because it means that nowwe canweigh anything in the Universe that has something orbiting around it. Forexample, sincewe also know that the Earth orbits the Sunwith a periodT =365.25daysatadistanceR=150millionkm,wecandeduce(usingtheformulainBox5)thattheSunisabout330,000timesmoremassivethantheEarth.Andwecankeepgoing.

Jupiter has a large collection of moons, the brightest of which werediscoveredbyGalileoin1610.Byobservingtheirorbits,wehavemeasuredthemassofJupitertobe1.898×1027kg;acolossalgasgiantworld318timesmoremassive than the Earth. We can observe clouds of gas, glowing in the radiospectrum, orbiting around distant galaxies, and this allows us to measure themassof thegalaxies.Andromeda, thenearestgalaxy to theMilkyWay, is1.5trilliontimesthemassoftheSun.Onlyasmallfractionofthismassisvisibletousintheformofstars,however,whichhasledastronomerstosuggestthatthereis an ocean of unseen dark matter, probably in the form of new, as yetundiscoveredsubatomicparticles,permeatingthegalaxies.Thereisnowagreatdealofindependentevidencefortheexistenceofdarkmatterfromstudiesoftheevolution of the Universe and of Cosmic Microwave Background radiation(we’ll lookat this inmuchmoredetail lateron),butweshouldpointoutherethatdarkmatterwasfirstdiscoveredusingNewton’slawsandourknowledgeofG.And, perhaps strangest of all, at the heart of theMilkyWay galaxy in thedirectionoftheconstellationofSagittarius,therearestarsknownastheS-starswithextremeorbitsaroundadense,compactobject.UsingNewton’s laws, theobjectismeasuredtobeover4milliontimesthemassofourSun.Astronomersbelieve this exotic object to be a supermassive black hole, 26,000 light yearsfromEarth,devouringdust andgas from the richclouds thatdrift through thedensecentralregionsandspewingradiationoutacrossthegalaxy.

Newton’slawsaretreasures.TheywerethefirstuniversallawsofNaturetobe discovered, and they have allowed us to discover unimaginable thingsunfathomablyfaraway,startingfromthebeachatOgmore-by-Sea.

4.THEDISTANCETOTHESTARS

Thestarsaretinyspecksoflightinthedarkness.Unliketheplanets,theyexhibitnoextravagantloopsonthesky,and,fortheoverwhelmingmajorityofthem,wehavenotelescopecapableofresolvinganydetailontheirsurfaces.BeyondthenightlycirculararcsacrosstheskyinducedbytheEarth’sspin,theyappeartobeimmobile, featurelesspoints.Andyetweknow thedistance toeachandeveryone. As we will see later in the book, this ability to map the cosmos withprecisionistheRosettaStonethatwillallowustoexploretheUniverse’soriginandevolution:theskiesareteemingwithinformationandwehavelearnedhowtodecodeit.

Before we cast our gaze beyond the solar system, we’d like to begin bymakinggoodonourpromise in the lastchapter tomeasure thedistance to themostdistantplanet inoursolarsystem:Neptune.Themethodwe’lluse isalsothemethodbywhichwedeterminethedistancestotheneareststars.Itiscalledtheparallaxmethod.

Parallaxisaneffectwithwhicheveryoneisfamiliar;itisthereasonhumanshave two eyes. If you hold one of your fingers up in front of your face, andalternately close one eye and then the other, your finger will appear to shiftagainst the background.The closer your finger is to your face, the greater theshift.As I (Jeff)amwriting this Ihavedecided touseparallax tomeasure thelength ofmy own arm. I (Brian) am not in the least surprised.Withmy armextended,Ilookatmyindexfinger,firstwithmylefteyeclosedandthenwithmy right eye closed. I notice that my finger moves by about 8 degrees. I’vedownloaded the imageofa largeprotractorontomy laptop for the job.Figure4.1showsthegeometry,and,knowingthatthedistancebetweenmyeyes–whichI measured with a ruler–is about 6.5 cm, I have deduced that my finger waslocated a distance 3.25/tan(4°) = 46 cm away. The parallax shift of nearbyobjects,discerniblebecausewehavetwoseparatedeyes,isoneofthepiecesofinformationourbrainsexploittoestimatedistances.Itisatwo-eyedabilitythathasbeenselectedforinmanyanimals.

Unfortunately,youcan’testimate thedistance toNeptunebywinking:bothbecauseyourhead isn’tbigenough, andbecauseNeptune isnotvisible to the

nakedeye. In fact, it is so faint that itwasn’tdiscovereduntil1846,when theFrenchmathematicianUrbain LeVerriermade a prediction of a new planet’sposition in the sky using Newton’s laws, based on his own observations ofirregularities in the orbit of Uranus. He sent the prediction to the Germanastronomer Johan Gottfried Galle, who on the evening of 23 September dulyfound it after only an hour of searching. Fortunately, Le Verrier was able tocircumventhislackofasufficientlylargehead–andsotoocanwe,asFigure4.2illustrates.

ThepositionofNeptune against the fixedbackground stars changesdue toparallaxastheEarthmovesaroundtheSun,providinguswithdifferentvantagepoints every evening. This figure shows the situation when Neptune is ‘inopposition’,whichmeansitislinedupwiththeEarthandtheSun.Wechosetomake the measurement close to opposition because it makes the mathematicseasier,but it isnothard toperform thecalculationat anyother time. In2014,this planetary alignment occurred on Friday 29 August, and we asked KevinKilburn, an amateur astronomerwith theManchesterAstronomicalSociety, tophotograph Neptune for us around the time of opposition. He obtained fourphotographs on 19 August, 29 August, 9 September and 21 September–veryprobably the only clear nights in late-summer Manchester that year. Wesuperimposed the photographs on top of each other, lining them up using thebackgroundstars;theresultingimageisshowninFigure4.3(p.85).

Figure4.1OverheadviewofJeff’sheadandfinger.Direction1correspondsto

thedirectioninwhichhisfingerappearswhenhislefteyeisclosed,direction2

whenhisrighteyeisclosed.

Figure4.2Justlikethepreviousfigure,excepttheeyesarereplacedbytwo

positionsoftheEarthinitsorbitaroundtheSunandthefingerisreplacedby

Neptune.BymeasuringhowmuchNeptuneshiftsitsangularpositionwith

respecttothedistantstars(theyaresofarawaythattheirangulardeflectionis

muchsmallerthanNeptune’s),wecandeducethedistancetoNeptune.

TheprocedurefordeterminingthedistancetoNeptuneispreciselythesameas the one Jeff used to calculate the length of his arm.We need to know the‘distance between the eyes’, which is the distance the Earthmoves from onephotographtothenext,andtheanglethroughwhichNeptuneshiftsagainstthestarrybackgroundduringthistime,whichwecangetfromthephotographs.Wedescribeeverystepof thisprocess indetail inBox6 (pp.85–7), andwehopeyoudecidetofollowitthrough,becauseit’salovely,verysimplemeasurementthat you might choose to perform for yourself. Our calculation, using onlyKevin’s photographs, gives the distance of Neptune from the Sun as 30.33Astronomical Units (AU). The official figures tell us that Neptune was at adistanceof29.96AUfromtheSunon29August2014,whichdiffersfromourmeasurementbyabout1%.

ThisparallaxmethodcanonlydeliverthedistancetoNeptuneintermsoftheEarth–Sun distance (which is 1 AU). If we want to express astronomicaldistancesintermsofmetresthenweneedtofigureoutwhat1AUactuallyisinmetres.Inthelastchapterwedealtwiththisbynotingthatwecanascertaintheradius of Venus’ orbit around the Sun by two different methods. First (asillustrated inFigure3.4)using trigonometry(the result isaround0.7AU)and,second, by bouncing radar off the planet (the result is 108million km). That

information is sufficient toestablish that1AU isaround150millionkm.TheAstronomicalUnitisoneofthemostimportantnumbersinastronomy,becauseit unlocks the whole distance ladder. What’s more, the pre-radar quest for aprecisionmeasurementofanabsolutedistancebetweenanytwocelestialobjectsisoneofthegreatstoriesinthehistoryofscience,spanningmanycenturies.

In determining distances using parallax, we use two vantage points on theEarth’ssurface,ortwopointsontheEarth’sorbitaroundtheSun–theequivalentof twoeyes.For example, on a clear night,we could take twophotographs atpreciselythesametimefromtwodifferentpointsonthesurfaceoftheEarth.Ifwe know the distance between the cameras, then we could use parallax todeterminethedistancetotheobjectwephotographed.ThiswouldbedifficulttodoforNeptune,though,becauseitistoofarawayandtheparallaxshiftwouldbeverysmall.For this reason, inourmeasurementof thedistance toNeptune,we increased the distance between the two vantage points by waiting for theEarthtomovearoundtheSun.ButmeasuringdistancesusingtheviewfromtwodifferentpointsonEarthisviableforlessdistantplanets.

In1672,whenMarswasinopposition,thegreatItalianastronomerGiovanniCassiniandhiscolleagueJeanRichermadeameasurement for theparallaxofMars. Richer observed the position of Mars from French Guiana in SouthAmerica,andCassiniworkedfromParis.Theyknewthe‘distancebetweentheeyes’with reasonableprecision, and from this theywere able to calculate thattheAstronomicalUnitisapproximately21,700Earthradii.Itwasaprettygoodattempt.Themodernvalueis23,455Earthradii.

There is a whole book to be written on the numerous expeditions theexplorers and astronomers of the seventeenth, eighteenth and nineteenthcenturies embarkedupon to refine themeasurement of theAstronomicalUnit.We aren’twriting that book, butwe cannot resistmentioning the story of theFrench astronomer Guillaume Le Gentil. One of the classic, high-precisionparallaxmeasurementscanbemadeduringatransitofVenus,whichiswhentheplanet crosses the face of the Sun as seen from Earth. Because of parallax,observations of the transit from two different points on the Earth will differ,somethingwhichcanbeexploitedtoascertainthedistancetoVenus.TransitsofVenuscome inpairs,8yearsapart, and thendonot repeat foroveracentury.ThemostrecenttransitoccurredinJune2012;therewillnotbeanotheroneuntilDecember2117.SoifyoureallywanttomakeaprecisionmeasurementoftheAstronomicalUnit, youdon’twant tomiss it–and, back in themid eighteenthcentury,GuillaumeLeGentilcertainlydidn’t.

In March 1760, Le Gentil left Paris and headed for the Indian city ofPondicherryasoneofaninternationalteamofoverahundredobserverssentoutacrosstheplanettomakemultipleobservationsoftheVenustransitof1761.HemadeittoMauritius,thenknownastheIsledeFrance,inJulyofthatyear,butoneoftheregularwarsbetweenFranceandBritainmadehisonwardjourneytoodangerous.Finally,inMarch1761hemanagedtoboardashipboundforIndia,andalthoughtheshipwasblownoffcoursehestillmadeittoPondicherrywithdaystospare.Unfortunately,theBritishhadoccupiedPondicherryandtheshipcouldn’tland,sothecaptainswungitaroundandheadedbacktoMauritius.Theday of the transit was beautifully clear, but with the ship still in open sea,pitchingandrolling,preciseastronomicalobservationswereimpossible.

Undaunted,LeGentil decided towait eight years in and around the IndianOceanforthetransitof1769,andafterspendingsometimemappingthecoastofMadagascarheheadedforManilainthePhilippines.ButtheSpanishinManilawere not helpful, so he returned to Pondicherry and built an observatory inreadinessforhisvitalmoment.Whenthemorningof4June1769dulyarrived,itturnedouttobetheonlycloudydayinweeks,andhemissedthetransit.Afterafew miserable months, he finally boarded a ship bound for home, but anoutbreakofdysenteryandseverestormsresultedinhimbeingdroppedoffatLaRéunion, off the eastern coast ofMadagascar.Hewas finally able to board aSpanish ship to take him back to Paris, where he arrived in October 1771 todiscover that he had been declared legally dead, his wife had remarried, hisrelativeshadsoldhisestate,andhehadlosthispositionintheRoyalAcademyofSciences,whichhaddispatchedhimontheexpeditioninthefirstplace.Thismustsurelyhavebeenthemomentforwhichthephrase‘Forfuck’ssake!’wasinvented. To put your mind at rest, Le Gentil was reinstated at the RoyalAcademyofSciences,remarriedandlivedahappylifeforafurthertwenty-oneyears.

This story goes to show the sheer commitment of these pioneers, whounderstoodtheimportanceofmeasuringthedistancebetweentheEarthandtheSun and thereby unlocking the distance scale that allows us to measure theUniverse today.Thesescientistsandexplorerswerebrilliant,dedicatedpeople,who–asthecaseofLeGentilshows–spent their livesattemptingtoacquiretheextremelyhard-wonknowledgeuponwhichourunderstandingof theUniversenowrests.

Using theparallaxmethod,wemadeaprecisemeasurementof thedistance to

Neptune, themost distant planet in the solar system.This raises the question:howfaroutintothecosmoscanwegowiththeparallaxmethod?Well,wecancertainlymeasuretheparallaxshiftsoftheneareststars.BecausetheEarthorbitsthe Sun, the closest stars will oscillate back and forth across a small angularregion of the sky, and the extent of that back-and-forth motion allows us todeterminehow far the star is away.With this inmind, themaximumpossiblebaseline1foraparallaxmeasurementisthediameteroftheEarth’sorbit(2AU),which corresponds tomakingmeasurements of the parallax shift of a star sixmonthsapart.The firstdeterminationof thedistance toa starwasmadeusingthis method in 1838 by the German mathematician and astronomer FriedrichBessel, who observed that the star 61 Cygni has a parallax angle of 0.314arcseconds(whichmeanstheextremesinitspositionintheskyareseparatedby0.628arcseconds,becauseastronomersdefine theparallaxangle tobehalf theangularshift).Using this,heconcluded that itsdistancefromtheEarth is10.4lightyears,2whichisclosetothemodernvalueof11.4lightyears.

BOX6.MEASURINGTHEDISTANCETONEPTUNE

KevinKilburntookthephotographsofNeptuneshowninFigure4.3using a 300 mm Zeiss Pentacon lens on a Canon 550D digitalcamera, attached to a Skywatcher EQ5 equatorial mount. Themount allows the camera to track with the rotation of the Earth,meaningthata20-secondexposurecanbemadewithouttheimagebeing blurred. The four photographs in the figurewere taken overthe course of a month in August and September 2014. We usedAdobe Photoshop to stack the photographs on top of each other,suchthatallofthebackgroundstarslineup.ThemotionofNeptuneacrosstheskyisevident:itistheonlythingthatmoves.Itispossibleto make parallax measurements of the brighter planets usingvirtuallyanydigitalcamera–andhere,anormalfixedphotographer’stripod can be used, because the exposure times can be muchshorter.Despite needing slightlymore expensive gear, though, it’sworth using Neptune; the maths is slightly simpler, becauseNeptune’smotionrelativetothebackgroundstarsismainlyaresultoftheEarth’smotionaroundtheSun.

Figure4.3Neptune(circled)movingacrosstheskyrelativetothestarsin

August/September2014.PhotographsbyKevinKilburnofthe

ManchesterAstronomicalSociety.

If we know the speed at whichNeptunemoves across the skywhen it is in opposition, thenwe can easilywork out how far it isaway from theSun.Toseehow this isdone, lookagainatFigure4.2.Forthemomentwewill ignorethefactthatNeptuneisorbitingaroundtheSun.Neptune’sorbitwon’taffectourmeasurementstoomuch,because it takes165years tomakeonecomplete circle sofromourperspectivewon’tmovemuchoverthecourseofamonth–butinanycasewewillcorrectforthislateron.ThefigureshowstheEarth at two points in its orbit around the Sun, marked A and B.

Thesecould represent twoof ourphotographs.ThecorrespondingshiftinthepositionofNeptuneagainstthestarsistheanglemarkedα.ThedistancebetweenpointsAandB isapproximatelyequal to(α/360°)×2πR,whereR is thedistancefromtheEarth toNeptune–whichofcourseiswhatwewanttoknow.Wecanalsocalculatethedistance between points A andB anotherway: it is approximatelyequal to thedistance theEarth travelsaround theSun in the timebetween photographs, i.e. (the circumference of the Earth’s orbitaroundtheSun)×(thetimebetweenmeasurements)/(1year).

Thismeansthat

R= (the timebetweenmeasurements) / (1year) / (theangleα bywhichNeptunemovedacrosstheskybetweenmeasurements/360degrees)×1AU

All that remains is todetermineα from thephotographs.The tablebelow lists the pixel positions ofNeptune, obtained by loading the5184×3456pixelimagesintoPhotoshop.Wewillfocusonthefirsttwomeasurements,takenon19and29August,thelatterbeingtheday when Neptune was in opposition. We need to convert thechange inpixel co-ordinates into theangle throughwhichNeptunemovedonthesky.Thechangeinthehorizontalpixelco-ordinateis(2950–2640)=310,andthechangeintheverticalpixelco-ordinateis(1756–1629)=127.Wecanconvertthisintoanangleifweknowthe image’s fieldofview,bywhichwemean theangularportionoftheskycoveredbyoneof thephotographs.Wecandetermine thefieldofviewasfollows.ThesensoronaCanon550Dcamerais22.3mm × 14.9 mm across. If you decide to make your ownmeasurement, you’ll need to look inyourcamera’susermanual tofindthesensorsize.Figure4.4showstworaystravellingfrompointsintheskyintothe300mmfocal-lengthlensandlandingontheverytopandverybottomofthesensor.

Thetangentoftheanglemarkedθ(theta)isequalto(14.9mm/2)(300 mm), which means θ = 1.42º. This means that the top andbottomofanyphotographtakenwiththesesettingshasanangularspreadof2.845ºdegrees.Wecando thesame for lightcoming infromtheleftandright.Inthatcasethetangentoftherelevantangleis (22.3mm2) (300mm),whichgivesanangular spreadequal to4.26º. Kevin’s photos therefore cover a patch of the sky equal to4.26×2.845squaredegrees,whichisapproximately13400thoftheentire sky. The images shown in Figure 4.3 correspond to azoomed-inportionofthis.Becausethechangeinangleissosmall,wecanusePythagoras’Theoremtodeterminethat,betweenthosetwodates,Neptunetravelledacrosstheskythroughanangleequalto degrees.

Figure4.4DeterminingthefieldofviewofKevin’scamera.

Armed with this angle we can use the underlined formulaoppositetodeducethatthedistancefromEarthtoNeptuneis1AU

×10.03days365.26days(0.2754/360)=35.9AU.Butwearebeingpremature.Wecanandshouldaccountforthe

fact thatNeptune creeps slowly across the skyas it tooorbits theSun. This corresponds to a steady movement across the sky inexactlytheoppositedirectiontothatgeneratedbytheEarth’smotionaroundtheSun.Neptunethereforemovesthroughanangleof360/164.8 /365.26=0.00598degreesperday.Thismustbeadded totheanglewecomputedabove,andmeansthatthecorrectparallaxangle is (0.2754+0.0598)=0.3352degrees.With this correction,we can use the underlined formula to deduce that the distance toNeptunewas29.49AU,whichmeans that, since theplanetswerealigned,thedistancefromtheSuntoNeptunewas30.49AU.

Itispossibletodoalittlebitbetterthanthis,andaccountforthefact that Neptune was not quite in opposition on 19 August 2014because the Earth was not yet in line. Doing the trigonometry toaccountforthischangesthe30.49AUto30.33AU.Wedonotdoit,but theoptimal useofKevin’s photoswould alsomakeuseof theothertwomeasurements.Ifyouwanttodosothenbeawarethat21September 2014 is over 3 weeks after opposition, so thetrigonometryisnotasstraightforward.

Parallaxmeasurements are so fundamental that astronomers have created ameasure of distance directly related to them. If a star has a parallax of 1arcsecond, it is defined as being 1 parsec away.These angular shifts are verysmall,whichiswhywedonotperceiveashiftintheshapeoftheconstellations.Aristotle used the lackof a perceivedparallax shift of the stars to argue for astationary Earth; in large part his logicwas sound, but he underestimated thedistances involved and instead of saying that ‘no parallax implies theEarth isstationary’, he ought to have considered the alternative hypothesis, that ‘noobserved parallax means that the stars are a long way away’. To put thesmallness of stellar parallax angles into context, the angular size of the fullMoonisapproximately2000arcsecondsandtheneareststar toEarth,ProximaCentauri,exhibitsaparallaxofjust0.762arcseconds,whichmeansitis1/0.762=1.31parsecsaway;1parsecisequalto3.26lightyears,whichisthedistancethat light travels in3.26years.3Even thenearest starsaremind-bogglingly faraway.

Withmoderntechnology,parallaxmeasurementsassmallas10millionthsofonearcsecondcanbemade.ThemostaccuratemeasurementstodatearebeingmadebytheEuropeanSpaceAgency’sGaiasatellite.ItwaslaunchedintoorbitaroundtheSuninDecember2013fromFrenchGuiana,thelocationfromwhichJean Richer helped to make the first accurate parallax measurement of Marsalmost350yearsearlier.Gaia’sprecisiontranslatesintoparallaxmeasurementsouttodistancesoftensofthousandsoflightyears,bringingalargefractionofthestarsintheMilkyWaywithinourreach.

Whenweconsiderdistances togalaxies lyingbeyondourownMilkyWay,we’ll have to deal with distances measured in megaparsecs, or Mpc. Amegaparsecisamillionparsecs:3.26millionlightyears.Theminusculeparallaxanglesassociatedwithsuchdistantobjectsare impossible tomeasure,so ifwewant to map the Universe beyond theMilkyWay, we need to develop othermethods. Even so, it remains the case that all distance measurements inastronomyultimatelyrelyonparallaxmethods–ifnotdirectly, thentocalibratethem.

Onmoonlessnightsawayfromlights,theskyisablazewithstars,andeveryonevisible to thenakedeyeis insideourgalaxy.Therearesomanyworlds inthe rich starfields sweeping through the constellations of the galactic plane–Cassiopeia,Perseus,Sagittarius–thatthelightfromabillionsunsmergesintoacontinuousglowingarchthat,whenviewedfromthetruedarknessandstillairofhighmountainordesert,deliversthesensationofstandingaloneontheedgeofagalaxy. The MilkyWay is vast beyond imagination. ‘Somewhere, somethingincredibleiswaitingtobeknown,’wroteCarlSagan.

Figure4.5TheMilkyWayfromtheAustraliandesert.

Close to the ‘W’ of Cassiopeia, in the constellation of Andromeda, theunaidedeyecan,just,catchafaintglimpseofamistypatchthatliesbeyond;itistheAndromeda galaxy, our closest large galactic neighbour. There are peoplealivetodaywhowerebornbeforethiswasknown.

Until themorningafter thenightof5October1923, itwaspossible to takethe view that our home galaxy constituted the entireUniverse, and that thosemistypatcheswerestructuresinsidetheMilkyWay.Wecandatewithprecisionthemomentwhenthelastvestigesofourpre-Copernicancentralitywerekickedaway, because on that evening, atMountWilson in California, the AmericanastronomerEdwinHubbletooka45-minuteexposureoftheAndromedaNebulawith the100-inchHooker telescope. In thenow-famousphotograph, shown inFigure4.6,Hubblerecognizedthreebrightstarsthathadnotbeenpresentinhisprevious photographs. He assumed that they were novae–bright flares fromsmall,densewhitedwarfstarscausedbytheaccretionofmatterfromanorbitingstellar companion–andmarked them on the photographic plate with the letter‘N’.Thefollowingday,checkingthroughtheobservatory’sarchivephotographsof Andromeda to confirm his result, he noticed that one of the novae wassometimespresentandsometimesnot;itappearedanddisappearedwithaperiodof approximately thirty-one days. He immediately realized that this wasn’t anova, but a type of star known as a Cepheid variable, which changes itsbrightness likeclockwork.Heexcitedlycrossedout the ‘N’onhisphotographandwrote‘VAR!’nexttothestar.

TounderstandwhyHubblewasexcitedenoughtouseanexclamationmark,wemuststepbackafurtherfifteenyears.In1908,anastronomerattheHarvardCollegeObservatorycalledHenriettaLeavittpublishedapaperwithherfellowastronomer Edward Charles Pickering, in which she reported a relationshipbetweenthebrightnessofCepheidvariablestarssittingintheSmallMagellanicCloud4and theirperiod.Theperiod is the lengthof timeoverwhich thestar’sbrightnessvaries, i.e. the timefrombright todullback tobrightagain.Leavittexpressedtherelationshipinsimple,preciseprose:‘It isworthyofnotethatinTable VI the brighter variables have longer periods. It is also noticeable thatthose having the longest periods appear to be as regular in their variations asthosewhichpass through theirchanges inadayor two.’Leavitt’sobservationbecamethebasisofwhat isnowknownas theperiod–luminosityrelation.Thekeyobservation is that theperiodofavariablestar is indicativeof its inherentbrightness.

Figure4.6ThephotographicplatetakenbyEdwinHubbleatMountWilson

Observatoryon5October1923.

Tounderstandwhythiswasoneofthemostimportantbreakthroughsinthehistoryofastronomy,recallourproblemwiththemeasurementofthedistancestoveryremoteobjects: theirparallaxshiftsaretoosmall tobedetectable.Thislookslikeaveryseriousproblematfirstsight.Starsarepointsoflightwithnodiscerniblestructure,andthedistantonesdonotappeartomove.Allwehaveistheirlight.Themostobviousdifferencebetweenstarsistheirbrightness.Ifonestar isbrighter thananother, twoobvious reasonsspring tomind: thestarmayactually be brighter, or it may be inherently less bright but closer to us. Tosimplifythelogic,imagineifallstarswereofthesameintrinsicbrightness:wecouldthenusetheobservedbrightnessofastartodeterminehowfarawayitis

relative to the other stars. For example, a star that was twice as far away asanother would be ¼ as bright.5 Now, if we used parallax to determine thedistancetojustonenearbystarthenwecoulddeterminethedistancetoanyotherstarbycomparingitsbrightnesstothenearbystar.

The trouble is that the inherentbrightnessofastar typicallydependson itsmassandage.Siriusisthebrighteststarinthenightsky.Itisonly8.6lightyearsaway,butitistwiceasmassiveand25timesmoreluminousthanourSun.Rigel,intheconstellationofOrion,istheseventhbrighteststarinthenightsky,butitisaround860lightyearsaway.Rigelisasupergiantstar,withadiameterarounda hundred times that of the Sun, and it is 200,000 timesmore luminous. ThemostluminousknownstarisintheLargeMagellanicCloud.ItisaWolf-RayetstarknownasRMC136a1.Itshineswiththebrightnessof8.7millionsuns,andis315timesmoremassive.Itcanbeseenwithasmalltelescope,eventhoughitis 163,000 light years away. So you see the problem with using observedbrightnessasaproxyfordistance.

TheimportanceofLeavitt’speriod–luminosityrelationisthatitsidestepsthisproblembyprovidingasampleofstarswithknownrelativebrightness.Thisissoimportantthatitisworthspellingitoutinsomedetail.ImaginethattherearetwoCepheidswithprecisely the sameperiod.According toLeavitt, theymustalsohave the same intrinsicbrightness.Nowsuppose that oneof them is fourtimesasbrightastheother,asseenfromEarth.Immediately,wecanconcludethatthebrighterstarmustbehalfasfarawayasthedimmerstar.UsingLeavitt’speriod–luminosity relation, we can therefore determine the distance to anyCepheid variable star in the sky, including the one Hubble identified inAndromeda,providedthatweknowthedistancetoatleastoneofthem.

But we already know how this can be done: we should look for nearbyCepheids whose distance can be determined by parallax. In 1913 the DanishastronomerEjnarHertzsprungmade the firstmeasurementof theparallaxof aCepheidvariable.ThestardeltaCephei,fromwhichCepheidvariablestaketheirname, has a parallax of 3.77 milli-arcseconds, which puts it at a distance ofapproximately890lightyears(thisisthemodernmeasurement).Ithasaperiodof5.366341days.Thisisthe‘standardcandle’,thestarthatHertzsprungusedtocalibrateLeavitt’sruler.

(A brief historical aside: scientists are only human. In his original paper,HertzsprunggottheparallaxtodeltaCepheicorrect,butmadeasimpleerrorinhis estimate of the distance to the Small Magellanic Cloud using Leavitt’srelation,whichhequotedasonly3000 lightyears.Thiswasa trivialmistake,

whichwasquicklynoticed,butforsomereasonHertzsprung,oneofthegreatestmodernastronomers,neverbotheredtocorrectitintheliterature.)

Edwin Hubble knew all about these measurements on the morning of 6October1923–hencehisexcitedlyscribbled‘VAR!’HisdiscoveryofaCepheidin the Andromeda nebula allowed him to determine the distance, which hecalculated tobea shockingly large900,000 lightyears.ModernmeasurementsofthedistancetoAndromeda,basedinpartonabetterunderstandingofCepheidvariable stars,6 put the giant spiral at an even greater distance of 2.54millionlight years. Historically, the factor of nearly three makes little differencebecause,whicheverwayyoulookatit,HubblehaddeterminedthatAndromedasits well outside of the Milky Way (which is only 100,000 light years indiameter).HehadshownconclusivelythatAndromedacouldnotbeanebula–acloudofstars,gasanddustinsideourgalaxy.Itisaseparateislandofstars,sodistantthatthelightreachingusnow,whichyoumaybeabletoglimpsetonightwithadecentpairofbinocularsfromyourbackgarden,beganitsjourneylongbeforetherewassuchathingasahumanbeingonplanetEarth.

Figure4.7TheHubbleeXtremeDeepField(XDF)image.Almosteverypatch

oflightisagalaxy,andthereareover5500galaxiesvisibleinthispicture.

Figure4.8M81,Bode’sGalaxy,andM82,theCigarGalaxy.Located12

millionlightyearsaway,theyarevisiblewithbinoculars.Intensestarformation

inM82iscausedbyitsinteractionwithM81:thetwogalaxiesareseparated

byonly150,000lightyears.

Ifyoudoventureout tonightwithyourbinoculars,youmaybeable toseeothernearbygalaxies;M81andM82inUrsaMajorareagoodstart,atadistanceofaround12millionlightyears,whichweknowthankstoCepheids.7

More powerful telescopes reveal greater numbers of galaxies. Figure 4.7shows the Hubble eXtreme Deep Field (XDF) image, a very long-exposurephotograph taken by the Hubble Space Telescope during 2003 and 2004. Itcorrespondstoatinypieceofsky(approximately1/30thofonemillionthoftheentiresky),andstillitcontainsover5500visiblegalaxies.TheXDFphotographcontainsawealthofinformation:dataaboutadeepsliceofspaceandtime.ItisarecordofthelightthatfellonHubble’smirror,virtuallyallofwhichoriginatedfromgalaxiesbeyondourown.

Now we have the distance to nearby galaxies, using Leavitt’s distance–luminosity relation for Cepheid variable stars, let’s do a quick back-of-the-envelopecalculationbasedonthenumberofgalaxiesvisibleintheHubbleXDF

image.IftheHubbleXDFistypicaloftheentiresky–andwehavenoreasontothinkotherwise–then thereareapproximately30,000,000×5500=165billiongalaxies in the observable Universe. If we further assume that the averagedistancebetweengalaxies is the sameas thedistancebetween theMilkyWayandAndromeda,thenwecanestimatethesizeoftheobservableUniversetobe(165×109)1/3×2.5millionlightyears=14billionlightyears.Thisisobviouslya very rough estimate,8 but it is one that gives us a sense of the scale of theUniverse.

Since light takes1billionyears to travel 1billion light years, our estimateimplies that theUniverse is probably at least 14billionyearsold, because thelight from themost distant galaxiesmust have had sufficient time tomake itacrosstheUniverseandontoHubble’smirror.SimplybycountingstarsandbymeasuringthedistancetoAndromeda,weareledtocontemplateaUniversethatisoldenoughtocontaintheSunandEarth,whichwehavealreadydatedtoover4billionyears inage.Obviously theUniversemustbeolder than theSunandEarth, but it is nice to see how very simple reasoning leads to the correctorderingoftheirages.

There are only a handful of measurable properties of the most distantgalaxies, because astronomers have only the light these galaxies emit toworkwith.For a relativelynearbygalaxy,we candetermine its size ifweknow itsangularsizeontheskyandhowfarawayitis(whichwecando,ifwe’vebeenabletofindaCepheidvariablestarorsomeotherobjectofknownbrightness).Forthemoredistantgalaxies,however,resolvingindividualstarsisnotpossible.ThismeansthattheCepheidmethodwillnotbeanyuseforgalaxiesthataretoofaraway.Fortheseweneedanothernewtrick.

Therearecertaintypesofastronomicalevents,knownasType1Asupernova,thatareverybrightandcanbeseenatverylargedistances.Tosaytheyareverybright issomethingofanunderstatement,becauseasingleType1Asupernovashinesbrighter thanentiregalaxiesof stars, albeit for just a few fleetingdays.Thistypeofsupernovaoccurswhenastargainsmassbyconsumingmatterfromanearbystar.Oncethestarbecomes40%moremassivethantheSunitstartstocollapseinonitselfundertheweightofitsowngravity.9ThefactthatallType1Asupernovaehappeninessentiallythesamewaymeansthattheyareallofthesameintrinsicbrightness.JustaswithCepheidvariables,thesecanthereforebeusedas‘standardcandles’todeterminehowfarthey(andtheirhostgalaxies)areaway.Thetroubleisthattheyarerareevents:thesupernovarateinanaveragegalaxy is estimated to be around one per century, and the last Type 1A

supernovatobeclearlyobservedintheMilkyWaywaswaybackin1604.Manymore have been seen beyond the MilkyWay, which is why they are a veryvaluabletoolformeasuringcosmicdistances.

ThereareotherwaystomeasurethedistancestogalaxiesthatdonotrelyontheraregoodfortuneofobservingaType1Asupernova.Wewillmeet twooftheseinChapter6,andusethemtohelpusdosomecosmology.Bothmethodsexploitthefactthatthereisextrainformationencodedinthelightfromagalaxy,beyondthesizeandshapeofitsimageonaphotograph.Thereisalsocolour:notjustthereds,bluesandgreensvisibleinphotographs,buttheprecise,finedetailsofthedistributionofthecoloursinthespectrum.Thismightnotseemlikemuch,butitis.

InBox7(pp.109–13),wegiveabriefexplanationofthewaylightisemittedandabsorbedbyatoms,whichshouldhelpinunderstandingthelastpartofthischapter.For thosewhodon’t fancyreading theBox, theexecutivesummary isthat the light from any star or galaxy contains a characteristic barcodewhichtellsuspreciselywhatthatstarorgalaxyismadeof.Thisbarcodeisuniversal:thechemicalelementsthatmakeupthestarsanddustintheAndromedagalaxyarethesameasthechemicalelementsthatmakeupyourbody.ThebarcodeoftheSun’satmosphere,takenduringasolareclipse,isshowninFigure4.9.Thisspectrumwasmadebypassingsunlightthroughaprism,ortobemorepreciseahigh-precision kind of prism known as a diffraction grating. The peaks in thespectrum signal the presence of different chemical elements in the Sun’satmosphere: hydrogen, helium, calcium, magnesium, iron and traces of otherelementscanbeseen.Thisisobviouslyaveryneatwaytoascertainwhatdistantstarsandgalaxiesaremadeof.

Figure4.9Thesolaremissionspectrum,takenduringthetimeofasolar

eclipse.NoticehowtheSunhasapropensitytoemitlightofveryspecific

colours.Thesecoloursindicatewhichatomsarepresent.

Figure4.10ThegalaxyNGC4535ataredshiftof0.00655.Thisimagewas

takenwiththeHale200-inchopticalreflectortelescopeatthePalomar

ObservatorylocatedinnorthSanDiegoCounty,California.

Nowlet’slookatagalaxy.We’vechosenabeautifulonecalledNGC4535,abarredspiralgalaxyintheconstellationofVirgo.ThephotographinFigure4.10was taken using the 200-inch Hale telescope at the Palomar Observatory inCalifornia. The telescope was completed in 1948, and remained the world’slargestuntil1993.Ifyouknowalittleaboutphotographyorengineering,you’llappreciatewhata superb instrument this iswhenwe tellyou that at theprimefocus of the 20 squaremetre mirror the focal length is 16.76metres with anapertureoff3.3.TheHubbleSpaceTelescopehasalsoobservedseveralCepheid

variable stars in NGC4535, and from these we know the distance to be 52millionlightyears.

JustasfortheSun,wecanexaminethespectrumoflightfromNGC4535.AportionofthatspectrumisshowninFigure4.11,wherewecanseefivestrongemission lines. The strongest is at a wavelength of 6606 ångstroms. Becausegalaxiesareexpectedtobepredominantlyhydrogengas,wemightsupposethatthis line is associatedwith hydrogen atoms. Indeed, in earth-bound laboratorymeasurements there isanemission lineclose to thiswavelength,called theH-alphaline,butonEarth thewavelength is6563ångstroms.This important linecanalsobeseeninthespectruminFigure4.13ofBox7.TheEarth-boundandgalacticwavelengthsdonotquiteagree.Shouldwedoubtourinterpretation?IsthebrightemissionlinefromNGC4535comingfromsomethingelse?

Tohelpunderstandwhat isgoingon, let’s lookat the two lines that lieoneithersideoftheproposedgalacticH-alphaline.Theyhavewavelengthsof6591and6627 ångstroms.They are also close to two similar lines that are familiarfromearth-basedexperiments:apairofemissionlinesfromnitrogenknownastheNIIlines,withlabwavelengthsof6548and6584ångstroms.Nowwehaveaclue: all of these three spectral lines havewavelengths that are around 0.65%bigger than their earthly counterparts. This systematic shift of wavelengths isconfirmedwhenwe study the two smaller lines at 6760 and 6775 ångstroms.Afteraccountingforthe0.65%shift,thesecorrespondtotwoEarth-boundlinesknownas theSII lines,whichareemittedbysulphuratoms,at6716and6731ångstroms. So, the spectrum from NGC4535 looks exactly like a terrestrialspectrum–except that all of thewavelengths have been increased by the samefactorof0.65%.

Wecouldconclude that there issomethingstrangeaboutgalaxyNGC4535.Couldthestructureofatomsbedifferentthere?It’shardtoseewhyorhowthiscouldbethecase.Themysterydeepenswhenwelookatthespectrafromothergalaxies. The barcode patterns from all galactic spectra correspond preciselywith the barcode patterns of atoms detected on Earth, but they are alwaysshifted. Furthermore, they are all shifted by different amounts–but theoverwhelmingmajority are shifted to higherwavelengths, just likeNGC4535.Thisshifttohigherwavelengthsisknownasaredshift.

