MASSACHUSETTSINSTITUTEOFTECHNOLOGY

PhysicsDepartment

Physics8.286:TheEarlyUniverse

September28,2016

Prof.AlanGuthR

EVIEW

PROBLEMSFOR

QUIZ1

QUIZDATE:Wednesday,October5,2016,duringthenormalclasstime.

QUIZCOVERAGE:LectureNotes1,2,and3;ProblemSets1,2,and3;Wein-

berg,Chapters1,2,and3;Ryden,Chapters1,2,and3.(WhileallofRyden's

Chapter3hasbeenassigned,questionsonthequizwillbelimitedtoSection

3.1.ThematerialinSections3.2and3.3willbediscussedinlecturelaterin

thecourse,andyouwillnotberesponsibleforituntilthen.Section3.4(for

the�=0case)mayhelpyouunderstandthecosmologicalDopplershift,also

discussedinLectureNotes2,buttherewillbenoquestionsspeci�callyfocused

onRyden'sdiscussion.)Oneoftheproblemsonthequizwillbetaken

verbatim

(oratleastalmostverbatim)from

eitherthehomework

assignments,orfrom

thestarredproblemsfrom

thissetofReview

Problems.ThestarredproblemsaretheonesthatIrecommendthatyou

reviewmostcarefully:Problems2,4,7,12,15,17,19,and22.Thestarred

problemsdonotincludeanyreadingquestions,butpartsofthereadingques-

tionsintheseReviewProblemsmayalsorecurontheupcomingquiz.

PURPOSE:Thesereviewproblemsarenottobehandedin,butarebeingmade

availabletohelpyoustudy.Theycomemainlyfromquizzesinpreviousyears.

Exceptforafewpartswhichareclearlymarked,theyareallproblemsthatI

wouldconsiderfairforthecomingquiz.Insomecasesthenumberofpoints

assignedtotheproblemonthequizislisted|

inallsuchcasesitisbasedon

100pointsforthefullquiz.

Inadditiontothissetofproblems,youwill�ndonthecoursewebpage

theactualquizzesthatweregivenin1994,1996,1998,2000,2002,2004,2005,

2007,2009,2011,and2013.Therelevantproblemsfromthosequizzeshave

mostlybeenincorporatedintothesereviewproblems,butyoustillmaybe

interestedinlookingattheoriginalquizzes,justtoseehowmuchmaterialhas

beenincludedineachquiz.Sincethescheduleandthenumberofquizzeshas

variedovertheyears,thecoverageofthisquizwillnotnecessarilybethesame

asQuiz1from

allpreviousyears.Infact,however,the�rstquizthisyear

coversessentiallythesamematerialasthe�rstquizineither2009,2011,or

2013.

REVIEW

SESSION:Tohelpyoustudyforthequiz,therewillbeareview

session.Detailstofollow.

FUTUREQUIZZES:TheotherquizdatesthistermwillbeWedneday,Novem-

ber9,andWednesday,December7,2016.

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INFORMATION

TO

BEGIVEN

ON

QUIZ:

Eachquizinthiscoursewillhaveasectionof\usefulinformation"attheback

ofthequiz.Forthe�rstquiz,thisusefulinformationwillbethefollowing:

DOPPLER

SHIFT(Formotionalongaline):

z=v=u

(nonrelativistic,sourcemoving)

z=

v=u

1�v=u

(nonrelativistic,observermoving)

z= s1+�

1���1

(specialrelativity,with�=v=c)

COSMOLOGICALREDSHIFT:

1+z��observed

�emitted

=a(tobserved )

a(temitted )

SPECIALRELATIVITY:

TimeDilationFactor:

�

1

p1��2

;

��v=c

Lorentz-FitzgeraldContractionFactor:

RelativityofSimultaneity:

Trailingclockreadslaterbyanamount�`0 =c.

KINEMATICSOF

A

HOMOGENEOUSLY

EXPANDING

UNIVERSE:

Hubble'sLaw:v=Hr,

wherev=recessionvelocityofadistantobject,H=Hubble

expansionrate,andr=distancetothedistantobject.

PresentValueofHubbleExpansionRate(Planck2015):

H0=67:7�0:5km-s �1-Mpc �1

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ScaleFactor:`p (t)=a(t)`c;

where`p (t)isthephysicaldistancebetweenanytwoobjects,

a(t)isthescalefactor,and`c

isthecoordinatedistance

betweentheobjects,alsocalledthecomovingdistance.

HubbleExpansionRate:H(t)=

1a(t)

da(t)

dt

.

LightRaysinComovingCoordinates:

Lightraystravelin

straightlineswithspeeddxd

t=

ca(t).

EVOLUTION

OFA

MATTER-DOMINATED

UNIVERSE:

H2= �_aa �2

=8�3

G��kc2

a2

;

�a=�4�3

G�a;

�(t)=a3(t

i )

a3(t)�(ti )

��=�c;where�c=3H2

8�G

:

Flat(k=0):a(t)/t2=3;=1

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PROBLEM

LIST

1.DidYouDotheReading(2007)?

............

5(Sol:23)

*2.TheSteady-StateUniverseTheory............

6(Sol:25)

3.DidYouDoTheReading?

...............

7(Sol:27)

*4.AnExponentiallyExpandingUniverse

..........

8(Sol:29)

5.DidYouDoTheReading?

...............

9(Sol:30)

6.AFlatUniverseWithUnusualTimeEvolution

......

10(Sol:31)

*7.AnotherFlatUniverseWithAnUnusualTimeEvolution

..

11(Sol:32)

8.DidYouDoTheReading?

...............

12(Sol:36)

9.AFlatUniverseWitha(t)/t3=5

............

13(Sol:37)

10.DidYouDoTheReading?

...............

14(Sol:41)

11.AnotherFlatUniverseWitha(t)/t3=5

..........

14(Sol:43)

*12.TheDecelerationParameter...............

15(Sol:47)

13.ARadiation-DominatedFlatUniverse

..........

16(Sol:47)

14.DidYouDoTheReading?

...............

16(Sol:48)

*15.SpecialRelativityDopplerShift.............

17(Sol:49)

16.DidYouDoTheReading?

...............

17(Sol:51)

*17.TracingALightPulseThroughARadiation-DominatedUniverse18(Sol:53)

18.TransverseDopplerShifts................

19(Sol:54)

*19.ATwo-LevelHigh-SpeedMerry-Go-Round

........

20(Sol:56)

20.SignalPropagationInAFlatMatter-DominatedUniverse

.

21(Sol:59)

21.DidYouDoTheReading?

...............

22(Sol:65)

*22.TheTrajectoryOfAPhotonOriginatingAtTheHorizon..

22(Sol:67)

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PROBLEM

1:DID

YOU

DO

THEREADING

(2007)?(35points)

ThefollowingproblemwasProblem1,Quiz1,2000.Thepartswereeachworth5

points.

a)TheDopplere�ectforbothsoundandlightwavesisnamedforJohannChris-

tianDoppler,aprofessorofmathematicsattheRealschuleinPrague.He

predictedthee�ectforbothtypesofwavesinxx42.Whatarethetwodigits

xx?

b)Whentheskyisveryclear(asitalmostneverisinBoston),onecanseeaband

oflightacrossthenightskythathasbeenknownsinceancienttimesasthe

MilkyWay.Explaininasentenceortwohowthisbandoflightisrelatedto

theshapeofthegalaxyinwhichwelive,whichisalsocalledtheMilkyWay.

c)Thestatementthatthedistantgalaxiesareonaveragerecedingfromuswith

aspeedproportionaltotheirdistancewas�rstpublishedbyEdwinHubblein

1929,andhasbecomeknownasHubble'slaw.WasHubble'soriginalpaper

basedonthestudyof2,18,180,or1,800galaxies?

d)Thefollowingdiagram,labeledHomogeneityandtheHubbleLaw,wasusedby

WeinbergtoexplainhowHubble'slawisconsistentwiththehomogeneityof

theuniverse:

Thearrowsandlabelsfromthe\VelocitiesseenbyB"andthe\Velocitiesseen

byC"rowshavebeendeletedfromthiscopyofthe�gure,anditisyourjob

tosketchthe�gureinyourexambookwiththesearrowsandlabelsincluded.

(Actually,inWeinberg'sdiagramthesearrowswerenotlabeled,butthelabels

arerequiredheresothatthegraderdoesnothavetojudgethepreciselength

ofhand-drawnarrows.)

e)Thehorizonisthepresentdistanceofthemostdistantobjectsfrom

which

lighthashadtimetoreachussincethebeginningoftheuniverse.Thehorizon

changeswithtime,butofcoursesodoesthesizeoftheuniverseasawhole.

Duringatimeintervalinwhichthelinearsizeoftheuniversegrowsby1%,

doesthehorizondistance

(i)growbymorethan1%,or

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(ii)growbylessthan1%,or

(iii)growbythesame1%?

f)Namethetwomenwhoin1964discoveredthecosmicbackgroundradiation.

WithwhatinstitutionweretheyaÆliated?

g)Atatemperatureof3000K,thenucleiandelectronsthat�lledtheuniverse

combinedtoformneutralatoms,whichinteractveryweaklywiththephotons

ofthebackgroundradiation.Afterthisprocess,knownas\recombination,"the

backgroundradiationexpandedfreely.Sincerecombination,howhaveeachof

thefollowingquantitiesvariedasthesizeoftheuniversehaschanged?(Your

answersshouldresemblestatementssuchas\proportionaltothesizeofthe

universe,"or\inverselyproportionaltothesquareofthesizeoftheuniverse".

Theword\size"willbeinterpretedtomeanlinearsize,notvolume.)

(i)theaveragedistancebetweenphotons

(ii)thetypicalwavelengthoftheradiation

(iii)thenumberdensityofphotonsintheradiation

(iv)theenergydensityoftheradiation

(v)thetemperatureoftheradiation

�

PROBLEM

2:

THE

STEADY-STATE

UNIVERSE

THEORY

(25

points)

ThefollowingproblemwasProblem2,Quiz1,2000.