Astronomers quantify the amount of redshift by the ratio of the shift inwavelength divided by the wavelength of the spectral line as it would bemeasured on Earth. This means that NGC4535 has a redshift of 0.0065.Redshifts can be much larger than this, though. In Figure 4.12 we show the

spectrumof light from3C273,a typeofactivegalaxyknownasaquasar.Theredshiftinthiscaseisfargreater,whichyoucanimmediatelyseebyjustlookingat the plot. In this case, the bright H-alpha line is somewhere closer to 7600ångstromsand the redshift is0.1583.This is confirmedby theother lines (wecanseeseveralotherhydrogenemissionlines).AsinthecaseofNGC4535,weseeabarcodepatternjustlikeonEarth–butthistimewitha15.8%shifttohigherwavelengths.

Fig4.11TheemissionspectrumofNGC4535.Thebiggestbumpisduetothe

presenceofhydrogenandthetwosmallerbumpseithersidearedueto

nitrogen.Thetwobumpsonthefarrightidentifythepresenceofsulphur.

Fig4.12Theemissionspectrumof3C273,whichisaquasaratredshiftof

0.1583.ThisspectrumwasobtainedusingtheHubbleSpaceTelescope.The

arrowsindicatetheextentoftheredshift.

There are two possible ways to explain redshift. One is that atoms aredifferent everywhere in theUniverse, andvirtually all of thememit lightwithlonger wavelengths than those in our galaxy. The other possibility is thatsomething happened to the light on its journey from the galaxies to ourtelescopesthatcausedthewavelengthstoincrease.

Itisoneofthemostremarkablefactsinthehistoryofsciencethatphysicistswere in possession of a theory that could explain the redshift of the galaxiesyearsbeforeHubblehadevenconfirmedthatgalaxiesexisted,andyearsbeforeany redshifts were observed. That theory is Einstein’s theory of GeneralRelativity,whichmanyphysicistsregardasthemostbeautifulphysicaltheoryofthemall.

BOX7.WHATISLIGHT?

Figure4.13Theelectromagneticspectrum.Theshortestwavelengthsare

knownasgammarays,andthelongestareradiowaves.Visiblelighthas

wavelengthsbetweenapproximately400and700nm.Violetandblue

lighthaveshorterwavelengths,andorangeandredhavelonger

wavelengths.

Lightcanbethoughtofasawave.Likeawaterwaveithaspeaksandtroughs.Butunlikewaterwaves,wherewecansee thepeaksand troughs very clearly, the peaks and troughs in a light wavecorrespond tovariations in thesizeofelectricandmagnetic fields,which is rathermore abstract.Wedo sometimes feel the effect ofthesevariationsbecause,ifthewavelengthisright,theelectricfieldspush theelectronsaround inoureyes togenerate thesignals thatour brains turn into images. Thedistancebetween twopeaks in alight wave is called the wavelength, and we interpret differentwavelengthsofvisiblelightasdifferentcolours.Weareabletoseelightwithwavelengthsbetweenaround400nmand700nm.Beyondourvisualrange,pasttheshort-wavelengthvioletlight,thereareX-rays and gamma rays. Beyond the longer-wavelength reds lie theradiowaves.TheelectromagneticspectrumissummarizedinFigure4.13.

Lightisemittedbyatomswhentheyareheatedup,andabsorbedbyatomswhen itshineson them.FromstudieshereonEarth,weknow that each kind of atom–each chemical element–emits orabsorbsonlyveryparticularwavelengthsoflight.Eachelementhasadistinctsignature,whichisultimatelydowntoitsindividualatomic

structure. Figure 4.14 shows the spectrum of light emitted andabsorbed by hydrogen atoms. The absorption spectrum is like arainbowwithpiecesmissing,anditismadeinmuchthesameway,by shining white light through hydrogen gas and then throughsomething likeaprism.Theprismacts like raindrops,splitting lightintoitscomponentcoloursbyspreadingthemout.Thedarkverticallinesacross the rainbow in thespectrumshown inFigure4.14areknownas absorption lines, and they are producedwhen light of aparticularcolour isabsorbedby thehydrogenatoms.Theemissionspectrumisproducedwhenwelookatthelightemittedbyahotgasofhydrogen.

The emission and absorption lines are at exactly the samewavelengths because emission and absorption are a result ofelectrons jumping around inside hydrogen atoms.Quantum theoryexplains why the electrons are only allowed to have specificenergies when they are confined inside atoms, and therefore whytheemissionandabsorptionlinesarediscrete.Wedon’tneedtogetintothedetailsofquantumtheoryhere,butitishelpfultoknowthatlightcanalsobethoughtofasastreamofparticlescalledphotons,andthateverytimeanelectroninanatomlosesenergy,aphotonisemitted fromnowherewithenergyexactlyequal to that lostby theelectron.Theseare thephotonswesee in theemissionspectrum.Likewise,anelectroncanraiseitsenergybyabsorbingaphotonof

justtherightenergy(thephotonthendisappears).Thisleadstotheabsence of photons we observe in the absorption spectrum. Theenergyofthephotonsisinverselyproportionaltothewavelengthofthe light, which means that higher energy photons correspond toshorterwavelengths.

Figure4.14Thecharacteristicfingerprintofhydrogenatoms.

The pattern of emission and absorption lines is always verydistinctive: it resembles a DNA fingerprint or, as we’ve seen, abarcode.Thisallowsustoanalysethelightcapturedbyatelescopefrom a star or galaxy and identify which chemical elements arepresent. Figure 4.15 shows the light from theSun, split up into itscomponent colours. The hundreds of black absorption linescrisscrossingtherainbowaretheatomicfingerprints,andthisishowwe know preciselywhat theSun ismade of, even though nobodyhaseverbeenthere.

Good science is often about paying attention to the smallest,seemingly insignificantdetailsofNature,andthere isonedetailwecan’t resist mentioning. Take a look again at Figure 4.9, whichshows the emission spectrum of the outermost parts of the Sun(called the chromosphere and corona) observed during a solareclipse. The eclipse helps us to observe an emission spectrum,becausetheMoonblocks thebackground light fromtherestof the

Sun.Thisisthelightemittedbyatomsinthesolaratmosphere.Theemission lines inFigure4.9canbepairedwith linesofexactly thesame wavelengths in the solar absorption spectrum, which weshowed in Figure 4.15. This is to be expected. The absorptionspectrum, which is the thing we usually see, is created when thewhite light from the surface of the Sun shines through the solaratmosphere.Thechemicalelements thenabsorb lightaccording totheir characteristic colours, creating the barcode. As we havealreadynoted,iftheatomsarepresenttoabsorblight,thentheyarealso present to emit it, but we don’t usually see the emissionspectrumbecausethebrightglowofthehotsurfaceoverwhelmsit.

Figure4.15Thesolarabsorptionspectrum.Thefingerprintsoftheatoms

inthesolaratmosphereareclearlyvisible,andthemostprominentones

arelabelled.TheEnglishchemistWilliamHydeWollaston,brotherofthe

manwhogaveJohnMichell’sapparatustoHenryCavendish,wasthe

firstpersontonotetheappearanceofthesedarkfeaturesinthesolar

spectrum,in1802.

However, if you look closely, you can see something odd. The

two linesmarkedHe5876ångstromsandHe4472ångstromsarenot visible in the solar absorption spectrum. The former is thesecondmost intenseline intheemissionspectrum,whichmeansitmustbeabundant intheouterportionsoftheSun.Theselinesareduetothepresenceoftheelementhelium.Helium,thesecondmostabundantelementintheUniverse,wasnotdiscoveredonEarthbutby studying the emission spectrum from theSun.During the solareclipse of 1868, the English astronomer Joseph Norman Lockyersawthe twobrightemission lines,whichat that timecorrespondedto no known terrestrial element. Appropriately enough, the newelementwaschristenedafterHelios,theGreeksungod:helium.

If,aswenowknow, theSun is27%heliumoutsideof thecore,why do we not see it in the much more visible solar absorptionspectrum?The reason is this: theSun’ssurfacemainlyglowsatarelativelycool6000degreescelsius,atwhichtemperatureheliumistransparenttovisible light,so,eventhoughthereareheliumatomspresent, noabsorption line is seen. It isoneof themost intriguingfacetsofourstartheSunthatitsoutermostatmosphereisfarhotterthan its surface, which is a result of the complex behaviour of itsmagnetic field. The corona has a temperature of around 3milliondegrees celsius, which is definitely hot enough to excite theelectronsinsideheliumatomsandmakethememitlight.UnlessthemuchmoreintenselightcomingfromthebulkoftheSunisblockedout(as it is inaneclipse), theseemittedphotonsaretoofewtobeeasily noticed. This is why helium is present in the emissionspectrum,butnotintheabsorptionspectrum.

5.EINSTEIN’STHEORYOFGRAVITY

All things fallwith the same acceleration under the influence of gravity.Thatstatementisprettywellknownanditdoesn’tsoundveryprofound,butitis.Intermsofanequation,youmayrecallthattheaccelerationofanobjectfallingtotheground isgivenbya=GM/r2.Here,G is the strengthof thegravitationalforce so skillfully pinned down by Henry Cavendish in the late eighteenthcentury,risthedistancefromthecentreoftheEarth,andMisthemassoftheEarth.Nowhereinthisequationisthemassofthefallingobjectpresent,whichisthe reason for theopening sentence to thischapter.Big,heavy things, like theMoon,acceleratetotheEarthatthesamerateaslightthings,likeamoteofdust.This is the famous result obtained by Galileo, which he is said–probablyapocryphally–tohavedemonstratedbydroppingtwoballswithdifferentmassesfromtheLeaningTowerofPisa.Newtonusedthisresulttoarguethattheforceofgravitymustbeproportional to themass thatappears inhisSecondLawofMotion, F = ma, which states that if you apply a force to something, itacceleratesbyanamountinverselyproportionaltoitsmass.TheminF=maisknown as the inertial mass, because it describes how hard it is to pushsomething.Itisbecauseofthepreciseequivalencebetweengravitationalmass,used in Newton’s Law of Gravitation, and inertial mass, used in Newton’sSecondLawofMotion,thattheaccelerationduetogravityisindependentofthemassofthefallingobject.Asweshallnowsee,theequivalenceofgravitationalandinertialmasshasprofoundconsequences.

Newton had an answer to why these two conceptually different masseshappen tobeequal: theyareequalbecause that is theway thingsare.Nobodydid any better than that for over two centuries, until a youngAlbert Einsteinrealizedthat,asheputit,‘foranobserverfreelyfallingfromtheroofofahouse,at least in his immediate surroundings, there exists no gravitational field.’Einsteinappreciated that freelyfallingobjects feelnoforce,whichmeans theydo not accelerate: this is why the paths they follow are independent of theirmass.Lookingbackin1920,Einsteindescribedthisas‘thehappiestthoughtofmy life’. To understand preciselywhy he said this–Einsteinwas not prone to

exaggeration–you need to understand what follows in this chapter. Anunexpectedspin-offofthishappythought,whichundoubtedlysurprisedEinsteinasmuchaseveryoneelse,wasthepredictionthattheremaywellhavebeen,touseGeorgeLemaître’spoeticphrase,adaywithoutayesterday.Inotherwords,Einstein’stheoryofgravitypredictstheBigBang.

Let’sthinkcarefullyaboutfree-fall.Imaginejumpingoffaroof.Asyoufall,accordingtoEinstein,itisasifsomebodyswitchesgravityoff.Thatsoundsodd.Hurtlingtowardstheground,itwouldbeabravesoulwhodarestoclaimthereisnogravity.

Inthetwenty-firstcentury,wedon’thavetoimaginethissituation,becausewe are all familiar with images of astronauts aboard the International SpaceStation(ISS).Fromtheperspectiveofanobserverontheground,theastronautsare falling, togetherwith the space station and everything aboard, towards thegrounddue to thegravitationalpullof theEarth.Theastronauts are,however,floating. It is a common misconception to imagine that the astronauts arefloatingin‘zero-G’becausetheyarea longwayfromEarth; theyarenot.ThealtitudeoftheISSisonly400km,andtheradiusoftheEarthis6370km.Thatcorrespondstoa10%reductioninthegravitationalpulloftheEarthrelativetothatfeltbyapersonfallingfromaroofattheEarth’ssurface.Thespacestationdoesn’thitthegroundwhenitfallsbecauseit’salsotravellingtangentiallyatavelocityof7.66km/s,whichmeansthatitiscontinuallymissingtheground.It’spreciselythesamefortheMoon.IttooisforeverfallingtotheEarthandforevermissing,asNewtonwellknew.Thisiswhatbeinginorbitmeans.Watchingthefootageoftheastronauts,it’sobviousthatgravityatleastappearstohavegoneaway.Ifanastronautletsgoofacup,orahammer,orevenaglobuleofwater,itstaysput.Awaterglobuledoesn’tmove relative to theastronaut,or the spacestation,orfalltowardstheEarthfasterorslower,despitethefactthatitsmassisrelativelytiny.Itjustfloats.Thisiswhatwecall‘zero-G’–butthereis,accordingtoNewton,plentyof‘G’present.

Einstein andNewton describe this behaviour in completely differentways.Newton puts the equivalence between inertial and gravitational mass centrestage, and this is his explanation for the fact that nothing moves relative toanythingelseontheISS.Einsteinsimplyassertsthatnothingmovesrelativetoanythingelsebecausethereisnoforceactingonanything.

Hopefully you can now see what Einstein is driving at. In a sense, theastronautsaboardtheISScan’ttellwhethertheyarefallingtowardstheEarthorsimplyfloating,farawayfromanygravitationalinfluenceininterstellarspace.If

we close off the option of looking outside, there is no experiment they canperformorobservationtheycanmakethatwillinformthemeitherway.Inwhichcase, assertsEinstein, there really is no difference between the two situations.People in free fall do not experience gravity because there isn’t any: ‘for anobserverfreelyfalling…atleastinhisimmediatesurroundings,thereexistsnogravitationalfield.’Ifwedescribetheworldfromthepointofviewofpeopleinfree-fall,theforceofgravitywillneverenterintothings.

Butsurelysomethingmoreneedssaying:afterall,gravityreallydoesexist.Ifyoujumpoffaroof,youdohittheground.Einsteinhasananswertothis.Yes,thedistancebetweenyouandthegrounddecreasestozerowhenyoujumpoffaroof,but that isbecausethegroundisacceleratinguptomeetyou.Again,youmay feel uncomfortablewith this statement, but youhave to admit that this iswhatbeingonthegroundactuallyfeels like.Asyousitreadingthisbook,youarebeingnot-so-gentlypressedintoyourchair.Itfeelsthesameaswhenyouarepressedbackintotheseatofanaircraftwhenyouacceleratedowntherunway.Thisisbecauseitisthesame,accordingtoEinstein.Yourweightisn’tanythingtodowiththeforceofgravityasNewtonunderstoodit.Rather,it’sthefeelingyougetbecauseyouareacceleratingupwardsasaresultoftheforceexertedonyouby theground, tomeet thepoorunsuspectingsodwho’smindinghisownbusinessfloatingaroundaftersteppingoffaroof.

Surelynot,youmaysay. Imagine twopeoplefallingfreelyat twodifferentplaces on the Earth; perhaps one is above England and the other is aboveAustralia.FollowingEinstein, theycouldeachlegitimatelyclaimtobefloatingfreelyinspacewhilstthegroundisrushinguptowardsthem.Thegroundcannotbeacceleratingtowardseachofthematthesametime,youmightsay,becausethiswouldimplytheEarthisgettingbigger.Butthatlogiciswrong.

Imagine twosmallballs, separatedbysomedistance, falling freely towardstheEarth.FromNewton’sperspective,eachballaccelerates towards thecentreoftheEarthbecausetheforceofgravityactsalongastraightlinedrawnbetweenthecentreoftheballandthecentreoftheEarth.Thismeansthatthetwoballsmustgetclosertogetherastheyfall,becausetheyeachfallalongstraightlinesthatmeetatthecentreoftheEarth.Iftheballsstartonlyametreapart,andfalloveradistanceofonlyafewmetres,thisisatinyeffect.Butimagineseparatingthe balls by a few thousand kilometres, and dropping them from a similaraltitude.Theywillcloseinoneachotherbyanappreciableamountastheyfall,asillustratedinFigure5.1.ThemostextremeexamplewouldbetoseparatetheballssowidelythatonefallsoverEnglandandtheotheroverAustralia,inwhich

casetheywouldhurtletowardseachother.Einstein’stheorymustbecapableofexplaining this, andofcourse it can.Sohowcan twoballsget closer togetherwhen they are experiencing no force? The explanation is the key tounderstandingGeneralRelativity.

We’vemet Newton’s Second Law,F =ma (the force acting on an objectequalsthatobject’smasstimesitsacceleration),butnothisfirst.Newton’sFirstLaw states that ‘every body remains in a state of rest or uniformmotion in astraight lineunlessacteduponbya force’.Wehave said thatour fallingballsexperience no force, because we’ve dispensed with gravity, and yet they getcloser to each other as they fall in the vicinity of theEarth.They aremovingtogetherwithoutaforce.Thisseemslikeacontradiction,butwe’vechosenourwords carefully. Here is an example of ‘moving together without a force’:imagine two explorers standing a small distance apart on the equator, andimagine they agree to journey due North, walking in perfect straight lines atconstantspeed.Therearenonetforcesacting,sotheywillcontinuetomoveinperfectstraightlinesinaccordwithNewton’sFirstLaw.ButtheywillgetclosertogetherandbumpintoeachotherattheNorthPole.Theymoveclosertogetherbecause straight lineson the surfaceof theEarthare linesof longitude,whichcross theequatorat rightangles,butconvergeoneachother towards thepole.Thereisnoforcepullingtheexplorerstogether;theygetclosertoeachotherasthey move north because they are moving in straight lines across the curvedsurfaceoftheEarth.Thegeometryofthespaceoverwhichtheyaremovingisnotflat,andbecauseofthistheyaredrawnclosertogetherastheymove.

Figure5.1TwoballsfallingtowardstheEarth.

This is why the two balls dropped above England and Australia movetowards each other as determined by someone standing on the surface of theEarth. Theymove towards each other just as the two explorersmove towardseachotherontheirwaytotheNorthPole.Eachfreelyfallingball(orexplorer)can‘claim’quitelegitimatelythattheotherisacceleratingtowardsthem,whilethey feelnoacceleration.Nowwecanunderstandwhy it ispossible for freelyfallingobserversinbothEnglandandAustraliatoeachclaimthatthegroundisacceleratingup tomeet them,andyet theEarth isnotgettinganybigger. It isbecause acceleration is not universal. To be explicit, we can go back to theexample of our two explorers. Imagine theywalk along paths that pass eithersideofIceland.Accordingtoeachexplorer,theypersonallyarenotaccelerating,becausetheyaremovingalongstraightlines,butIcelandisgettingclosertoeach

ofthemastheyjourneypastitontheirwaytothepole.Iceland,theyeachclaim,mustbeacceleratingtowardsthem,inthesensethatitismovingwithachangingspeedanddirectionrelativetothem.Icelandisacceleratingfromtheperspectiveofbothexplorers,butitobviouslydoesn’tchangeshapeinreality.

AccordingtoEinstein,Newton’sforceofgravityisasfictitiousasthe‘force’pullingthosetwoexplorerstogether.Wearemisledintosuspectingthepresenceofanattractiveforcethatcausesthingstofallwhen,inreality,theyaremovingin straight lines over a curved surface. Butwhat is the surface in the case ofobjects falling in space? This iswhere it becomes harder to visualizewhat isgoing on in our mind’s eye because the surface is not a surface of twodimensions, like the surface of theEarth. InEinstein’s theory, the ‘surface’ isbothspaceandtime.ThemathematicallysimplebutintuitivelytrickyideaofasurfaceinmorethantwodimensionsisexploredinBox8(p.123).

BOX8.HIGHERDIMENSIONALSPACE

Figure5.2Theshadowofatesseract.

Whenwespeakofspacetimeasa‘surface’,weareusingtheword

in amathematical sense, in order tomakeaparallelwith the two-dimensionalexampleoftheEarth’ssurfaceapparent.Butspacetimeis four-dimensional. It has threedimensionsof spaceplus theonedimensionof time.Donotgetalarmedby the fact that thesurfacewe are speaking of is four-dimensional and that one of thosedimensions is time. You will certainly not be able to picture this;nobody can. Fortunately, Nature is not restricted to things thathuman beings can picture: Nature is richer than that. And, alsofortunately, human beings have discovered mathematics, whichallowsthemtodeducethingsthattheycannotpicture.

In mathematics, if we’re allowed a 2-dimensional surface, thenwe’realsoalloweda3,4,5orn-dimensionalsurface.Allweneedtodo is add more numbers in order to specify the co-ordinates ofpointsthatlieonthesurface;wealsoneedtospecifysomewayofcalculatingdistancesonthesurface.Youcanpicturethesurfaceofa ball without any problem, but in an imaginary flat-world, wherethere is no ‘up’ and ‘down’, the inhabitants would really struggle.They could still do mathematics concerning the surfaces of balls,though. They might speak of a ball as the mysterious two-dimensionalsurface thatgeneralizes theone-dimensionalnotionofacircle(acircleissomethingtheywouldbeabletopicture).Figure5.2showsapictureofa four-dimensionalcube–wellnotquite. It isthe shadow cast in three dimensions by a four-dimensional cube,drawninprojectionintwodimensions(i.e.onasheetofpaper).Asyoucansee,we’reabletogetaglimpseofwhathigherdimensionalobjects look like by looking at the shadow they cast in the lowerdimensions, just as your shadow would be something that a flat-worldinhabitantcouldvisualize.

Let’ssummarizewhatwehavesofar,becausethis iscounter-intuitive.Justas people walking due north walk ‘over the Earth’ and get drawn together‘becauseit iscurved’,sotwoballsinfree-fallmove‘overspaceandtime’andgetdrawntogether‘becauseitiscurved’.The‘it’werefertoisa‘surface’madefromspaceandtime,a.k.a.spacetime.Theforceofgravityisafictitiousforce;itisamanifestationofthefactthatspacetimeiscurved.

The word ‘genius’ is certainly overused and, in science, it gives the

impressionthatprogressistheresultofaseriesofeurekamomentsexperiencedby a handful of intellectual freaks. This is not usually the case. Science is acollectiveendeavour.Havingsaidthat,ifiteverwasappropriatetousetheword,thenEinstein’sdevelopmentofGeneralRelativitymustqualify.IttookEinsteinagreatdealoftimeandefforttoarriveathisexplanationfortheforceofgravity.Hehadtheideathattherearenoforcesactingonanobjectinfree-fallsometimeinlate1907;itwaslate1915when,havingworkedouttheconsequencesforanewtheoryofgravity,hepublishedthemashisGeneralTheoryofRelativity.

ThebeautyofEinstein’sGeneralTheoryofRelativity is in part due to thesimplicityofitscoreidea:thatobjectsmoveoveracurvedspacetimeinstraightlines.The curvature of spacetime does all thework ofwhat used to be called‘force’.Inthisway,theEarthandtheMoonareboundtogetherinorbit,andyetboth travel in perfect straight lines over spacetime. It is so brilliant andenchantinganideathatmanymoderntheoreticalphysicistshopethatallof theforcesofNaturecanbedispensedwithandreplacedbygeometry,usingGeneralRelativityasthetemplate.

The part of General Relativity that tells us how things move over curvedspacetimeisonlyonepieceofthetheory.Theother,verynecessary,parttellsushowspacetimebecomescurved in the firstplace.Obviously, in thecaseofanobjectfallingneartheEarth,itmustbethattheEarthisresponsibleforwarpingthe spacetime around it. More precisely, and more generally, the presence ofmatterandenergycurvesspacetime. It isprettyobvious that thishas tobe thecase,becausegravityissomethingwenaturallyassociatewithplanetsandstars,and bigger things exert more gravitational pull than little ones (think ofastronautsleapingontheMoon).GeneralRelativitycontainsasetofequationsknownas theEinsteinFieldEquations,whichprovide themathematicalmeanstodeterminetheshapeofspacetimeifweknowthewaythatmatterandenergyarespreadabout.Heretheyare:1

Einstein’s equationsmay look foreboding–but, likemanyof theequations thatdescribe how Nature behaves, they tell us stories. The term Tµv on the rightcontainsthedetailsofthematterandenergydistributioninspaceandtime.Ifwewanted to use Einstein’s equations to describe the solar system, we’d put a

sphericalblobinherewiththemassof theSun.Thetermsontheleftdescribethe resulting geometry of spacetime, which tells planets how to move in thevicinity of the Sun. As an aside, the symbolΛ is known as the cosmologicalconstant,andwe’llmeetitlateron.Youwillrecognizeatleastonesymbol:G,Newton’s gravitational constant, which describes the strength of gravity. InEinstein’s equations, it tells us how much spacetime curves in response to agivendistributionofmatterandenergy.

WecanuseEinstein’sequationstocomputethewarpingofspacetimecloseto the Earth’s surface. This gives results that are almost identical to thosepredictedbyNewton.Wesay‘almost’,becausetherearesomedifferences.Forexample,Einstein’sequationspredictthattimepassesmorequicklyatthetopofamountainthanatthebottom;remember,bothspaceandtimearewarpedbythepresence of the Earth, and not just space. This effect is so tiny that it wouldprobablyhaveremainedundetecteduntilthetwenty-firstcenturyhadpeoplenotbothered to look for it, but it is a large enough effect that, today, it must beaccountedforintechnologythatrequireshigh-precisiontiming,suchastheGPSsystem.

Things get more dramatic when we look beyond the Earth. Soon afterEinsteinpublishedhistheoryin1915,twoofhispredictionsthatdifferedfromNewton’swere tested.One concerned the orbit ofMercury,which is notwelldescribed by Newton’s laws and had been a known problem for almost twocenturies. Einstein’s theory, it turned out, correctly predicted the observedmotionofMercury.TheotherwasEinstein’spredictionfortheamountbywhichlightisdeflectedinagravitationalfield,whichdifferedfromnaïveNewtonian-inspired predictions by perhaps themost famous factor of 2 in all of physics.Einstein’sresultwasconfirmedduringthesolareclipseof1919,whenstarlightcouldbeobservedpassingclosetotheSun,andthiswasthemomenthebecameworld-famous.Today,Einstein’stheoryhasbeentestedtoremarkableprecisioninavarietyofastrophysicalsystems,systemsoftenbizarreandviolentbeyondimagination.

Figure5.3Asimulationofapairofcollidingblackholes,liketheonesthat

producedthegravitationalwavesobservedbythetwoLIGOdetectors.

The most spectacular recent triumph was the discovery of gravitationalwaves: ripples in the fabricof spacetime.At5.51 a.m.EST,on14September2015,thetwinLIGOdetectors,locatedintheUnitedStatesinWashingtonStateand Louisiana, detected gravitationalwaves as they passed through the Earth.Thesewaveswerecausedbythecollisionoftwoblackholes,themselvesalsoapredictionofEinstein’stheory.Theblackholes,29and36timesthemassoftheSun,spiralledintoeachotherinlessthantwotenthsofasecond,duringwhichtimetheirclosingspeedchangedfromonethirdtoalmosttwothirdsofthespeedof light. The collision resulted in a peak power output fifty times that of the

entire observable Universe, which distorted space and time enough for theeffects tobemeasured1.3billion lightyears awayonEarth.Einstein’s theorypredictedwithprecisiontheobservedsignal.Whatmorecouldyouaskfor?

So there we have it: Einstein’s theory of gravity, General Relativity. Itsupersedes Newton’s Law of Gravitation because it delivers a more accuratedescriptionofNature.ThatisnottodenythatNewton’slawremainsexcellentinmany instances, and has the advantage of being far easier to deal withmathematically.Butthehardevidenceindicates,withoutashadowofdoubt,thatEinstein’stheoryismorecorrectthanNewton’s.

Wecannowbealittlemoreambitious.We’veseenthatEinstein’stheoryisextremelygoodatdescribingthecurvatureofspacetimearoundsphericalblobsof matter, like planets, suns and even black holes. We might therefore betemptedtoaskwhetheritcouldalsobeappliedtoalargerdistributionofmatter:the entire Universe, for instance. This is audacious to say the least–and,naturally, the thoughthadalreadyoccurred toEinstein. In1917hepublishedapaper entitled ‘Cosmological Considerations of the General Theory ofRelativity’, the audacity of which did not escape him. ‘I have… againperpetratedsomethingaboutgravitationtheorywhichsomewhatexposesmetothedangerofbeingconfined inamadhouse,’hewrote ina letter tohis friendPaul Ehrenfest. Let us,with due humility, follow inEinstein’s footsteps. Thiswill be well worth it, because what follows is a triumph. Einstein’s theorypredictstheexistenceoftheredshiftsofthegalaxiesthatweencounteredinthelastchapter.AndthatisbecauseitalsopredictstheexistenceoftheBigBang.

BOX9.GRAVITATIONALWAVES

Figure5.4ThetwoLIGOdetectors.AtthetopisthedetectoratHanford,

WashingtonState,andbelowisthedetectoratLivingston,Louisiana.

Thescienceiscarriedoutbyacollaborationofhundredsofpeoplefrom

countriesacrosstheworld.

Until the LIGO detection of gravitational waves, astronomerswerelimitedtousingelectromagneticwavestoobservethecosmos–nowthey have a whole new way of seeing. The LIGO detectors arestaggering in their sensitivity and they are the modern dayequivalent of HenryCavendish’s experiment (see pp. 68–72). The

twoLIGOdetectorsareshowninFigure5.4andtheyeachconsistoftwo 4 km long high-vacuum ‘arms’ with heavy mirrors hanging ateitherend.Themirrorsaresuspended likependulaon fusedsilicafibres.Youcanseethelongarmsonthephotos.Asagravitationalwavepasses, it pushes and pulls themirrors so as to change thedistancebetweenthem,andthescientistsareable tomeasure thechangeinlengthofonearmwithrespecttotheother(byarrangingthearmsat right-anglesonearm tends to increase in lengthwhenthe other decreases). They do this using a technique called laserinterferometry, which involves sending laser light into each arm,whereitbouncesbackandforthmanytimesoffthehangingmirrorsbefore re-emerging. The strength of the combined output light issensitive to variations in the roundtrip time for the light to travelalongeacharm–soifthelengthschangethentheoutputlaserlightwillreflectthatchange.

Figure 5.5 shows the amount by which the difference in lengthbetween the two arms of the detector changed as the wave ofSeptember 2015 passed by. This is labelled ‘strain’, and it is thefractionalamountbywhichthearmschangedinlength.Noticehowminiscule the strain is. It means that the scientists were able tomeasure the change in length to better than 1 part in 1021. Thatrelativeprecisionis likemeasuringthedistancetoourneareststar,Proxima Centauri, to the thickness of a human hair: it is mind-blowingly precise. In terms of absolute distance, it corresponds tomeasuringthechangeindistancebetweenthemirrorstoaboutonethousandthofthesizeofasingleproton.Youmightwelldoubtthatsuch a precision is possible, not least because the surface of themirrors has variations that are much bigger than that. But thebrilliance of the measurement sidesteps the problem because itmeasures an average distance. Each photon that enters theapparatus is, inasense,making itsownmeasurement,andoneofLIGO’s key features is its ability to harness the power of vastnumbersofphotons.Tounderstandhowpowerfulthisaveragingis,imagine a totally fictitious wave that has the effect of makingeveryone on Earth increase in height by one hundredth of amillimetre. Ifyouknewwhenthewavewasdue topass,couldyou

detect it?Well, you certainly couldn’t tellwithany confidence frommeasuring the height of any one person before and after thesupposedpassingofthewave,becauseyousimplywouldn’tbeabletomeasuretheirheightaccuratelyenough.Butyoucoulddoitifyoumeasured the difference in height (before and after the wavepasses)ofeveryoneand then took themean.That isbecause theuncertainty on the mean decreases as the number of peopleincreases1andso,withenoughpeople,youcouldspotwhetherornot their average height has increased slightly after thewavewasdue. This is similar to what is done to measure that minusculechange in the arm lengths in LIGO. The main limitations on theprecisionoftheexperimentareduetothelimitednumberofphotonstheyhaveattheirdisposalandtothesmallvibrationsinducedinthehangingmirrors due to seismic effects and the fact that a laser isbouncingoffthem.Passingtrains,oceanwavesandtheweatherallinducerelatively largevariationsbut theycanallbeeliminatedtoasufficientdegreewithdiligenceandattentiontodetail.ThisisbrilliantsciencestraightoutoftheHenryCavendishschool.

Figure5.5Theall-importantgraphsdemonstratingthediscoveryof

gravitationalwaves.Thetwoleft-handgraphsshowthemeasurement

fromHanfordandthecorrespondingtheoreticalprediction.Theright-

handgraphsarethesamebutforthedetectoratLivingston.

Figure5.5notonlyshowshowthestrainchangedwithtimeoverthefractionofasecondwhenthewavewaspassing, italsoshowsthe theoretical prediction of what the signal ought to look like if itcamefromapairofcollidingblackholes. It iswonderful toseetheextentoftheagreement.Thespeedingupofthewigglesreflectsthespeedingupoftheblackholesastheyspiralledtowardseachother,and the final part of the curve, when the wiggles eventually dieaway, corresponds to the final phaseof themerger,when the twoblackholeshavebecomeone.

Averyexcitingpostscripttothestoryisthaton15June2016theLIGO teamannounced their seconddetectionof apair of collidingblackholes.TheeventoccurredonBoxingDay2015and involvedsmallerblackholes(8and14timesthemassoftheSun).Again,theobserved signal agrees perfectly with the calculations based onEinstein’stheory.Asecondobservationsosoonafterthefirstisveryencouraging because it means that it is likely there will be muchmore to see in the years to come, and that we are seeing theseevents for the first time now becausewe have finallymanaged todevelopthetechnologytodoso.Thissurelyheraldsanewdawninastronomy.

IfweknewallaboutthedistributionofmatterandenergyintheUniverse,wecouldputthatinformationintoEinstein’sequationsandobtainthegeometryofspacetime.Wecould thenask if thereareanyobservableconsequencesof thisgeometry,andlooktotestthetheory.Thisiswhatwewillnowdo.

There is, of course, no way we can know where all the material in theUniverseislocated.Itishardenoughtokeeptrackofapairofsocks.Wearenotstymiedthough.Whatfollowsisanexampleofgoodscience.Wearegoingtomakeasimplifyingassumption.Asever,ifourassumptioniswrong,weexpecttofindoutwhenwecompareourpredictionswithexperimentsorobservationsofNature.

Figure 5.6 shows amap of the galaxies visible fromEarth, compiled fromseveral databases by Thomas Jarrett, an astronomer at Caltech. It’s worthlooking for amoment at this picture, because it is remarkable and humbling.Therearesomanygalaxiesthatforthemostpartonlyclustersandsuperclustersof galaxies are labelled. TheVirgoCluster alone contains over 1300 galaxiesthat lie around 53 million light years from Earth. A deep photograph of theclusterfromtheEuropeanSouthernObservatoryinChileisshowninFigure5.7,revealing a snowstorm of galaxies: delicate clouds of light from billions ofworlds.

Theskyisfilledwithgalaxies,scatteredinwhatlookslikearandompattern.Theyappearuniformlydistributedonthelargestdistancescales,andthisistheclueweneed tomakeour assumption.Wewill suppose that thematter in theUniverseisscatteredevenly.Thisisofcoursenottrueoversmalldistances:thesolarsystemisverylumpy,withallthemassbeingconcentratedintheSunandafewplanets.Butwearegoingtoassumethatover‘sufficientlylarge’distances,theUniverse is smooth. Inotherwords, ifwe imaginecounting thenumberofgalaxies inside an imaginary box, sufficiently large so as to contain manygalaxies,thenthatnumberwillnotvarymuchifwechooseanotherequallysizedboxsomewhereelseintheUniverse.

In the jargon, this iscalledahomogenousand isotropicmatterdistribution:thesameeverywhereandthesameineverydirection.It’stheultimatestatementoftheCopernicanprinciple.WedonotoccupyaspecialplaceintheUniverse,andneitherdoesanyoneelse.

Figure5.6Anall-skyviewoftheobservableUniversefromtheXSCcatalogue

of1.5milliongalaxiesandthePSCcatalogueof0.5billionstarsintheMilky

Way,whichcanbeseenacrossthecentreofthemap.Thegalaxiesare

colour-codedbytheirredshifts.

Figure5.7AdeepimageoftheVirgoClustertakenbytheEuropeanSouthern

Observatory.Thedarkspotsarenotinteresting–theyareduetotheremoval

ofbrightforegroundstarsfromtheimage.

Ifwewant tocalculate theshapeofallofspacetime,weknowwhat todo.WeplugourhomogenousandisotropicmatterdistributionintotherightsideofEinstein’sfieldequations,andseewhatcomesoutontheleftside.Fortunately,we can immediately anticipate some of the resultswithout actually doing anymathematics.

A perfectly smooth matter distribution throughout the Universe leads to aperfectlysmoothspacetime.Galaxies,starsandplanetswillcausethespacetimeintheirvicinitytobedistortedcomparedtotheperfectlysmoothaverage.Thinkof it like a golf ball,whichwe can picture as a perfectly smooth spherewithsmall dimples that constitute local, small distortions on the surface. The localdimplingofthegolfballisanalogoustothewayspacetimeis‘dimpled’bylocalclustering of mass. The Sun will make a tiny dimple. A galaxy will make abiggerone.Whenwewanttoconsiderthelarge-scalestructureoftheUniverse,

wearenotinterestedinthedimples.WeareinterestedinthesmoothspacetimethatrepresentstheUniverseatlarge.2

We can now think about what a ‘perfectly smooth spacetime’ might looklike.Itturnsoutthatthereareonlythreepossibilities,whichweshallrefertoas‘flat’, ‘spherical’ and ‘hyperbolic’ (youneed toknowsomemathematics tobeable toprovethat therearenomorepossibilities).Remember thatspacetimeisfour-dimensional, with three spatial dimensions, so we can’t picture it veryeasily. Fortunately, the same three possibilities also arise in two spatialdimensions,andtheyaresketchedinFigure5.8.Wemadetheinitialassumptionthat thereareno specialplacesandno specialdirections in theUniverse.ThismustimplythattheshapesofspacetimewegetoutofEinstein’sequationshaveno special places and no special directions. It is easy to see that there are nospecialpointsordirectionsonaflatsheet.Likewise,therearenospecialpointsor directions on the surface of a sphere, the second of the three possiblegeometries. The third and final possibility is a saddle,which is rather like aninside-out sphere. The curvature of the surface is the same everywhere, butinsteadofbeingconvexlikeasphere,itisconcave.Thethreepossibilitiesinthefour dimensions of spacetime are analogous to these, and the analogy is goodenoughforustodaretodrawsomegalaxiesonthesurfacesinthefigure.Theseare the three possible universes under the assumptions of homogeneity andisotropy, and it is correct to thinkof themas showinghow space is curved atsomeinstantoftime.