Thesteady-statetheoryoftheuniversewasproposedinthelate1940sbyHer-

mannBondi,ThomasGold,andFredHoyle,andwasconsideredaviablemodelfor

theuniverseuntilthecosmicbackgroundradiationwasdiscoveredanditsproperties

werecon�rmed.Asthenamesuggests,thistheoryisbasedonthehypothesisthat

thelarge-scalepropertiesoftheuniversedonotchangewithtime.Theexpansion

oftheuniversewasanestablishedfactwhenthesteady-statetheorywasinvented,

butthesteady-statetheoryreconcilestheexpansionwithasteady-statedensityof

matterbyproposingthatnewmatteriscreatedastheuniverseexpands,sothat

thematterdensitydoesnotfall.Liketheconventionaltheory,thesteady-statethe-

orydescribesahomogeneous,isotropic,expandinguniverse,sothesamecomoving

coordinateformulationcanbeused.

a)(10points)Thesteady-statetheoryproposesthattheHubbleconstant,like

othercosmologicalparameters,doesnotchangewithtime,soH(t)=H0 .Find

themostgeneralformforthescalefactorfunctiona(t)whichisconsistentwith

thishypothesis.

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b)(15points)Supposethatthemassdensityoftheuniverseis�0 ,whichofcourse

doesnotchangewithtime.Intermsofthegeneralformfora(t)thatyoufound

inpart(a),calculatetherateatwhichnewmattermustbecreatedfor�0

to

remainconstantastheuniverseexpands.Youranswershouldhavetheunitsof

massperunitvolumeperunittime.[Ifyoufailedtoanswerpart(a),youwill

stillreceivefullcredithereifyoucorrectlyanswerthequestionforanarbitrary

scalefactorfunctiona(t).]

PROBLEM

3:DID

YOU

DO

THEREADING?(25points)

ThefollowingproblemwasProblem1onQuiz1,2007,whereeachofthe5questions

wasworth5points:

(a)Inthe1940's,threeastrophysicistsproposeda\steadystate"theoryofcos-

mology,inwhichtheuniversehasalwayslookedaboutthesameasitdoes

now.Statethelastnameofatleastoneoftheseauthors.(Bonuspoints:you

canearn1pointeachfornamingtheothertwoauthors,andhenceupto2

additionalpoints,but1pointwillbetakeno�foreachincorrectanswer.)

(b)In1917,aDutchastronomernamedWillem

deSitterdidwhichoneofthe

followingaccomplishments:

(i)measuredthesizeoftheMilkyWaygalaxy,�ndingittobeaboutone

billionlight-yearsindiameter.

(ii)resolvedCepheidvariablestarsinAndromedaandtherebyobtainedper-

suasiveevidencethatAndromedaisnotwithinourowngalaxy,butis

apparentlyanothergalaxylikeourown.

(iii)publishedacatalog,NebulaeandStarClusters,listing103objectsthat

astronomersshouldavoidwhenlookingforcomets.

(iv)publishedamodelfortheuniverse,basedongeneralrelativity,which

appearedtobestaticbutwhichproducedaredshiftproportionaltothe

distance.

(v)discoveredthattheorbitalperiodsoftheplanetsareproportionaltothe

3/2powerofthesemi-majoraxisoftheirellipticalorbits.

(c)In1964{65,ArnoA.PenziasandRobertW.Wilsonobserveda uxofmi-

crowaveradiationcomingfromalldirectionsinthesky,whichwasinterpreted

byagroupofphysicistsataneighboringinstitutionasthecosmicbackground

radiationleftoverfromthebigbang.Circlethetwoitemsonthefollowinglist

thatwerenotpartofthestorybehindthisspectaculardiscovery:

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(i)BellTelephoneLaboratory

(ii)MIT

(iii)PrincetonUniversity

(iv)pigeons

(v)groundhogs

(vi)Hubble'sconstant

(vii)liquidhelium

(viii)7.35cm

(Grading:3ptsfor1correctanswer,5for2correctanswers,and-2foreach

incorrectanswer,buttheminimumscoreiszero.)

(d)ImportantpredictionsoftheCopernicantheorywerecon�rmedbythediscov-

eryoftheaberrationofstarlight(whichshowedthatthevelocityoftheEarth

hasthetime-dependenceexpectedforrotationabouttheSun)andbythebe-

havioroftheFoucaultpendulum(whichshowedthattheEarthrotates).These

discoveriesweremade

(i)duringCopernicus'lifetime.

(ii)approximatelytwoandthreedecadesafterCopernicus'death,respectively.

(iii)aboutonehundredyearsafterCopernicus'death.

(iv)approximatelytwoandthreecenturiesafterCopernicus'death,respec-

tively.

(e)IfoneaveragesoversuÆcientlylargescales,theuniverseappearstobeho-

mogeneousandisotropic.Howlargemusttheaveragingscalebebeforethis

homogeneityandisotropysetin?

(i)1AU(1AU=1:496�1011m).

(ii)100kpc(1kpc=1000pc,1pc=3:086�1016m=3.262light-year).

(iii)1Mpc(1Mpc=106pc).

(iv)10Mpc.

(v)100Mpc.

(vi)1000Mpc.

�

PROBLEM

4:ANEXPONENTIALLYEXPANDINGUNIVERSE(20

points)

ThefollowingproblemwasProblem2,Quiz2,1994,andhadalsoappearedonthe

1994ReviewProblems.Asisthecasethisyear,itwasannouncedthatoneofthe

problemsonthequizwouldcomefromeitherthehomeworkortheReviewProblems.