Itisimpressivethatwecanmakestatementsaboutthepossiblegeometriesofspacetimeusingsuchgeneral reasoning,butwecangofurther.LookatFigure5.9,whichshowsahypotheticalflatuniverseatthreedifferenttimes(wecouldequallywellhavechosenoneoftheothertwogeometries).Thefigureindicatesthatourmodeluniverse isbeingevenlystretchedas timeadvances–justasonemight stretch a rubber sheet by pulling its edges equally in all directions.Crucially, if the space is to remain flat, spherical or hyperbolic, then the onlyconceivablethingitcandoastimeadvancesistoshrinkorexpand.Atthispointinournarrative,wehavenoideawhetherthespaceweliveinexpands,contractsor stays the same.Remember,Einsteinhasencouragedus to thinkof spaceassomethingmalleable:thepresenceofmatterandenergycananddoesdistortit.Thisistheoriginofplanetaryorbitsandthereasonthingshitthegroundwhentheyaredropped.Theideathatitcanstretchaswellaswarp,inresponsetothematterandenergywithinit,shouldnotbesobizarre.

Figure5.8Thethreepossiblegeometriesin2D,withsomegalaxiesdrawnon.

Figure5.9Theflatuniversefromthepreviousfigureatthreedifferentdegrees

ofstretching.

NowisthetimetoaskwhatthefullmachineryofGeneralRelativityhastosay.WewouldliketoknowexactlyhowtheUniversestretchesorshrinkswithtime.Onceweknowthat,weknoweverythingabouthowspaceevolvesinaperfectlyuniform universe. Other questions we might have will be concerned withdeviations from the ‘same everywhere’ assumption; we’ll come to that inChapter8.

ImaginetwopointsintheUniversethatareacertaindistanceapartatsomeinstant in time. Einstein’s equations tell us how that distance changes as timeadvances. In other words, they tell us how the Universe is stretching orshrinking.Not surprisingly, thedetailedanswerdependsonwhichof the threegeometries we live in and on what kind of stuff fills the Universe, but oneconclusion is pretty hard to circumvent: in Einstein’s theory of GeneralRelativity, the distance does tend to change. This is extremely important,becauseitimmediatelyimpliesthataUniversefullofstarsandgalaxieslikeourscouldwellbeexpandingorcontracting.

If you did not know any better, the idea of an expanding Universe issurprising,andonemightaskwhethertheequationsalsoallowforthepossibilityof a static universe. Einstein asked this question. In 1917 he showed that auniversewithoutstretchingorshrinkingis(justabout)conceivable,buttodoithehadtointroducethecosmologicalconstant,whichwebrieflymentionedwhenwemethisfieldequations.Einstein includedthecosmologicalconstantsimplybecause there isnoobvious reasonwhy it shouldbeexcluded,andeven todaythereisnogoodunderstandingofwhatitmightrepresentatadeeperlevel.Oneway to think about a cosmological constant is to regard it as aweird formofmatter.Ordinarymatter,ofthetypethatmakesupstarsandplanetsandgalaxiesandpeople, ismadeupofparticles. Ifwecapture someof theseparticles inabox,theyziparoundandcollidewiththewallsandexertapressurethatactstotryandexpandthebox.Einstein’sweirdmattercanbethoughtofasdoingtheexact opposite: itwould suck thewalls of the box in rather than push againstthem. This might sound contrived, and in fact Einstein later dismissed theintroduction of the cosmological constant as his ‘greatest blunder’. HistoriansargueaboutwhatEinsteinmeantbythis.Onepossibilityis thathedescribeditthatwaybecause,inspendingsomuchtimetryingtoforceGeneralRelativitytodescribeastaticuniverse,hemissedthefactthathisequationswerescreamingout to him to entertain the idea of a Universe that is expanding. ThisuncharacteristicblindnessmeanthemissedthechancetopredicttheBigBang.

But two of Einstein’s contemporaries, Alexander Friedmann in the Soviet

Union and the Belgian Georges Lemaître, did not miss this message.Independently, they correctly deduced that Einstein’s General Theory ofRelativity,when applied to aUniverse such as ours, predicts that it should beexpandingorcontracting.

Alexander Friedmann was the first to entertain the idea of an expanding3Universe.He usedEinstein’s equations towrite down an equation forwhat isknownasthescalefactoroftheUniverse.Thescalefactorisanumberthattellsus howmuch space is stretched or contracted relative to its size today–in anexpanding universe, the scale factor would be smaller than 1 in the past andbigger than 1 in the future. Used by cosmologists today, the equation is,unsurprisingly,calledtheFriedmannequation.WeexploretheequationinBox10 (pp. 144–5). Friedmann made a calculation for the spherical geometry ofspacetimein1922andforthehyperbolicgeometryin1924.Hedidnotdiscussthe possibility of a flat geometry, nor did he comment on astronomicalimplications:rather,hewasmoreinterestedinpointingoutthatEinstein’sstaticsolution(correspondingtoascalefactorthatdoesnotchangewithtime)wasnotnecessarily correct. Lemaître re-discovered the Friedmann equation in 1927,although, in focusing on a spherical geometry he too missed the flat-spacesolution. Lemaître’s solution of 1927 described an infinitely old sphericaluniversewithanever-increasingscalefactor.

Despitethesedecisivecontributions,bothFriedmannandLemaîtrewerenotsowellknowninthescientificcommunity,andtheirworkwaslargelyignoredwhenitwasfirstpublished–partlybecauseoftheprevailingprejudicetowardsastaticuniverse,supportedbythevenerableEinstein,butalsoformoremundanereasons.Lemaître’spaperappearedinFrenchin therelativelyobscureAnnalesde la Société Scientifique de Bruxelles, and was not widely read; Friedmann,meanwhile,diedoftyphoidin1925,shortlyafterpublishinghiswork.In1930,however,thewell-knownBritishphysicistArthurEddingtonbegantopublicizeLemaître’s work, when he came to realize that Einstein’s static universe wasimplausiblebecauseeventhetiniestdeviationfromtheidealizedconditionsthatEinstein postulated would cause the universe to expand or contract. The dustfinallysettledin1933,whenPrincetonphysicistHowardPercyRobertsonwrotea beautiful paper that carefully catalogued all of the possible mathematicalsolutionstotheFriedmannequation.Hecountedeighteenofthem,spanningthevariouslogicalpossibilitiescorrespondingtoflat,sphericalorhyperbolicspaceandvaryingamountsofcosmologicalconstant.Robertson’sconclusionwasthat,usingonlyEinstein’sequationsandpurelogic,auniversecontainingmattercan

onlybe:

(i)ofafiniteageandexpandingforeverintothefuture;(ii) of a finite age and expanding for some period of time before

contractingbackagain;(iii)ofinfiniteage,foreverexpanding(theexpansionratewouldneedto

tendtowardszerointhedistantpast);(iv)ofinfiniteagewithnoexpansionorcontraction;(v)ofinfiniteage,firstcontractingandthenexpanding.

In those solutionswhere theuniversehas a finite age, theremust havebeen atimeinthepastwhentheuniversewasinfinitelydense,meaningthattheaveragedistancebetweenanytwoparticleswaszero.Theequationsstarttofailwhenthedistancebetweentheparticlesbecomestoosmall,butthenotionthattherewasatimewhentheUniversewasextremelydenselypackedwithparticlesisarobustprediction.ThatspecialtimeintheUniverse’shistoryiswhatwenowrefertoasthe Big Bang. Scenario (iii) is the solution found by Lemaître in 1927 andscenario (iv) isEinstein’s static solution. Scenario (ii) is interesting because ithas a ‘BigCrunch’ at some point in the future, and scenario (i) is interestingbecause,aswewillseenext,itistheUniverseweactuallylivein.

BOX10.THEFRIEDMANNEQUATION

The Friedmann equation is without a doubt the most importantequationincosmology.Hereitis:

Theequationtellsushowfastspaceexpandswhenthescalefactorof theUniverse isequal toa.Wehavewrittentheequationsothatthe scale factor a = 1 at the present time. If the Universe isexpanding,awassmallerthan1inthepastanditwillbebiggerthan

1 in the future. The Friedmann equation tells us precisely how achangesastimechanges.Let’sgothrougheachbitoftheequationinturn.OntheleftoftheequalssignisH,whichistheHubblerate:ittellsushowfastspaceisexpandingatsomemomentintimeanditis equal to the fractional rate of change of the scale factor a. InChapter6,wewillmakeourownmeasurementof thepresentdayvalueofH,andwe’ll find it isequal toapproximately70km/s/Mpc,whichmeansthat(today)spaceisexpandingsuchthattwoobjectsthat are 1 megaparsec apart are moving apart at a speed of 70kilometrespersecond.Becausespace isassumed(onaverage)tobe the same everywhere, this means that objects that are 2megaparsecsapartarecurrentlyrecedingfromeachotheratarateof 140 kilometres per second, and so on. If we know howH haschangedinthepast,andifweknowhowitwillchangeinthefuture,thenwewill also knowhow the scale factora has changed in thepast and how it will change in the future. In otherswords,wewillknoweverythingthereistoknowabouttheexpansionhistoryofthecosmos:oneoftheholygrailsofcosmology.1

Now let’s turn to the right of the equals sign. The first termdepends on Newton’s gravitational constant, G, and also on theaverage mass density of the Universe, denoted by ρ. This is theamountofmassineverycubicmetreofspace,averagedacrosstheUniverse.Thesecond term iswhere thegeometryofspaceentersintothings.ThesymbolK justkeepstrackofthesign: it isequalto+1 (corresponding to a spherical geometry) or -1 (hyperbolicgeometry) or, if this term is absent, it is 0 (corresponding to a flatgeometry). c is the speed of light and R is the curvature of theUniverse,whichneedsspecifying if theUniverse iseithersphericalor hyperbolic, because we need to know how big the sphere orsaddleis(atwo-dimensionalhyperbolicspacelookslikeasaddle).

Theequationthereforetellsusthatspaceisexpandingataratethatisfixedbyhowmuchstuffitcontains,anditsshape.ThisistheessenceofEinstein’sequation.

If youarekeen toplayaboutwith theFriedmannequation thenyouwillwanttoknowhowtheaveragemassdensitydependsonthescalefactor,becauseclearlyitwilldependonit.Ifyouarenotkeenthen you can safely skip the rest of this paragraph. In the caseof

matterparticles,likeprotonsordarkmatter,thenumberofparticlesperunit volumewill decreaseasspacestretches.Thismeans thatthedensityisequaltothepresent-daydensitydividedbya3(thinkofmeasuringthedensitybymeasuringhowmuchmatterisinacube–and that the cube shrinks or grows by a factor a; its volume willcorrespondinglyshrinkorgrowbyafactora3).Thecaseoflight2isalittlemoresubtle; itcontributestoρbyvirtueofhavingenergy,andtheappropriatemassdensityisobtainedbydividingtheenergyperunitvolumebyc2.ThisishowEinstein’stheoryofgravitydiffersfromNewton’s:Einsteinunderstoodthatbothmassandenergycontributetogravity.Youmightthinkthatthepresent-daycontributiontoρfromlightwouldalsoneed tobedividedbya3 inorder todetermine thecontributionatsomeothertime.Butinfactitneedstobedividedbya4.This isbecause lightbecomesredshiftedasspaceexpands(orblueshiftedasitcontracts),whichmeansitsenergydensityrisesorfalls in inverse proportion to the scale factor. The last remainingcontributiontoρcomesfromtheenergythatmaybestoredinemptyspace, such asmight be associatedwith a cosmological constant.This source of mass density does not change as the Universeexpandsor contracts, so it contributesa constant value toρ.Nowyou can go ahead and solve the Friedmann equation using acomputer(theintegralyouneedtodoistoohardtodobyhand): ifyou know the present-day mass densities associated with light,matterandthecosmologicalconstantthentheaboveinformationisenoughforyoutocomputethevalueofthescalefactoraforalltime.Wewilldeterminethepresent-daymassdensitiesinChapter7.

6.THEBIGBANG

Part ofwhatmakesEinstein’sGeneralRelativity so astonishing is that itwas,more or less, an exercise in pure thought, triggered by his musings on theuniversality of free-fall. It’s wonderful that we can trace the development ofEinstein’shappiest thoughtall thewaythroughtoRobertson’sclassificationofthe possible histories of theUniverse; to follow this simplest,most elegant ofideasintounchartedandunexpectedlyrichterrain.

This,ofcourse,isabookabouttheUniverseweliveinandnotanimaginaryone, howeverbeautiful.Today,GeneralRelativityhasbeen tested to exquisiteprecision–theLIGOdetectionofgravitationalwavesthatwediscussedinthelastchapter being themost recent andmost striking example.Now, our goal is toestablish the extent to which the Universe as a whole fits into Einstein’sframework.WewanttogatherevidenceinsupportoftheideathatweliveinanexpandingUniversethatstartedwithaBigBang.

Our first task is toestablishwhether theUniverse isexpandingnowand tomeasuretherateofthatexpansion.WehavealreadyseenonepieceofevidencethatindicatesourUniverseisexpanding:thewaythatlightfromdistantgalaxiesisredshifted.AccordingtoGeneralRelativity,thisoccursbecausetheexpansionof the Universe causes successive crests and troughs of light waves to moveapart.ThismeansthatthewavelengthoflightarrivingattheMilkyWayfromadistantgalaxyshouldbeincreased(i.e.redshifted)bythesamefractionthatthedistancebetweenthetwogalaxieshasincreasedduringthetimethelighttooktomakethejourney.Spacestretches,andthelightstretcheswithit.IftheredshiftisduetotheexpansionoftheUniverse,thenwecanpredictthatlightfromthemost distant galaxies should have the biggest redshifts, because the light hasbeen travelling for longer,meaning that spacewill have stretchedmore. Let’sseeifthesefeaturesareborneout,bytakingalookatsomerealdata.

We’llusetheNASAExtragalacticDatabase,whichisfreelyavailableontheweb, and explore how the redshifts of a sample of galaxies vary with theirdistances from Earth. So far in the book, we have tried to make our ownmeasurements,solookinginadatabasemightsoundlikecheating,butitisonlya little cheat. It is possible for an amateur astronomerwith a reasonably sized

telescope costing a few thousand pounds, a spectrograph, a digital imagingsystemandalaptoptomakegalaxyredshiftmeasurementsforgalaxiesoutasfaras100Mpc.Theabsurdlysuspiciousorcommendablyenthusiastic readerwhodoesn’t trust thedatabasesisencouragedtotakethisroute.Indeed, ifforsomereason you don’t believe the redshift data, then you should take this route,becauseitwillinformyouthatyouropinioniswrongandthedatabasesareright.Takingdelight inbeingshowntobewrong isoneof themost importantskillsanyhumanbeing,letaloneascientist,shoulddevelop.

Figure6.1TheAndromedagalaxy–ournearestgalacticneighbour.

Inordertotesthowredshiftdependsonhowfarawayagalaxyis,weneedprecisedistancemeasurementstofar-awaygalaxies.Thesearemoredifficultforanamateur toperform.Aswehaveseen, theCepheidmethodcanbeused forgalaxies close enough for individual stars to be resolved and, formoredistantgalaxies, supernovae can be used, but only for those galaxies that happen tocontain a supernova that we are lucky enough to have seen. To make a

comprehensive map of the distances to the galaxies, we really could do withanotherwaytomeasurehowfarawaydistantgalaxiesare.

There are several otherwidelyusedmethodsofmeasuring the distances togalaxies.Oneof themostaccuratewasdeveloped in1977byNorthAmericanastronomersR.Brent Tully andRick Fisher. The Tully-Fishermethod can beusedtomeasuredistancestospiralgalaxies,whicharesonamedbecauseoftheirbeautifulspiralshapes.TheMilkyWayandAndromedaarebothspiralgalaxies–inFigure6.1weshowaphotoofAndromeda;itisadisc-likeassemblyofstarsthatisrotatingaboutanaxisthroughitscentre.TullyandFisher’smethodusesthemeasuredbrightnessofaspiralgalaxyandthe‘width’ofitsspectrallines1todeterminehowfarawayitis,anditisexplainedinBox11(p.151).Inthespiritof this book, however,wewould also like a simple ‘Ogmore-by-Sea’methodthatwill allow us to check that themore sophisticated distancemeasurementsmakesense.

Hereisoursimplemethod.Sinceallspiralgalaxieslookquitesimilartoeachother,wewill be so bold as to assert that they are all roughly the same size.Underthisassumption,wecanestimatethedistancestoallofthemifweknowthe distance to just one. For example, if a galaxy is half the size of theAndromedagalaxy,asviewedfromEarth,we’llassumeitistwiceasfaraway,and so on. It may or may not be a good approximation to say that all spiralgalaxiesareexactlythesamesize,butwemightreasonablyexpectthevariationin sizes to be not so great as to spoil things. In any case, since we have theaccurate distancemeasurements to hand in theNASAdatabase,we can checkwhetherourmethodstandsupunder the illuminatingspotlightof theprecisiondata.TheangularsizeoftheAndromedagalaxyontheskyis6200arcseconds,2and the distance to Andromeda from Earth, obtained from Cepheid variablemeasurements, is 780,000 parsecs, or 2.5 million light years. Our old friendNGC4535hasanangularsizeof410arcsecondsand,ifweassumeit’sthesamesizeasAndromeda,thismeansitmustbe15timesfurtheraway,whichputsitatadistanceof12Mpc.Intotal,we’veselectedsixteenspiralgalaxiesatrandomfromtheNASAdatabase;thedistancesandredshiftstothesegalaxiesarelistedin Table 6.1. Photographs of the galaxies are shown in Figure 6.2. There isnothing special about these sixteen galaxies except that they are all spirals atredshiftsbetween0.001and0.02.

BOX11.THETULLY-FISHERRELATION

BOX11.THETULLY-FISHERRELATION

TheTully-Fisherrelationcanbeusedtodeterminehowfarawayadistantspiralgalaxy is. Itexploitsacorrelationbetweenhowfastaspiralgalaxyisspinningonitsaxisandhowbrightitis.AccordingtoNewton’sLawofGravitation, thespeedatwhichadistant starwillorbitaroundthecentreofagalaxyisdeterminedbythemassofthegalaxy, see Box 5 (p. 64). This makes sense, because moremassivegalaxiesexertabiggergravitationalpull,whichmeansstarscancirclearoundthegalaxyathigherspeedswithoutflyingoff intospace. But the wavelength of the light emitted from a star andobserved on Earth depends partly on how fast the star ismovingrelative to theEarth.Specifically, if the star ismoving towards theEarth the light will be slightly blueshifted and if it is moving awayfrom the Earth the light will be slightly redshifted. This change incolour(wavelength)isnotduetotheexpansionofspace.Rather,itarisesformuchthesamereasonthatahigh-pitchedsirenonafast-movingambulancedropstoalowerpitchastheambulancepassesby.ThisiscalledtheDopplereffect,anditcomesaboutbecausethesound waves emitted by the siren are compressed (shorterwavelength) when the ambulance approaches and stretched out(longerwavelength)whentheambulancerecedes.Becauselight isalsoawavemotion,thereisaDopplereffectforlight.Inthecaseofaspiralgalaxy,thissmallshiftinwavelengthmeansthatthespectrallinesweobserveonEartharenotprecisely fixedat oneparticularwavelength: they are spread out a little bit, with some light beingredshiftedandsomeblueshiftedrelativetotheaverage.Thesizeofthespreadisdeterminedbyhowfastthestarsmoveand,aswejustsaid,thisiscorrelatedtothemassofthegalaxy.Inadditiontothis,the amount of light a galaxy emits depends on itsmass, which isprettyobvious,becausemoremassivegalaxiescontainmorestars.Since theDopplerbroadeningof thespectral linesand theamountlightagalaxyemitsbothdependonthemassofthegalaxy,itfollowsthat there must be a correlation between the amount of Dopplerbroadeningand theamountof light thatagalaxyemits, i.e. faster-spinninggalaxieswillhavebroaderspectral linesandtheywillalsobe more massive. This correlation can be used to determine the

distancetoagalaxybymeasuringthegalaxy’sbrightnessonEarth(more distant galaxies will be dimmer), provided that we firstcalibrate the method by measuring the distances to a few spiralgalaxiesusingan independentmeasurement,suchas theCepheidvariablemethod.

Wechosetheupperlimitof0.02becausewewantedtobeabletoidentifythegalaxiesasspiralsbyeye,whichallowsustousethe‘Ogmore-by-Sea’methodtoestimatetheirdistancesfromtheMilkyWay.Aswe’veseen,alargerredshiftmeansagreaterdistance,andthereforeasmallergalaxyonthesky.IfyoulookatFigure6.2,you’llseethatourhighestredshiftgalaxy,UGC6533,isjustaboutidentifiable as a spiral, but we might struggle to identify spirals at greaterdistances and, as a result, we might bias ourselves by selecting bigger thanaveragespiralsthatareeasiertosee.

Therearealsospiralgalaxieswithredshiftssmaller than0.001. Indeed,ourclosestneighbourAndromedahasanegativeredshiftof−0.001.Thismeansthatthe light is shifted towards the blue part of the spectrum, signifying thatAndromedaishurtlingtowardsusandwillhitusinaround4billionyears.ThisblueshiftofAndromedaiscausedbytheDopplereffect,whichweexplainedinBox 11 (p. 151). We can understand the blueshift of Andromeda once weappreciate that galaxies are not only carried along with the expansion of the

Universe.Theyalsomovearoundbecauseoflocalgravitationalinteractions,justastheEarthorbitstheSunandtheSunorbitsthecentreoftheMilkyWay.So,wehavetofactorinthis‘propermotion’ofagalaxy,aswellasitsmotionduetotheexpansionofspace:accountingforitwouldleadtoanadditionalredorblueDoppler shift superimposed on the redshift caused by the expansion of theUniverse.AsfarastheAndromedagalaxyisconcerned,theblueshiftcompletelydominates,becauseitissoclosethattheeffectoftheexpandingUniverseonitslight isverysmall.Forsufficientlydistantgalaxies,however, thecosmologicalredshifts are large enough to overwhelm any Doppler shifts caused by theirproper motions–which is why we only selected galaxies at distances above 4Mpc,correspondingtoredshiftsabove0.001.

Figure6.2Thesixteenspiralgalaxiesusedtomeasuretherateatwhich

spaceisexpanding.

TheapproximatedistancestotheothergalaxiesinTable6.1areworkedoutinthe sameway thatweworkedout thedistance toNGC4535andappear in thecolumn labelled ‘Estimated distance’. These are to be compared with thenumbers in the ‘Measured distance’ column, which were obtained by theprofessionals.

TheclosestgalaxyinthelistisNGC1313,atadistanceofaround4Mpc,andaredshiftof0.00157.ThemostdistantisUGC6533,whoseredshiftisafactorof10 largerat0.0179.Lookingat theTable, themostdistantgalaxyexhibits thelargestredshift,andtheothergalaxiesappeartobesuchthatlargerredshiftsareassociatedwithmoredistantgalaxies.ThisisinlinewithwhatweexpectforanexpandingUniverse.Tomakeourobservationmorevisible,weshouldplotthedataonagraph,andthisiswhatwe’vedoneinFigure6.3.

Immediately, the pattern in the data springs into view: all the points arescatteredaroundastraightline.InFigure6.4weshowEdwinHubble’svarianton this type of graph, which he produced in 1929. Hubble’s vertical axis isslightlydifferentfromours,becausehechose topresenthisresults in termsofthe‘recessionalvelocity’ofthegalaxiesratherthantheredshift.Aswewillseeinamoment, the recessionalvelocity isequal to the redshiftmultipliedby thespeedoflight,soHubble’sdataextendtoredshiftsaslargeas0.003(1000km/sdividedby3x105km/s).

OurgraphsandHubble’shavestraight linesdrawn through thedatapoints.Thesecorrespondtothe‘best-fit’lines,whichmeanstheyaretheoptimallinesthat can be drawn through the data. The data from the sixteen galaxies arescattered in the vicinity of the line but, because themeasured distances to thegalaxies are uncertain, the data are consistent with the line being the truedescriptionofwhat ishappening.By thiswemean that, ifweareconfident intheredshiftvalueofagalaxy,thenthelinewillgiveusabettermeasurementofthe distance than the Tully-Fisher measurement does. For example, the tableshows thatNGC0011has a redshift of0.0146and itsmeasureddistance is 55Mpc. But from the left-hand graph in Figure 6.3, you can use the line toconclude that this distance is probably abit too low, and that the actual valuemightbemore like65Mpc.Todo that youneed tonotice that thedatapointcorrespondingtoNGC0011isthesixthpointfromtheright.3

The line in the left-handgraphcorresponds to theequationz=2.23×10−4Mpc−1×d,wheredisthedistanceandzistheredshift.Noticethatthisallowsusto determine the distance to any astronomical object simply by observing itsredshift–that’stosay,taketheredshiftanddivideitby2.23×10−4todetermine

the distance in megaparsecs (this gives 65.5 for NGC0011). This techniqueprovides astronomerswithyet anotherway todeterminedistances to far-awayobjects. Strictly speaking, we should only use the equation for objects whoseredshifts lie between 0.001 and 0.02, because we haven’t plotted any dataoutside this range.4 The number 2.23 × 10−4Mpc−1 is called the gradient, orslope, of the graph. This is a very important number because, apart fromcalibratingtheredshift–distancerelation,ittellsushowfastspaceisexpanding.Let’snowunderstandwhyitshoulddoso.

Figure6.3Theredshiftofthespiralgalaxiesinoursample,plottedagainst

theirdistancesfromEarth.ThegraphontheleftistheHubbleplotmadeusing

distancesasdeterminedbyprofessionalastronomers,andthegraphonthe

rightistheHubbleplotmadeusingestimateddistancestothegalaxiesunder

theassumptionthattheyarethesamesize.

For a galaxy that is a distance d away, the light takes a time d/c to reachEarth,wherecisthespeedoflight.Duringthisintervaloftime,theUniversehasstretchedbythesamefactorasthelight,whichistheredshiftz.ThismeansthatthegalaxyhasrecededfromEarthbyadistancezdinatimed/c,i.e.thegalaxyis receding from theEarthat a speedv=zd/(d/c)=cz.This is the recessionalvelocitythatHubbleplottedontheverticalaxisinsteadoftheredshift.Hubble’sgraphdemonstrateswhatisnowknownasHubble’sLaw,whichstatesthattherecessional velocity of a galaxy v = Hd, where H is equal to the slope ofHubble’splot.Naturallyenough,H isknownas theHubbleconstant, and it isone of the most important numbers in physics because, as we will see in amoment,itisthenumberthattellsushowfastspaceisstretching.AccordingtoHubble’s original data, H had a value of approximately 500 km/s/Mpc(somethingyou can readdirectly off his graph).According to our graph,H isequaltothespeedoflightmultipliedbytheslopeofourline,i.e.3×105km/s×2.23 × 10−4Mpc−1 = 67 km/s/Mpc. Clearly our result is not compatiblewithHubble’s. The reason for the discrepancy is that Hubble underestimated thedistances to his galaxies, just as he did when he correctly identified thatAndromedaisfaroutsidetheMilkyWay.Notwithstandingthis,hisplotwasthefirstexperimentalevidencetoindicatethattheUniverseisexpanding.

Figure6.4EdwinHubble’soriginalgraph.Thevelocitieswithwhichnearby

galaxiesrecedefromEarthconstitutethey-axisandtheirdistancesfromEarth

areshownonthex-axis.(Thewoefullylabelledy-axisshouldreadkm/s,not

km.)

AHubbleconstantof67km/s/Mpcmeansthatagalaxythatis1Mpcawayfrom Earth ismoving away from the Earth at a speed of 67 km/s due to theexpansionofspace(andagalaxyat10Mpcismovingawayat670km/s).Thislinear relationship between redshift (or recessional velocity) and distance ispreciselywhatwewouldexpectifspacehasbeenexpandingataconstantrate.Thismeansour spiral galaxydata confirm that spacehasbeen expanding at auniformrate,atleastforthepast300millionyears(becauseweonlylookedatgalaxiesoutto100Mpc,andthelighttraveltimefromthesegalaxiesisjustover300millionyears).

A niceway to visualize howHubble’s law comes about is to draw lots ofspotsonthesurfaceofaballoonandtheninflatetheballoon.Thedotsallrushaway from each other as the fabric of the balloon stretches; the dots that arefurther apart rush away from each other faster. If the expansion rate of theballoonstaysconstant, theballoonHubbleconstantwillbemeasured tobe thesame for each dot, as seen from any other dot, and we’ll get a ‘straight lineballoonHubbleplot’.Substitute‘galaxy’for‘dot’andthisiswhatweseeinourspiral galaxy data. The surface of a balloon is a two-dimensional example; athree-dimensionalexampleis thecaseofacakecontainingraisinsbeingbaked

inside an oven.As the cake expands, the raisinsmove away from each other.The Big Bangwould then be like themomentwhenwe start to cook a huge(possibly infinitely big) blob of dense dough. So, our Universe is more likebakingacakethaninflatingaballoon.Thisilluminatesoneofthemorecommonmisunderstandingsinmoderncosmology.TheobservationalfactthatallgalaxiesbeyondournearestneighboursappeartoberushingawayfromusdoesnotmeanweareatthecentreoftheUniverse,inthesamewaythatnodotonthesurfaceof a balloon can be said to be at the balloon’s centre. Rather, the data lendsupport to the picture, which stems from General Relativity, in which thegalaxiesareridingalonginaUniversewhosespaceisexpanding.

Tofinishoffthisanalysis, let’stakeanotherlookat theright-handgraphinFigure 6.3, which ismade using our simple estimate of distances to galaxies.Using it, we obtain aHubble constantH = 79 km/s/Mpc. This is satisfyinglyclose to the more accurate result, and while the data are far more scatteredaround the line than for the left-hand graph, the general trend towards higherredshiftsasthedistancesincreaseisstillapparent.Thebroadagreementbetweenthe two approaches indicates that our assumption that all spiral galaxies areroughly the samesizewasnot toobadafter all. It is satisfyingwhena simpleapproach is able to produce an approximate result that corroborates a moresophisticated analysis–it is a kind of sanity check. It is even more satisfyingwhenwelearnthatthemostprecisedeterminationoftheHubbleconstant,madeusingdatacollectedbytheEuropeanSpaceAgency’sPlancksatellite,is67.8±0.9 km/s/Mpc.This is terrific physics: anybody, standing in their back gardenwithareasonablysizedamateurtelescopeandafewthousandpoundsworthofkit,canprovethatweliveinanexpandingUniverseandmeasuretherateoftheUniverse’sexpansion.5

Let’srecapforamoment.Wehaveusedobservationaldataonsixteenspiralgalaxiesatdistancesbetween4and84MpctodeterminethattheUniversehasbeenexpandingataconstantrateforthepast300millionyearsorso.WeknowthisbecausethedataonourHubbleplotfallonastraightline.ThisfitswithourexpectationsbasedonGeneralRelativity.Whileweusedadatabaseratherthanmeasurementswemadeourselves,wehopethatitdoesn’tbotheryoutoomuchbythisstage.AsourexplorationoftheUniversegetsmoreambitious,wemustinevitably reach the point where the measurements we need to make requiremorethanjustacameraandamapofOgmore-by-Sea.Ifthisweren’tthecase,wewouldn’t need to spend billions of pounds on sophisticated telescopes andLarge Hadron Colliders. The age of the lone experimental scientist is almost

over, certainly in particle physics and cosmology, because all the simplemeasurementshavebeenmade,andthereisadmittedlyacertainlossofromanceinthis.Attheendoftheeighteenthcentury,WordsworthwroteofthestatueofNewtonattheentrancetoTrinityCollege,Cambridge:

Andfrommypillow,lookingforthbylightOfmoonorfavouringstars.IcouldbeholdTheAntechapelwheretheStatuestoodOfNewton,withhisprismandhissilentface,ThemarbleindexofaMindforeverVoyagingthroughstrangeseasofThought,alone.

Nevertheless,theromanticdeficitisnottotal.Itisstillopentoustounderstand,oreventoparticipatein,thelargeexperimentalcollaborationsthathavereplacedthe individual in gathering precise data about theUniverse. Ifwe spend sometime understanding how the datawere acquired, and satisfy ourselves thatweunderstandthemeasurements,thereisnoreasonwhyweshouldn’tusethedatatovoyagealonethroughstrangeseasofthoughtandtoconvinceourselvesoftheveracity of the remarkable picture of reality thatmodern science delivers. So,fromnowon,wewillrelyheavilyonthedatacollectedbymodern-dayteamsofastronomersasweheadrapidlytowardsthefrontiersofcurrentunderstanding.

WearestillquitesomewayfromestablishingtheexistenceoftheBigBang.That’s because, although we have managed to confirm that the Universe hasbeen expanding at the same rate for at least the last 300millionyears, such aconfirmation only rules out scenario (iv) from Robertson’s list (p. 142): thestatic, eternalUniverse initially favouredbyEinsteinhimself. It’s timenow toturntotheevidencethattheUniversehasbeenexpandingfornearly14billionyears.

WearegoingtodothisbysupposingthereactuallywasaBigBangandthenexploringwhat the consequencesof that suppositionmightbe.Of course, it isprettywell known that there is a good deal of evidence in support of theBigBang–buttheprocessofmakingaguessandseeingwheretheguesstakesusissomethingthatunderpinshowcutting-edgescienceisoftendone.AstheiconictheoreticalphysicistRichardFeynmanputit,whendescribingthesearchfornewlawsofNature:

Ingeneralwelookforanewlawbythefollowingprocess.Firstweguessit.Thenwecomputetheconsequencesoftheguesstoseewhatwouldbeimpliedifthislawthatweguessedisright.Thenwecomparetheresultofthe computation to nature, with experiment or experience, compare itdirectly with observation, to see if it works. If it disagrees withexperiment it iswrong. In thatsimplestatement is thekey toscience. Itdoes not make any difference how beautiful your guess is. It does notmakeanydifferencehowsmartyouare,whomadetheguess,orwhathisnameis–ifitdisagreeswithexperimentitiswrong.Thatisallthereistoit.

Here,ourguessisthatweliveinanexpandingUniverseoffiniteage,whichistosaythattherewasonceaBigBang.Inotherwords,weareguessingthatourUniverse iseither scenario (i) (expand forever into the future)or scenario (ii)(expandandthencontracttoaBigCrunch)onRobertson’slist.OurguesswillforceustomakeapredictionfortherelativeamountsofhydrogenandheliumintheUniverse,andwecanthencomparetheresultofthecomputationtoNature,asFeynmanputit,toseeifitworks.

IfourUniversestartedwithaBigBangandhasbeenexpandingeversince,therewouldhavebeenatimeinthedistantpastwhenanytwopointswereveryclosetogether.Theparticlesthatmakeupthemostdistantstarswouldhavebeenwithin a centimetre of the particles that make up your body. Of course, theywouldoncehavebeenevencloser thana centimetre–but that is alreadymind-bogglingenough.Thepointisthat,inauniversethathasbeenexpandingforitsentirelifetime,therewillhavebeenatimewhenthingsweresquashedtogetherintoastateofveryhighdensity.Imaginesquashingalloftheplanetsinthesolarsystemintoaregionthesizeofapea,nevermindallthestarsintheMilkyWaygalaxy and all of the galaxies in the sky. As far as particle physicists canascertain, the particles that make up atoms are of no discernible size, so thenotionthatallthematterintheobservableUniversewasoncecontainedwithinatiny region is not as crazy as it sounds. In any case,we don’t need to let ourimaginationrunquitesowildforthetimebeing.Wejustneedtosupposethatatsometime,longago,theUniversewasofamuchhigherdensitythanitistoday.Thatideaaloneisgoingtoproveveryfruitful.

Incidentally,sayingthattheentireobservableUniversewasoncecompressedintoatinyregionofspaceisnotthesameassayingthat,atthetimeoftheBigBang, the Universe burst forth from a tiny region of space. The word

‘observable’ in ‘observableUniverse’ iscrucial inmaking thisdistinction.TheobservableUniverseisthecollectionofthingsthatweknowtoexistbecausewecanseethem.6TheremaybemoretotheUniversethanthis,butwecan’tseetheother stuff because light travels at ‘only’ 300,000 kilometres per second andhasn’thadtimetoreachusfromverydistantobjects.WethereforedonotknowhowbigtheUniverseis–anditmaybeinfinitelybig,inwhichcaseitwouldhavebeeninfinitelybigatthetimeoftheBigBang.Tovisualizethis,wecanreturntothe cake-baking analogy. Just after the Big Bang, the particles were closetogether;as timepasses, the ‘cake’stretchesand theparticles (like the raisins)moveawayfromeachother.ThisanalogyinvitesustopicturetheBigBangassomethingthathappenedeverywhereinspace,andperhapsinvitesustorenameittheBigStretch.What’smore,thereisnoreasonwhythe‘cake’couldn’thavestarted out infinitely big: it can still expand and the particles will still beobservedtomoveawayfromeachotherastimepasses.

Nowlet’sfocusourattentiononthephysicalconditionswhentheUniversewas very much denser than it is today, and all the particles were very closetogether.Wecanimmediatelysaythatitwasveryhot.Theideathatagasheatsupwhen it is compressed into a smaller region of space is familiar to anyonewhohasusedabicyclepumptoinflateatyre.Let’simagineajourneybackintime.Theclumpingofgasandrocksintostarsandplanetsandtheclusteringofstarsintogalaxieswillbeundone.Ifwegofarenoughback,evenatoms–whichare clumpsofprotons, neutrons andelectrons–will break apart as theymelt inthesearingtemperatures.Infact,let’sgobacktoatimewhentheUniversewasahot, dense, featureless gas of elementary particles; protons, neutrons, photons,electrons andneutrinos.The temperature is nowaround1billiondegrees, andtheUniverse isonlyamatterofsecondsold.7 It is likely that thereweresomedark matter particles too, but we know that these interact very weakly withordinarymatter,meaningthattheywerenotmajorplayersinthedramaweareabouttodescribe.Ifyoudon’tbuythat,whichisfairenoughbecausewehaven’tpresentedanyevidenceyet,thenwecanguessthatthedarkmatterispassiveandseewhatpredictionsemerge.

WehavechosentowindtheclockbacktowhentheUniversewasabilliondegreesbecauseweareinterestedinthequestionoftheoriginandabundanceofchemicalelementslikeheliumintheUniverse,andthisisthetimewhenthefirstatomic elements were assembled from the primordial protons and neutrons.AlthoughweareguessingthatsuchconditionswereoncepresentintheUniverse(ahypothesis suggestedbyourpriorobservation that theUniverse iscurrently

expanding),wemostcertainlydonothavetoguessaboutthephysicsthattakesplaceatsuchtemperatures.Everythingthatweareabouttodescribeisverywelltriedandtestednuclearphysics.

Isolatedneutronsdonotliveforverylong.Oneofthemhasatypicallifetimeof about 10 minutes, after which it decays into a proton, an electron and aneutrino.Theword‘isolated’isimportant,becauseneutronscanliveforeveriftheybind togetherwithprotons tomakeatomicnuclei. In thedense,1-billion-degree-hotsoup,aneutroncancollidewithaprotonbeforeithastimetodecayandthetwowillsticktogetherthroughtheactionofthestrongnuclearforcetoform a deuterium nucleus. If the soupwasmuch hotter, the deuteriumwouldquicklybreakupagain,butattemperaturesbelow1billiondegreesitisstable.Deuteriumnucleicanthenfusewithmoreprotonsandotherdeuteriumnucleitoformheliumand,eventually,verysmallamountsoflithium.