TheproblemalsoappearedasProblem2onQuiz1,2007.

Considera at(i.e.,ak=0,oraEuclidean)universewithscalefactorgiven

by

a(t)=a0 e�t;

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wherea0and�areconstants.

(a)(5points)FindtheHubbleconstantH

atanarbitrarytimet.

(b)(5points)Let(x;y;z;t)bethecoordinatesofacomovingcoordinatesystem.

Supposethatatt=0agalaxylocatedattheoriginofthissystememitsalight

pulsealongthepositivex-axis.Findthetrajectoryx(t)whichthelightpulse

follows.

(c)(5points)Supposethatwearelivingonagalaxyalongthepositivex-axis,and

thatwereceivethislightpulseatsomelatertime.Weanalyzethespectrumof

thepulseanddeterminetheredshiftz.Expressthetimetratwhichwereceive

thepulseintermsofz,�,andanyrelevantphysicalconstants.

(d)(5points)Atthetimeofreception,whatisthephysicaldistancebetweenour

galaxyandthegalaxywhichemittedthepulse?Expressyouranswerinterms

ofz,�,andanyrelevantphysicalconstants.

PROBLEM

5:DID

YOU

DO

THEREADING?

(a)Theassumptionsofhomogeneityandisotropygreatlysimplifythedescription

ofouruniverse.We�ndthattherearethreepossibilitiesforahomogeneous

andisotropicuniverse:anopenuniverse,a atuniverse,andacloseduni-

verse.Whatquantityorconditiondistinguishesbetweenthesethreecases:the

temperatureofthemicrowavebackground,thevalueof=�=�c ,mattervs.

radiationdomination,orredshift?

(b)Whatisthetemperature,inKelvin,ofthecosmicmicrowavebackgroundto-

day?

(c)Whichofthefollowingsupportsthehypothesisthattheuniverseisisotropic:

thedistancestonearbyclusters,observationsofthecosmicmicrowaveback-

ground,clusteringofgalaxiesonlargescales,ortheageanddistributionof

globularclusters?

(d)IsthedistancetotheAndromedaNebula(roughly)10kpc,5billionlightyears,

2millionlightyears,or3lightyears?

(e)DidHubblediscoverthelawwhichbearshisnamein1862,1880,1906,1929,

or1948?

(f)WhenHubblemeasuredthevalueofhisconstant,hefoundH�1�100million

years,2billionyears,10billionyears,or20billionyears?

(g)Cepheidvariablesareimportanttocosmologybecausetheycanbeusedtoesti-

matethedistancestothenearbygalaxies.WhatpropertyofCepheidvariables

makesthemusefulforthispurpose,andhowaretheyused?

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(h)Cepheidvariablestarscanbeusedasestimatorsofdistancefordistancesup

toabout100light-years,104

light-years,107

lightyears,or1010

light-years?

[Notefor2011:thisquestionwasbasedonthereadingfromJosephSilk'sThe

BigBang,andthereforewouldbenotbeafairquestionforthisyear.]

(i)Namethetwomenwhoin1964discoveredthecosmicbackgroundradiation.

WithwhatinstitutionweretheyaÆliated?

(j)Atthetimeofthediscoveryofthecosmicmicrowavebackground,anactivebut

independente�ortwastakingplaceelsewhere.P.J.E.Peebleshadestimated

thattheuniversemustcontainbackgroundradiationwithatemperatureofat

least10 ÆK,andRobertH.Dicke,P.G.Roll,andD.T.Wilkinsonhadmounted

anexperimenttolookforit.Atwhatinstitutionwerethesepeopleworking?

PROBLEM

6:AFLATUNIVERSEWITH

UNUSUALTIMEEVOLU-

TION

ThefollowingproblemwasProblem3,Quiz2,1988:

Considera atuniverse�lledwithanewandpeculiarformofmatter,witha

Robertson{Walkerscalefactorthatbehavesas

a(t)=bt1=3:

Herebdenotesaconstant.

(a)Ifalightpulseisemittedattimeteandobservedattimeto ,�ndthephys-

icalseparation`p (to )betweentheemitterandtheobserver,atthetimeof

observation.

(b)Againassumingthatteandtoaregiven,�ndtheobservedredshiftz.

(c)Findthephysicaldistance`p (to )whichseparatestheemitterandobserverat

thetimeofobservation,expressedintermsofc,to ,andz(i.e.,withoutte

appearing).

(d)Atanarbitrarytimetintheintervalte<t<to ,�ndthephysicaldistance

`p (t)betweenthelightpulseandtheobserver.Expressyouranswerinterms

ofc,t,andto .

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�

PROBLEM

7:ANOTHERFLATUNIVERSEWITHAN

UNUSUAL

TIMEEVOLUTION

(40points)

ThefollowingproblemwasProblem3,Quiz1,2000.

Considera atuniversewhichis�lledwithsomepeculiarformofmatter,so

thattheRobertson{Walkerscalefactorbehavesas

a(t)=bt ;

whereband areconstants.[Thisuniversedi�ersfrom

thematter-dominated

universedescribedinthelecturenotesinthat�isnotproportionalto1=a3(t).Such

behaviorispossiblewhenpressuresarelarge,becauseagasexpandingunderpressure

canloseenergy(andhencemass)duringtheexpansion.]Forthefollowingquestions,

anyoftheanswersmaydependon ,whetheritismentionedexplicitlyornot.

a)(5points)Lett0denotethepresenttime,andlettedenotethetimeatwhich

thelightthatwearecurrentlyreceivingwasemittedbyadistantobject.In

termsofthesequantities,�ndthevalueoftheredshiftparameterzwithwhich

thelightisreceived.

b)(5points)Findthe\look-back"timeasafunctionofzandt0 .Thelook-back

timeisde�nedasthelengthoftheintervalincosmictimebetweentheemission

andobservationofthelight.

c)(10points)Expressthepresentvalueofthephysicaldistancetotheobjectas

afunctionofH0 ,z,and .

d)(10points)Atthetimeofemission,thedistantobjecthadapoweroutputP

(measured,say,inergs/sec)whichwasradiateduniformlyinalldirections,in

theformofphotons.Whatistheradiationenergy uxJfromthisobjectat

theearthtoday?ExpressyouranswerintermsofP,H0 ,z,and .[Energy

ux(whichmightbemeasuredinerg-cm�2-sec �1)isde�nedastheenergyper

unitareaperunittimestrikingasurfacethatisorthogonaltothedirectionof

energy ow.]

e)(10points)Supposethatthedistantobjectisagalaxy,movingwiththeHubble

expansion.Withinthegalaxyasupernovaexplosionhashurledajetofmaterial

directlytowardsEarthwithaspeedv,measuredrelativetothegalaxy,which

iscomparabletothespeedoflightc.Assume,however,thatthedistancethe

jethastraveledfrom

thegalaxyissosmallthatitcanbeneglected.With

whatredshiftzJ

wouldweobservethelightcomingfrom

thisjet?Express

youranswerintermsofallorsomeofthevariablesv,z(theredshiftofthe

galaxy),t0 ,H0 ,and ,andtheconstantc.

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PROBLEM

8:DID

YOU

DO

THEREADING?(25points)

ThefollowingproblemwasProblem1,Quiz1,1996:

Thefollowingquestionsareworth5pointseach.

a)In1814-1815,theMunichopticianJosephFrauenhoferallowedlightfromthe

suntopassthroughaslitandthenthroughaglassprism.Thelightwasspread

intoaspectrumofcolors,showinglinesthatcouldbeidenti�edwithknown

elements|

sodium,iron,magnesium,calcium,andchromium.Werethese

linesdark,orbright(2points)?Why(3points)?

b)TheAndromedaNebulawasshownconclusivelytolieoutsideourowngalaxy

whenastronomersacquiredtelescopespowerfulenoughtoresolvetheindivid-

ualstarsofAndromeda.WasthisfeataccomplishedbyGalileoin1609,by

ImmanuelKantin1755,byHenriettaSwanLeavittin1912,byEdwinHubble

in1923,orbyWalterBaadeandAllanSandageinthe1950s?

c)Someoftheearliestmeasurementsofthecosmicbackgroundradiationwere

madeindirectly,byobservinginterstellarcloudsofamoleculecalledcyanogen

(CN).Statewhethereachofthefollowingstatementsistrueorfalse(1point

each):

(i)The�rstmeasurementsofthetemperatureoftheinterstellarcyanogen

weremadeovertwentyyearsbeforethecosmicbackgroundradiationwas

directlyobserved.

(ii)Cyanogenhelpstomeasurethecosmicbackgroundradiationbyre ecting

ittowardtheearth,sothatitcanbereceivedwithmicrowavedetectors.

(iii)Onereasonwhythecyanogenobservationswereimportantwasthatthey

gavethe�rstmeasurementsoftheequivalenttemperatureofthecosmic

backgroundradiationatwavelengthsshorterthanthepeakoftheblack-

bodyspectrum.

(iv)Bymeasuringthespectrum

ofvisiblestarlightthatpassesthroughthe

cyanogenclouds,astronomerscaninfertheintensityofthemicrowave

radiationthatbathestheclouds.

(v)Byobservingchemicalreactionsinthecyanogenclouds,astronomerscan

inferthetemperatureofthemicrowaveradiationinwhichtheyarebathed.

d)Inabout280B.C.,aGreekphilosopherproposedthattheEarthandtheother

planetsrevolvearoundthesun.Whatwasthenameofthisperson?[Notefor

2011:thisquestionwasbasedonreadingsfromJosephSilk'sTheBigBang,

andthereforeisnotappropriateforQuiz1ofthisyear.]

e)In1832HeinrichWilhelm

Olberspresentedwhatwenowknowas\Olbers'

Paradox,"althoughasimilarargumenthadbeendiscussedasearlyas1610by

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JohannesKepler.Olbersarguedthatiftheuniverseweretransparent,static,

in�nitelyold,andwaspopulatedbyauniformdensityofstarssimilartoour

sun,thenoneofthefollowingconsequenceswouldresult:

(i)Thebrightnessofthenightskywouldbein�nite.