Around3minutesaftertheBigBang,astheUniverseexpandedandcooled,the temperatures fell below those at which such nuclear fusion reactions canoccur,andthesynthesisofthefirstelementsstopped.ThisprocessisknownasBigBangNucleosynthesis.Textbooknuclearphysicscalculationspredictwhatitproduced:approximately25%bymassheliumand75%hydrogenwithtracesofdeuterium and lithium, the amounts of which can be calculated provided weknow the ratio of the number of photons to the total number of protons andneutrons in the Universe. Turning this round, the calculations allow us todetermine the photon-to-proton ratio, if we can measure the amount ofdeuterium.

ThefactthattheSuncontainsroughly27%helium,excludingtheheliumthathas been produced in its core, is an encouraging start. However, the task ofmaking precise comparisons between the nuclear physics calculations and theastronomical observations is not entirely straightforward, mainly because thestarsfusehydrogenintheircorestomakeheavierelements–and,whenstarsrunout of fuel and explode, these newly minted nuclei can get scattered around,polluting the conditions that pertained after Big Bang Nucleosynthesis. Tocircumvent the uncertainties that arise from not knowing precisely howmuchpollutionthere isfromdeadstars,astronomers identifyregionsin theUniverseinwhichtheactionofstarshasnothadalargeeffect.Onewaytodothisistolookattheabundanceofelementssuchasoxygen,whichcouldonlyhavebeenmade in stars. Ifvery littleoxygen ispresent in some regionof space,wecaninferthattheproductsofthenuclearreactionsinsidestarshavenotyetpollutedthe local environment.ThereareplacesknownasHII regions (pronounced ‘H

two’), particularly inside dwarf galaxies, which satisfy this criterion.Astronomerscanalsolookatverydistant,brightobjectsknownasquasars,someofwhich are over 10 billion light years away.We are seeing these objects astheywerelessthan4billionyearsaftertheBigBang,andthisisnotsufficienttime for much stellar nucleosynthesis, so the quasars exist in a more pristineenvironment. Just as predicted, these regions contain 25% helium and 75%hydrogen.

The most accurate measurements of the deuterium content of the earlyUniverse come from these quasars, and they tell us that there are around 30deuterium nuclei for every million hydrogen nuclei. Primordial elementalabundancescanalsobemeasuredbyobservingthelightemittedfromtheouterlayersofveryoldstars,whichwouldreflect thecompositionof theprimordialgascloudsoutofwhichtheyformed.Therearestars in theMilkyWaygalaxythatarewellover10billionyearsold.Someofthese,knownas‘lithium-plateau’stars, are used tomeasure the lithium content of the earlyUniverse. There isaroundonelithiumnucleusforevery10billionhydrogennuclei in thesestars’outerlayers.

Thegoodagreementbetweentheobservedandpredictedrelativeamountsofprimordial helium and hydrogen provides our first piece of evidence that theUniversehasbeenexpanding,notonlyforthepast300millionyears(whichiswhatourHubbleplotdemonstrated),butat leastsinceitwasahotplasmaatatemperature of a billion degrees (a plasma is a hot gas of electrically chargedparticles). The theoretical calculations are also able to explain the primordialabundances of deuterium and lithium.8We said that these abundances can bepredicted only if we know the photon-to-proton ratio: a ratio of 1.7 billionphotonsforeveryprotonorneutron,averagedacrosstheUniverse,isthevaluethatmakes the predictions accordwith observations.Wewill be able to crosscheckthis,usingatotallydifferentmethod,inChapter8.

Big Bang Nucleosynthesis ended after a few minutes, and laid down theprimordialelementsthatarespreadacrosstheUniversetoday,usuallyascloudsofatomsbutoccasionallyasstars.Here,weoughttosayawordortwoabouttheoriginoftheother,heavier,elements.ThesewereproducedmuchlateroninthehistoryoftheUniverse,asaresultofnuclearprocessesintheheartsofstars.Thefirststarswereformedoutof theprimordialhydrogenandheliumand,as theyburned, they fused the nuclei in their cores tomake heavier elements such ascarbonandoxygen.Elementsasmassiveasironcanbemadethisway,buttheheavier elements, including gold and silver, were formed in even more

spectacularfashion.Afterafewbillionyears,massivestarsrunoutoffuelanddie.Themostmassiveexplode,producingsupernovae;thefurnacesinwhichtheheaviestelementsareforged.Theyalsoscatter thenewlysynthesizedelementsacrossspace,seedingtheeventualformationofplanetssuchasours.Thisiswhythe outer layers of the Sun contain a slightly higher helium content than theoldest stars in the Milky Way: the stars that lived and died before the Sunformed have contaminated it with extra helium, not to mention the oxygen,nitrogen, iron and other elements beyond hydrogen that we see in the solarspectrum.Again,weseeconsistencyinourdescriptionofNature.

Let’snowfollow theevolutionof theUniverse forward in time from thosefirstfewminutesandseeifthereisanythingelsewecandeducethatmayleadtoobservableconsequencesandgiveusstillmoreconfidenceintheideaoftheBigBang. The next major landmark occurred when the Universe had cooledsufficiently for the electrons and primordial atomic nuclei to stick together tomakeatoms.Thishappensata temperatureofafewthousanddegrees,andweunderstandtheprocessverywell indeedfromlaboratoriesonEarth.BeforetheUniversecooled to this temperature, itwas largelyfeaturelessplasmamadeupmainly of photons and neutrinos, with a tiny admixture of light nuclei andelectrons.Atearlyenoughtimes,whentheplasmawastoohot,theelectronsandnucleiwere zipping around too rapidly to stick together and formatoms (theycansticktogetherasaresultoftheirmutualelectromagneticattractionbecausethey have opposite electric charge). But, as theUniverse cooled, the particlesmoved around less energetically and it became increasingly likely that anelectron found itself bound in orbit around a nucleus to make an atom. Thislandmarkevent–theformationofatomichydrogenandheliumatoms–hadaverydramaticimpactonthephotonsintheUniverse.

Everythingchangedwhentheatomsformed.Beforethistime,photonswereunabletotravelveryfarbeforehittinganelectronoranatomicnucleus.Thatisbecause photons scatter strongly off electrically charged particles.9 But withvirtuallyallthechargedparticlescombiningtomakeelectricallyneutralatoms,this state of things altered: photons do not scatter so easily off atoms. TheUniverse thereforemade a transition from being opaque to being transparent.Occurring 380,000 years after nucleosynthesis, this is known as the time ofrecombination.10ItisakeymomentinthehistoryofourUniverseandithasleftagloriousimprintonthesky.

Figure6.5showsthreesnapshotsof thehistoryof theUniverse. In thefirstframe,wetracethepathsofphotonsastheybouncearoundoffchargedparticles

inthehotplasma.Inthesecondframewehavewoundtheclockforwardpastthetime of recombination: the photons are now travelling in straight lines. Thesephotonscarryonuntiltheycollidewithsomething,and–becausetheUniverseispretty empty and filled with an ever-diluting gas of neutral hydrogen andhelium–most of them keep on travelling until the present day. The last of thethree frames shows someof theseancientphotons arrivingatEarth,now.Thepossibleexistenceof thesephotons isavery striking idea. In thewordsof thePrincetonphysicistsRobertDicke,JimPeebles,PeterRollandDavidWilkinson,back in 1964: ‘Could the universe have been filledwith black-body radiationfrom[a]possiblehigh-temperaturestate?’

Figure6.5TheoriginoftheCosmicMicrowaveBackground(CMB)radiation.

Dickeandhiscolleaguesmadetheseobservationshavingjustheardthenewsthat, away at the Bell Laboratories in New Jersey, Arno Penzias and RobertWilson had detected an unexpected background of microwaves through atelescope. The fact that the photons from theCosmicMicrowaveBackground(CMB)shouldtodaybeobservedasmicrowavesisexactlyasexpected,becausetheexpansionofspacesincethetimeofrecombinationredshiftedthephotonstomicrowavewavelengths.Bythemid-1960s,theevidenceforwhatisnowknownas the CMB had become compelling. The Earth, in other words, is bathed inmicrowaveradiationofpreciselythekindanticipatediftheUniversehadindeedpassedthroughahotphasebeforetheformationofatoms.

Interestinglyenough, the idea that theUniverse should todaybe filledwithCMB radiation goes back decades before Penzias andWilson’s Nobel Prize-winningdiscovery.In1948,inatremendousburstofcreativity,GeorgeGamow,RalphAlpher,RobertHermanandHansBethewroteaseriesofelevenresearchpapers in which they developed the theory of Big Bang Nucleosynthesis.11Gamowandcolleagues’ideaswereaheadoftheirtimeandlaylargelydormantuntilthediscoveryoftheCMBin1964.

The CMB and its properties have been, and continue to be, examined indetail. This oldest light is a treasure trove of information, because it providesearth-bound astronomers with an opportunity to garner information on whathappened shortly after the Big Bang.We’ll be delving much further into thetreasures of theCMB inChapter 8, but for nowwe aregoing to focuson thebroadfeaturesofthoseold,stretchedphotonsfromthebeginningoftime,whichwenowdetectonEarthasmicrowaves.

Figure6.6showsameasurementoftheCMBmadeintheearly1990sbytheCosmicBackgroundExplorer (COBE) satellite. It shows how themicrowavesarriving at the Earth are distributed in wavelength. The height of the curvemeasures the brightness of themicrowaves and the shape tells us how this isdistributed across different wavelengths. We see that the most commonlyoccurringwavelengthisaround2mm.Whatisclearfromthegraphisthat themeasureddatapointsare inperfectagreementwith the theoreticalexpectation,whichisthesmoothsolidline.

Thatsmoothlineistheresultofacalculationoriginallyperformedinthelatenineteenth century by Max Planck, the brilliant German physicist who alsoplayedacentralroleintheearlydevelopmentofquantumphysics.Itisthecurvecorresponding to the spectrum of light emitted by an object cooled to atemperatureof2.73kelvin,which is justoverminus270degreescelsius.This

amazing agreement between Max Planck’s calculation and the COBEmeasurements is what Dicke and colleagues anticipated when they spoke of‘black-bodyradiation’.Here,thetechnicalterm‘black-body’isusedtodescribehowtheenergyofagasissharedoutbetweenitscomponentparticleswhenallparts of the gas are at the same temperature. Because the CMB photonsoriginated from the primordial plasma, they ought to exhibit a near-perfectblack-bodyspectrum,andthisisexactlywhatwasseenbyCOBE.

Figure6.6TheCOBEmeasurementoftheCosmicMicrowaveBackground

radiation,whichshowsthattheEarthisbathedinmicrowavesata

temperatureof2.74kelvin.

The predicted observation of a near-perfect black-body spectrum ofmicrowave radiation is our second, very compelling, piece of evidence insupportoftheBigBang.

In a nutshell, then, Big Bang Nucleosynthesis and the existence of theCosmicMicrowaveBackgroundprovidecompellingevidenceinsupportof theideathattheUniversewasonce,longago,inahot,densestate.However,wecandaretogomuchfurtherbecauseEinstein’sequationsarealsoabletoprovideuswith a precise description of the evolution of the Universe, from its earliest

momentstothedistantfuture,ifwecanmeasuretheconstituentsoftheUniversetoday.Inordertodothis,wemustendeavourtoweightheUniverse.

7.WEIGHINGTHEUNIVERSE

Wenow turn to the taskofmakingan inventoryof theUniverse.Wewant toknowwhattypesofmatterexistinitandinwhatproportions,howmuchenergyis carried by electromagnetic waves, and whether there is anything else outthere–anythingthatisespeciallyhardtodetect.ThisinventoryiscrucialbecauseFriedmann’s equation tells us precisely how these different sorts of materialcontribute,via theirgravitationaleffects, to theexpansionof theUniverse.Wewillbeabletocomputehowthescalefactorhaschangedwithtime,andhowitwillchangeinthefuture.ThiswilltelluspreciselywhentheBigBanghappenedandwhatthefutureholds;willtheUniverseexpandforeverorwilliteventuallycollapse? Answering these twomomentous questions will in turn allow us toidentify which of Robertson’s imagined universes we actually live in.Understanding thehistoryof the expansionof theUniverse is nowour focus–but,asissooftenthecaseinscience,wewillencountersomesurprisesalongtheway.

WedescribedtheFriedmannequationinBox10(pp.144–5),andit’sworthalook now if you haven’t already done so. You don’t have to understand theequation in order to understand the rest of this chapter. All you need toappreciateisthattheprecisewayinwhichtheUniverseexpandsisdeterminedbythetypeandamountofstuffitcontains.

From the observation that theCosmicMicrowaveBackground (CMB) is agasofphotonsatatemperatureof2.73kelvin,wecanusesomeundergraduate-level statistical mechanics to deduce that, today, there are an average of 410CMBphotons ineverycubiccentimetreof space. In turn, this implies that thetotalenergycarriedbythesephotonsisjustover40millionthsofajouleinsideeach1kilometrecube(4×10−14J/m3).Thisisatinyenergydensitybyeverydaystandards:a10wattlightbulbemits10joulesoflightenergyeverysecond.1ThegravitationalinfluenceofthisenergydensityissuchthatittendstoslowdowntherateatwhichtheUniverseisexpanding.

For accounting purposes, we will convert this energy density into massdensity, by dividing by the speed of light squared: doing this gives a mass

densityequalto4.5×10−31kg/m3.Themeaningofanaveragemassdensityisclearforparticlesthatactuallydohavemass(likeprotonsorelectrons):wejustneedtocountupthetotalmassofparticlesinsidesomeregionofspaceandthendivide by the volume of that region. Photons are different in that they carryenergy but do not have any mass. This peculiarity is a feature of Einstein’sSpecialTheoryofRelativity, thedetailsofwhichareunimportanthere:allweneedtoknowisthattheenergycarriedbythephotonshasagravitationalimpacton how the Universe evolves, and that its impact can be quantified by acontributiontothetotalmassdensity2oftheUniversethatistodayequalto4.5×10−31kg/m3.

AswesawinFigure5.8,therearethreepossibilitiesforthegeometryoftheUniverseundertheassumptionthatmatterandenergyareevenlydistributed(wecalledthisahomogeneousandisotropicUniverse).Iftheaveragemassdensityis precisely equal to a very special value, known as the critical density, theFriedmann equation tells us that space is flat. If the average mass densityexceedsthecriticaldensitythentheUniversecurvesintoasphericalgeometry,and if it issmaller than thecriticaldensity then theUniverse ishyperbolic.3 IfwetakethevalueoftheHubbleconstantthatweobtainedfromouranalysisofspiralgalaxies,H=70km/s/Mpc, then thecriticaldensity is9×10−27kg/m3.There’s a niceway to think about this, because a protonweighs in at 1.67 ×10−27kg:thecriticaldensitycorrespondstoanaverageofjustover5protonsineverycubicmetreof space (there’snothingspecialaboutusingprotons in thiscomparison:itislikesayingthatatypicalAsianbullelephanthasamassequaltothatofaround30humanbeings).Asitstands,thephotonsaccountforfarlessthan the critical density, which means that if there was nothing else in theUniverse besides photons we would be living in a Universe with hyperbolicgeometry,whichwouldgoonexpandingforever.But,ofcourse,theUniverseismadeofmorethanjustlight.

Because we know how much energy in the Universe is carried by thephotons,wecanimmediatelydeducehowmuchiscarriedbyneutrinos.Thisisbecause in the seconds-old Universe the neutrinos and photons were bothbouncingaroundatthesametemperature,togetherwiththerestoftheparticlesintheUniverse.Theneutrinoenergydensityisslightlylessthanthatduetothephotons,forreasonswedescribeinBox12(p.184).Together,theneutrinosandphotons account for a present-daymass density of approximately 7.5 × 10−31kg/m3.

WenotedinthelastchapterthattheBigBangNucleosynthesiscalculations

agreewiththeobserveddataonlyifthereare1.7billionphotonsintheUniversetoday for every proton or neutron. From our study of theCMB,we have justdiscoveredthatthereare410photonsineverycubiccentimetreofspace.Takentogether, these pieces of information mean that, on average, there should beapproximately 1 proton or neutron in every four cubicmetres of space.Now,protons and neutrons should dominate the mass in the Universe arising fromordinarymatter,becausetheyareeacharound2000timesheavierthantheonlyother ordinary matter particle; the electron.4 Suddenly, we find ourselvesarrivingataprediction:thematterintheUniverseoutofwhicheverystar,gascloudandgalaxyismade,shouldamounttoagrandtotalof4×10-28kg/m3.Ifthis is not found to be the case then the Big Bang theory we have beendescribingwillfail.

Thisisprettyimpressivestuff:wearelayingclaimtoknowingpreciselyhowmanyphotons,neutrinos,protons,neutronsandelectronsarepresentonaverageineverycubicmetreof theUniverse today.Moreover,weclaim toknowhowthoseprotonsandneutronscombinedtomakehydrogen,deuteriumandheliumwithasmidgeonoflithium.Thesearetheingredientsthathaveclumpedtogethertomakeeverythingwearefamiliarwith.Ofcourse,wewouldlovetobeabletocorroboratethesenumbersusingsomeentirelydifferentpieceofscience,anditis one of the highlights of modern cosmology that–as we’ll see in the nextchapter–weareabletodoso.Fornow,wemustpresson,becausetherethereismoretotheUniversethanjustlightandordinarymatter.

BOX12.THEENERGYDENSITYOFNEUTRINOS

As the Universe cooled, neutrinos stopped interacting witheverythingelsebeforephotonsdidbecausetheweaknuclearforceacts over much shorter distances than the electromagnetic forcedoes. So, as the Universe expanded, it became sufficiently dilutethat the neutrinos no longer had the opportunity to interact. Thishappenedatatemperatureofaround10billionkelvin,around10−5

secondsafter theBigBang.Even though theystopped interacting,thetemperatureoftheneutrinosfellastheUniverseexpanded,justas the photon temperature fell. Today, the neutrino temperature isslightly less than the photon temperature because, after the

neutrinosstopped interacting, thephotongasgotheatedupa littlebitasaresultofaprocesscalledelectron-positronannihilation.Thisistheprocessbywhichanelectronencountersapositron(ananti-matter electron) and both disappear, to be replaced by twophotons.1 In this way, some extra photons were produced whichheatedupthephotonsalittlebitcomparedtotheneutrinos.Ifyou’rewonderinghow thephotonsandneutrinoscouldhave twodifferenttemperatures, then bear in mind that this is possible because theneutrinos have effectively stopped interacting with anything, whichmeans that there is thennoway for themtoshareenergywith theother particles in the Universe. The result is that the primordialneutrino temperature today is slightly less than 2 kelvin. Theneutrinos form a Cosmic Neutrino Background on the sky, at aredshiftofaround1010,anditwouldbefantastictobeabletobuildadetector to see it. Unfortunately, that is way beyond currentcapabilities.

Theearlieststrongevidencethat thereissomeotherformofmatterbeyondthat visible through telescopes was presented way back in 1933, in a paperwrittenbytheSwissastronomerFritzZwicky.HenoticedthatthegalaxiesintheComagalaxycluster5weremovingmuchfasterthanexpected.Thegalaxiesthatmakeupaclusterareinorbitaroundeachotherduetotheirmutualgravitationalattraction,justastheEarthorbitstheSun,exceptthatthedistancesinvolvedarevastlygreater: typically thegalaxies are a fewmillion lightyears apart.UsingNewton’slaws,wecanestimatethetotalmassoftheclusterifweknowhowfasteach galaxy ismoving. This is rather like the examplewe encountered in thepreviouschapterwhenwediscussed theTully-Fisher relationforstarsorbitingaround a galaxy. It’s also similar to thewaywe inferred themass of the SunusingNewton’slawsandobservationsoftheplanetaryorbitsinChapter3.Ifwereplacetheword‘star’by‘galaxy’and‘galaxy’by‘cluster’inBox11(p.151),we can deduce the speed of an average galaxy in a cluster by observing theDopplershiftofthelightitemits.Oncewehavethespeedofthegalaxy,wecanestimatethemassof thecluster.Zwickydiscoveredthat themassof theComaclusterisfarbiggerthancanbeaccountedforbyitsvisiblecontents.Thelightemittedby thecluster is roughlyequivalent to thatemittedby30 trillionsuns,whiletheinferredmassisequivalenttothatof4500trillionsuns.Thatisquitea

difference.Astronomers speak of the ‘mass-to-light ratio’, which is the total mass

divided by the total light output. In the case of the Coma cluster, this isapproximately 4500/30= 150.OurSun has amass-to-light ratio of 1 in theseunits.6Suchalargemass-to-lightratioimmediatelytellsusthatmostofthemassoftheComaclusterisnotlocatedinstuffthatemitslight.Itisnaturaltosupposethatthismightbeintheformofdarkobjectslikedeadstarsorblackholes,butwewillseeinamomentthatthiscannotbethewholestory.Whateverthecase,thereisalotofdarkmaterialtoaccountfor–andthiswasasurprisetoZwicky.What’smore,wenowknowthatthereisnothingspecialabouttheComacluster.Thelargestclustershavemass-to-lightratiosofaround250.

The mass-to-light ratio is a particularly valuable number. If we know itsvalueaveragedacrosstheentireUniversewecanuseitinconjunctionwiththemeasurement of the average luminosity of space to infer the average massdensity of space. The average luminosity of space corresponds to 130millionsunspercubicmegaparsec.So,ifthemass-to-lightratiofortheUniverseis250,itfollowsthatthematterdensityisequivalentto250×130millionsunsinsideeach1Mpccube.GiventhatourSunhasamassof2×1030kg, thisgivesanaveragematterdensityof22×10−28kg/m3.Youmayreasonablyobjectthatthemass-to-lightratioofaclusterofgalaxiesisnotthesamethingasthemass-to-light ratioof theUniverse at large,whichwe cannotmeasuredirectly.This istrue: indeed, for our solar system the mass-to-light ratio is very close to 1,because the Sun carries most of the mass and gives off most of the light.Conversely, in thevicinityofablackholeordeadstar, themass to light ratiowillbeverylargebecausethereisalotofmassbutnotmuchlight.However,thelarger the region of space over which we average, the more we expect it toapproximate the behaviour of theUniverse at large, so by looking at the verylargest galaxy clusters we expect to obtain a decent estimate of the averagemass-to-lightratiooftheUniverse.

Amass density of 22 × 10−28 kg/m3 is more than 5 times larger than thenumberwedetermined for themassdensityof ordinarymatter,which impliesthat the material content of the Universe is predominantly composed ofsomething other than the products of Big Bang Nucleosynthesis. This isimportantbecause it removes thepossibility that theunseenmattermightarisefrom dead stars or black holes or other non-luminous stuffmade from atoms,because the mass density of ordinary matter that we estimated from the BigBangNucleosynthesiswasnotjustrestrictedtothematterthatwecansee.We

callthisnewstuffdarkmatter.

Figure7.1TheBulletClustershowshowthehotgas(red)isdisplacedfrom

themajorityofthemassinferredfromgravitationallensing(blue).

This is, of course, a rather bold conclusion to draw. We seem to haveinventedanewformofmattersimplybecauseourcalculationsandobservationsdon’tmatch.Thiswouldbeavalidcriticismifitweren’tforthefactthatthereare many other ways of measuring the amount of matter contained withingalaxiesandgalaxyclusters–andtheyallgivethesameresult.Duringthe1970s,a series of observations pioneered by the American astronomer Vera Rubinrevealedthat,forverymanygalaxies,starsandgasareorbitingmuchfasterthanwouldbeexpectediftheonlymassinthegalaxywereluminous.Thephysicsisthesameasforgalaxyclusters–excepthereitisstars,ratherthanentiregalaxies,thatappeartobemovingtoofast.AnalysisofRubin’s‘galaxyrotationcurves’indicatesthatasmuchas90%ofthemassinagalaxyisnotluminous,anduntilquiterecentlythiswastheprincipalevidencefordarkmatter.

The most dramatic evidence for the existence of dark matter comes from

observations of the aftermath of a violent cosmic collision, as two clusters ofgalaxiesploughed througheachotherat severalmillionmilesperhour.Figure7.1 shows several images of what is now known as the ‘Bullet Cluster’superimposedontopofeachother.Thebackgroundisacompositeimagetakenin visible light by the 6.5metreMagellan telescopes inChile and theHubbleSpaceTelescope.TheredclumpsareanimageofthesamesystemtakenbytheChandra X-ray observatory and they show where the hot gas resides (bydetecting theX-rays they emit). In theBulletCluster, this gas is glowing at atemperatureofaround100milliondegrees.Theblueclumpsshowthelocationof themajorityof themass in thecluster, inferredusinga techniqueknownasgravitationallensing,aningeniousapplicationofGeneralRelativity.SpaceandtimearedistortedbythepresenceofthematterintheBulletCluster.ThismeansthattheimageofanythingbehindtheBulletCluster,asviewedfromEarth,willbedistortedbecausethelighthastopassthroughthedistortedspacetimeonitswaytous.Figure7.2showsamoredramaticexampleofgravitational lensing.ThegalaxyclusterAbell2218isdistortingtheimagesofmoredistantgalaxieslying behind it, which are visible as smeared-out arcs of light. The effect isratherlikeviewingsomethingthroughthebottomofawineglass.Justaswecaninfertheshapeofawineglassfromthewayitdistortsanimage,sowecaninfertheshapeofspacetimebythewayitdistortsanimage.GeneralRelativity,then,allows us to determine the distribution of matter in the intervening galaxycluster,becauseit tellsuswhatdistributionisnecessarytodistortspacetimeinthe required way. Incidentally, the masses of galaxy clusters can also bedeterminedusinggravitational lensing,andtheresultsagreeverywellwiththemasses inferred using the orbital speeds of their constituent galaxies in themanner first used by Zwicky in the 1930s. As ever, it is good to be able tomeasurethesamethingindifferentways.

ItisimmediatelyobviousfromFigure7.1thatthemajorityofthemassintheBulletCluster isnot in thevicinityof thehotgas.This is inexplicablewithoutinvokingdarkmatter,becausethebulkofthemassthatwecanseeislocatedinthevicinityofthehotgas–meaningthat,unlessdarkmatterispresent,theblueandredregionsshouldlieontopofeachother.7Clearly,therightmostclusterofgalaxieshasplougheditswayfromlefttoright:youcanseetheshockwaveofX-rayslyinginitswake.Duringthecollision,thehotgasofchargedparticlesineach cluster was slowed down, which is expected because the hot gas iscomposedof electricallychargedparticles that interact stronglyand scatteroffeachother.Bycontrast,mostofthemassintheclusterswasevidentlymuchless

affectedbythecollision.Indeedithardlyseemstohavefeltthecollisionatall;ithas continued on its journey through space, passing straight through anothergalaxy cluster at over a million miles an hour. The collision has caused themajority of the ordinary matter in the clusters, contained in the hot gas, tobecome separated from themajority of themass, which exists in the form ofweaklyinteractingdarkmatter.

Figure7.2TheHubbleSpaceTelescopeimageoftheAbell2218galaxy

cluster,includingthegravitationallydistortedimagesofmoredistantobjects

lyingbehindit.

Today, the existence of dark matter is very well established, although wecertainlydonotknowwhatitactuallyis.Thesimplestideawouldbetosupposethat there is a new type of particle,which hasn’t been observed onEarth yet.This is a very reasonable possibility: it would be verging on the arrogant tosupposethattheonlyparticlesthatexistarethosewehavealreadyobserved.Aswehaveseen, thedarkmatteronlyappears to interactappreciablyviagravity,whichiswhyitisdark.Itcouldconceivablyhavesomeweak,non-gravitationalinteraction with other particles, which might make it possible to eventuallyproduceand/ordetectitinparticlephysicslaboratoriesonEarth.Thatpossibilityhasbeenconsideredveryseriously,notleastbecausesomeofthemorepopular

ideas in particle physics do predict the existence of just such a dark matterparticle, and those ideas do not draw on anymotivation from astronomy andcosmology.Aswewillseeinthenextchapter,darkmattercanalsobeinferredfromthewaythatgalaxiesclumptogetheracross theentireUniverseandfromananalysisofthefinedetailsoftheCosmicMicrowaveBackground.

NowisagoodtimetopauseforbreathandrecapwhereweareinourquesttopindownthematerialcontentsoftheUniverse.Wehaveworkedoutthat,atthepresenttime:

(i)photonsandneutrinosgive rise toanaveragemassdensityof7.5×10−31kg/m3;

(ii)thereareapproximately4×10−28kg/m3ofordinarymatter,whichismadeupofatomicnucleiandelectrons;

(iii)thesumtotalofthedarkmatterandtheordinarymatteraveragesto22×10−28kg/m3,whichmeansthatthedarkmatteralonecontributesaround18×10−28kg/m3.

Becausethesumtotalofthesemassdensitiesisabout30%ofthecriticaldensity(9×10−27 kg/m3), thepossibility thatwe live inahyperbolicUniverse is stilllooming.However, there issomethingverycurious tonote: thecriticaldensitycorrespondstoaround5protonspercubicmetre,whilethesumtotalofthedarkandordinarymatterchecksinaround1.5protonspercubicmetre.Whyarethesenumbers so similar? Without any prior bias, we would have had no troubleimagining that these numbersmight bewildly different: a trillion protons percubicmetre,forexample,ormaybeoneprotoneverymegaparsec.Thefactthattheobserveddensityissoclosetothecriticaldensitysmacksofcoincidence–orperhaps it isaclue.Whatever thecase, it certainlymeans thatourUniverse isnotsofarfrombeingflat.

Thisapparentcoincidenceisallthemoresurprisingbecause,accordingtotheFriedmann equation, the difference between thematter density and the criticaldensity ought to get bigger as time passes.8 Thismeans that if the density isclosetocriticalnow,whichitis,thenitmusthavebeenevenclosertocriticalinthepast.Thisisanexampleofwhatphysicistscalla‘finetuning’problem.Itisas if thematterdensity in theUniversewasadjustedverypreciselyat theBigBanginordertogiverisetotheUniverseweseetoday.Toputitanotherway,for us to find ourselves in a nearly flat Universe now, spacemust have been

‘unfeasibly close to flat’ at the Big Bang. Cosmologists refer to this as ‘theflatnessproblem’.

ManycosmologistswouldmuchrathercontemplateaperfectlyflatUniverseinwhichthedensityisequaltothecriticaldensitythananearlyflatone.Atfirstsight,thismightsoundlikeaprejudice;whywouldascientistbehappywithaUniverseofpreciselythecriticaldensityratherthan30%ofit?TheexplanationfortheapparentprejudiceisthatthereisaverysimplereasonwhytheUniversemightappeartobeflat.Specifically,evenacurveduniversewouldappearflatifitissufficientlybig.

The link between flatness and the size of theUniverse is easy to grasp.ApersonrestrictedtoroamoverasmallportionoftheEarth’ssurfacecouldeasilybefooledintothinkingthattheEarthisflatsimplybecauseitisbig.Likewise,ifwearedoingourastronomyinasmallportionofavastlybiggerUniverse,thenwecouldeasilybedrawntoconcludethatitisflat.YoucanseethisideaatplayintheFriedmannequation(takealookatBox10again):makingRlargerhastheeffect ofmaking the term that dependson thegeometryof space smaller, andonecanimagineitbeingsolargeastomakeitseffectirrelevant.Inthatcase,theUniversewould appear flat even though it isn’t flat on the largest scales.Thequestion of why the density of the Universe should be close to the criticaldensity thengets replacedwith thequestionofwhy theUniverse issobig.Aswe’vesuggestedinChapter1,ahugeUniverseisthenaturalconsequenceofthetheory of inflation. Perhaps you now see the reasonwhy a cosmologistmightprefertosupposethatweliveinaveryflatUniverse,whosemassdensityisveryclose to the critical density, rather than one where it is 30% of the criticaldensity.Inthesuper-flatcase,thetaskis‘simply’tofindatheory(likeinflation)thatdeliversaUniversethatisvastlybiggerthantheobservableUniverse.Thisisrathereasiertocontemplatethantryingtofindatheorythatdeliversa‘justso’Universe,where the radiusR just happens to be roughly the same size as theobservable Universe. Obviously, the trouble with supposing that we live in averyflatUniverseinsteadofanearlyflatoneisthatweneedtoaccountforthemissing 70% of themass density. Remarkably, the theoretical prejudice for averyflatUniverseappearstobecorrect,althoughtheevidencetoproveitcomesfrom a quite unexpected direction: towards the end of the twentieth century,Einstein’scosmologicalconstantrosephoenix-likefromtheashes.

Figure7.3(left)M3isoneofthebrightestglobularclustersinthesky.Atadistanceofover30thousandlightyears,itisvisibleinthenorthern

hemisphereusingbinocularsandcontainsaroundhalfamillionstars;(right)TheOmegaCentauriGlobularCluster,thelargestglobularclusterintheMilky

Way.

WemetthecosmologicalconstantbackinChapter5,whenweobservedthatEinsteinintroduceditintohisequationsofGeneralRelativityinanunsuccessfulattempttoconstructaviablemodelofanon-expandinguniverse.AfterEinsteindismisseditashisgreatestblunder,thepossibilitythattheUniversemighttodaybeendowedwithacosmologicalconstantdidnotfeaturehighontheprioritiesofmostcosmologists–notuntilthe1990s,thatis.

TheFriedmannequationallowsustocalculatetheageoftheUniverse(oncewe know the average mass densities of the different types of matter and thepresent-dayHubble constant). The trouble is that a density ofmatter equal toonly30%ofthecriticaldensityandaHubbleconstantof70km/s/MpcsuggeststhattheUniverseisyoungerthansomeofitscontents,whichisobviouslyabitofadisaster.Theproblemcomesfromthedatingofclustersofstarsknownasglobularclusters.

Astronomers love globular clusters because they provide some of themostspectacularsightsinthenightskythatcanbeviewedthroughasmalltelescope.Theyaregreatballsofstars,andarenumerousintheMilkyWay.Theleft-handimageinFigure7.3isaphotographofMessier3inthenorthernconstellationofCanesVenatici. This picturewas taken by a friend of ours,BillChamberlain,whorecentlytookupastronomyasahobbyatthetenderageof70.Oneofthemost spectacular and well-studied globular clusters is Omega Centauri, thelargest globular cluster in theMilkyWay. It is only 15,800 light years awayfromEarth,andcontainstenmillionstars,whichmakesitabrightandbeautifulobject.AquitestunningphotographofOmegaCentauri,takenbytheEuropeanSouthern Observatory’s La Silla Observatory, is also shown in Figure 7.3.AstronomerscandeterminetheageofclusterslikeM3bystudyingthevariationincolourandbrightnessofthestarsthatmakeupthecluster.InthecaseofM3,theyhavefiguredoutthatitisaround11.4billionyearsold.

To study systems of stars, astronomers arrange the stars into a diagramaccordingtotheircolour(whichisdirectlyrelatedtotheirsurfacetemperature)andbrightness.TheseareknownasHerzsprung-Russell(HR)diagrams:9Figure7.4shows theHRdiagramsforOmegaCentauri (right)and thePleiades(left).These make for very pretty diagrams because the stars are obviously notscatteredaboutatrandom.InthecaseofthePleiades,themajorityofthestarslieonthesame,sweepingcurve,withthehot,bluestarsbeingthebrightestandthecooler,redstarsbeingthedimmest.Thiscurveisknownasthe‘mainsequence’.ThemainsequenceislessprominentinthecaseofOmegaCentauri.Instead,thepatternofstarscurvesbackonitself,indicatingthatOmegaCentauricontainsasignificantpopulationofbright,redstars.Thesestarsareknownasredgiants.

Figure7.4Herzsprung-Russelldiagramscanbeusedtodatestarclusters.

TheleftdiagramisforthePleiadesandtherightdiagramisforOmega

Centauri.

ThebasicphysicsoftheHRdiagramissimpletograsp.Mainsequencestarslike our Sun are busy fusing hydrogen into helium in their cores. The energyreleasedinthesefusionreactionscreatesthepressurethatresiststheinwardpullofgravity,allowingthestartoexistinastablestateaslongasithasfueltoburn.More massive stars have to generate more heat energy to create the higherpressureneededtoresisttheinwardpullofgravity;theyarethereforehotterandbluer,attheexpenseofhavingtoburntheirfuelmorequickly.Incontrast,lessmassivestarshavetogeneratelessenergy,andarethereforedimmer,redderandburntheirfuelmoreslowly.Asstarsrunoutofhydrogenfuel,theycontractandtheircoresheatup,allowingthemtoinitiateheliumburning.Thisishowsomeof the heavier elements, including oxygen and carbon, aremade. Their super-heatedcorescausetheouterlayersofthestartoexpandandcool,resultinginthecoolbutbrightstars–redgiants–whichcanbeseentowards the toprightof theHRdiagrams.Thestoryofstellarevolution,then,islaidoutonanHRdiagram.Forus,thekeyobservationisthatwhenastarrunsoutofhydrogenfuelitwillmoveoffthemainsequenceandontotheredgiantbranch,whichismuchmore

prominentinthecaseofOmegaCentaurithanitisinthePleiades.Bright blue-white starswillmove off themain sequence first, followed by

yellowstars likeourSun, leavingonly thevery long-liveddimreddwarfstarsbehind: thesehaveenough fuel tocarryonshining formany times thecurrentageof theUniverse.Inarelativelyshort time, theredgiantswillexhaust theirfusionfuelcompletely,atwhichpointtheircoreswillcollapseintoeitherawhitedwarfor,forthemostmassivestars,aneutronstarorablackhole.Whitedwarfsaredim,blue-whitestellarremnants,andsitinthelowerleftoftheHRdiagram.PerhapsyoucannowseehowanHRdiagrammightallowastronomerstodateastarcluster.Olderclusterswillcontainalargernumberofredgiantsandwhitedwarfs,andfewerstarsonthemainsequence;whereas,forveryyoungclusters,allthestarswillstillbeonthemainsequenceandtherewillbenoredgiants.

With only this information, it is clear that the Pleiades is significantlyyounger than Omega Centauri because it contains no red giants, and a largepopulationofyoung,brightbluestars;meanwhile,reddwarves,redgiantsandasmatteringofwhitedwarvesarecommoninOmegaCentauri.Astronomerscandomuch better than simply determining the relative ages of clusters, becausetheyunderstandstellarevolution.InChapter2,weworkedouttheageoftheSunusingourknowledgeofnuclearphysics,andwecandothesamewiththestarsinclusters.IfyoulookattheHRdiagramforOmegaCentauri,forexample,youseethattherearenoyellowSun-likestarsleftonthemainsequence.Becauseweknowthatyellowmain-sequencestarslikeourSunhavealifetimeofaround10billionyears,thismustmeanthatOmegaCentauriisolderthan10billionyears;itis,infact,11.5billionyearsold.

This age is interesting because it is marginally older than the age of auniverse that is currently expanding at 70 km/s/Mpc and contains onlymatterwith a density equal to 30% of the critical density–using the Friedmannequation,suchauniversewouldbe11.3billionyearsold.Theproblemisevenmoreseriousbecausethereareotherglobularclustersthathaveagesinexcessof12billionyears.Atastroke,theglobularclusterageproblemcanbesolvedbysupposing that the Universe is endowedwith a cosmological constant, whosesize corresponds to amass density equal to 70% of the criticalmass density.ThishasthetwinadvantagesofincreasingthepredictedageoftheUniversetocloserto14billionyearsandincreasingthetotalmassdensityto100%–asbefitsa flat Universe. By the mid-1990s other data, such as that coming from thetheoryofgalaxyformationthatwewillencounterinthenextchapter,wasalsoencouragingcosmologiststotakeseriouslythepossibilitythattheremightbea

cosmological constant after all. However, for many cosmologists the balancefinallytippedinfavouroftheexistenceofacosmologicalconstantin1998.

Inthatyear,twoteamsofastronomerspresentedtheirresultsontheredshift-distance relationship for Type 1A supernovae. In other words, they made aHubble plot just like we did to measure the current expansion rate of theUniverseinthelastchapter,butinsteadofusingspiralgalaxiestheastronomersused Type 1A supernovae. Recall that these exploding stars are especiallyvaluabletoastronomersbecausetheyallexplodeinthesameway,whichmeanswecanmeasurehowfarawaytheyareusingtheirobservedbrightness.Becausesupernovaecanbeseenatverylargedistances,theycanbeusedtoobservehowtheUniverse has been expanding over the past fewbillion years,which is farbetterthanwemanagedusingspiralgalaxies.

TheSupernovaCosmologyProjectandtheHigh-ZSupernovaSearchTeampublishedtheirobservationsafterseveralyearsofcollectingdataonafewtensof supernovae. They found, independently, that most distant supernovae weresignificantlydimmerthanexpected,whichistosaythattheywerefurtherawaythantheyshouldbeinauniversedominatedbymatteralone.ThebestfittothedatainfactsuggestedthattheUniverseisnotdeceleratingbutacceleratinginitsexpansion. In a universe containing only ordinary and dark matter, thegravitational attraction of the matter tends to cause the expansion rate tograduallyslowdown,sodecelerationwouldbethenorm–anacceleratedrateofexpansion is thehallmarkofacosmologicalconstant.Withoutdoubt, themostremarkableaspectofthesupernovaemeasurementsisthattheycanbeexplainedif themassdensityassociatedwithacosmologicalconstant isequal to70%ofthecriticaldensity.Inotherwords,theUniversereallydoesappeartobeflat–thetheoreticalhunchwasright.

Thecosmologicalconstantnowoftengoesbythenameofdarkenergy.Themoremystical-soundingnameissuggestiveofthepossibilitythattheremightbeadeeperexplanationforwhatcausestheacceleratedexpansionoftheUniverse–perhaps something exists that mimics the effect of a cosmological constant.Many ideas abound, but so far there are no compelling explanations forwhatdarkenergyis–itcouldbenothingmorethanacosmologicalconstant.Oneideaisthatemptyspaceitself isasourceofenergy(calledvacuumenergy)–it isanideathatisveryfamiliartoparticlephysicists,andwewillmeetitagaininthenextchapter.Butthereisaseriousproblemwiththenotionofvacuumenergy:theparticlephysicscalculations tend topredictacosmologicalconstant that isvastly bigger than observed. For this reason, particle physicists tended to

suppose that the cosmological constant is actually equal to zero and that theirunderstandingofvacuumenergy is flawed.Thepreference for avalueof zerocomes about because it is easier for a theorist to imagine that some presentlyunknown physics magically cancels away all of the large numbers leavingpreciselynothingthanitistoimaginesomepresentlyunknownphysicsthatevenmore magically cancels away almost all of the large numbers leaving almostnothing. Admittedly that isn’t the most compelling logic–but the inability oftheoriststomakesenseofthecosmologicalconstantmeantthattheywereverywaryofclaimsthatitmightactuallybepresentinNature.Thefactthatthedatadoseemtorequireanon-zerovalueofthecosmologicalconstantforcedthemtofacetheirdemons.ThetheoreticalprejudiceagainstacosmologicalconstantwasinitiallysuchthattheleaderoftheHigh-ZTeam,BrianSchmidt,whosharedthe2011NobelPrizefor thediscovery,hassaid thathe thought thepublicationoftheresultwouldbetheendofhiscareer,becausehefeltitmustbewrong.

Here, then, is the present-day inventory of the Universe that we’veestablished in this chapter: the mass density is (in percentages of the criticaldensity)dividedinto5%matter,25%darkmatterand70%darkenergy.WecannowsettlethequestionastowhichoneofRobertson’suniversesweinhabit.Welive inuniverse type (i):ourUniverse isofa finiteage,and itwillexpandforeverintothefuture.

Now that we know how the mass and energy in the Universe are sharedbetweenitsdifferentcomponentswecan,usingFriedmann’sequation,computetherateatwhichtheUniverseexpands,foralltimesaftertheBigBang.Figure7.5showshowthescalefactor10varieswithtimeforfourconceivableuniverses,withvarying amounts ofmatter anddark energy: ourUniverse corresponds totheblackcurve.

Thepresentdaycorrespondstoascalefactorof1,andwecanthereforereadoffthatourUniverseisjustlessthan14billionyearsold.Tobeprecise–andweneed tobe forwhat iscoming in thenextchapter–wemean that this lengthoftimehaspassedsincethescalefactorwasverysmall.Wedon’tactuallybelievethattheFriedmannequationiscorrectwhenthescalefactorbecomestoosmall,butwedotrustittothepointwherethetemperatureoftheUniversecorrespondstotheenergiesbeingprobedatthehighestenergyparticlecollideronEarth,theLargeHadronCollider. These energies correspond to a temperature of around1016 degrees celsius, which occurred at a tiny scale factor of 10−16. Thiscorresponds toa timewhen thecurrentvisibleUniversewasabout10−16×10billionlightyearsacross(=10millionkm).Althoughweappeartobebacking

awayfromclaiminganunderstandingoftheveryoriginsoftheUniverse,wearestillmakingtheaudaciousclaimthatwecannowdescribeitsevolutionstartingout from a time when all the matter necessary to make all the hundreds ofbillionsofgalaxiesintheobservableUniversewouldhavebeencontainedinaspherethatwouldsitcomfortablyinsidetheEarth’sorbitaroundtheSun.Butwearebeingtoomodest–wecandobetterthanthat.

Oh–butbeforewedo,wealmostforgotaboutweighingtheUniverse.Whileacosmologist would be content with saying that the Universe is at the criticaldensity,wecandothecalculationourselves.TheFriedmannequationallowsusto ascertain that theobservableUniverse is contained in a sphereof radius47billion light years, which is bigger than 14 billion light years because of theexpansionofspace.11ThatmeansthevolumeoftheobservableUniverseisjustunder 4 × 1080m3.We have ascertained that the averagemass density is 9 ×10−27kg/m3 (ifweincludethecontributionfromdarkenergy),andsothetotalmassoftheobservableUniverseisjustover3×1054kilograms.12Numbersthissize are very difficult to picture–if it helps, wemight say that the observableUniverseweighsasmuchas5×1054pintsofbitter.

Figure7.5Thevariationofthescalefactorwithtime.Thewhitecurveisour

Universe,i.e.30%matterand70%darkenergy.Theredcurveisthesame

butwithnodarkenergy,andtheblueisforauniversecontaining100%

matter.Thegreenisforauniversewithnodarkenergyandamatterdensity

10timesthecriticaldensity–ascanbeseen,itisauniversethatendswitha

BigCrunch.

Figure8.1AmapoftheUniversefromtheSloanDigitalSkySurvey(SDSS).

Thesurveycoversaroundonethirdoftheskyandwasmadeusinga2.5-

metreopticaltelescopeinNewMexico.Eachtinydotinthefigurecorresponds

toagalaxy(eachonecontainingafewhundredbillionstars).Thegalaxies

clustertogetherintowispyregions,leavinggreatvoids.Perhapsthegreatest

triumphofmoderncosmologyisitsabilitytoexplain,indetail,theSDSS

galaxymapandtheripplesintheCosmicMicrowaveBackground(CMB),

showninFigure8.3,usingthesametheory.

8.WHATHAPPENEDBEFORETHEBIGBANG?

The Universe is not a homogeneous mulch of matter; it is full of intriguingstructure.Starsaregravitationallyboundintoabeautifulvarietyofgalaxies,andthe galaxies are woven into filamentary networks spanning many millions oflightyears(Figure8.1).Whatwewanttodonowistoexplorehowstructuresliketheseemergedoutoftheprimordialplasma,theproductsoftheBigBang–this will lead us to focus on a time when the entire visible Universe wascompressedintoaregionofspacemuchsmallerthanthenucleusofanatom.

Aswehaveseen,formostofthefirst380,000yearsaftertheBigBang,theUniverse was filled with an almost featureless hot plasma composed ofelectrons, atomic nuclei (mainly protons) and photons.1We know the plasmawas nearly featureless, because the radiation we detect from this time–theCosmic Microwave Background (CMB)–is almost perfectly uniform in alldirections. It couldnothavebeenabsolutely featureless,however,otherwise itwouldhaveremainedsoforall time:galaxieswouldnothavebeenformed.Insomeplaces theUniversemusthavebeeneversoslightlydenser than inotherregions,andacloserinspectionoftheCMBmicrowavesdoesindeedrevealthatthey are not exactly the same across the sky. The European Space Agency’sPlanck satellite (seeFigure 8.2)marks the pinnacle in our observations of theCosmicMicrowaveBackground;Figure8.3showsthemagnificentphotographithastakenofthelightcomingfromthetimeofrecombination.

Usingcomputersimulations,wecangoaheadandtracktheevolutionoftheUniverse starting from the time of recombination all the way through to thepresentday.Duringthisperiodofevolution,thedarkmatterintheUniversetookcentre stage. Those regionswhere the darkmatterwas of slightly higher thanaverage density tended to attract more matter from their surroundings.ThroughouttheUniverse’shistory,thisgravitationaltendencyfordarkmattertoclumphascompetedwiththeexpansionofspace,whichhastheoppositeeffectofdilutingmatter.Initially,duringthefirst50,000yearsaftertheBigBang,theexpansionwonoutbecause spacewasexpanding tooquickly for thematter toclump:afterthistime,thoselittlelumpsofdarkmatterstartedtogrow,gradually

pullingintheordinarymatteraswell.Inthisway,largecloudsofatomicnucleiformedthat, in thesubsequentfewbillionyears,collapsedtomakegalaxiesofstars. This evolution of the Universe, from an almost-smooth distribution ofmatterintoalumpyone,isanunavoidableconsequenceofgravity.

Figure8.4showstheresultsoftwosuchsimulations.Theimageontheleft,provided by our colleague Scott Kay, shows how the dark matter should bedistributed today across a large portion of a Universe like ours (i.e. with thesameamountsofdarkmatter,darkenergyandordinarymatter).The imageontherightisasimulationperformedaspartoftheEagleProject,whichinvolvescosmologists from across Europe. This image shows how the ordinarymatter(mainlyhydrogengas)isspreadacrossamuchsmallerportionoftheUniverse;it’sazoomed-inversionoftheleft-handimage,inwhichthewispynatureofthematter distribution is more evident. Astonishingly, the Eagle computersimulation is able to describe the formation of realistic-looking individualgalaxies.

Figure8.2TheEuropeanSpaceAgency’sPlanckspacetelescopewas

launchedon14May2009fromtheGuianaSpaceCentre,onanAriane5

rocket,andswitchedoffon23October2013.Duringthosefewyearsit

collecteddatathatimprovedonthealreadystunningmeasurementsmadeby

NASA’sWMAPspacetelescope,whichoperatedfrom2001until2010.

WMAPprovidedthefirstdetailedmeasurementsoftheripplesintheCMB.

Today,thoseripplesprovidecosmologistswithanopportunitytoexplorewhat

happenedatthebirthofourUniverse.

Figure8.3ThePlanckpictureofthemicrowavesthatbathetheEarth.These

originatedatthetimeofrecombination,whichiswhentheyounghotUniverse

suddenlybecametransparenttolight.ThisisaphotographoftheUniverse

whenitwasjustafewhundredthousandyearsold.Thecoloursrepresentthe

temperatureofthesky;thecooled,fadedglowoftheplasma.Hotterregions

arered,andcoolerregionsareblue.Themapshowsdeviationsfromthe

averagetemperature,whichis2.726kelvin,or2.726degreesaboveabsolute

zero.Thehotregionsarearound100millionthsofadegreehotterthanthe

average,andthecoolregionsaround100millionthsofadegreecooler.These

variationsintemperaturearetinyandtheyaredirectlyrelatedtodensity

fluctuationsintheplasma.

We can compare these simulations to the map of the galaxies in the realUniverse from the SloanDigital Sky Survey, shown in Figure 8.1. The samewispydistributionofmatterisevident.Noticealsohowtheordinarymatter(thestuff of galaxies) tends to clump in networks that resemble those in the dark-matterdistribution.Aswenotedabove, this isbecausethedarkmatter ismoreabundant and its gravity tends to attract the ordinary matter. The agreementbetween simulation and observation looks good to the naked eye, but it alsostands up to more rigorous mathematical analysis, meaning that we can beconfidentinouruseofthesesimulationstobetterunderstandtherealUniverse.

Thesimulationsrevealthatthedarkmatterinteractsprimarilyviagravity(aswe suspected), but also that it shouldbe ‘cold’: cosmologist’s jargonmeaningthat it is composed of particles that are not zipping around at high speeds.Simulations inwhich thedarkmatter is ‘hot’ (aswouldbe thecase if thedarkmatterwascomposedoffast-movingneutrinos)donotlookanythinglikeFigure8.1; they even fail to produce galaxies. Through simulations like those of theEagleProject,wehaveagoodunderstandingofhowthelumpinessintheCMBrelates to the distribution of the galaxies in the Universe today. Our nextchallengeistounderstandwherethisinitiallumpinesscamefrom.

Figure8.4Ontheleftisacomputersimulationofhowthedarkmatteris

spreadaboutacrossa3200Mpcpatchofsimulateduniverse(32Mpcdeep).

Thebrighterredregionscorrespondtomoredarkmatterandthedarker

regionstovoids.OntherightisacomputersimulationfromtheEagleProject.

Itshowshowtheordinarymatterisdistributedoverasmaller100Mpcpatch

ofsimulateduniverse(20Mpcdeep).Theintensityindicatesthedensityof

gas,whilethecolourindicatesitstemperature(redishotter).Thezoomsallow

ustoseethatthefinestdetailincludesaccuratelysimulatedsinglegalaxies.

Indoingso,perhapsunsurprisingly,wearegoing to return to the theoryofinflationthatweencounteredbrieflyinthelastchapterandinChapter1.First,though,we should stress that the intellectual origins of the theory of inflationhadnothingtodowithcosmologists’attemptstodescribetheoriginofstructureintheUniverse.That’stosay,we’renotworkingbackwardsfromtheUniversewesee todayand inventinga theory toexplain thestructureswithin it.Rather,we’re using a pre-existing theory that was constructed for entirely differentreasons.

Partoftheoriginalmotivationforthetheoryofinflationwasthatitshouldbeabletosolvetwopuzzlesthatappearatfirstsighttobecompletelyunrelated:the‘flatness’ and ‘horizon’ problems. We met the flatness problem in the lastchapter, and found that it may not be a problem at all, if the portion of theUniversewearecurrentlyabletoexploreprovestobejustatiny(andrelatively

flat) part of amuchbigger (and curved)Universe. Inflation delivers this hugeUniversebycausingspacetoexpandataphenomenalrate–wewillexplainhowitdidsoshortly.Regardlessofthedetails,weknowthattheperiodofinflationmusthavehappenedbeforethetimewhenthefirstatomicnucleiformed,whichyouwillrecallwaswhentheUniversewasatatemperatureofaroundabilliondegrees.Ifinflationhappenedduringorafternucleosynthesis,thenucleiwouldhave become diluted and this would spoil the good agreement between thetheoreticalpredictionsandobservationsoftheirabundancesthatwediscussedinChapter6.

Figure8.5Illustratingthehorizonproblem.

ThehorizonproblemcanbeappreciatedbythinkingabouttheuniformityoftheCMB–thenear-perfectblackbodyspectrumwedescribedinChapter6.Anygasof interactingparticleswill tend to shareout theavailableenergybetweentheparticles as a result of collisions.Faster particleswill lose energywhereassloweroneswillgainenergy.Afteraperiodoftime, thegaswillsettle intoanunchanging state, known to physicists as a state of thermal equilibrium.For agasinequilibrium,thetemperatureisameasureoftheaveragekineticenergyofthegasparticles. In the timebefore theCMBphotonsstartedon theirstraight-

linejourneysacrossspace,theUniversewasstillahotplasmaandthephotonswere a part of it, jostling the electrons and protons to share out the availableenergy.ThetemperaturerecordedbythePlancksatellite isadirectmeasureofthe temperatureof thehotgas in those finalmomentsbefore thephotons tookflight.

Sofarsogood.Nowlet’suncoverthehorizonproblem,whichisillustratedin Figure 8.5. The CMB photons arriving today at Earth all started out frompoints lying on the surface of a sphere,whose radius is the distance light cantravelinalittleunder14billionyears.Thisisthebluecircleinthefigureandisreferredtobycosmologistsasthe‘surfaceoflastscattering’.Now,considertwophotonsarrivingattheEarthfromoppositepointsinthesky.ThesetwophotonswerereleasedfromthehotplasmaatpointsAandBinthefigure.ThePlanckdatarecordsthephotonscomingfromAandBandinformsusthattheseregionsintheplasmawereatthesametemperature,toonepartinahundredthousand.Thequestion is,howdidAandBget tobeat sonearly thesame temperaturewhen theyare so faraway fromeachotherandat first sightcouldneverhavebeen in contactwith each other? This is the horizon problem, and it isworthdescribinginalittlemoredetail.

If, before the time of recombination, the expansion of the Universe wasalwaysgovernedbytheordinaryanddarkmatteritcontains2then,accordingtotheFriedmannequation,thefollowingproblemarises.Thinkaboutthepointinthe primordial plasma labelledA. In the 380,000 years that the Universe hadbeen expanding before recombination, this part of the plasma could only everhaveinfluencedotherpartsoftheplasmathatliewithinaspherewhoseradiusisthe distance that light could travel in those 380,000 years. The small circlearoundpointAshowsthesizeofA’spossiblesphereofinfluenceatthetimeofrecombination.Thesameis trueforpointB.Theparticles insideeachof thesespheres could conceivably have had sufficient time to jostle their neighbours,shareout their energyand reacha common temperatureby the time theCMBphotonswerereleased.Theproblemisthatthetwospheresofinfluencearenotevenclosetooverlapping,whichmeansthatthegasneartoAneverjostledwiththegasnear toB.Thismakesuswonderwhat caused them tobe at the sametemperature.

One explanation might be that ‘whatever created the Universe in the firstplacediditinsuchawaythat,atthemomentoftheBigBang,everythingwascreatedwithnearperfectuniformityintemperature.’That’sfineasfarasitgoes,but it would be nice to have some kind of explanation beyond this. Inflation

provides such an explanation by supposing that, at some time prior torecombination, the Universe underwent a freakish period of very rapidexpansion,suchthatthespheresofinfluenceofAandBweremuchbiggerthanindicatedinthefigure.Withafastenoughexpansion,theywouldbebigenoughto overlap.The horizon problem is a little bit like the case of two alienswhoknow each other despite appearing to be too far away to have evermet. Thepuzzle is solvedonceweknow that the spacebetween the aliensunderwent arapidexpansionsometimeinthepast,sothatthealienswereonceneighbours.

Ofcourseitiseasyenoughtosaythatarapidperiodofexpansion3earlyonsoundslikeagreatidea,but‘soundinglikeagreatidea’doesn’tcountformuch.Wemustcheck that inflation isaviable theory indetailaswellas inabroad,hand-waving sweep.To do this,we need amechanism for inflation and somepredictions to compare with data. One way to arrange for a rapid burst ofaccelerated expansion is to add a sufficiently large cosmological constant intoEinstein’sequations.Wealreadyknowthatacosmologicalconstantcauses theexpansionoftheUniversetoaccelerate,andthebiggeritsvalue,thebiggertheacceleration (if you know a little maths, you can see this by solving theFriedmann equation in Box 10 (pp. 144–5) for the scale factor, with only acosmological constant and neglecting all other terms). As we saw in the lastchapter,theevidenceevenpointstotherebeingacosmologicalconstanttoday.The trouble is that the present-day cosmological constant is way too small toproduce the dramatic expansion needed to solve the horizon and flatnessproblems. Fortunately, there is a way forward: General Relativity can becombinedwithamainstreamideainparticlephysicstoproducesomethingthatisexactlylikeacosmologicalconstant.Theideafromparticlephysicsisthatofa‘scalarfield’.

Fields are familiar things to us all. You can map out the magnetic fieldproducedbytheEarthusingacompass,andmobilephonesworkbytransmittingandreceivingwavesintheelectromagneticfield.The‘scalarfield’thatweshallconsider is very similar to the electromagnetic field, the important differencebeingthatitcanpervadethewholeofspaceasakindofbackdroptoeverythingelsethathappens;picturethescalarfieldasastilloceanfillingspace.AccordingtoGeneralRelativity, if anocean-likescalar fieldwaspresentat some time inthe history of the Universe, the energy stored within it could act like acosmological constant andmake theUniverseexpandverymuchmore rapidlythanwouldotherwisebe the case. In this ‘particlephysics’wayof thinking, aperiodofinflationisaverynatural thing,becauseit iswhattendstohappenif

scalar fieldsexist inNature. InBox13 (pp.225–7),weexplore the ideaofanall-pervasivescalarfieldinmoredetail;weexplainthatfundamentalfieldsalsopredicttheexistenceofcorrespondingparticles,andwetalkaboutascalarfieldthatweknowtoexistbecausewediscovereditattheLargeHadronCollider:theHiggsfield.

TheconjecturedscalarfieldthatcausedtheyoungUniversetoacceleratesophenomenallyquicklyisknownasthe‘inflaton’field.Asfaraswecantell,theinflatondoesn’tplaymuchofaroleinourUniversetoday.Thereisnotraceofany inflaton particle production at the Large Hadron Collider, for example.Given that scalar fields can cause accelerated expansion, we might well askwhether thepresent-daycosmological constant isdue to a scalar field, dubbedthe‘quintessencefield’.Weshan’tfollowthatideainthisbook,butitispopularamong cosmologists. Nor shall weworry about whether theremight be somerelationship between these two postulated and one observed scalar fields; theHiggsfield, theinflatonfieldandthequintessencefield.Theresearchonthesethingsistoospeculativeatthemoment.Here,wewanttofocusontheinflatonfield.

To summarize the basic idea: if the inflaton field was once present in theUniverse,spacecouldhaveexpandedveryrapidly,becausetheenergystoredinthe field acted like a large cosmological constant. This would have caused asmall portion of the pre-existing space to grow rapidly to cosmic proportions,driving any particles present in the Universe before inflation to very largeseparations and diluting them. As time passed, the energy driving theinflationary expansion diminished until it became too small to generate anyfurtherinflation.Inthatway,inflationcametoanend.

As this timeof rapidexpansiondrew toaclose, the inflaton fieldmorphedfrombeinglikeastillocean,whoseenergydrovetheacceleratedexpansion,intoacoldgasofinflatonparticles.Theseinflatonparticlesthendecayedtoproduceawhole bunch of lighter particles, including those that populate theUniversetoday.Thedecayofheavyparticlesintolighteroneswiththereleaseofenergyisthenorm inNature: all knownparticles tend todecay into lighterones if theypossibly can. We’ve already encountered one example in Chapter 6: isolatedneutrons decay into protons in a few minutes. Top quarks decay into lighterquarks in aminuscule 10−25 seconds, and theHiggs particle decays in around10−22seconds.Giventhefleetingnatureoftheirexistence,it isimpressivethatparticlephysicistshavemanagedtodiscoverthetopquarkandtheHiggsboson,letalonemeasuretheirproperties.

BOX13.FUNDAMENTALFIELDSANDTHEHIGGSBOSON

When physicists think of a field, they imagine some quantifiablething that is spreadoutacrossspace.Avery simpleexampleofascalar field is the temperature field ina room.Ateachpoint in theroom we can associate a temperature, and so can represent thefieldbyalistofnumbers,oneforeachpointinspace.Ifyouwanttoknowthetemperaturesomewherethenyoujustneedtolookupthecorrectnumberfromthelist.Obviouslythis isdull.Theutilityofthefieldconceptcomeswhenwewanttodomoresophisticatedthings,liketrackhowthetemperaturevarieswithtime.If thereisasourceofheatintheroom,thentheairtemperaturewouldchangewithtimeinacomplicatedway:theairclosetotheheatsourcewouldbecomewarmer and rise upwards by convection and this would dispersethroughtheroom,andsoon.Ifwehadtherightequations,wecouldattempt tocomputehow the temperature field throughout the roomwouldvaryinthepresenceoftheheatsource.Inthiscase,thefieldis a concrete mathematical representation of something very real,anditcanbemanipulatedusingmathematicstoallowustocomputethetemperatureintheroomatanyplaceandatanyparticulartime.This isagood illustrationofhowphysicistsusemathematics: theyrepresentrealphysicalthingsbyabstractmathematicalentities(likefields), and then they process those entities using mathematicaloperations in order to answer concrete questions about the realworld.

Thetemperaturefield isascalar fieldbecausewecanspecify itby giving just one number at each point in space. Theelectromagnetic field isnotascalar field; it isavector field.Vectorfieldsarespecifiedbygivinganumberandadirectionateachpointinspace.Asimpleexampleistheflowoftheairinaroom.Ateachpoint we should say how fast the air is flowing and also in whichdirection it is heading. Taken together, these two pieces ofinformation (stated for each point in the room) correspond to acompleterepresentationoftheair-flowfield.Thedirectionalpieceof

the electromagnetic vector field is referred to as its polarization–exploited, for instance, in thedesignofpolaroidsunglasses,whichselectively block out electromagnetic light waves of a particularpolarization.Roughlyspeaking,thepolarizationofalightwavetellsusthedirectioninwhichthewaveiswaving.Thinkofshakingwavesontoarope:thedirectioninwhichyourhandisshakingdeterminesthe polarization of the wave. Where the temperature and air-flowfieldsdiffer fromtheelectromagnetic,Higgsor inflatonfields is thatin the former it is very clear that the fields are representingsomethingmorefundamental:theyaretrackingfeaturesoftheairinthe room. In the case of the electromagnetic, Higgs and inflatonfields, however, we do not know whether they are composed ofsomethingmore fundamental. Perhaps experiments such as thoseat the LargeHadronCollider will reveal that theHiggs field is notfundamental,but,fornow,ourunderstandingofNatureistoocrudeto be able to resolve this either way. As far as these fields areconcerned, we are in a similar position to those who studied theworldbefore theyknew thatatomsexisted.Now,ofcourse,wenolonger regard fire, earth and water as elemental–but who knowswhetherornotwearereallymuchcloser thanourpredecessorstofindingthe‘truenatureofthings’?

Today, physicists have identified a set of what appear to befundamental fields. The Higgs and electromagnetic fields are two,but there are also six different quark fields, six lepton fields(including an electron field and three neutrino fields), a gluon fieldand the weak interaction fields. All these can be described by asinglemathematicalframeworkcalledtheStandardModelofparticlephysics, constructed during the course of the 1960s by SheldonGlashow, Abdus Salam and Steven Weinberg.1 The laws ofquantum mechanics, when applied to these fields, imply that foreachfieldthereisacorrespondingparticle.Fortheelectromagneticfield,theparticleisthephotonandfortheHiggsfielditistheHiggsboson.AlltheknowntypesofparticleinNatureareassociatedwiththe quantum behaviour of a corresponding field, and the apparentwave-likenatureofelectromagneticwavescanbeunderstoodasthebehaviour of a large number of photons, all propagating in accordwith the laws of quantum physics. The Standard Model is

astonishinglysuccessfulbecauseitcandescribethebehaviourofallthe particleswe see in Nature, with the notable exception of darkmatter,andthewaysthattheyinteractwitheachother.ArmedwiththeStandardModel,wecanexplainhowatomswork,theoriginsofradioactivityandthefusionprocessesthatmakestarsburn.Infact,theStandardModelencouragesustothinkoftheentireUniverseasbeingbuiltupoutofahugenumberofparticlesthathoparoundandinteract with each other in precisely calculable ways. Only thegravitational interactionbetweenparticles isnotentirelyunderstoodinthisway.Weunderstandthisinteractionquitewell–we’vespentalarge part of this book talking about General Relativity–but thisunderstanding failswhenwetryandworkoutwhathappensat theshortest distances, or in the proximity of huge masses at highdensitiessuchasthosepresentattheheartofablackhole.Figuringout what dark matter is, and understanding how to computequantumgravitationaleffectsatveryshortdistances,aretwoofthebigquestionsthatphysicistsarecurrentlywrestlingwith.

The particle physics way of tackling these questions is theultimate in reductionism, because everything boils down to tinyparticlesandtheirinteractions.Butunderstandingwhattheparticlesare and how they interact is not the same as understanding therichnessoftheUniverse.Presumably,thecomplexbehaviouroflifereallydoesemergefromtherulesencodedintheStandardModelofparticlephysics–butitwouldbeafoolwhothoughtthatbiologyisinsomesensediminishedbythefact.Mostofuscanlearntherulesofchess, but very few can play the gameaswell as a grandmaster.Butenoughofthosemusings…let’sgetbacktoscalarfields.

In themain text (p. 223),we said that theelectromagnetic fielddoes not pervade thewhole of space in theway that scalar fieldsmight.Roughlyspeaking,thismeansthatelectromagneticfieldswillbe largest in the vicinity of light bulbs or other sources of jigglingelectricallychargedparticles;but, faraway fromsuchsources theywillbe toall intentsandpurposesabsent. Incontrast, scalar fieldscanbesubstantialacrosshugeswathesoftheUniverse.Beforeweget on to the hypothesized-but-not-yet-detected inflaton field, thecaseoftheHiggsfieldisworthconsideringinmoredetail.

PeterHiggssharedthe2013NobelPrizeinphysicswithFrançois

Englert for their prediction of the existence of the Higgs boson,which was discovered the previous year at the Large HadronCollider. In the 1960s Higgs and (independently) Englert–workingwith another Belgian physicist, Robert Brout–argued for theexistenceofanall-pervasivescalar field inorder tosolveapuzzleconcerning how the particles in Nature get to have mass. Beforethey came up with the Higgs field (or, more correctly, the Brout-Englert-Higgs field), the mathematics governing how elementaryparticlesbehaveonlymadesenseiftheparticleshadzeromassandzipped around at the speed of light, which is obviously nonsense.Ratherthanditchthewholemathematicalframework,Brout,EnglertandHiggsrescuedthingsbysupposingthatthereisanall-pervasivescalar field, spreaduniformlyacross theUniverse.This fieldwouldinteractindifferentwayswiththevarioustypesofparticleand,insodoing,would give us the impression that they have different, non-zeromasses.Thissmartideawascentraltotheconstruction,afewyearslater,oftheStandardModel.Wearen’t immediatelyawareofthisall-pervasivescalarfieldbecausewehavebeenlivingwithitallofourlives:itisthebackdroptoourmaterialexistence.

If thisall soundsabit likeahotchpotchof ideas, it isn’t.Scalarfieldscanveryeasilyfillupallofspaceinaprocesssimilartohowwatercondensesoutoftheairtoproducedew.Theequationsrevealthat empty spacemay prefer not to be empty, but rather to fill upwith ‘Higgs field condensation’, because condensation is lower inenergythan‘empty’.Thismightsoundweird,becauseitseemslikesomething is coming from nothing–but the quantum nature of theworldmeansthatemptyspaceisneverasimplething;insteaditisaseethingbrothofparticlesforeverhoppinginandoutofexistence.

Incontrast,theelectromagneticfieldcannotcondense,becauseitdoes not interact with itself. To illustrate the point, notice that thelightcarryingtheinformationfromapageofthisbooktoyoureyesisevidentlynotknockedoffcoursebylighttravellingsideways–or,putanother way, photons do not interactmuchwith other photons. Incontrast,Higgsparticlesdo interactwithotherHiggsparticles,andthisself-interactioncreates theconditions thatallow theHiggs fieldtocondenseoutofnothing.Scalarfieldsdonothaveamonopolyonbeingabletocondenseintothevacuum;thequarkandgluonfields

can also condense. Today, understanding ‘nothing’ is one of themostinterestingproblemsinthewholeofphysics.

Unlike all of the other fields we have just been discussing, theinflatonfieldisnotwellestablishedbyexperiment.Aswewillsee(p.242), the evidence for inflation is strong–but it is not yetoverwhelming.Moreover,wemight entertain the idea that inflationdidhappen,butbymeansother thanvia theexistenceofascalarfield.However,againaswewillsee,theexistenceofaninflatonfieldcan also explain the origins of the primordial lumpiness in theUniverse.This is the real reasonwhy the inflaton field is taken soseriouslybysomanycosmologists.

Inthemaintextwesaidthattheenergystoredupinascalarfieldcan boost the expansion of the Universe. You might well askwhethertheenergystoredupintheHiggsfield(andthequarkandgluoncondensates) ispresentlydoingjustthat.Youmightalsoaskwhether this could conceivably be the source of the cosmologicalconstant, which we encountered in the last chapter and which iscurrentlycausingtheexpansiontoaccelerate.Not toput toofineapointonit,thesituationisatheoreticaldisasterzone.Takenatfacevalue,theenergystoredupintheHiggsfieldshouldbecausingtheUniversetoexplodeapart.That isevidentlynotwhat ishappening,whichmeanswedonotunderstandhowtoaccountforthevacuumenergythatpertainstoday.Ourlackofunderstandingaboutwhytheobservedcosmologicalconstant issosmall isperhapsthegreatestunsolvedprobleminphysics.

After the inflaton particles decayed, what remained was a Universe filledwiththestuffoftheBigBang.Thisisveryneat:theinflatonfielddrivesatinypatchofspaceintoarapidburstofexpansion;then,asinflationends, thefielddecaystofill thenowmuchlargerUniversewiththestuff thatwasdestinedtoformeverythingintheUniversetoday.

Each idea in thissequencefollowsfromthe lastone inaprettycompellingway.Nothing is contrived: every piece is built on known or at least plausiblephysics,givenwhatweunderstandaboutparticles,fieldsandGeneralRelativity.Aswellasdoingawaywiththehorizonandflatnessproblems,inflationprovidesavividdescriptionofhowtheBigBangcameabout,becauseitexplainswhere

alloftheparticlesintheUniversecamefrom.Ifthiswereallthereistoinflation,however,itwouldstillbeconfinedtotheclassoffancyscientificideasthataredoomedtoremainthe idlemusingsof theoreticalphysicists.Weneedconcretepredictionsthatcanbetestedbymeasuringthings.Andthisiswheretheideaofinflationdelivers.ItpredictsthecorrectfeaturesoftheripplesintheCMBandthedistributionofgalaxiesacrossthesky.

Asan idea, inflation tookoff in theearly1980s followingapaperbyMITcosmologist Alan Guth, who drew attention to the virtues of working on theassumptionthattherewasarapidperiodofexpansionearlyonintheUniverse’shistory. Although Guth introduced a scalar field and obtained an inflatingUniverse, his original version of events did not lead to inflation ending withinflaton-particledecayandaresultantBigBang.Guthrealizedtheproblemandconcluded: ‘I am publishing this paper in the hope that it will highlight theexistenceoftheseproblemsandencourageotherstofindsomewaytoavoidtheundesirablefeaturesoftheinflationaryscenario.’Withthisremark,hehadlaiddown the gauntlet. Around the same time, and quite independently,Moscow-based physicists Alexei Starobinsky at the Landau Institute for TheoreticalPhysicsandAndreiLindeattheLebedevInstitutealsobegantoexploretheideaof an inflationary phase in the early Universe. In 1982, Gary Gibbons andStephen Hawking convened a now-famous meeting of Soviet and Westernphysicists in Cambridge. By the end of that year, cosmologists had not onlyestablishedaworkingmodelof inflation,4 theyhadalso realized thatquantumfluctuationsintheinflatonfieldcouldgeneratetheinitiallumpinessthatseededthegrowthofstructureintheUniverse.5

According to the laws of quantum physics, the scalar field cannot beperfectly smooth over all of space–it cannot be a perfectly still ocean.Heisenberg’sUncertaintyPrinciple,whichisaconsequenceofthebasiclawsofquantum physics, says that the field has to fluctuate up and down slightly byvarying amounts fromplace to place.6Oneof the implications is that ‘empty’spacecomprisesa seethingbrothofparticlesandanti-particles thatpopoutofnothinginpairsbeforecomingbacktogetheranddisappearingafleetinginstantlater. In ordinary, non-inflating space these quantum effects, which arenecessarilypresentinallfields,leadtofluctuationsthataverageouttozero.Wecan glimpse them, however: the fleeting production of electron-positron pairsleads to fluctuations in the photon field that can be detected bymaking veryprecisemeasurementsofthespectrallinesemittedbyhydrogenatoms.(In1965,RichardFeynman,JulianSchwingerandSin-ItiroTomonagareceivedtheNobel

PrizeinPhysicsforworkingouthowtoperformthiscalculation.)While the effect of these ‘vacuum fluctuations’ averages to zero in non-

inflatingspace,somethingveryinterestinghappensifthespaceisexpandingfastenough.TheUncertaintyPrinciple says that the particle and anti-particle pairscanexistonlyforalimitedamountoftimebeforetheyslipbackintonothing.Ifspaceisinflating,however,itispossiblefortheparticleandanti-particletogetswept so far apart that theycannotcomeback together toannihilate.Theyarecarried out of each other’s horizons before they can return their energy to thevacuum. If this happens, they have no option but to become real materialparticles. It is as if the expanding space has provided the means by whichparticles emitted out of nothing can escape the death-grip of Heisenberg’sUncertaintyPrinciple.Emptyspaceinarapidlyexpandinguniverseglowswithparticles.7Astheparticlesbecomerealtheyaddlittlewavestotheformerlystilloceanof thescalar field. InBox14 (p.231)wego intoa littlemoredetailonhow it is that real particles can be produced from nothing in an inflatingUniverse.