(ii)Anypatchofthenightskywouldlookasbrightasthesurfaceofthe

sun.

(iii)Thetotalenergy uxfromthenightskywouldbeaboutequaltothe

totalenergy uxfromthesun.

(iv)Anypatchofthenightskywouldlookasbrightasthesurfaceofthe

moon.

WhichoneofthesestatementsisthecorrectstatementofOlbers'paradox?

PROBLEM

9:A

FLATUNIVERSEWITH

a(t)/

t3=5

ThefollowingproblemwasProblem3,Quiz1,1996:

Considera atuniversewhichis�lledwithsomepeculiarformofmatter,so

thattheRobertson{Walkerscalefactorbehavesas

a(t)=bt3=5;

wherebisaconstant.

a)(5points)FindtheHubbleconstantH

atanarbitrarytimet.

b)(5points)Whatisthephysicalhorizondistanceattimet?

c)(5points)SupposealightpulseleavesgalaxyAattimetA

andarrivesatgalaxy

BattimetB.Whatisthecoordinatedistancebetweenthesetwogalaxies?

d)(5points)WhatisthephysicalseparationbetweengalaxyAandgalaxyBat

timetA?AttimetB?

e)(5points)Atwhattimeisthelightpulseequidistantfromthetwogalaxies?

f)(5points)WhatisthespeedofBrelativetoAatthetimetA?(By\speed,"I

meantherateofchangeofthephysicaldistancewithrespecttocosmictime,

d`p =dt.)

g)(5points)Forobservationsmadeattimet,whatisthepresentvalueofthe

physicaldistanceasafunctionoftheredshiftz(andthetimet)?Whatphysical

distancecorrespondstoz=

1?

Howdoesthiscomparewiththehorizon

distance?

(NotethatthisquestiondoesnotrefertothegalaxiesAandB

discussedintheearlierparts.Inparticular,youshouldnotassumethatthe

lightpulseleftitssourceattimetA.)

h)(5points)ReturningtothediscussionofthegalaxiesAandBwhichwere

consideredinparts(c)-(f),supposetheradiationfrom

galaxyAisemitted

withtotalpowerP.WhatisthepowerperareareceivedatgalaxyB?

i)(5points)WhenthelightpulseisreceivedbygalaxyB,apulseisimmediately

sentbacktowardgalaxyA.Atwhattimedoesthissecondpulsearriveatgalaxy

A?

8.286QUIZ1REVIEW

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PROBLEM

10:DID

YOU

DO

THEREADING?(20points)

ThefollowingquestionsweretakenfromProblem1,Quiz1,1998:

Thefollowingquestionsareworth5pointseach.

a)In1917,Einsteinintroducedamodeloftheuniversewhichwasbasedonhis

newlydevelopedgeneralrelativity,butwhichcontainedanextraterminthe

equationswhichhecalledthe\cosmologicalterm."

(ThecoeÆcientofthis

termiscalledthe\cosmologicalconstant.")WhatwasEinstein'smotivation

forintroducingthisterm?

b)Whentheredshiftofdistantgalaxieswas�rstdiscovered,theearliestobserva-

tionswereanalyzedaccordingtoacosmologicalmodelinventedbytheDutch

astronomerW.deSitterin1917.Atthetimeofitsdiscovery,wasthismodel

thoughttobestaticorexpanding?Fromthemodernperspective,isthemodel

thoughttobestaticorexpanding?

c)Theearlyuniverseisbelievedtohavebeen�lledwiththermal,orblack-body,

radiation.Forsuchradiationthenumberdensityofphotonsandtheenergy

densityareeachproportionaltopowersoftheabsolutetemperatureT.Say

Numberdensity/Tn1

Energydensity/Tn2

Givethevaluesoftheexponentsn1andn2 .

d)Atabout3,000Kthematterintheuniverseunderwentacertainchemical

changeinitsform,achangethatwasnecessarytoallowthedi�erentiationof

matterintogalaxiesandstars.Whatwasthenatureofthischange?

PROBLEM

11:ANOTHER

FLAT

UNIVERSEWITH

a(t)/

t3=5

(40

points)

ThefollowingwasProblem3,Quiz1,1998:

Considera atuniversewhichis�lledwithsomepeculiarformofmatter,so

thattheRobertson{Walkerscalefactorbehavesas

a(t)=bt3=5;

wherebisaconstant.

a)(5points)FindtheHubbleconstantH

atanarbitrarytimet.

b)(10points)Supposeamessageistransmittedbyradiosignal(travelingatthe

speedoflightc)fromgalaxyAtogalaxyB.Themessageissentatcosmic

timet1 ,whenthephysicaldistancebetweenthegalaxiesis`0 .Atwhatcosmic

8.286QUIZ1REVIEW

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timet2isthemessagereceivedatgalaxyB?(Expressyouranswerintermsof

`0 ,t1 ,andc.)

c)(5points)Uponreceiptofthemessage,thecreaturesongalaxyBimmediately

sendbackanacknowledgement,byradiosignal,thatthemessagehasbeen

received.Atwhatcosmictimet3istheacknowledgmentreceivedongalaxyA?