BOX14.DESITTERSPACE

If theUniverse isexpandingunder the influenceofa cosmologicalconstant (andnootherappreciablesourceofenergy),wesay thatthecorrespondingspace–oneofRobertson’spossibleuniverses–is‘de Sitter space’, named after the Dutch mathematical physicistWillemdeSitter.DeSitterspacehasahorizon, inthesamesensethatapointon theEarth’ssurfacehasahorizonbeyondwhichwecannotsee.OnEarth,thehorizonisafeatureofthecurvedsurface.In de Sitter space, the horizon is a feature of the Universe’sacceleratingexpansion.Toseethis,imagineyouarestandingindeSitterspaceandwatchingalight-bulbrecedeasspaceexpands.Atsomepoint,thelight-bulbwillbemovingawaysofastthatthelightitemits will never reach you and, at that point, we say it hasdisappeared beyond your horizon.Real particles and anti-particlesareproduced if, after thepair haspoppedoutofnothing, theyaresubsequentlysweptoutsideofeachother’shorizonbeforetheygetthechancetorecombineagain.

Now let’s think about how this new idea impacts on our picture of theexpandingUniverseduringthetimeofinflation.ConsiderthepatchofUniversethat is destined to grow into our visible Universe. If the inflaton field variesslightlyfromplacetoplaceacrossthepatchthentheamountofenergyavailabletoinflatetheUniversewillvarytoo,becausetheenergydrivingtheexpansioniscontrolled by the size of the field. This means that the quantum-fluctuations-made-realwillcausesomeregionsofthepatchtoinflateforslightlylongerthanotherregions–andthishasdramaticconsequences.

Wehaveseenthatwheninflationends,spacefillsupwithparticles.Becausetheparticlesarefillingupaspacethathasbeenstretchedbydifferentamounts,thedensityofparticleswillalsovaryfromplacetoplace.Theparticleswillbemoredenselypopulatedinregionsthathavenotinflatedasmuch.Crucially,thismeansthattherewillbeavariationinthedensityofparticlesthatisthesameforallofthedifferenttypesofparticle.Ifaregionhasbeenstretched1%morebyvolume than a neighbouring region, the density of photons, protons, neutrinosanddarkmatterparticleswillallbe1%lower.Cosmologistsrefertothistypeofdeviation from a smooth, perfectly uniform distribution of particles, as a‘curvature perturbation’ or ‘an adiabatic perturbation’. Figure 8.6 illustrates acurvatureperturbationintwodimensions,andthegridlinesallowustoseehowdifferent regions of space have been stretched by different amounts. Thecorrespondingvariationsinthedensityofparticlescreatedattheendofinflationare the seeds that generate the clumpingofmatterwe see across theUniversetoday.

Figure8.6Picturingacurvatureperturbation.Therippledsurfacecorresponds

toatwo-dimensionalspacethathasbeenstretchedbydifferentamountsin

differentplaces.ThespaceofourUniverseattheendofinflationislikea

three-dimensionalversionofthis.

Figure8.7TheevolutionoftheinflatonfieldinourpatchoftheUniverse.The

topleftisatatimewhentheentirevisibleUniverseis10−26metresacross.

Timeincreasesfromtoplefttobottomright.

Totestwhetherthiswayofgeneratingsmallnon-uniformitiesinthedistributionof matter and energy at the time of the Big Bang is correct, we need tounderstandthedetailsoftheripplesintheinflatonfield,andhowtheseleadtomeasurableproperties in theCMBand thedistributionof thegalaxies.This issomething we can certainly do. The way the inflaton field changes as timeadvances,inresponsetothequantumfluctuations,isbestillustratedinamovie,andinFigure8.7wehaveshownstillsfromsuchamovie.ThetinyrippleinthetopleftimagerepresentstheinflatonfieldwhenourvisibleUniversewasaboutthe same size as thedeSitter horizon (seeBox14, p. 231), atwhich time thefield would have been fairly smooth over the patch.8 For typical inflationaryscenarios, the de Sitter horizonmight be something like 10−26 metres across,which ismind-bogglingly small (it is100billion times smaller thanaproton).ThesecondimageshowstheUniversea little lateron,whentheoriginalpatchhasbeenstretchedandsomenewrippleshavebeencreated.Inthefirstimageonthe second row, the ripples have stretched somemore, andmore new rippleshave appeared. The creation of new ripples as the old get stretched continuesoverandoveragain,atasteadyrate,untiltheendofinflation.

Thesestillsprovideawaytovisualizetheripplesintheinflatonfieldandthewaytheygrowwithtime.Tosimplifythings,theyshowripplesonaflatsurface,likeripplesonthesurfaceofapond,whereasintherealworldtheyareripplesina three-dimensional space, which are harder to visualize.We’ve also taken alittle artistic licence to make things easier to see, by representing the regionsurrounding thegrowingpatchasbeingcompletely flat.This isn’t correct: it’snot really flat, because it too will have ripples that were generated at timesbeforewestartedthemovie.We’vedrawnthingsthiswaybecausewewanttotrack how the ripples in one part of space evolvewith time, and it’s easier toidentifythepatchweareinterestedinifweleavethesurroundingspaceoutofit.Inreality, thefirststill inthesequenceoughttolookmorelikethelastone, inwhichcaseourentirevisibleUniversemightbeoneofthesmallripplessittingontopofastackofpreviouslyformedandstretchedones.

Aparticularlynoteworthyfeatureofthepatternofripplesformedinthiswayisthatitiswhatphysicistsrefertoas‘scaleinvariant’.Thismeansthatwecanzoominoroutof theimageanditwill lookthesame.Obviouslythis isn’t thecasethewaywe’vedrawnit,particularlyin theearlierpictures–butremember,the flat regions are only there for clarity. The scale invariance becomesmoreevident in the final few images in the sequence. The net result of this steadyproductionandstretchingofripplesisapictureofripplesontopofripplesthat

lookslikeabristlinghiveofactivityonanunchangingtheme.Thepatternmightlookrandom,butbecauseitisscaleinvariantitisalsoverydistinctive.

Figure8.8Asnapshotofauniverseduringinflation.Theblueregions

correspondtoplaceswherethefieldissmaller.Thezoomed-inportion

illustratesthatthepatternisscaleinvariant.

ThetheoryofinflationthereforedeliversapredictionforwhattheUniverseoughttolooklikewheninflationdrawstoaclose.Figure8.8showsadifferentwayofvisualizingtheinflatonfieldinFigure8.7,whichfocusesourattentionontheveryimportantpredictionofscaleinvariance.Theblueregionsarewherethe

inflaton field is smallest, and the red regions are where it is biggest. Scaleinvariancemeansthatifwelookataparticularpatchoftheimageandzoominorout,wewillseepatternsthatareindistinguishablefromeachother.SincetheinflatonfieldshapesthespacethatwillbefilledbytheparticlesproducedintheBig Bang, we might expect this scale invariance to impact on the observedtemperaturefluctuationsintheCMB,andhereaglanceatthePlanckphotographof the CMB is at least suggestive: Figures 8.8 and 8.3 do have certainsimilarities. Of course, we need to do far better than making such vagueobservations. We need to understand how the initially scale-invariant plasmachangedasitevolved;ifwecandoso,wewillthenbeabletopredictwhattheCMBshouldlooklikeindetail.

First, we need to mention that there is a subtle prediction from inflationregardingthescaleinvarianceoftheripplepattern.Theexpansionrateofspacemust have slowed during the course of inflation, not least because inflationeventually had to end. As the expansion rate slowed, new ripples were notcreated quite so rapidly, and this slowing-down leads to a slight deficit ofsmaller-sizedripplescomparedtoexpectationsbasedontheideaofperfectscaleinvariance.This point is important, because it is a prettygenericpredictionofinflation. Those small deviations from scale invariance leave a measurableimprintontheCMB.

Ourgoalnow is toworkoutwhat theobservable consequences are for theCMB and the galactic structure we see in the Universe today. The theory ofinflationwillstandorfallbasedonthiscomparison.Perhapssurprisingly,thisisnotasdifficultasitsounds.Thewholebusinessisratherliketryingtofigureoutwhathappensifyoukickabucketofwater.Thekickwillcausethewatertobeperturbedinsomewayand,ifweknowpreciselyhowitisperturbed,wecanusetheequationsoffluiddynamicstoevolvetheperturbationsforwardandpredicthowthewaterinthebucketwilllookatalatertime.Inflation,likethekick,laysdowntheinitialperturbations;therestisdictatedbyequationssimilartothoseoffluiddynamics.Insteadofabucketofwater,theearlyUniversewasahotsoupofelementaryparticles,but theideaisverysimilar.Infact,strangeas itmightseem,theequationsfortheUniversearefarsimplertosolvethantheequationsforabucket,becauseofthefeeblenessofgravityandthesmallsizeoftheinitialripples. It’s not too far off themark to say that tracking the evolution of theUniverse,startingfromatimewheneverythingwascompressedintoaspacefarsmallerthanthenucleusofanatom,isaneasierjobthantrackingtheevolutionofwavesinakickedbucket.

Figure 8.9 is possibly themost astonishing graph in thewhole of physics.ThedatapointsarederiveddirectlyfromthePlanckmeasurementoftheCMB.They are amathematical representation of what our Universe actually lookedlike, 380,000 years after the Big Bang. The solid curve is a theoreticalprediction,andyoumustbeimpressedatitsaccuracy;itsitsbangontopofthedata.Youshouldbeevenmoreimpressedwhenwetellyouthatthetheoreticalcurve is produced from a set of initial perturbations in an otherwise smoothplasma that have their origin in curvatureperturbations that arenearlybut notquite scale invariant, precisely as predicted by inflation. These perturbationswere evolved forward to the time of recombination, when the CMB wasreleased.Moreover, the theoretical curve is for a universe that today contains68%darkenergy,27%darkmatterand5%ordinarymatter.Thismeanswealsohave independentconfirmationofall the resultsweestablishedbyavarietyofother methods in the previous chapter! F. Scott Fitzgerald said that using anexclamationmarkislikelaughingatyourownjoke,butitissurelyappropriatehere.Notonlydoes thisgraphsupport the idea that thedensity fluctuations intheCMBhad theirorigin inalmostscale-invariant ripples in the inflatonfield,but it also provides support for our previous estimates of the amount of darkmatter,darkenergyandordinarymatter in theUniverse,whichhadnothing todowiththeobservationsoftheripplesintheCMB.We’llnowdescribehowallthesekeynumbersdescribingourUniversewereextractedfromthissingleplot.9

Figure8.9ThetemperaturefluctuationsintheCMBasmeasuredbythe

Plancksatellite.

First of all, we need to understand what the wiggles, or–rather morescientifically–thepeaksand troughs, in thegraphrepresent.Cosmologists refertotheseas‘acousticpeaks’,becausetheywereproducedbysoundwavesintheprimordialplasma.TheUniverseranglikeabell,withtheinitialstrikedeliveredby the curvature perturbation at the end of inflation: Figure 8.9 is a visualrepresentationofthatsound.

Ananalogymightbehelpful.Imaginetossingahandfulofpebblesontothesurface of a pond so they all land at the same time. This creates a set ofdisturbances in thewater,which gradually evolve into a series of overlappingcircles. Like water, the primordial plasma was a medium that supported thepropagation of waves, and the disturbances generated by the curvatureperturbationevolvedinmuchthesameway.Onthetwo-dimensionalsurfaceofthewater,thepebbled-inducedwavesmakecircles,butinthethree-dimensionalplasmathewavesaresphericalshells.EachandeverypointinFigure8.8actedasaninitialsourceforanoutwardlypropagatingsphericalshell:youcanthinkofthe shells as soundwaves forging through theplasma.Soundwaves in air aremovingvariations indensity,andsowere thewaves in theprimordialplasma,butinthiscaseitisthedensityofphotons,electronsandnucleithatvariesasthewave travels. Under-and over-dense spots in the original plasma produceddensitywaves,whilespotsofaveragedensityproducednowavesatall.

Crucially,becausetheperturbationsintheinitialplasmawereseededatthesametimeat theendof inflation(i.e.at theBigBang), thesphericalwavesallhadthesameradius380,000yearslater,atthetimeofrecombination.Contrastthiswithwhatwouldhappenifyouthrewpebblesintoapondoneatatime:thepebbles tossed in first would make waves with a larger radius than pebblestossedinlater,andthenetresultwouldbeawholebunchofcircularwaves,allof different radii. The peaks in Figure 8.9 are visible because the wavesgeneratedafterinflationendedwereallreleasedatthesametime.

The details of the peaks are sensitive to how the plasma was originallydisturbed.We have pictured the initial disturbance–the kick of the bucket–byimaginingaplasmathat iscreated,squashedandstretchedfrompoint topoint,asaresultofthecurvatureperturbationgeneratedbyquantumfluctuationsintheinflatonfield.Thiswayofinitiatingthedensitywaveswithintheplasmaisakin

topluckingthestringonamusicalinstrumentbypullingittoonesideandthenreleasing it. The waves in the plasma were initially released ‘from rest’. Theotherwayofpluckingastringistohitit,whichgivesitakickandsetsitmovingaway from an initially undisturbed position. This kind of thing couldconceivably have happened to the plasma. In that case, there would not havebeenanyinitialvariationinthedensityoftheplasma.Rather,theperturbationswouldhavebeeninitiatedbyplasmaflowingintooroutofeachregionofspace.Thisway of creating a disturbance in the plasma is known as an isocurvatureperturbation.Curvatureandisocurvatureperturbationsproducedifferentsounds,just as plucking or striking a stringed instrument produces different sounds.Generally speaking, any kind of disturbance can be characterized as somemixture of curvature and isocurvature perturbations, and different theories fortheoriginof thewaves in theplasmapredictdifferentmixtures.Themodelofinflation we have been describing is particular in selecting only curvatureperturbations;itpredictsthattheUniversewasplucked.10Itispossibletocreatemoresophisticatedmodelsofinflationthatgenerateinitialconditionsthatareamixture of curvature and isocurvature perturbations. However, the location ofthe peaks in the Planck data indicates that curvature perturbations weredominant.

Hopefully, you are starting to develop a feeling for what happened in theplasma. It really is remarkably simple physics, and just goes to show that,althoughitmayhavehappenedaverylongtimeago,whentheUniversewasavery different place, it is not beyond our ken. In many ways, it is the worldaroundustodaythatiscomplicatedandhardtounderstand,nottheUniverseatitsbirth.ThechallengefacingusistodigasmuchinformationaspossibleoutoftheprimordialsoundwavescapturedinFigure8.9.

To do this we’ll need to know more about how Figure 8.9 was actuallyproduced. Stickingwith our pond analogy for amoment, if you threw a largenumberofpebblesintoapond,andthentookaphotographofthepondatsomelatertime,itcouldbeprettyhardtospotthat theresultingpatternwasactuallyproducedbyaseriesofsuperimposedcircularwavesofequalradii.ThisiswhywecannotseeanythinglikecirclesinthePlanckphotographbyeye.ThingsaremadeevenmorecomplicatedbythefactthatPlanckobservesatwo-dimensionalspherical slice through a three-dimensional snapshot of the myriad sphericalwavesintheplasmaatthetimeofrecombination.Fortunately,astronomershavedevelopedtechniquestosortthismessout.TheresultisFigure8.9.

Figure8.10ThetemperaturefluctuationsintheCosmicMicrowave

Backgroundarehighlysensitivetothekeynumbersgoverningthe

compositionoftheUniverse.Noticehowmuchthecurveschangeaswevary

theamountsofordinarymatter,darkmatteranddarkenergy.Intheright-hand

graph,theredandgreencurveshavelessthan100%ofthecriticalenergy

density,whichmeanstheycorrespondtoanon-flatgeometryfortheUniverse.

Inallcases,exceptforthegreencurveintheright-handgraph,thetypicalsize

oftheinitialripplesarisingafterinflationisthesame.Forthisgreencurve,the

initialrippleswerechosentobesmaller,otherwisethecurvewouldbewaytoo

big.

ThefirststepinappreciatingthedetailsbehindFigure8.9istoknowthatitisderivedfromsomethingcalledthetwo-pointcorrelationfunction.Thisfunctiontellsushowcorrelatedthehotandcoldregionsareonthesky.Forexample,ifthehotregionsalonewereallspacedby1degree,thecorrelationfunctionwouldbelargeandpositiveatthisangle.Or,ifthehotregionsandcoldregionswereallspacedby1degreethenthecorrelationfunctionwouldbelargeandnegativeat this angle. If there is no correlation between the temperatures at differentpointsonthesky,thecorrelationfunctionwouldbeequaltozero.Youcanseethat the correlation function might provide a good way to spot whether theplasmawasperturbedbyawholebunchofsuperimposedsphericalsoundwaves,andthatitcouldinformusoftheirradii.Indeed,thisisthecase,asthepositionsof thepeaks inFigure8.9are related to theradiusof thespheres.11 InBox15(pp.254–7)wegiveamuchmoredetaileddescriptionofthephysicsresponsiblefor Figure 8.9. It is particularly important to emphasize that if the spherical

shellswere not all of the same size, therewould not be any peaks. ThemereexistenceofthepeaksinFigure8.9informsusthatthestructureintheUniversewas laid down once and for all at the timewhen the soundwaves were firstlaunched. Inflation provides a means to orchestrate this grand opening to theUniverse.

There is even more information hiding in the details of Figure 8.9. Thephotons received by Planck all began their journey from the surface of lastscattering almost 14billionyears ago, as illustrated inFigure8.5.Thismeansthat we are looking from a vast distance at a pattern built up from lots ofsphericalwaves.Thesizeof thesphereswesee in theCMBfromourvantagepoint onEarth therefore depends on two separate things. Firstly, the observedsize obviously depends on the actual size of the spheres at the timewhen theCMBphotonswerereleasedfromtheplasma,whichisdeterminedbythespeedatwhichsoundwavesmovedthroughtheplasma.And,secondly,theirobservedsizedependsonthedistancetothesurfaceoflastscattering,whichisgivenbytheexpansionhistoryoftheUniverse:thatis,themoredistantthesurfaceoflastscattering, the smaller the spheres will appear. As we’ve seen, this distancedependsonhowmuch theUniversehasexpandedduring the time thephotonshavebeen travelling toEarth,and this is related to theamountofmatter,darkmatter anddark energy in theUniverse.The fact that theobserved sizeof thesphericalsoundwavesdependsontheexpansionhistoryoftheUniverseisonereasonwhyFigure 8.9 can be used to help us determine howmuch andwhattypesofmatterthereareintheUniverse.Ifyouwanttoknowmore,thentakealookatBox15.

We can get a good feeling for just how sensitive Figure 8.9 is to thecompositionoftheUniversebyshowingsomecurvesforwhatitwouldlooklikeif we changed the amounts ofmatter in the Universe. The left-hand graph inFigure8.10showshowthetheoreticalpredictionchangesastherelativeamountsof ordinarymatter and darkmatter are altered (the sumof the twobeing heldfixed).IfyouworkedthroughtheBoxthenyoumightbeabletofigureoutwhythecurves lookas theydo.The right-handgraphemphasizeshowhard it is tomakethetheoryagreewiththedata,andthereforehowimpressiveitisthatthesweet spot of near-perfect agreement occurs using the same numbers weharvestedinthelastchapterusingotherastrophysicalobservables.Iftherewassomethingwrongwithourunderstandingofthisvastswatheofphysics,itisveryhardtoseehowwewouldgetsuchbeautifulagreement.

We’ve been focusing a lot on the way that the initial perturbations in theprimordialplasmaled to themicrowavebackground,but,aswesaid, thesameinitialperturbationsalsoledtotheformationofthegalacticstructuresweseeintheSDSSmap(Figure8.1).ThefactthatthesimulationslikethoseinFigure8.4agreewithobservationsisstrongevidencethatthemodelisgood.Thereisalsoavery striking way to see the imprint of those sound waves in the primordialplasma.

We have seen that, at the time of recombination, the plasma tended to beover-dense in spherical shells as the sound waves travelled outwards frominitially over-dense regions. This means that, at the time of recombination,protonstendedtohaveahigher-than-averagechanceofbeinginthesesphericalshells.Thesearethesameprotonsthat,muchlaterinthehistoryoftheUniverse,formedthehydrogengasthatcollapsedtoformgalaxies.Theoriginalsphericalshells, slightly richer in protons than the surrounding regions, grew with theexpansionoftheUniverse,and(usingtheFriedmannequationagain)theyhadaradiusofaround150Mpcwhenthefirstgalaxieswereformed.ThesituationisrepresentedschematicallyinFigure8.11.Hereisadirectprediction:ifweplaythesamegameaswedidfortheCMB,andconstructacorrelationfunction,thistimeforgalaxies,thenweshouldseethatthereisaslighttendencyforpairsofgalaxies to be separated by a distance of 150 Mpc. Figure 8.12 shows theobservational data and, quite remarkably, we can see that the prediction iscorrect;thereisapeakinthecorrelationfunctionat150Mpc.PerhapswereallydounderstandtheevolutionoftheUniverse,fromatimebeforetheBigBangallthewaythroughtothepresentday.

Figure8.11BaryonAcousticOscillationsmeanthatthereisahigherthan

averageprobabilityoffindingpairsofgalaxiesthatareseparatedbyaround

150Mpc.

Figure8.12Thetwo-pointcorrelationfunction,whichshowsthatgalaxiesare

mainlyfoundtobeclosetogether(hencetheriseatlowdistances).Farmore

interestingisthelittlebumpat150Mpc.Thisisexactlywhatwouldbe

expectedifthegalaxiesformedpreferentiallyonover-denseregions

correspondingtotheexpandedshellswhichthattheimprintoftheearlier

plasmasoundwaves.Thesolidcurveisthepredictionusingthesame

parametersaswereusedtoproducethecurveinFigure8.9.

BOX15.SOUNDWAVESINTHEPRIMORDIALPLASMA

Figure8.13:Addingtogetherthreeplane-wavedisturbancesinacubeof

plasmatoproducethepatternofripplesindicatedinthefourthcube.

Absolutelyanypatternofplasmadisturbancescanbebuiltbyadding

togetherplane-wavedisturbancesindifferentcombinations.

WewanttounderstandhowthepeaksinthePlanckgraphrelatetothe radius of the spherical shells. To do this, we are going tointroduceanentirelydifferent,buttotallyequivalent,wayofthinkingabout waves in the plasma (or in a tank of water, or in the air).Figure 8.13 shows how it is possible to think of a pattern ofdisturbances as being due to plane waves in combination. Planewavesarespecialwaves,likethethreetotheleftoftheequalssigninFigure8.13.Itisclearwhytheyarecalledplanewaves:theylooklikestackedplanes.Thelightanddarkregionscorrespondtoplaceswhere the wave is big and small. The waves we have in mindcorrespond to variations in the density of the plasma in the earlyUniverse–but, on a more down-to-earth level, they could bevariationsinthedensityoftheairinaroomasasoundwavetravelsthrough it.The threeplanewaves inFigure8.13 justhappenall tohave the same wavelength (the distance between successiveplanes,indicatedbyλinthefigure)andtheyareallarrangedatrightangles toeachother.This iswhy the resultantpatternobtainedbyadding them together isso regular.But there isnothing tostopusbuilding patterns by adding together plane waves of differentwavelengthsandorientations. Itdoesn’t take toomuch imaginationtoappreciatethatitmightbepossibletoconstructanyparticularsetofdisturbancesintheplasmabyaddingtogetheralargenumberofplanewaves.The ideaof building ripples in theplasmabyaddingtogetherabunchofplanewavesissimplyforourconvenience;itisameanstoanend.Thevirtueofthinkinglikethisbecomesevidentwhen we consider what happens to one particular plane wave astimeadvances.

Figure 8.14 shows a slice through a plane wave, and thesinusoidal wave below it indicates how the density in the plasmavariesalongthewave.Weareimaginingthatthisparticularwaveisoneofmanyplanewaves thatmustbeaddedtogether todescribe

the plasma as it is delivered to us at the end of inflation. At timezero,thiswaveisreleasedintotheplasma.Bythiswemeanthat,attimezero,theplasmawassquashedandsqueezedinthepatternofaplanewave.Ashorttimeafter,theparticlesintheplasmawillstarttomoveas theyare pushedaway from thehigher-density regionswhere the pressure is greatest. This happens across the entirewave,thenetresultbeingthatwecanthinkofwhathappensastheformationoftwowaves,eachhalfthesizeoftheoriginalwave.Oneof thesewavesmoves to the left at the speedof sound; theothermovestotheright.Wemustaddtogetherthetwowavestoascertaintheneteffect.Thisisillustratedinthefigurefortwomomentsshortlyaftertimezero.Noticethatthetresultantwaveisthesameshapeastheoriginalone;all that ischanging is itssize. Inotherwords, theoriginalwavesimplyoscillatesinsize,firstgettingsmallerandthengrowing back again. For example, therewill be a timewhen eachhalf-wavehas travelledexactlyonequarterofawavelength.Whenthathappens,thepeaksinonewavewillcoincidewiththetroughsinthe other and the two will exactly cancel each other out. At thismoment, the plasma will be perfectly uniform. However, this is afleetingmomentbecausetheparticlesintheplasmaaremovingandtheyovershootperfectuniformity.At the timewhen the twowaveshavetravelledexactlyhalfofonewavelength,theinitialplanewavewillbeinverted;initiallyhigh-densityregionswillnowbelow-densityregions, and vice versa. The net result of all of this is that, if youwere locatedsomewhere in theplasma, theplasma inyourvicinitywouldoscillateperiodicallybetweenhighandlowdensity.Thetimebetweensuccessivepeaksinintensityisthetimeittakesthosetwohalf-wavestotravelonefullwavelength(thentheywillhaveexactlypassedthrougheachotheronce),i.e.theperiodofoscillationofthestandingwaveisequaltoT=λ/v,wherevisthespeedofsoundintheplasma(thespeedatwhichwavestravelthroughit).

The Planck image is a snapshot of the plasma at the time ofrecombination,asviewedfromtheEarth.Toobtainthebehaviourofthe plasma, wemust combine the effect of many plane waves ofdifferent wavelength. Using what we have just learned, we candetermine which standing waves will correspond to the biggestdisturbances in the plasma at recombination. Although all the

standingwaveswereoriginallyasbigastheycanbe(i.e.allwavesstarted out by diminishing in size–the Universe was ‘plucked’, asillustratedinFigure8.14),eachoscillateswithadifferenttimeperiod,andonlywavesofcertainwavelengthswillbebigagainatthetimeof recombination.Specifically, thebiggestwaveswill be those thathave undergone either an integer or a half-integer number ofoscillationsat recombination.Standingwaves thathaveundergoneanintegernumberofoscillationslookexactlyliketheoriginalwave,while those that have undergone a half-integer number ofoscillationslookprettymuchthesame,exceptthatthehigh-densityregionsandlow-densityregionsareinterchanged.Wavesthathaveundergoneaquarter-integernumberofoscillationswillnotproduceanydisturbanceat the timeof recombination. Thismeans that thebiggest waves at recombination will have a wavelength equal to2vT/n,whereTisthetimeofrecombinationandnisaninteger(n=1,2,3,…).1

Figure8.14:Howstandingwavesareproducedintheprimordialplasma.

Figure8.14:Howstandingwavesareproducedintheprimordialplasma.

Thetopimagerepresentsaslicethroughaplanewave.Thegraph

immediatelybelowitshowshowthedensityvariesalongthewave.The

graphbelowthatiswhathappensalittlelater,whentheoriginal

disturbancehasstartedtopropagatethroughtheplasma.Onehalfofit

headsofftotheleft,theotherhalftotheright.Thebottomgraphshows

thesituationstilllater,whenthetwohalf-waveshavetravelledthrough

eachotherevenmore.Thethreegraphsontherightaretheresultant

wavesformedbyaddingtogetherthehalf-waves.Theresultantwaveis

thesameshapeastheoriginalbutthesizeisdiminishing.Thisisa

standingwave.

Now,wecanmakethelinktoFigure8.9,thegraphmappingouttheripplesinplasmaatthetimeofrecombination.ThemathematicalprocessingoftherawPlanckphotographthat leadstothisfigureisspecifically designed to help us seewhich planewaves are big atthe timeof recombination.Tobeprecise, theanglemarkedon thehorizontal axis is the angle between successive peaks in planewaves that were oriented across the line of sight at the time ofrecombination. This means that waves of shorter wavelengthcontribute toFigure8.9atsmallerangles.The firstpeakoccursatan angle of close to 1 degree, which tells us that 1/360 = λ/2πR,where R is the distance the CMB photons have had to travel toreachEarth.Givenwhatwejustdidinthelastparagraph,weknowthatλ=2vT(correspondington=1),whichmeansthatthepositionofthefirstpeaktellsusthatthespeedofsoundthroughtheplasmaisequaltoπ/360×(R/T).ThevaluesofRandTare influencedbythe amount and type of the stuff that is causing the Universe toexpand,becausethatiswhatcontrolswhenrecombinationhappens(whichfixesT),andhowfartheCMBphotonshavetotravelbeforetheyreachourtelescopes(whichfixesR).Youshouldnowbeableto appreciate why the CMB graph in Figure 8.9 can be used toextract information on how fast the plasmawaves travel (which isgovernedbythecompositionofthemediuminwhichtheyexist)andhow much the Universe has expanded since the CMB photonsstarted their journey (because increasing theamount of expansionincreasesthedistancethephotonsmusttraveltoreachEarth,whichinturnreducestheangleanyparticularplanewavesubtendsonthe

sky).Sofar,ourfocushasbeenonthepositionsofthepeaks,butthe

heightsofthepeakscontaininformationtoo.Thegeneraltrendisforthe peaks to diminish in height as the angle decreases, whichmeans that the standing waves of smaller wavelength were lessprominentthanthelonger-wavelengthonesatrecombination.Thisisbecause the photons bounce around inside the plasma and thisproduces a blurring of the standing waves, that is, peaks andtroughs are smeared out by a distance determined by how far atypical photon travels between successive collisions with theelectronsandprotons.Ifthisdistanceislargerthanthewavelengthofastandingwave,thenthatwavewillbewashedout.Thedeclineinsizeoftheacousticpeaksisbecauseofthisgradualdissolutionofthe standing waves. This logic also implies that there will be lessdampingof theshort-wavelengthwaves if therearemoreelectronsandprotonsintheplasma,becausethisiswhatcontrolsthetypicaldistance a photon travels between collisions: if there are very fewcharged particles, the original waves will dissolve as the photonssimply stream away from the regions of high density. That’s nicethen: theheightsof thepeaksaresensitive to thechargedparticledensityintheplasma.

Thecherryon thecake is the fact that careful inspectionof thegraphrevealsthat theodd-numberedpeaksareboostedrelativetotheiradjacenteven-numberedpeaks.Givenwhatwesaidinthelastparagraph,wemightexpecttheheightsofthepeakstofallsteadilyas the angle decreases, corresponding to shorter wavelengths.However, this isn’t quite what happens. This failure of the peakheights to steadily reduce is particularly visible if we look at thesecond and third peaks, which are almost the same size. Thisboostingoftheodd-numberedpeaksisadirectconsequenceofthefact that the plasma is oscillating in a background of darkmatter,whichtendstogravitationallyattractthechargedparticlestowardsit.This has no effect on the standing wave pattern after each fulloscillation, because each standingwave always rebounds back tothesamestartingpoint.Butitdoeshaveaneffectonhowdeeplytheplasmaisabletocompressbeforeitreboundsback.Increasingthedensity of the charged particles in the plasma causes the half-

integer oscillations to be more intense, which boosts those odd-numberedpeaks.

Wehaveonlytouchedonthemostimportantideashere–butitisdeeply satisfying that every feature of the CMB graph can beaccountedforwithbasicphysics.

9.OURPLACE

Wehave travelled a longway from those idle contemplations on the beach atOgmore-by-Sea.We have followed in the footsteps ofmany ordinary people,who took theirmusings seriously enough to act on them and spent their livestrying to understand. Standing on the seashore, Mike Seymour noticedsomethingthatpiquedhiscuriosity.Thatinquisitiveness,thatspiritofenquiry,isthethingthatleadsustobecomescientistsandtakeseriouslythequestionswehaveabouttheworld.Backintheeighteenthcentury,HenryCavendish’surgetoquantify gravity and measure the mass of the Earth unlocked the power ofNewton’slaws.Hedaredtoimagine,buthealsodaredtoconductanexperimentthat required painstaking effort, meticulous craftsmanship and integrity. Heundertooktoquestioneveryaspectofhismeasurementsandwasnevergoingtoallowhimselftobeswayedbyhisownpreconceptionsorbypressurefromhisrivals.Today,bigadvancesareoftenmadebylargecollaborationsofpeople,butthe motivation is just the same. The Planck satellite project and the LargeHadronColliderprojectatCERNinvolvethousandsofscientistsratherthanone,but the only real difference is that, because of the complexity of themeasurementstheyneedtomake,abunchofenthusiastshadtoclubtogethertobuild the experiments andanalyse thedata.Thepowerfuldesire tounderstandbrings people together from all over the world in these large internationalcollaborations.National boundaries dissolve and are replaced by an unfetteredspiritofco-operation.Thisspiritisoneofthegreatjoysofscience,andbecauseofitgreatthingshappen.

Throughout this book we have shown how people have gone aboutmeasuring things and, in so doing, demonstrated away to accumulate reliableknowledge.OurthoughtshaveledusoutwardsfromtheEarthtothestars,andtothe galaxies beyond. Eventually, we have reached a point where we canseriously consider the origins of the Universe. If we lift our gaze from theparochialandanthropocentric,thecosmosawaitsusinhumbling,awe-inspiringglory.

It is staggering to suggest that the entire observableUniverse came fromasubatomic-sizedpatchofspace,butthetheoryofinflationaccountsforhowsuch

apatchevolvedintotheUniverseweinhabit.ThetheoryleadstoaBigBangandgenerates a pattern of initial perturbations that has been confirmed by precisemeasurements of the Cosmic Microwave Background and the clustering ofgalaxies.What’smore, inflationpredicts the existenceof primordial ripples inspacetimethatmaybeobservabletodayasgravitationalwaves:themeasurementofthesewould,formanypeople,betheclinchingpieceofevidenceconfirmingthatinflationreallydidhappen.Worktowardsthatgoalisunderway.

Inflationcertainlyprovidesavividaccountof theoriginsof theobservableUniverse. It is a daring theory with high ambition. However, what we havepresented so far is positively prosaic when compared to inflationary theory’smost astonishing prediction: that our visible Universe might be just one of apossiblyinfinitenumberofuniverses.

In the lastchapter,wesawsomemoviestillsofourpatchofUniverseas itevolved during inflation.Our particular patch started out tiny, and there is noreasontothinkthatitwasanythingotherthanonesmallpartofamuchbiggerspace,whichmay be infinite in extent.We chose to start themovie sequencewhen the observableUniversewas big enough to start developing ripples but,presumably, the rest of the space had alreadybeen inflating for some timebythen,andwejoinedthestorypartwaythrough.Overthefollowingfewmillionmillionmillionmillionmillionmillionthsofasecond,theinflatonfieldcausedourpatchtogrowtothesizeofamelon,beforetheexponentialexpansioncametoanendastheenergydrivingitranout.

Let’s now consider what was happening in the rest of the inflationaryUniverse.We have said that the inflaton field gradually faded as it drove theexpansionofourpatchandultimatelydecayedawaycompletelyattheBigBang.This implies that theinflatonfieldwouldhavebeenbigger in thepast,andtheaccelerationof the expansionof spacemore rapid.This is an ‘on theaverage’statement, because the size of the inflaton field actually varied from place toplacebecauseithadquantum-inducedripplesinit,andtheplaceswherethefieldwas bigger than average would have inflated more rapidly. In our patch, theripplesweresmall; theirmaineffectwastosowtheseedsthatgaverise to theperturbationsintheCMBandthenetworkofgalaxiesthatfilltheskytoday.

Now, here is a crucial piece of information: when the inflaton field waslarger, the rippleswere larger too.Aswegoback in time,wesee the inflatonfield fluctuating more and more wildly; the tiny ripples in a still ocean wereprecededbyhugewavesonastormysea.Wecanthinkofthesituationanotherway, using an idea from the last chapter. During inflation, space glows with

particles, and the glowbecomesmore intense aswe look further into the pastandtheaccelerationofthespaceincreases.

Figure9.1TheMultiverse.Notethatthescalehereiswrong:thetypical

distancebetweenbubblesisvastlybiggerthantheirsizes.Ourtypeof

universemightberelativelyrareinbeingoldenoughandbigenoughto

supporttheevolutionofintelligentlife.

We have said that the size of the inflaton field steadily falls as inflationprogresses and thatwhen it falls belowa certain level inflation ends.At earlyenoughtimes,however,thiswasnotnecessarilythecase,becausetheripplesintheinflatonfieldweresodramaticthatsomeregionsreceivedfluctuationslargeenough to cause the inflaton field to increase in size, notwithstanding thedilution during the expansion. In those regions, inflation would have sped upinsteadofslowingdown.Ifregionslikethisarecreatedatasufficientrate,therewillalwaysbesomepartsofspacethatareinflatingandinflationcarriesonforever.Therewillstillberegionswherethefieldisnotsobig,andinflationwillendintheseregions,givingrisetoBigBangs.OurobservableUniverseemergedfromonesuchregion.Inthisscenario,theentiretyofspacecontainsbubblesof

universewhereinflationhasceased, isolatedfromeachotherbyregionswhereinflationisongoing.ThisisknownastheinflationaryMultiverse.

IntheMultiverse,universesarebeingcreatedoutofnothing,andthisseemswrong.Foronethingitappearstoviolatethelawofenergyconservation,whichstatesthatthetotalamountofenergyinaclosedsystemdoesnotchange.ThisisnotaprobleminGeneralRelativity;thereisnothingwrongwiththeideaofanexpandingspaceinwhichthetotalenergycarriedbyitscontentschanges.Thisidea ismanifest in our observableUniverse; as space expands, the photons itcontains are redshifted. This is the effect that we used tomeasure the rate atwhichspaceisexpandinginChapter6.Quantumtheorytellsusthattheenergyof a photon is inversely proportional to its wavelength, so, as the photon’swavelengthisstretchedbytheexpansionofspace,itsenergyfalls.Thisiswhythe CosmicMicrowave Background temperature is only 3 kelvin today, eventhough it was emitted at a temperature of 3000 kelvin at the time ofrecombination.Evidently, the totalenergyofanexpandinguniversecontainingonly photons falls. The opposite is true for a universe expanding due to acosmological constant. In this case, the energy density in space stays constanteven though space is expanding,whichmeans the total energy in theuniverseincreases.

Abriefrecapisinorder.Weintroducedtheinflatonfieldinordertosolvethehorizonand flatnessproblems inamannerconsistentwith the lawsofparticlephysicsasweunderstandthemtoday.AllweinitiallywantedtodowastocreateatheorycapableofdescribingarapidexpansionintheearlyUniverse,andjustlookwhereithastakenus.Inflation’slogichaspushedusintoanexplanationfortheoriginoftheBigBangandthedistributionofgalaxiesinourUniverse,nottomention the detailed structure of the CMB. Now we discover that the sametheory appears to predict that thiswhole process probably happened over andoveragain, littering theMultiversewithavastnumberofbubbleuniverses,ofwhich ours is just one. It may be that we will never be able to checkexperimentallyiftherereallyisaMultiverse,althoughwemaybelucky.Somecosmologistshavespeculatedthatthebubbleuniversesmightcollidewitheachotherearlyintheirevolution,andthatthismayhaveleftafaintsignaturewaitingfor us in the CMB. Perhaps there are other, as yet unimagined, theoreticaldevelopmentsormeasurementsthatmightallowustoinfertheexistenceoftheMultiverse.

There is another popular idea in theoretical physics that, at first sight, haslittle to offer us in our contemplations of theMultiverse butwhich, on closer

inspection,turnsouttoaddaremarkablemetaphysicaltwist.Foralargepartofourscientificlives,StringTheoryhasbeendominantintheholy-grailquestforaTheoryofEverything,bringingtogetherquantumtheoryandGeneralRelativityintoabeautifulandconsistentwhole.TheideaisthateverythingintheUniverseisconstructedoutoftinyloopsandstrandsofvibrating‘string’.1Thetypicalsizeofapieceofstringisaround10−35m,whichisahundredbillionbilliontimessmallerthanaproton–thissmallnessexplainswhywemighthavehithertobeenfooled into thinking that everything is made out of particles. String Theorygatheredmomentum in themid-1980s, especially following thework ofMikeGreen, then ofQueenMaryCollege in London, andCaltech’s John Schwarz.ThiswaswhenStringTheorystartedtobetakenseriouslyasapossibleTheoryofEverything–theoristsrealizedthatnotonlydidStringTheorycontainGeneralRelativity,italsogaverisetophysicsresemblingthewell-establishedStandardModelofparticlephysics.Stringtheoristsbegantodreamthattheremaybeonlyonelogicallypossibletheory,andthatthistheorywouldpredictallofthelawsofphysics aswe experience them. Itwould explain the values of all the particlemasses,thecosmologicalconstantandthestrengthsoftheforces.Theseductiveidea is that theUniversewe live in is theonly logicallypossibleuniverse,andthat underlying everything is a perfect, unique mathematics. Subsequently, agooddealofprogresswasmade towardsshowinghowthe lawsofphysicsweobservetodaycouldemergeoutofStringTheory,butnobodymanagedtofindtheholygrail–theonetheorytorulethemall.

In practice, String Theory is complicated, because its mathematicalconsistencydemandsthestringsshouldvibrateinaten-dimensionalspacetime.Since we only experience four dimensions, the other dimensions need to behiddenfromusinsomeway.Onewaytodothatistohavetheextradimensionscurlupintotinygeometricalshapesateverypointinourspace;sotinythatwesimplycan’tseethem.Afamiliaranalogywouldbetoimagineahosepipeatthebottom of a long garden. From sufficiently far away, it looks like a one-dimensional line,even though it is in facta two-dimensional surface rolledupintoacylinder.Insuchapicture,ourexperienceoftheworldistobeseenasacoarse,‘largedistance’one,andthelawsofphysicsweencounter todayaretobe regarded as emergent ‘low energy’ approximations to the true theory.Naturallyenough,scientiststookgreatencouragementfromthefactthatlawsofNatureresemblingthoseweknowemergefromStringTheory.

In the early years of the twenty-first century, however, the optimism wastemperedbythegradualrealizationthatStringTheorydidn’tpredictauniqueset

of low-energy laws. Instead, it seemed to predict a grand array of possibleuniverses,eachwithadifferentsetofemergentlawsthataredistinguishedfromeachotherbythedifferentwaysthatthoseextradimensionsinspacearecurledup. This promptedLeonard Susskind, of StanfordUniversity, to introduce theidea of a String Landscape. You might picture a vast landscape of hills andvalleys,stretchingoffasfarastheeyecansee,witheveryvalleycorrespondingto a different possible universe with different low-energy physical laws. Thismighthavebeenfineiftherewereonlyahandfulofpossibilities,butitbecameclear that therecouldbeasmanyas10500 low-energymanifestationsofStringTheory, and we appear to live in a universe described by one of them. In aremarkableaboutturn,manystringtheoristswentfromseekingauniqueTheoryofEverything toexploringa theory thathas thepotential toenumerateanear-infinitevarietyofpossibleworlds.

Atfirstsight,thismayseemlikeaterrificdisappointment–aretherereallyavastnumberofpossibleuniverses?And if so,howcanwepossibly figureoutwhy we live in the one we do? These questions echo those articulated byGottfriedvonLeibniz,inhis1710workThéodicée.Leibnizconsideredtheideaofavastnumberofpossibleworldsandbelievedthatweinhabit‘thebestofallpossible worlds’.2 However, the ambiguous plethora of the String Landscaperaisesafascinatingalternative–mightallofthesepossibleuniversesactuallyberealizedinthevastMultiverseofeternalinflation?

Thecentral ideaof eternal inflation is that therewill alwaysbeportionsofspacethatareacceleratingrapidlybecauseofunavoidableupwardfluctuationsinthe inflaton field. In these regions, the largervalueof the inflaton fieldcausesthe energy density to increase and, at very high energy densities, we needsomething like String Theory to compute what happens. Crucially, at suchenergydensities,itseemstobepossibleforthelawsofphysicstobere-set,asthe Universe makes a transition from one valley in the String landscape toanother.Roughlyspeaking,thetiny,curledupextradimensionsunravelandre-adjust;itisasifspaceismeltingandre-crystalizinginadifferentconfiguration.Thismechanism formoving about the StringLandscapemakes it possible fordifferentbubbleuniversesintheMultiversetohavedifferentlow-energylawsofphysics.

Viewed this way, String Theory and the inflationary Multiverse fit neatlytogether. String Theory allows for the possibility that there are very manydifferent ways of arranging 10-dimensional spacetime to produce differentemergentlawsofphysics,andtheMultiverseprovidesamechanismforrealizing

themall.Accordingtothetheoryaswehavejustdescribedit,althoughdifferentbubble universeswill generally exhibit different low-energy laws, they are allstill governed by the same overarching laws of String Theory. Low-energyobserverslikeusaretrappedinoneparticularvalleyinthelandscape,andthatiswhat shapes our view of the Universe. Other vantage points in the landscapewould correspond to vastly different physics: different elementary particles,differentforcesofNature,andevendifferentdimensionalitiesofspace.Acrossthe Multiverse, this means there will be universes with stronger gravity andvastlylargercosmologicalconstants,universeswithnoatomsorstars,universesfilledwithblackholesanduniversesthatarealmost,butnotquite,thesameasours.Inalllikelihood,therewillbevastlymoreuniversesthanthereareatomsinourobservableUniverse.

Ifwhatwejustdescribed is really thewaythingsare, then the implicationsforhowweviewourplaceinthecosmosareclearlyprofound.Wemustaddthecaveat, though, that the evidence for inflation itself is not yet absolutelycompelling,andwehavenofirmevidencefor thevalidityofStringTheoryortheMultiverse.Nevertheless, the case for theMultiverse isnotpluckedoutofthe head of an imaginative dreamer. It is built on a chain of reasoning that ismore or less compelling, depending on who you ask. And, of course, thefrontiers of science must always lie in the realm of speculation–collectingevidencecanbealengthyanddifficulttask.Oneofthemajorchallengesfacingscientists today is to uncover the evidence that is needed before we can beconfidentinthetheoreticalideaspresentedinthischapter.

ThequestionoftheoriginofourUniverseisobviouslyofimmenseculturalsignificance.Arewetheresultofanintelligentcreator?Isthereareasonfortheexistence of the Universe? At first sight, modern physics offers some of thestrongest evidence in favour of the idea that the Universe was designed. Thefundamental lawsofNatureareastonishinglycompact,powerfulandbeautiful.TheStandardModelofparticlephysics,whichdescribeshowalloftheparticlesin the Universe interact with each other, is possessed of a high degree ofsymmetry. When you work with the mathematics behind the physics it isimpossible not to be touched by its elegance; the equations make snowflake-beautiful patterns that encode the laws of Nature. It is as though a brilliantmathematician set up the Universe. This ‘unreasonable effectiveness ofmathematics’3mightbeinvokedtoappealtoahigherintelligence,asisdonebytheReverendJohnPolkinghornewhenhe states that4 ‘[t]heworld that sciencedescribesseemstome,withitsorder,intelligibility,potentiality,andtightlyknit

character,tobeonethatisconsonantwiththeideathatitistheexpressionofthewillofaCreator.’Polkinghorneknowsthemathematicswell;hewasaparticlephysicistandprofessorinmathematicalphysicsattheUniversityofCambridgebeforejoiningthepriesthood.Eventhoughmanytheoreticalphysicistsdonotgoso far as to invoke the idea of aCreator, they are still deeply affected by theremarkablebeautyinherentinthefundamentalequationsinphysics.

The argument for a Creator also appears to be bolstered by anotherremarkableaspectofthenaturalworld:thelawsofphysicsseemtobeperfectlyadjusted inorder toproduceaUniverse that ishospitable to life.Toapply thefundamental laws of physics, as encoded in the StandardModel and GeneralRelativity, it is necessary to first specify the values of around thirty numbers,whichincludethestrengthsoftheforces,themassesoftheparticlesandthesizeofthecosmologicalconstant.Onlyoncethesehavebeenfixedcantheequationsbe used to predict the outcomes of experiments and observations. Changingthesenumbers,oftenbyjustafewpercent,givesrisetotheoreticaluniversesthathave no chance of supporting life. It is very easy to end upwith a theory inwhich stars never form, or burn out in millions rather than billions of years,leaving no time for biological evolution. The strengths of the forces seemparticularlywelladjustedtoavoidingthesedisasterscenarios.Itisalsoveryeasytoendupwithatheoryinwhichthechemicalelements–thenecessarybuildingblocksofallcomplexstructures–donotexistinanythingliketheformweknow.Theperiodic table isadelicatebalancinggame,anditappears tobeextremelydifficult to pick values for the strengths of the forces and the particlemassessuch that the heavier elements, including carbon and oxygen, are produced instars and remain stable against radioactive decay. The expansion rate of theUniverseisalso‘justso’:itwouldbeeasytomakeauniverseinwhichmatterneverclumpedtogether,which iswhathappens if thecosmologicalconstant istoo big. Therewould also be no stars or galaxies if therewere too little darkmatterortoomuchlight.Eventhewaythatsupernovaeexplode,scatteringtheheavy elements necessary for life across interstellar space, would besignificantly affected if the weak nuclear force was just a little weaker orstrongerthanitis.Takenatfacevalue,ourUniversehasabespokefeeltoit.InthewordsofthegreattheoreticalphysicistFreemanDyson:‘AswelookoutintotheUniverseandidentifythemanyaccidentsofphysicsandastronomythathaveworkedtogethertoourbenefit,italmostseemsasiftheUniversemustinsomesensehaveknownwewerecoming.’

Obviously the numbers that characterize our Universe–such as the particle

massesandtheforcestrengths–mustdescribeauniversefitforlife,becauseweexist.That is not at issue.What is at issue is thenatureof themechanism forselecting thosenumbers:did themechanismpossessacreator-like foresightornot?TheMultiversesays‘not’,becauseitdeliversanalmostinconceivablyrichvarietyofbubbleuniverses thatmanifestallof thepossible lawsofphysics. Itdoes this randomly, with no foresight, and it guarantees the existence of anapparently unique universe such as ours. Viewed this way, our existence isinevitable–alongwiththeexistenceofeveryotherconceivableuniverse.

TheMultiverseideaclearlyunderminestheargumentforaCreatorbasedonthefine-tuningofthelawsofNature.However,itdoesnotquiteunderminetheargument for what we might call a Creator-Mathematician. String Theorypotentiallyprovidesaverybeautifulmathematicalconstruction thatoverarchestheMultiverse,andtheoriginsofthattheoryremainmysterious.Perhapssomeintelligence is responsible. But if you would still like to posit a Creator, theMultiverse idea paints a striking picture of their methods. It seems that thearchitectofourUniversesetabouttheirtaskbycreatinguniverseafteruniverse.Foreachuniverse,thelawsofphysicswereselectedatrandom–asifbyrollingdice.Thiskindof thingisreminiscentofwhatscientistsdowhentheywant tosimulatesystemstheydonotunderstandandwanttounderstandbetter.Theyputtheequationsthatgeneratethesystemontoacomputerandallowthesystemtoevolve while they watch what happens, often choosing the key numbers thatdeterminetheevolutionofthesystematrandomtogenerateawidediversityofoutcomes. The simulations of chunks of universe we presented in Chapter 8werecreatedlikethis.

Today, the cosmologists responsible for those simulations arehamperedbyinsufficientcomputingpower,whichmeansthat theycanonlyproduceasmallnumberofsimulations,eachwithdifferentvaluesforafewkeyparameters,likethe amount of dark matter and the nature of the primordial perturbationsdeliveredat theendof inflation.But imagine that therearesuper-cosmologistswhoknowtheStringTheorythatdescribestheinflationaryMultiverse.Imaginethat they run a simulation in their mighty computers–would the simulatedcreatureslivingwithinoneofthesimulatedbubbleuniversesbeabletotellthattheywereinasimulationofcosmicproportions?

Sciencehasmadeanastonishingamountofprogressoverthepast500years.Eachnewgenerationofscientistshasbenefitedhugelyfromtheoftenherculeanefforts of those who went before, and today we find ourselves in the veryprivilegedpositionofbeingabletocontemplateandtocomputewhathappened

at the birth of our Universe. The process of learning new things is notmysterious, but there is romance in the endeavour. There is something trulywonderfulaboutthedeterminedpursuitofseeminglyinsignificantdetails.Atitsheart,scienceisaboutconnectingwiththeworld;itisalivingcelebrationoftheUniverse.Itisaboutreachingoutintotheunknownandexploringtheunchartedlandscapeofideas.Wearepartofthegreatestofmysteries,and,forus, thatisenough.

APPENDIX

Thefollowingisalistofsomebasicmathsandphysicsthatmaybehelpful.

POWERSOF10

In cosmology and particle physicswe often encounter numbers that are eitherverybigorverysmall.Tohelpwritesuchnumbersweuseexponentialnotation.For example, 1million,which is equal to 1,000,000, can bewritten 106. Thisshouldbereadas‘10tothepowerof6’,whichmeansitisequalto10multipliedby itself 6 times. Tiny numbers are written with negative powers, so onebillionth,whichisequalto1/1,000,000,000=0.000000001,iswritten10−9.

UNITS

Veryoftenwedealwithquantitiesthatcarryunits.Thesimplestexamplemightbeadistance,suchas1kilometre=0.621miles.Asinthecaseofkilometresandmiles, we always have the freedom to choose the units we want to use tomeasure something. Although metres and kilometres are convenient units forstating the typical distanceswe encounter in our daily lives, they are not veryconvenientincosmologyandparticlephysics.Morecommonlywewillwanttouse light years, megaparsecs, ångstroms and nanometres. These can all beconvertedintometresasfollows:

1lightyear=9.46×1015metres1megaparsec=1Mpc=3.26×106lightyears1nanometre=10ångstroms=10−9metres1femtometre=10−15metres

Likewise, in everyday circumstances it makes sense to measure energies injoules(e.g.a20wattlight-bulbradiatesenergyatarateof20joulespersecond),

butwhendiscussingparticlesofatom-sizeandsmaller itmakesmoresense tousetheelectronvolt(denotedeV),whichcanbeconvertedintojoulesusing:

1eV=1.60x10-19joules

Wewilloftenabbreviateunits,e.g.1nm=1nanometreor1km=1kilometreor1mega-electronvolt=1MeV.

Numbers that carry with them an associated unit are called dimensionfulnumbers, and it is common to want to combine dimensionful numbers bymultiplyingordividing them.The simplest exampleof this iswhenwe takeadistanceand thendivide itbya time inorder togetsomething that isaspeed.For example, a car moves 100 km in 2 hours, therefore its speed is 100kilometresdividedby2hours=100km/2h=100×1km/2/(1h)=50×1km/1h = 50 km/h.Obviously you could see that the carmoves at 50 km per hourstraight away–but we chose to show the various intermediate ways we couldhave written the ratio because simple manipulations like these are sometimesperformedinthetext.

Weencountertwoparticularlyelaboratedimensionfulnumbersinthebook:theGravitational constant, G = 6.67 m3/s2/kg and the Hubble constant, H = 68km/s/Mpc. These units might look abstract, but in fact they are easy tocomprehend.Forexample,divideGbyadistancesquaredandthenmultiply itbyamassandyouendupwithanumberwiththeunitsofacceleration(m/s2):ifthedistanceistheradiusoftheEarthandthemassisthemassoftheEarth,thenthe corresponding acceleration is the acceleration due to gravity, for an objectdroppedclosetotheEarth’ssurface.Likewise,theunitsoftheHubbleconstanttellusthatagalaxy1MpcawayrecedesfromtheEarthataspeedof68km/s.

Occasionallywemakeuseofelementaryalgebra.Forexample, thecalculationof the speed of the car above could be carried out using the formula v = d/twhere d = 100 km and t = 2 hours, to give v = 50 km/h. A formula can betransformedintoanotherequallyvalidonebyperformingthesameoperationoneach side of the equals sign, e.g. v × t = vt = d/t × t = d. In this case, wemultipliedbothsidesof theequationby t todeliveranequation tellingus thatthedistancedisequaltotheproductofthespeed,v,andthetime,t.Noticehow

we canwrite a product of two numbers either using an explicitmultiplicationsymbol(v×t)or,moresimply,asvt.Wealwaysdenotedivisionbyabackslashsymbol(e.g.d/t×tmeans‘ddividedbytmultipliedbyt’,whichissimplyequaltod).

ELEMENTARYPARTICLES

Atoms are the building blocks of the ordinarymatter we encounter on Earth.Theyareapproximatelyanångstromacross,andmostoftheirmassresidesinatiny central nucleusbuilt fromprotons andneutrons.Protons are only about 1femtometreindiameterandtheycarrypositiveelectriccharge.Orbitingaroundthenucleusaretheelectrons,whichcarrynegativeelectricchargesuchthattheentireatomisneutral.Thesimplestatomiscalledhydrogenanditconsistsofasingleprotonandasingleelectron–it is themostabundant typeofatomin theUniverse.Theway that the electrons are arranged around the nucleus governsthe way an atom interacts with other atoms, e.g. to produce molecules. TheentirelistofatomscanbecollatedinthePeriodicTable(pp.42–3).

We now know that protons and neutrons are themselves built from smallerparticlescalledquarksandgluons.Thegluonsmediatethestrongnuclearforce,whichbinds thequarks together. In total therearesix typesofquark,althoughonly the lightest two of these are used in building protons and neutrons. Theelectrons are also part of a bigger family of particles called leptons–themuonandtauleptonsarelikeheavierversionsoftheelectron.Theremainingleptonsarethreeelectricallyneutralneutrinos.

Apart from the electromagnetic force, which causes particles with oppositeelectricchargetoattracteachother,andthestrongnuclearforce,thereisalsotheweaknuclearforce.Thisforce ismuchweaker thantheotherforces,except inveryhigh-energyinteractions,andit isable tomakeneutronsturnintoprotonswiththeemissionofanelectronandaneutrino.ThisfeatureplaysakeyroleinthenuclearfusionprocessesthattakeplaceinthecentreoftheSun,causingittoburn.

Justasgluonsmediatethestrongforce,theelectromagneticforceismediatedbyphotons,whichcanalsoberegardedasparticlesoflight.Theweaknuclearforce

ismediatedbytheWandZparticles.

The Standard Model of particle physics is a very precise mathematicalframeworkbasedonquantumtheorythatdescribeshowallofthese(i.e.thesixquarks, the six leptons, and their anti-matter partners) elementary particlesinteractwitheachotherthroughtheexchangeofphotons,gluonsandWandZparticles. The StandardModel involves onemore particle, the Higgs particle,whoseinteractionswiththeotherparticlesareresponsiblefortheirhavingmass.Photonsandgluonsdonot interactwith theHiggsparticleand theyhavezeromass.

TheStandardModeldoesnotincludethegravitationalforcesbetweenparticles,and itdoesnot includedarkmatter in its listofparticles.Theseare twoof thereasonswhyitisgenerallyregardedasbeingincomplete.

EVOLUTIONOFTHEUNIVERSE

ILLUSTRATIONCREDITS

Gratefulacknowledgementisgiventothefollowingforpermissiontoreproduceoradaptmaterialforthefiguresinthisbook:

2.1:NationalGeographic

2.2:datafromMITHaystack,seewww.haystack.mit.edu/edu/pcr/downloads/

2.4: Elliot Lim and Jesse Varner, CIRES & NOAA/NCEI, seewww.ngdc.noaa.gov/mgg/image/crustalimages.html.R.D.Müller,M.Sdrolias,C.GainaandW.R.Roest(2008),GeochemistryGeophysicsGeosystems,vol.9,Q04006

2.6: Data from G. Brent Dalrymple (1991), The Age of the Earth, StanfordUniversity Press. Based on original data from (upper) J.-F.Minster and C. J.Allègre (1979),Earth and Planetary Science Letters, vol. 42, 333; (lower) S.Moorbathetal.(1977),Nature,vol.270,43

2.8: Meteor Crater, Northern Arizona, USA, www.fossweb.com/delegate/ssi-foss-ucm/Contribution%20Folders/FOSS/multimedia/Planetary_Science/binders/earth/earth_craters/barringer_crater_1.html2.9:ProducedusingBP2000StandardSolarModeldatafromJ.N.BahcallandM.H.Pinsonneault,seewww.sns.ias.edu/~jnb/SNdata/sndata.html2.12:AndreyV. Zotov and Alexander A. Saranin, seehttp://eng.thesaurus.rusnano.com/wiki/article939.Based onM. F.Crommie,C.P.LutzandD.M.Eigler(1993),Science,vol.262,218

3.3:photographsbyBobSeymour

3.7:FromH.Cavendish(1798),PhilosophicalTransactionsoftheRoyalSocietyofLondon,vol.88,469

4.3:photographsbyKevinKilburn

4.5:MichaelGoh,http://apod.nasa.gov/apod/ap160217.html

4.6: Carnegie Observatories,www.aip.org/history/exhibits/cosmology/ideas/larger-image-pages/pic-island-m31.htm4.7:hubblesite.org/NASA,http://xdf.ucolick.org/xdf.html

4.8: Miodrag Sekulic,commons.wikimedia.org/wiki/File:Bode%27s_Galaxy_Cigar_Galaxy_m81_m82.jpg4.9: Based onhttp://dailysolar.weebly.com/uploads/3/4/8/5/3485153/9810884.jpg

4.10:EuropeanSouthernObservatory,www.eso.org/public/images/potw1312a/

4.11:BasedonthespectrumavailableattheNASAExtragalacticDatabase,seehttps://ned.ipac.caltech.edu/.OriginalreferenceL.C.Ho,A.V.FilippenkoandW. L. Sargent (1995), The Astrophysical Journal Supplement Series, vol. 98,477

4.12:BasedonthespectruminR.Walker,‘Quasar3C273OpticalSpectrumandDeterminationoftheRedshift,seewww.ursusmajor.ch/downloads/the-spectrum-of-quasar-3c273-1.2.pdf 4.15: N. A. Sharp, NOAO/NSO/Kitt PeakFTS/AURA/NSF, http://noao.edu/image_gallery/html/im0600.html; NationalOptical Astronomy Obervatory/Association of Universities for Research inAstronomy/National Science Foundation; calibrated axes by Larry McNish,http://members.shaw.ca/rlmcnish/

5.3: S. Ossokine and A. Buonanno (Max Planck Institute for GravitationalPhysics), Simulating eXtreme Spacetime project; scientific visualization: D.Steinhauser (Airborne Hydro Mapping GmbH),www.nature.com/news/gravitational-waves-how-ligo-forged-the-path-to-victory-

1.19382

5.4: Courtesy Caltech/MIT/LIGO Lab,www.ligo.caltech.edu/image/ligo20150731f andwww.ligo.caltech.edu/image/ligo20150731c 5.5: Based on a figure in B. P.Abbottet al. (LIGOScientificCollaboration andVirgoCollaboration) (2016),PhysicalReviewLetters,vol.116,061102

5.6: IPAC/Caltech, by Thomas Jarrett,https://commons.wikimedia.org/wiki/File:2MASS_LSS_chart-NEW_Nasa.jpg5.7: Chris Mihos, Case Western Reserve University/ European SouthernObservatory, https://en.wikipedia.org/wiki/Virgo_Cluster#/media/File:ESO-M87.jpg 6.1: Adam Evans,https://en.wikipedia.org/wiki/Andromeda_Galaxy#/media/File:Andromeda_Galaxy_(with_h-alpha).jpg6.2:NGC5055(Sunflower),GertGottschalk;NGC7171,TheNGC/ICproject; NGC1073, Adam Block/Mount Lemmon SkyCenter/University ofArizona; NGC3887, copyright © 2011 The Carnegie-Irvine Galaxy Survey;NGC3684, NGC7280, NGC0150 and UGC6533, Sloan Digital Sky Survey;NGC0011, NGC0251 and NGC0337, Donald Pelletier; NGC4535, AdamBlock/Mount Lemmon SkyCenter/University of Arizona; NGC4712, RobertGendler (www.robgendlerastropics.com), Subaru Telescope (NAOJ), HubbleLegacy Archive, Colour data courtesy Adam Block(http://skycenter.arizona.edu/gallery), Bob Franke (http://bf-astro.com) andMaurice Toet (www.dutchdeepsky.com/index.html); NGC3668, courtesy ofCourtneySeligman6.4:FromE.P.Hubble(1929),ProceedingsoftheNationalAcademyofSciencesoftheUSA,vol.15,168

6.6:Adapted from J.C.Mather et al. (1990),TheAstrophysical Journal, vol.354,May10,L37

7.1: X-ray: NASA/CXC/CfA/M.Markevitch et al.; Optical: NASA/STScI;Magellan/U.Arizona/D.Cloweet al.;LensingMap:NASA/STScI;ESOWFI;Magellan/U. Arizona/D. Clowe et al.,http://chandra.harvard.edu/photo/2006/1e0657/index.html 7.2: NASA,http://hubblesite.org/gallery/album/galaxy/cluster/pr2000007b/

7.3:photographbyBillChamberlain;(right)

European Southern Observatory,https://en.wikipedia.org/wiki/Omega_Centauri#/media/File:Omega_Centauri_by_ESO.jpg

8.1:M.BlantonandSloanDigitalSkySurvey

8.2: NASA,http://imagine.gsfc.nasa.gov/science/toolbox/emspectrum_observatories1.html8.3:DamienP.George,http://thecmb.org/

8.4:ScottKayandtheVirgoConsortiumforCosmologicalSimulations;(right)fromJ.Schayeetal.(2015),MonthlyNoticesoftheRoyalAstronomicalSociety,vol.446,521.Eaglesimulationscanbeobtainedhttp://eagle.strw.leidenuniv.nl/

8.9: Based on http://sci.esa.int/planck/51555-planck-power-spectrum-of-temperature-fluctuations-in-the-cosmic-microwave-background/. The 2015results of the PlanckCollaboration are summarized inR.Adam et al. (2015),arXiv:1502.01582.

8.11:imagebyZosiaRostomian,LawrenceBerkeleyNationalLaboratory

8.12: Ariel Sánchez. Based on figure in A.G. Sánchez et al. (2014),MonthlyNoticesoftheRoyalAstronomicalSociety,vol.440,2692.

INDEX

‘2MASS’measurements,150n

Abell2218galaxycluster,190,191figacceleration–ofexpandingUniverse,205,222–3,224,227box,231box–andgravity,62–3,64box,65,114–16,117–18,120–21,205–asnotuniversal,121–ofspaceininflatonfield,260,262,267

Africa,13–18ageoftheUniverse,3–4,100–101,198,203–4,207Albrecht,Andreas,229nalphadecay,nuclear,44boxAlpher,Ralph,176americium-241(alphaemitter),44box,46boxAndromeda,74,102,148fig,149,185n–blueshiftof,153–distancefromEarth,99,100,150,162–distancefromMilkyWay,100–Hubbleidentifies,92–4,93fig,96–9,162

ApolloMoonmissions,33Aristotle,55–6,88AstronomicalUnits(AU),80–81,82–4AtlanticOcean,11–24,12fig,15box,15fig,20boxatomicnumbers,46boxatoms,41–6box,44fig–formingoffirst,6,173–4–fusionandstarformation,6–7,172–andlight,102,103fig,105,106–8,109–13box–nuclei,3,5–6,41–5box,169–70,173,193,211,212,218–20–periodictable,41box,42–3fig,46box,270

–radioactivedecay,24–35,26fig,29fig,40,44–5box,45fig–sizeof,47–8box,167

Barium-137atoms,45boxBarringerCrater,Arizona,34figBaryonAcousticOscillations,251figBellLaboratories,NewJersey,176Bessel,Friedrich,84–8betadecay,nuclear,44–5boxBethe,Hans,176BigBang–cake-bakinganalogy,163,168–anddecayofinflatonfield,4,228,229,243,260–definitionof,3–Einstein’stheorypredicts,115,131,140–41,143–evidenceinsupportof,146,165,166–79–extremedensityofUniverseat,3,5,143,166–7,168,194–and‘flatnessproblem’,194–andHiggsfield,5–nucleosynthesis,5,36,169–70,172,176,183,187,218–20–sizeofUniverseat,167–8–andtemperatureuniformity,221–2–Universebefore,3–4;seealsoinflation,theoryof–whenithappened,3,10,180–seealsoplasma,primordial

‘BigCrunch’concept,143,166,208figbiochemistry,7blackholes,3,14,74–5,131,202,226box–collisionof,126fig,127,128–30box,138fig–Hawkingradiationfrom,232n–andmass-to-lightratios,186,187

blueshift,145box,151box,153blue-skiesresearch,46boxBrahe,Tycho,61Brout,Robert,226–7boxBulletCluster,188fig,189–92

caesium-137atoms,45boxCambridgemeetingoninflation(1982),229cameras,digital,61,85–7boxCanesVenatici,198–9carbon,7,10,47box,172,202,270Cassini,Giovanni,81–2Cassiopeia,89,92Cavendish,Henry,66,68–73,69fig,114,128box,258CavendishLaboratory,Cambridge,68Cepheistardelta,96–9Cepheidvariablemethod,92–4,93fig,96–101,99n,105,149,150,151boxChamberlain,Bill,198–9ChandraX-rayobservatory,189Chibisov,Gennady,229nchlorine,46boxComagalaxycluster,185,186computersimulations,212–18,216–17figs,272continents,11–24,16–17figcontractingUniverseconcept,136–41Copernicus,Nicolaus,57–61,60fig,92,132CosmicBackgroundExplorer(COBE)satellite,177–9,178figcosmicdarkages,3–7CosmicMicrowaveBackground(CMB),6,174–9,175fig–anddarkmatter,74,193,246–7figs,249–50–‘horizon’problem,218,219fig,220–22,223,228–nearuniformityof,211–12,220,236–photonenergydensity,181–3,193,214fig,263–photonsreleasedfromplasma,173–9,220–21,243,249,256–7box,263–Plancksatellitephotographof,212,214fig,220–21,239,240–53,254–7box–quantum-induced ripples in, 210fig, 213fig, 228, 236, 239–51, 246–7figs,254–7box,259,260–62

–‘surfaceoflastscattering’,220,249–temperaturefluctuationgraph(Plancksatellite),240–53,241fig,254–7box–temperatureof,177–9,178fig,181,214fig,220–22,239,241fig,246–7figs,263

CosmicNeutrinoBackground,184boxcosmological constant, 124, 140, 142, 145box, 208fig, 221n, 222–3, 231box,

263,270–asdarkenergy,205,246–7figs,249–50–asEinstein’s‘greatestblunder’,140,199–revivalofin1990s,198,203–6,223–and‘scalarfield’concept,223,224,227box–andStringTheory,265,267–theoreticalprejudiceagainst,206

Creator,268–72criticaldensity,182,193–8,203–4,205,206,207–9,208figcurvature/adiabaticperturbation,232–6,233fig,242–53,259

darkenergy,205,208fig,242,246–7figs,249–50–seealsocosmologicalconstant

darkmatter–andCMB,74,193,246–7figs,249–50–computersimulations,212–18,216–17figs,272–anddecayofheavierparticles,4–evidenceforexistenceof,74,185–93–andFriedmannequation,145box,221–andgalaxyformation,6,7,193–gravitationalclumpingof,5–6,7,193,212,215–18,257box,270–andgravity,6,7,192–3,205,212,215,257box–andmass-to-lightratios,186–7–notexplainedbyStandardModel,226box,278–aspartof‘observable’Universe,167n–aspassiveinearlyUniverse,168–9,211n–Rubin’s‘galaxyrotationcurves’,189–sumtotalof,193,206,242–theoriesonnatureof,74,192–3–Zwicky’sdiscoveries,185–6,190

deuteriumnuclei,5,170,171,172,185Dicke,Robert,174–6,177–9dinosaurs,extinctionof,10Dopplereffect,151box,153,185–6Dyson,Freeman,271

EagleProject,215,217fig,218

Earth–ageof,10,11–33,40,101–atomiccompositionof,41box–densityof,56,68–73–distancetoMoon,65–distancetoSun,39,40,59fig,80–81,82–4–formationof,7,33–gravitationalforceof,65,116–17–magneticfield,20box–massof,56,63,65–74,67fig,69fig,258–orbitof,39,57–61,61tab,73–4–planets’distancefrom,59fig,76–83,78–9fig,84,85–7box,89–pre-Copernicancentrality,57,58,92,132–radiusof,50–56,52fig–rotationof,57,85box

Eddington,Arthur,142Ehrenfest,Paul,131Einstein,Albert,50n,269n–‘CosmologicalConsiderations of theGeneral –Theory ofRelativity’ (1917paper),131

–E=mc2formula,46box–specialtheoryofrelativity,58,181–andstaticUniverse,139–40,141,142,143,165,198–theoryofgravity,115–18,119fig,120–31,145box–seealsoGeneralTheoryofRelativity,Einstein’s

electromagneticforce,5,173,180,184box,223–4,225–7box,277electromagneticspectrum,109box,109figelectrons,4,5n,45–6box,113box,193,277–andatomicnumbers,46box–andbetadecay,44–5box–distancefromproton,48–9box–electronfield,225box–electron-positronannihilation,38fig,45box,184box,230–32,231box–andformationoffirstatoms,6,173–4–massof,41box,46box,48–9box,181,183–inprimordialplasma,6,168,169,173,211,220,243,257box–andquantumtheory,41box,44box,48–9box,110–11box

elements,7,10,170,172–3,202,270–seealsoentriesforindividualelements

energyconservation,lawof,262Englert,François,226–7boxEuropeanSouthernObservatory,Chile,132,134fig,199EuropeanSpaceAgency–Gaiasatellite,89–Planck satellite, 163–4, 212, 213–14figs, 220–21, 239, 240–53, 254–7box,258

evolutionbynaturalselection,10,20boxexpandingUniverse–asaccelerating,205,222–3,224,227box,231box–average mass density, 144–5box, 181–5, 186–7, 193–8, 203–4, 205, 206,207–9

–cake-bakinganalogy,163,168–coolingof,5,6,170,173,184box–deSitterhorizon,231box,236–offiniteageandexpandingforever,142,143,166,180,206–first50,000yearsafterBigBang,212–and Friedmann equation, 141, 142, 144–5box, 168–9n, 180–82, 195–8,203–4,206–9,221–3

–and general theory of relativity, 136–41, 145box, 146–9, 150–64, 151box,158–9figs

–andEdwinHubble,2,156,158–9figs,160–62,161fig,163–4,172,182,199–Hubbleconstant,162–4,182,199,275–6–Hubblerate,144box–Hubble’s original graph/data, 156, 158–9figs, 160–62, 161fig, 163–4, 172,182,199

–pre-BigBangphaseseeinflation,theoryof–primordialelements,171–2–redshiftevidence,145box,146–9,150–63,151box,152tab,158–9figs,163,263

Feynman,Richard,41box,166,174n,230fission,nuclear,44boxFitzgerald,F.Scott,242flatuniverse,138fig,141,142,182,193–8,204,205,218

–asoneofthreepossibilities,135–6‘flatness’problem,195–8,218,222n,223,228,263fluorine,45–6boxforce,114,115,116,118–20–electromagnetic,5,173,180,184box,223–4,225–7box,277–Newton’sFirstLaw,120–Newton’sSecondLaw,62–3,64box,65,114–15,118–andspacetimeconcept,121–strongnuclear,169–70,277–weaknuclear,5,44–5box,184box,270,277–seealsogravity

FrenchGuiana,82,89Friedmann,Alexander,140–42Friedmann equation, 141, 142, 144–5box, 168–9n, 180–82, 195–8, 203–4,206–9,221–3

fundamentalfields,225–7box–Higgsfield,5,5n,223–4,225box,226–7box–‘scalarfield’concept,223–8,225box,226–7box,229–32–seealsoinflatonfield

fusion,nuclear–inearlyUniverse,5,35,169–72,183–5–ofheavyelements,7,10,170,172–3,202,270–hydrogenintohelium,7,34–40,37fig,38fig,199–202–andstarformation,6–7,172–intheSun,35–40,37fig,38fig,44box,199–202,277

Gaiasatellite,ESA,89galaxies–3C273(quasar),107fig,108–averagesupernovaratein,101–2–clumpingtogetherof,7,193,210fig,211,215,218,250–53,251fig,252fig,259

–clustercollisions,188fig,189–92,191fig–clustersandsuperclusters,132,134fig,185–7,185n,188fig,189–92,191fig,210fig,259

–distancesto,89,99,100,101–2,147,149–50,151box,158–9figs–distributionof,132,135,210fig,215,228,236,250–53,251fig,252fig,259,

263–dwarf,171–formationof,6,7,193,204,212,250–53–giantfilamentarywebs,7,211–HubbleeXtremeDeepField(XDF)image,97fig,99–100–HubbleidentifiesAndromeda,92–4,93fig,96–9,162–M81(Bode’sGalaxy),98fig,99–M82(TheCigarGalaxy),98fig,99–massof,74,151box,185–6,190–NGC0011(spiral),152tab,157–NGC1313(spiral),156–NGC4535(barredspiral),104fig,105–6,107fig,108,150,152tab,156–numberofinobservableUniverse,100–‘propermotion’of,153–andripplesintheinflatonfield,5,228,260–Rubin’s‘galaxyrotationcurves’,189–sizeof,101,150n,163–spectraoflightfrom,102,105–8,107fig,111box,146–7,149n,151box–speedofinclusters,185–6–spiral,149–50,151box,154–5fig,158–9figs,163–4,185n–UGC6533(spiral),152tab,153,156–visiblefromEarth,132,133fig–wispydistribution,210fig,215–seealsoAndromeda;MilkyWay