(Expressyouranswerintermsof`0 ,t1 ,t2 ,andc.)

d)(10points)ThecreaturesongalaxyBspendsometimetryingtodecodethe

message,�nallydecidingthatitisanadvertisementforKellogg'sCornFlakes

(whateverthatis).Atatime�tafterthereceiptofthemessage,asmeasured

ontheirclocks,theysendbackaresponse,requestingfurtherexplanation.At

whatcosmictimet4

istheresponsereceivedongalaxyA?Inansweringthis

part,youshouldnotassumethat�tisnecessarilysmall.(Expressyouranswer

intermsof`0 ,t1 ,t2 ,t3 ,�t,andc.)

e)(5points)WhentheresponseisreceivedbygalaxyA,theradiowaveswillbe

redshiftedbyafactor1+z.Giveanexpressionforz.(Expressyouranswerin

termsof`0 ,t1 ,t2 ,t3 ,t4 ,�t,andc.)

f)(5points;Nopartialcredit)Ifthetime�tintroducedinpart(d)issmall,the

timedi�erencet4 �t3canbeexpandedto�rstorderin�t.Calculatet4 �t3

to�rstorderaccuracyin�t.(Expressyouranswerintermsof`0 ,t1 ,t2 ,t3 ,

t4 ,�t,andc.)[Hint:whilethispartcanbeansweredbyusingbruteforceto

expandtheanswerinpart(d),thereisaneasierway.]

�

PROBLEM

12:THEDECELERATION

PARAMETER

ThefollowingproblemwasProblem2,Quiz2,1992,whereitcounted10pointsout

of100.

Manystandardreferencesincosmologyde�neaquantitycalledthedeceler-

ationparameterq,whichisadirectmeasureoftheslowingdownofthecosmic

expansion.Theparameterisde�nedby

q���a(t)a(t)

_a2(t):

Findtherelationshipbetweenqandforamatter-dominateduniverse.[Incase

youhaveforgotten,isde�nedby

=�=�c;

where�isthemassdensityand�c

isthecriticalmassdensity(i.e.,thatmass

densitywhichcorrespondstok=0).]

8.286QUIZ1REVIEW

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PROBLEM

13:A

RADIATION-DOMINATED

FLATUNIVERSE

Wehavelearnedthatamatter-dominatedhomogeneousandisotropicuniverse

canbedescribedbyascalefactora(t)obeyingtheequation

�_aa �2

=8�3

G��kc2

a2

:

Thisequationinfactappliestoanyformofmassdensity,sowecanapplyittoa

universeinwhichthemassdensityisdominatedbytheenergyofphotons.Recall

thatthemassdensityofnonrelativisticmatterfallso�as1=a3(t)astheuniverse

expands;themassofeachparticleremainsconstant,andthedensityofparticles

fallso�as1=a3(t)becausethevolumeincreasesasa3(t).Forthephoton-dominated

universe,thedensityofphotonsfallsofas1=a3(t),butinadditionthefrequency

(andhencetheenergy)ofeachphotonredshiftsinproportionto1=a(t).Sincemass

andenergyareequivalent,themassdensityofthegasofphotonsfallso�as1=a4(t).

Fora at(i.e.,k=0)matter-dominateduniversewelearnedthatthescale

factora(t)isproportionaltot2=3.Howdoesa(t)behaveforaphoton-dominated

universe?

PROBLEM

14:DID

YOU

DO

THEREADING?

Thefollowingproblem

wastakenfrom

Problem1,Quiz1,2004,whereeachpart

counted5points,foratotalof25points.Thereadingassignmentincludedthe�rst

threechaptersofRyden,IntroductiontoCosmology,andthe�rstthreechapters

ofWeinberg,TheFirstThreeMinutes.

(a)In1826,theastronomerHeinrichOlberwroteapaperonaparadoxregarding

thenightsky.WhatisOlber'sparadox?Whatistheprimaryresolutionofit?

(b)WhatisthevalueoftheNewtoniangravitationalconstantGinPlanckunits?

ThePlancklengthisoftheorderof10 �35m,10 �15m,1015m,or1035m?

(c)WhatistheCosmologicalPrinciple?IstheHubbleexpansionoftheuniverse

consistentwithit?(Forthelatterquestion,asimple\yes"or\no"willsuÆce.)

(d)Inthe\StandardModel"oftheuniverse,whentheuniversecooledtoabout

3�10aK,itbecametransparenttophotons,andtodayweobservetheseasthe

CosmicMicrowaveBackground(CMB)atatemperatureofabout3�10bK.

Whataretheintegersaandb?

(e)Whatdidtheuniverseprimarilyconsistofatabout1/100thofasecondafter

theBigBang?Includeanyconstituentthatisbelievedtohavemadeupmore

than1%ofthemassdensityoftheuniverse.

8.286QUIZ1REVIEW

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�

PROBLEM

15:SPECIALRELATIVITY

DOPPLER

SHIFT

Thefollowingproblemwastakenfrom

Problem2,Quiz1,2004,whereitcounted

20points.

ConsidertheDopplershiftofradiowaves,foracaseinwhichboththesource

andtheobserveraremoving.Supposethesourceisaspaceshipmovingwithaspeed

vsrelativetothespacestationAlpha-7,whiletheobserverisonanotherspaceship,

movingintheoppositedirectionfromAlpha-7withspeedvorelativetoAlpha-7.