GalileoGalilei,74,114Galle,JohanGottfried,77Gamow,George,176GaryGibbons,229GeneralTheoryofRelativity,Einstein’s,58,108,118,146,226box,228–andblackholes,127,128–30box,131–andcontractinguniverseconcept,136–41–andCreatorconcept,270–andcurvatureofspacetime,121–5,127–31,135–9–EinsteinFieldEquations,124–5,131,135,136,139–41,142–3,179–andexpandinguniverse,139–43,146–9,150–64,151box,158–9figs–andgravitationallensing,188fig,189–90,191fig–lightdeflectioningravitationalfield,125

–andtheMultiverse,262–3–predictsBigBang,115,131,140–41,143–andredshift,108,131,146–9,150–63,151box,158–9figs–andstaticUniverse,139–40,141,142,143,165,198–andStringTheory,264–5–assuccessortoNewton’stheory,127,145box–seealsocosmologicalconstant

geoscience,19fig,20box,21–3Giant’sCauseway,Ireland,10Glashow,Sheldon,226boxglobularclusters,196–7fig,198–9,200–201figs,203–4GlomarChallenger,20boxgluons,4,5,44box,225box,227box,277,278gneiss,granitoid,31–2gold,7,172Goldstoneobservatory,MojaveDesert,59figGPStechnology,14,15box,55,125gravitationallensing,188fig,189–90,191figgravitationalwaves,126fig,127,128–30box,138fig,146,259gravity–andacceleration,62–3,64box,65,114–16,117–18,120–21,205–clumpingofmatter,6,185,204–5,212,215–18–andCMB,181–anddarkmatter,5–6,7,192–3,205,212,215–18,257box,270–Einstein’stheoryof,115–18,119fig,120–31,145box;–seealsoGeneralTheoryofRelativity,Einstein’s–GravitationalConstant(G),62,65–6,67fig,73–4,114,124,144box–gravitationalmass,114–15–andinertialmass,114–15–andStandardModelofparticlephysics,226box,278–andstarformation,6–7–UniversalLawof,62–6,64box,67fig,114–15,116,125,127,151box,185–6–‘zero-G’,116

Green,Mike,264Greenland,26fig,31–2Guth,Alan,229

Haletelescope,PalomarObservatory,104fig,105half-life,24–35,26fig,29fig,40,44–5box,45figHawking,Stephen,229heartpacemakers,44boxHeisenberg’sUncertaintyPrinciple,230–32helioseismology,36,37fighelium,6,41box,111–13box,166,173–andalphadecay,44box–firstatomsof,173–4–fusionfromhydrogen,7,35–40,37fig,38fig,199–202–primordial,5,169,170,171,172,185

Herman,Robert,176Hertzsprung,Ejnar,96Herzsprung-Russell(HR)diagrams,199–203,200figHiggs,Peter,226–7boxHiggsboson(Higgsparticle),5n,226–7box,228,278Higgsfield,5,5n,223–4,225box,226–7boxHigh-ZSupernovaSearchTeam,204–5,206HIIregions,171homogenousmatterdistribution,132–5,136,139,144boxHookertelescope,MountWilson,92‘horizon’problem,218,219fig,220–22,223,228,263Hubble,Edwin–andexpandingUniverse,2,156,158–9figs,160–62,161fig,163–4,172,182,199

–identifiesAndromeda,92–4,93fig,96–9,162–redshiftdata,156–60,158–9figs,163

HubbleSpaceTelescope,97fig,99–100,105,107fig,189,191figHubble’sLaw,162–4,199,204Hutton,Charles,68,72–3hydrogen,3,6,7,41box,166,215,253,276–7–absorptionspectrum,109–11box,110fig,113box–emissionspectrum,111fig–firstatomsof,173–4–fusionintohelium,7,35–40,37fig,38fig,199–202–H-alphaline,103fig,105–6,107fig,108,110fig,111fig,112fig–primordial,170,171,172,185

sizeofanatomof,48–9boxhyperbolicUniverseconcept,135,136,141,142,145box,182,193

inflation,theoryof,3,4–concretepredictionsof,228–9,239–42,259–andcosmologicalconstant,222–4,231box–asdeliveringhugeuniverse,3–4,195,218,224,228,229–and energy density, 232–6, 243–4, 246–7figs, 250–53, 252fig, 254–7box,263,267

–eternalinflationconcept,262,263–4,267–8–‘flatness’problem,195–8,218,222n,223,228,263–graphshowingripplesinCMB(Plancksatellite),240–53,241fig,254–7box–Guth’sworkon,229–‘horizon’problem,218,219fig,220–22,223,228,263–inflationaryMultiverse,261fig,262,263–4,267–8,271–2–andnucleosynthesis,218–20–andrateofexpansion,222n–realparticlesproducedfromnothing,230–32–soundwavesinprimordialplasma,243–53,252fig,254–7box–‘stonesthrowninwater’analogy,242n,243–4

inflatonfield–accelerationofspace,260,262,267–decayofandBigBang,4,228,229,243,260–decayofparticles,224–8,229–definitionof,4–asnotwellestablishedbyexperiment,227box–quantum induced ripples,4,5,6,228–51,233fig,234–5fig,238fig,241fig,246–7figs,254–7box,260–62

–ripplesas‘scaleinvariant’,237–40,238fig–as‘scalarfield’,223–8,225box,227box,229–32

InternationalSpaceStation(ISS),116–17iron,7,10,172,173isochronmethod,25–33,26fig,29figisocurvatureperturbations,244–5isotopes,25–33,26fig,29fig,46boxisotropicmatterdistribution,132–5,136,139,144boxIsua,Greenland,26fig,31–2

JackHills,WesternAustralia,32Jarrett,Thomas,132Jupiter,61,61tab,74

Kay,Scott,212–15Kepler,Johannes,61–2,63,64boxKilburn,Kevin,80,85–7box

Labradorcoast,31LargeHadronCollider,4–5,207,223–4,225box,226box,258LargeMagellanicCloud,95laserinterferometry,128–30boxLeGentil,Guillaume,82–4LeVerrier,Urbain,77lead(element),32lead-isotopicdating,33Leavitt,Henrietta,94,95–6Leibniz,Gottfriedvon,266Lemaître,Georges,140–42,143leptons,5n,225box,277,278Lewis,David,OnthePluralityofWorlds(1986),266nlight–absorptionspectrum,109–13box,110fig–averageluminosityofspace,186–7–coloursinthespectrum,102–5,103fig,104fig,109–13box,149n–emissionspectrum,110–13box–andmassdensityofUniverse,144–5box,186–7–mass-to-lightratios,186–7–solarabsorptionspectrum,111–13box–solaremissionspectrum,103fig,111–13box–spectrallines,102–8,103fig,107figs,109–13box,149–50–wavelengthof,109–13box,151box,176–7,178fig–seealsophotons;redshift

LIGOdetectors,126fig,127,128–30box,138fig,146Linde,Andrei,229,229nlithium,5,170,171,172,185livingthings,10,11,20box

LocalGroupcluster,185nLockyer,JosephNorman,113box

Magellantelescopes,Chile,189Manhattanproject,44boxMars,61tab,81–2,89Maskelyne,Nevil,66,68,71–2mass-to-lightratios,186–7mathematics, 48–9box, 62, 64box, 142, 225–7box, 240–42, 256–7box, 269,274–6,278–andspacetimeconcept,123box,124–5–andStringTheory,265,271–2

Mercury,58,59fig,61tab,125Messier3,CanesVenatici,196–7fig,198–9metamorphicrock,31–2,33meteorites,26fig,32,33,34fig,35,40Michell,ReverendJohn,70,71Mid-AtlanticRidge,12fig,18–23,20boxMilkyWay,1,90–91fig,92,100,102,133fig,147–formationofourSun,7–globularclustersin,196–7fig,198–9–inLocalGroupcluster,185n–oldeststarsin,171,173–asspiralgalaxy,149–S-starsin,74–5–Sun’sorbitofcentreof,158–vastnessof,89–92–seealsoEarth;solarsystem,Earth’s;theSun

‘modalrealism’concept,266nMoon,33,35,40,57,65,88MountWilson,California,92,93figMukhanov,Viatcheslav,229nMultiverse,261fig,262,263–4,267–9,271–2

NASAExtragalacticDatabase,147,150nebulae,7,33,99Neptune,2,58–61,61tab,76,77–80,78–9fig,84,85–7box

neutrinos,4,45box,225box,277–emergingfromtheSun,39,44box–energydensityof,183,184box,193,236–inprimordialplasma,168,169,173,182–3,184box,211n

neutronstars,202neutrons,5,168,169,183–5,276,277–andbetadecay,44box,45box–andBigBangnucleosynthesis,170,172,183–isolated,169,228–andisotopes,25–7,46box–massof,46box,183,193–numberinatom,46box

Newton,Isaac–F=maequation,62–3,64box,114–15,118,185–6–FirstLawofMotion,120–lawsofasuniversal,63–8,74–5,77,258–PrincipiaMathematica(1687),7–8,62–3–SecondLawofMotion,62–3,64box,65,114–15,118,185–6–statueofatTrinityCollege,Cambridge,164–5–UniversalLawofGravity, 62–6,64box,67fig, 77, 114–15, 116, 125, 127,151box,185–6

nitrogen,7,107fig,173NorthAtlanticcraton,31–2nuclearforce–strong,169–70,277–weak,5,44–5box,184box,270,277

nucleosynthesis,BigBang,5,36,169–70,172,176,187,218–20photon-to-protonratio,170,172,183

observableUniverse–all-skyviewof,132,133fig–definitionof,3–4,167–8–edgeoftoday,209n–andinfinitenumberofuniverses,259–60,266n,267–8–andinflationaryMultiverse,261fig,262,263–4,267–8,271–2–sizeofattimeofrecombination,6–assizeofmelonatBigBang,4,260

–sizeofnow,100–weightof,207–9

oceans–ageofrocksonseabed,19fig,20box,21–3–atomiccompositionof,41box–evolutionof,11–24

Ogmore-by-Sea,Wales,50–56,51fig,52fig,258–FairyBuoy,50–53,51fig,52fig,54–5,54fig

oil,47boxOmegaCentauriGlobularCluster,196–7fig,199,201fig,202,203Orion,constellationof,95oxygen,7,10,41box,45–6box,171,172,173,202

paleontologicaldatingmethods,20boxPalomarObservatory,California,104fig,105parallaxmethod,65n,76–83,78–9fig,84–9,85–7box–andbrightnessofstars,94–5,96–9–megaparsecs(Mpc),89parsecs,88

particlephysicsaccelerators,4–5,46box,207,223–4,225box,226box,258particles,elementary,41–8box,276–8–averagemassdensity,145box,181,182–3,184box,193–4,232–6–anddarkmatter,192–3,215–electron-positronannihilation,38fig,45box,184box,230–32,231box–formingoffirstatoms,6,173–4–andHiggsfield,5,5n,223–4,225box,226–7box–interactions and collisions, 5n, 6, 44box, 140, 169–70, 190–92, 220, 221,226box,278

–massesof,5,5n,181,183,226–7box,265,270,271,276,278–inprimordialplasma,3,4–5,6,142–3,166–74,168–9n,211,240,254–7box–producedfromnothingduringinflation,230–32–andQuantumElectrodynamics(QED),174n–releaseofatBigBang,3,4–5,166–8,168–9n,184box,239–StandardModel of particle physics, 5n,44box,225box,227box, 265, 269,270

–andStringTheory,264,265–seealsofundamentalfields;quantumphysics

Peebles,Jim,174–6,177–9Penzias,Arno,176periodictable,41box,42–3fig,46box,270period–luminosityrelation,94,95–6Perthshire,66–8,71–2,73PETscanners,45–6boxphotons,4,45–6box,173,278–andelectromagneticfield,226box,227box,230,277–andelectron-positronannihilation,38fig,45box,184box,230–32,231box–inemissionspectrum,110–13box–energydensityinCMB,181–3,193,214fig,263–andLIGOdetectors,128–30box–photon-to-protonratios,170,172,183–in primordial plasma, 6, 168, 169–70, 172, 173–9, 211, 220–21, 243, 249,256–7box,263

–andtransparentUniverse,6,7,174,214fig–travelinstraightlines,6,174–7,220–21–wavelengthof,109–13box,176,263–zeromassof,181,278–seealsolight;redshift

Pickering,EdwardCharles,94Planck,Max,177planetarymotion,57–63,64box,65,73–4,81,125planets–distancefromEarth,59fig,76–83,78–9fig,84,85–7box,89–distancestoSun,58–61,59fig,80–81–formationof,33–massof,62–3,64box,74–inopposition,60fig,61,77–80,81–2,85box,86box,87box–orbitalperiodsof,60fig,61–3,61tab–orbitsof,57–63,60fig,64box,81,125–planetarymotion,2,57–63,60fig,64box,65,73–4,81,125

plasma,primordial,3,172,173,174,179,211,214fig,220–21,239,240–coolingof,5,6,170,173,184box,214fig–denserregionsof,6–7,211–12,243,244–5,250–53,252fig–densityof,254–7box–electronsin,6,168,169,173,211,220,243,257box

–graphshowingripplesin(Plancksatellite),240–53,241fig,254–7box–neutrinosin,168,169,173,183,184box,211n–photonsin,6,168,211,220–21,249,256–7box,263–photonsreleasedfrom,173–9,220–21,243,249,256–7box,263–planewavesin,254–7box–protonsin,5,168,169–70,172,211,220,250–53,257box–quantum-inducedripplesin,4,5,6,228–51,233fig,234–5fig,238fig,241fig,246–7figs,254–7box,260–62

–sound/densitywavesin,243–53,252fig,254–7box–standingwavesin,255–7box–two-pointcorrelationfunction,248,252fig,253

Playfair,John,72PleiadesGlobularCluster,199,200fig,202,203plutonium-238(alphaemitter),44boxPolkinghorne,ReverendJohn,269positrons(antielectrons),38fig,45–6box,184boxpotassium,25,28Pouillet,Claude,36–9protons,276–7–andatomicnumbers,46box–andaveragemassdensity,144–5box,181,182,183–5,193–4–andbetadecay,44–5box–massof,46box,48–9box,182,183–andnuclearfusion,38fig,169–70,183–5,277–photon-to-protonratios,170,172,183–inprimordialplasma,5,168,169–70,172,211,220,250–53,257box

ProximaCentauri,88,129boxPtolemy,57Pythagoras’Theorem,52fig,87box

QuantumElectrodynamics(QED),174nquantumphysics,24,41box,110–11box,177,229–alphadecay,44box–betadecay,44–5box–andelectrons,41box,44box,48–9box,110–11box–andfundamentalfields,226box,227box,229–32–andgravitationaleffects,226box

–‘Higgsfieldcondensation’,227box–randomnessasdefiningfeature,45box–ripplesininflatonfield,4,5,6,228–51,234–5fig,238fig,241fig,246–7figs,254–7box,260–62

–StandardModel of particle physics, 5n,44box,225box,227box, 265, 269,270

–andStringTheory,264–5quantumtunnelling,44boxquarks,4,5,5n,44box,277,278–quarkfields,225box,227box–topquarkparticle,228

quasars,14,15box,107fig,108,171‘quintessencefield’,224

radiotelescopes,14,15box,15figradioactivedecay,24–andhalf-life,24–35,26fig,29fig,40,44–5box,45fig–randomnessof,45box

radiometricdatingmethods,20box,24–35,26fig,29fig,40Rayleigh,Lord,47boxreddwarfstars,202redgiantstars,199,200–201figs,202redshift,104fig,105–8,131,133fig,146–9,150–63,151box,152tab,158–9figs,176,263–distancerelationship(1998results),204–5,206–Hubbledata,156–60,158–9figs,163,204

Richer,Jean,81–2,89Rigel(star),95rivervalleys,13RMC136a1(star),95Robertson,HowardPercy,142–3,146,165,166,180,206,231boxRoll,Peter,174–6,177–9rubidium-87(87Rb),25–32,26fig,29figRubin,Vera,189

Sagan,Carl,89Salam,Abdus,226box

satellitelaserranging,15boxSaturn,58,61tab‘scalarfield’concept,223–8,225box,226–7box,229–32scalefactorofUniverse,141,144–5box,180,222–3–variationwithtime,206–7,208fig

Schiehallion(Perthshiremountain),66–8,71–2,73Schmidt,Brian,206Schwarz,John,264Schwinger,Julian,174n,230science–accumulationofevidence,11,22–4–ageofloneexperimentalscientist,82–4,164–5,258–assumptions, guesses and estimates, 13–14, 18–21, 56, 100, 132, 149–53,151box,163,165–6,169

–awarenessofuncertainty,53–4,55–blue-skiesresearch,46box–asacollectiveendeavour,122,165,258–9–andCreatorconcept,268–72–anddelightinbeingwrong,147–determining of distance, 58–61, 59–60figs, 65, 76–89, 78–9figs, 85–7box,94–102,149–62,158–9figs

–determining the age of things, 11–23, 19–20box, 24–40, 26fig, 29fig,100–101,198–207

–determiningthesizeofthings,47–8box,50–56,100–101–determining the weight/density of things, 56–7, 65–75, 69fig, 181–9,193–209

–everydayapplications,11,24,44–6box–‘finetuning’problems,194–incrementalapproach,11–interconnectionofdisciplines,23–4,57–logicalconsistency,22–4,157,173–‘order-of-magnitude’estimates,56–romanceandwonder,272–3

Scotland,31,66–8,71–2,73Seymour,Bob,54–5,71,73Seymour,Mike,50–56,51fig,52fig,71,73,258silicon,41box

silver,7,172Sirius,95deSitter,Willem,231box61Cygni(star),84–8Slavecraton,northwestCanada,32SloanDigitalSkySurvey(SDSS),210fig,215,250–53SmallMagellanicCloud,94,96Smallwood,John,73smokedetectors,44boxsolarconstant,39solareclipses,102–5,103fig,111box,113box,125solarsystem,Earth’s,2,61tab,64box,76,124–5,132,167–mass-to-lightratio,187–planetaryformation,33–seealsoEarth;planets;theSun

soundwavesinprimordialplasma,243–53,252fig,254–7boxSouthAmerica,13–18spacetime,121–4–curvatureof,121–5,127–31,135–9–EinsteinFieldEquations,124–5,131,135,136,139–41,142–3,179–asfour-dimensional.,123box,135–6–geometryof,123box,131–43,137fig,138fig–andgravitationalwaves,127,259–isotropicandhomogenousmatterdistribution,132–5,136,139,144box–andmathematics,123box,124–5–andStringTheory,265–threepossiblegeometriesof,135–6,137fig,141,142,145box,182,193–8,204,205

sphericalgeometryUniverse,135,136,137fig,141,142,145box,182‘standardcandles’,96,101–2StandardModelofparticlephysics,5n,44box,225box,227box,265,269,270Starobinsky,Alexei,229stars–brightnessof,94–9,101–2–Cepheidvariable,92–4,93fig,96–101,99n,105,149,150,151box–anddarkmatter,74–dead,7,10,170–71,173,186,187

–distancesto,58–61,76,84–9,94–6,129box–earliest,6–7,171,172–formationof,4,6–7,35,98fig,172,212–globularclusters,196–7fig,198–9,200–201figs,203–4–andgravity,124,135,151box,202,211–heliumburningin,202–Herzsprung-Russell(HR)diagrams,199–203,200–201figs–lifespansof,7,10,172–3,199–204–‘lithium-plateau’,171–massive,7,10,172–3,202–neutronstars,202–nuclearfusionin,35–40,37fig,38fig,170–71,172,199–202,226box,270–pollutionfromexplosionsof,170–71–reddwarves,202–redgiants,199,200–201figs,202–spectraoflightfrom,102,105–8,107fig,111–13box,146–7,149n,151box–speedof,151box,189–S-stars,74–5–‘standardcandles’,96,101–2–stellarevolution,199–204,200–201figs–supergiants,95–useofinparallaxmethod,77–80,78–9fig,85–6box–whitedwarves,92,202–3–Wolf-Rayet,95–seealsogalaxies;theSun

staticUniverseconcept,139–40,141,142,143,165,199Steinhardt,Paul,229nStringLandscapeconcept,266–8StringTheory,264–8,271–2strongnuclearforce,169–70,277strontium-86,25–32,26fig,29figstrontium-87,25–32,26fig,29figsulphur,106,107figtheSun–ageof,10,35–40,101,203–atomiccompositionof,35–40,37fig,38fig,–41box,170,173–formationof,7,35

–gravitationalforceof,62–3–massof,74,187–mass-to-lightratio,186–andnuclearfusion,35–40,37fig,38fig,199–202,277–planets’distancefrom,58–61,59fig,80–81–solarabsorptionspectrum,111–13box–solarconstant,39–solaremissionspectrum,102–5,103fig,111–13box–totalenergyoutput,36–40,38fig

SupernovaCosmologyProject,204–5supernovae,7,149,172–3,270–Type1A,99n,101–2,204–5

Susskind,Leonard,266

Tegmark,Max,OurMathematicalUniverse(2014),266nthermalequilibrium,220thorium,32,44boxTieschitz,chondritemeteorite,26fig,33Tomonaga,Sin-Itiro,174n,230transparentUniverse,6,7,174,214figTrinityHouse,55Tully-Fishermethod,149–50,151box,152tab,157,185‘2MASS’measurements,150n

Universe,inventoryof,180–94,198–209Universe,origins/history/evolution,1,3–8,259–coolingofprimordialplasma,5,6,170,173,184box,214fig–andCreatorconcept,268–72–earlysecondsof,5,168–9–first380,000years,6,174,211,221,241fig,242,243–formingoffirstatoms,6,173–4–fusioninearlyminutesof,5,35,169–72,183–5–galaxyformation,6,7,193,204,212,250–53–highdensity/hotphase,1,3,166–7,168–73,176,179,207,211,232,234fig,236–7,259,260

–originofheavierelements,172–3,202,270–starformation,4,6–7,35,98fig,172,212

–timeofrecombination,6,174,176,212,221,242,–250–53,255–6box–seealsoBigBang;inflation,theoryof;inflatonfield;plasma,primordial

uranium,32,44boxUranus,61tab,77UrsaMajor,97–8figs,99Ussher,BishopJames,22–3

vacuumenergynotion,205,227boxvectorfields,225boxVenus,58,59fig,61tab,81–transitof,82–4

VeryLongBaselineInterferometry,15boxVirgo,constellation/cluster,105,132,134figvolcanicactivity,18–21

WandZbosons,5n,278water,47–8boxweaknuclearforce,5,44–5box,184box,270,277Wegener,Alfred,13weight,63nWeinberg,Steven,226boxwhitedwarfstars,92,202–3WhiteMountains,California,10Wigner,Eugene,269nWilkinson,David,176Wilson,Robert,176Wollaston,ReverendFrancisJohnHyde,70Wollaston,WilliamHyde,113figWordsworth,William,164–5

zircon,32Zwicky,Fritz,185–6,190

1Forabriefresuméontheknownelementaryparticles,seetheAppendix.2Thisisthe27kmcircularundergroundtunnelontheFranco-SwissbordernearGeneva,inwhichprotonstravellingwithinawhiskerofthespeedoflightaremadetocollide.ThedebrisfromthosecollisionstellsushowparticlesinteractatenergiesrelevantforstudiesintotheBigBang.3 Specifically, in the StandardModel of particle physics, theHiggs field givesmass to the quarks, theelectricallychargedleptons(theelectronisoneofthese)andthecarriersoftheweakforce(theWandZbosons).WithouttheHiggsfield,theseparticleswouldhavezeromassandziparoundatthespeedoflight.1Thisisdonebyobservinghowthesolarspectrum,whichwewillmeetinthenextchapter,changesovertimeperiodsofseveralminutes.2MeVisaconvenientunitofenergyused inatomicandnuclearphysics.Foraprimeronunits, see theAppendix.3Moreprecisecalculationsleadtoanageofaround4.6billionyears.1NeutrinosareproducedinvastnumbersintheSun,butmostofthempassthroughordinarymatterasifitdoesnotexist.Sincetheyaresoelusive,ittookuntil1956beforetheywerefinallydetected.Todaythereareseveralneutrinolaboratoriesaroundtheworld.1Thisisbothgrammaticallyambiguousand,accordingtoEinstein,physicallyill-defined.2 If thebaseof thebuoywasexactlyon thehorizon then,usinga littlegeometry, thehorizonwouldbedippedbelowthehorizontallevelofMike’seyesbyanangleof4km/5000km=8×10-4radians.BecauseMike’seyesarenotperfect,theactualdiptothehorizoncouldbeassmallas3×10-4radiansorasbigas13×10-4 radians.Now,abitmoregeometry tellsus that the radiusof theEarth,R, is related to thedipanglebyR=2h/(dipangle)2.PuttinginthetwoextremedipanglesgivesRintherange2000kmto36,000km.3 Trinity House, which dates back to the reign of Henry VIII, is responsible for the provision andmaintenanceoflighthousesandbuoysinEngland,Wales,theChannelIslandsandGibraltar.4Strictlyspeaking,wewanttodeterminethemassoftheEarth.Weightisameasureofhowmuchamassispulledundergravity.YourweightwouldbedifferentonEarthorontheMoon,eventhoughyourmassisthesame.5Historically the distance to theMoonwas obtained by the parallaxmethoddescribed in the followingchapter,buttodayit’sdonetosub-millimetreprecisionbybouncinglaserlightoffmirrorsleftontheMoonbytheApolloastronauts.6ThisanglecanbeobtainedusingtheformulainthefigurecaptionifweanticipatethattheEarth’smassis6 × 1024 kg. Of course the logic is the other way around, i.e. we are proposing to use the measureddeflectiontoinfertheEarth’smass.7Theforcecanbededucedbymeasuringthewire’sresistancetobeingtwisted.Thisisdonebytimingtheback-and-forthoscillationsofthebeamthatoccurwhenitistwistedwithoutthepresenceoftheheavyballs.1Thisisthe‘distancebetweentheeyes’,butwearegettingabitmoreprofessionalwithourlanguage.2Ifyouhavebeenfollowingthedetailsyoucanworkthisoutforyourself:thetangentof0.314arcseconds

isequaltotheratiooftheEarth–Sundistancetothedistancetothestar.Sincetheangleisverysmall,thetangentisapproximatelyequaltotheangleexpressedinradians.Thismeansthedistanceisequalto1AUdividedby0.3143600×π180.3Astar1parsecawayisinfact149.6×106(13600×π/180)km=3.1×1013kmaway.Lighttravelsat3×108m/s,sothisdistanceisalsoequalto3.26lightyears.4ItisnowknowntobeasatellitegalaxyoftheMilkyWay,some199,000lightyearsfromEarth.5Thisisbecausethebrightnessisameasureofthelightpower,andthisfallsawaywiththesquareofthedistancefromthesource.Forexample,thepowerincidentona1cm2lightdetectorplaced1metrefroma10-wattlightsourcewillequal10watts×1cm2/(4π×1m2).Thisisjustastatementofthefactthat(toagoodapproximation)thelightbulbemitslightequallyineverydirectionandthedetectorreceivesitsshareofthetotal.6Itwasrealizedinthe1940sthattherearetwomainclassesofCepheid,withdifferentperiod–luminosityrelationships.Notknowingthisledtosomeconfusionintheearlydays.7AType1AsupernovawasobservedinM82inJanuary2014,whichmakesforanaccuratedeterminationofhowfaritisaway(seebelow).8ItdoesignoretheexpansionassociatedwiththeBigBang,ofwhichtherewillbemoreinthenexttwochapters.9Wecomputedthis40%figureinourearlierbook,TheQuantumUniverse.1Althoughthislookslikeonlyoneequation,theGreeksubscriptscanvary(theylabelprojectionsinspaceandtime),givingrisetoseveraldistinctequations.1Thepercentageerroronthemeanfallsas1/√NwhereNisthenumberofmeasurements.2WewillgetseriouslyinterestedinthedimplesinChapter8.3Although‘expandingUniverse’isthecommonparlance,‘stretchingUniverse’isprobablybetter.1Ifyouknowsomeelementarycalculusthenit’seasierjusttosaythatH= /awhere =da/dtandthenweobtainaasafunctionofthetimetbysquarerootingeachsideoftheequationandintegrating.2Oranymattermovingatspeedsclosetothespeedoflight,infact.1Recall that the spectral lines emittedby agalaxycorrespond to thebarcodepatternof light they emit.Eachlinecorrespondstoaparticularatomictransitionthat leads to theemissionorabsorptionof lightofoneparticularwavelength.Eachspectral line isstronglypeakedat thatparticularwavelength,but itdoesstillhavea‘width’,whichmeansthereisasmallspreadofwavelengthsaboutthepeakvalue.Youcanseethissmallspreadin thespectraweshowedin theplotsat theendofChapter4(thespikesarenotsuper-sharp).Thiswidthcontainsimportantinformation,asTullyandFisherrealized.2Itisnotentirelyobvioushowtomeasurethesizeofagalaxy,becauseyouhavetodecidepreciselyhowtheedgeofthegalaxyisdefined,andwhichwavelengthsoflightareusedtomakethemeasurement.Weusetheso-called‘2MASS’measurements,whicharemadeinthenear-infraredpartofthespectrum.Forourpurposes,itdoesn’tmatterwhichmethodwechoose,aslongasweareconsistentandusethesamemethodforeverygalaxy.Formoredetails,seehttp://iopscience.iop.org/article/10.1086/345794/pdf.

3Don’tbeconcerned that thequoteduncertaintyfor thedistance toNGC0011 isonly4Mpc;a10Mpcerror is only2.5 times theuncertainty,which isnot sounlikely that it shouldworryus (especially sincethereisalsosomeuncertaintyonthebest-fitline).Thereisawell-establishedprocedureforunderstandinghowtointerprettheuncertaintiesonmeasurements,butwedonotproposetogointoithere.4Infact,ifwewentouttohigherredshifts(around1.0andbigger),theequationeventuallyfailsbecausethepointsno longer fall in thevicinityofa straight line.Even then,astronomerscanstilluse redshift toascertaindistance.Itwouldtakeustoofarofftracktodiscusshereexactlyhowtheydothat–sufficetosaythat ifweknow theaveragematerialcompositionof theUniverse,whichwewillby theendof thenextchapter,theFriedmannequationallowsustotellhowthescalefactoroftheUniversehasvariedwithtime.Thisinformationissufficienttodeterminethedistancetoanyobjectwhoseredshiftisknown.5Allofthesixteenspiralgalaxiesweselectedcanbeobservedandmeasuredbyakeenamateur.6 By this we mean we can know of their existence by using some clever measuring device and ourintellects.Eyesareonlyoneofarangeof‘seeing’devicesavailabletoscientists.SothedarkmattercountsasbeingpartoftheobservableUniverse.7TofigurethisoutweneedtosolvetheFriedmannequationforthescalefactorandusemeasurementslikethosewedescribeinthenextchaptertoquantifyhowmanyparticlesofeachtypethereareperunitvolumeat thepresent time.SayingtheUniverse……is1secondoldreallymeansthat1secondearlierallof theparticleswouldbedirectlyontopofeachotherunlesssomenewphysicsstepsintochangethings.AswewillseeintheChapter8,perhapssomethingdidstepintochangethings,butattimesearlierthanweareconsideringinthischapter.8Thetheoreticalpredictionisactuallyalittlehighforlithium,whencomparedtothedata.Butitisnotaneasythingtomeasureandthereisquitealargeuncertaintyonthephysics,soitisn’tregardedastooseriousaproblem.9PreciselyhowphotonsinteractwithchargedparticlesisthestuffofQuantumElectrodynamics(QED),thetheorydevelopedbyRichardFeynman,JulianSchwingerandSin-ItiroTomonagainthelate1940s.QEDisexploredinourbookTheQuantumUniverse.10Asforthecaseofnucleosynthesis,tocalculatethistimeweneedtosolvetheFriedmannequation.Wealso need to use simple atomic physics to determine the temperature atwhich electrons tend to stick tonucleiandmakeatoms.11 For a fascinating but technical review see P. J. E. Peebles, ‘Discovery of the Hot Big Bang:WhatHappenedin1948’,EuropeanPhysicalJournal,H39(2014),pp.205–23.1TheCMBphotonenergydensityismuchlargerthanthatforphotonsofnon-CMBorigin,suchasthosefromstars.2ReferringbacktoBox10(pp.144–5),thephotonmassdensitycontributestothequantitydenotedbyρontheright-handsideoftheFriedmannequation.3 You can see how the density of the Universe controls its geometry by studying Box 10 again; forexample,ifρisbiggerthan3/(8πG)H2thenKmustbeapositivenumber,whichcorrespondstoaspherical

geometry.Youcanputthenumbersintodeterminethenumericalvalueofthiscriticaldensityandcheckthatitisthevaluequotedinthetext.4Weneedtheword‘ordinary’because,aswewillshortlysee, there’smore to theUniverse thanatoms,electrons,photonsandneutrinos.1Youmight recall fromChapter 2 that electron-positron annihilationwas responsible for 1.02MeV ofenergyinthefusionchainthatleadstoheliumproductioninsidetheSun.5TheMilkyWayisamemberofadifferentgalaxycluster,knownastheLocalGroup.Itcontainsaround50smallgalaxiesplus2bigspiralgalaxies(theMilkyWayandAndromeda).6Thiscorrespondsto5100kg/watt.7Youmayhavespottedthattherearenotmanygalaxiesintheregionofthehotgas(thegalaxiesareintheblueregionsinfact).Donotbeconfused,though:itisverywellknowntoastronomersthatthebulkoftheordinarymass(i.e.excludingdarkmatter)intheUniverseisingasandnotinstars.ThebrightnessoftheX-raysallowsastronomerstodeterminehowmuchmassresidesinthegas(moregasmeansmoreX-rays),andthis isconsiderablymore thanresides in thegalaxiesofstars.Thegalaxies inFigure7.1 followthedarkmatterbecause,unlikethegas,theyareunlikelytointeractwitheachothermuch.ThegalaxiesarenottheimportantpartsoftheBulletClusterphoto.8Youcan’tspotthisjustbylookingattheequation;youneedtosolveit.Roughlyspeaking,astimepasseswe become increasingly aware of any curvature to space (i.e. the density increasingly deviates from thecriticaldensity)–soifwecannotdiscernanycurvaturenowthenwemostcertainlycouldn’tdiscernanyinthepast(i.e.thedensitywasclosertocriticalinthepast).9TheHRdiagramisnamedafterEjnarHerzsprungandHenryNorrisRussell,whofirstpresentedstellardatainthisway,independentlyfromeachother,in1911.10Recallthisisthenumberthatsayshowmuchspaceisstretchedorshrunkrelativetoitssizetoday.11Thefactthat47billionlightyearsisbiggerthanthetimesincetheBigBangmultipliedbythespeedoflight (= 14 billion light years) is because the 47 billion years refers to how far away the edge of theobservableUniverseisnow.Inthepast,thisdistancewassmallerowingtothefactthatspaceexpands.12WeworkedoutthevolumeoftheobservableUniverseusingtheformula forthevolumeofasphereofradiusr,andweworkedoutthetotalmassbymultiplyingthisvolumebythemassdensity.1Darkmatterandneutrinoswerealsopresentatthistime,buttheydidnotinteractmuchwiththeparticlesintheplasma.2ThecosmologicalconstantplayedaveryminorroleearlyonintheUniverse’sexpansion.3Thehorizonandflatnessproblemsareboth‘justsolved’iftheUniversegrewduringitsperiodofinflationbyatleastasmuchasithasgrownsincetheinflationended.Thisisn’tobviouslythecaseandtoshowitwouldbetoomuchofadetourforthisbook.ItmeanstheUniverseinflatedattheveryleastbyafactorofaround500million,whichistheamounttheUniversehasexpandedfromthetimeofnucleosynthesisuntiltoday. In practice, most theorists expect that it inflated for vastly more than this–a factor of 1026 iscommonlyregardedastheminimumamountofinflation.

1AverybriefintroductiontotheparticlesoftheStandardModelcanbefoundintheAppendixattheendofthebook.4 The pioneering work was done by Andrei Linde and, independently, Andreas Albrecht and PaulSteinhardt,thenoftheUniversityofPennsylvaniaintheUSA.5Infact,thefirstcalculationsofstructureformationweremadebeforetheCambridgemeeting,in1981,byViatcheslavMukhanovandGennadyChibisov,alsooftheLebedevInstitute.6WeshowhowtheUncertaintyPrinciplecomesabout,startingfromthelawsofquantumphysics,inourbookTheQuantumUniverse.7ThisissimilartotheproductionofHawkingradiationfromablackhole.8Becausethequantum-inducedrippleshaveasizebiggerthanthedeSitterhorizon.9 To be precise, the CMB cannot be used to extract unique values for all of the key cosmologicalparameters. In other words, there are other combinations of parameters that also agreewith the data inFigure8.9.The remarkable thing is thatagreementwith thedata is foundusing the sameparameterswefoundinthelastchapter.10 The ‘stones thrown in water’ analogy has its limitations, because really we should think of theperturbationsgeneratedbyinflationmoreasapressingdownorliftingupthesurfaceofthewaterandthenreleasing it from rest. The stones analogy is helpful, however, in distinguishing between the twopossibilities that theperturbationsweregenerated‘allatonceandeverywhere’or that theyaregeneratedprogressivelyovertime.Theformercasecorrespondstothestonesallenteringthewateratthesametimeandleadstowavesofequalradii.11 For the more mathematically inclined, Figure 8.9 is closely related to the Fourier transform of thecorrelationfunction.1 Because the Universe is expanding all the time, we need to be a little careful in defining times anddistances.Ifwemeasuredistances(likethewavelengthλ)usingrulersthatstretchwiththeexpansionoftheUniverse (the grid lines onFigure 8.6 could be used tomeasure distances in thisway), then the timeTshouldbewhat cosmologists call the conformal time,which is thedistance anunhinderedbeamof lightwouldtravelinthatintervaloftimedividedbythespeedoflight.1 The possibility of strings is a natural extension of the idea of a particle. While particles are zero-dimensionalobjects,i.e.pointsinspace,stringsareone-dimensionalobjects,i.e.linesinspace.2Morerecently,theideathat‘theworldwearepartofisbutoneofapluralityofworlds’isakeyideainthe‘modalrealism’advocatedbythelateDavidLewisinhis1986bookOnthePluralityofWorlds.MaxTegmark’s bookOur Mathematical Universe (2014) provides a thought-provoking introduction to howmodernphysicsleadstowhathereferstoasthefourlevelsoftheMultiverse.3 This is the title of theoretical physicist Eugene Wigner’s 1960 essay that ends, ‘The miracle of theappropriatenessof the languageofmathematicsfor theformulationof the lawsofphysics isawonderfulgift which we neither understand nor deserve.’ It echoes Einstein’s quote: ‘Themost incomprehensiblethingabouttheuniverseisthatitiscomprehensible.’

4InOneWorld:TheInteractionofScienceandTheology(2007).