(a)(10points)CalculatetheDopplershiftzoftheradiowaveasreceivedbythe

observer.(Recallthatradiowavesareelectromagneticwaves,justlikelight

exceptthatthewavelengthislonger.)

(b)(10points)Usetheresultsofpart(a)todeterminevtot ,thevelocityofthe

sourcespaceshipasitwouldbemeasuredbytheobserverspaceship.(8points

willbegivenforthebasicidea,whetherornotyouhavetherightanswerfor

part(a),and2pointswillbegivenforthealgebra.)

PROBLEM

16:DID

YOU

DO

THEREADING?

ThefollowingquestionwastakenfromProblem1,Quiz1,2005,whereitcounted

25points.

(a)(4points)Whatwasthe�rstexternalgalaxythatwasshowntobeatadistance

signi�cantlygreaterthanthemostdistantknownobjectsinourgalaxy?How

wasthedistanceestimated?

(b)(5points)Whatisrecombination?Didgalaxiesbegintoformbeforeorafter

recombination?Why?

(c)(4points)InChapterIVofhisbook,Weinbergdevelopsa\recipeforahot

universe,"inwhichthematteroftheuniverseisdescribedasagasinthermal

equilbriumataveryhightemperature,inthevicinityof109K(severalthou-

sandmilliondegreesKelvin).Suchathermalequilibrium

gasiscompletely

describedbyspecifyingitstemperatureandthedensityoftheconservedquan-

tities.Whichofthefollowingisonthislistofconservedquantities?Circleas

manyasapply.

8.286QUIZ1REVIEW

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(i)baryonnumber

(ii)energyperparticle

(iii)protonnumber

(iv)electriccharge

(v)pressure

(d)(4points)ThewavelengthcorrespondingtothemeanenergyofaCMB(cosmic

microwavebackground)photontodayisapproximatelyequaltowhichofthe

followingquantities?(Youmaywishtolookupthevaluesofvariousphysical

constantsattheendofthequiz.)

(i)2fm(2�10 �15m)

(ii)2microns(2�10 �6m)

(iii)2mm(2�10 �3m)

(iv)2m.

(e)(4points)Whatistheequivalenceprinciple?

(f)(4points)WhyisitdiÆcultforEarth-basedexperimentstolookatthesmall

wavelengthportionofthegraphofCMBenergydensityperwavelengthvs.

wavelength?

�

PROBLEM

17:

TRACING

A

LIGHT

PULSE

THROUGH

A

RADIATION-DOMINATED

UNIVERSE

Thefollowingproblemwastakenfrom

Problem3,Quiz1,2005,whereitcounted

25points.

Considera atuniversethatexpandswithascalefactor

a(t)=bt1=2;

wherebisaconstant.Wewilllearnlaterthatthisisthebehaviorofthescalefactor

foraradiation-dominateduniverse.

(a)(5points)Atanarbitrarytimet=tf,whatisthephysicalhorizondistance?

(By\physical,"Imeanasusualthedistanceinphysicalunits,suchasmeters

orcentimeters,asmeasuredbyasequenceofrulers,eachofwhichisatrest

relativetothecomovingmatterinitsvicinity.)

(b)(3points)Supposethataphotonarrivesattheorigin,attimetf,fromadistant

pieceofmatterthatispreciselyatthehorizondistanceattimetf.Whatis

thetimeteatwhichthephotonwasemitted?

(c)(2points)Whatisthecoordinatedistancefromtheorigintothepointfrom

whichthephotonwasemitted?

(d)(10points)Foranarbitrarytimetintheintervalte �t�tf,whilethephoton

istraveling,whatisthephysicaldistance`p (t)fromtheorigintothelocation

ofthephoton?

(e)(5points)Atwhattimetmax

isthephysicaldistanceofthephotonfromthe

originatitslargestvalue?

8.286QUIZ1REVIEW

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p.19

PROBLEM

18:TRANSVERSEDOPPLER

SHIFTS

Thefollowingproblemwastakenfrom

Problem4,Quiz1,2005,whereitcounted

20points.

(a)(8points)Supposethespaceship

Xanthu

is

at

rest

at

location

(x=0;y=a;z=0)inaCartesianco-

ordinatesystem.(Weassumethat

thespaceisEuclidean,andthatthe

distancescalesintheproblem

are

smallenoughsothattheexpansion

oftheuniversecanbeneglected.)

ThespaceshipEmmeracismoving

atspeedv0

alongthex-axisinthe

positivedirection,asshowninthe

diagram,wherev0iscomparableto

thespeedoflight.AstheEmmerac

crossestheorigin,itreceivesara-

diosignalthathadbeensentsome

timeearlierfrom

theXanthu.

Is

theradiationreceivedredshiftedorblueshifted?Whatistheredshiftz(where

negativevaluesofzcanbeusedtodescribeblueshifts)?

(b)(7points)Now

supposethatthe

Emmeracisatrestattheorigin,

whiletheXanthuismovinginthe

negativex-direction,aty=aand

z=

0,asshowninthediagram.

Thatis,thetrajectoryoftheXan-

thucanbetakenas

(x=�v0 t;y=a;z=0):

Att=0theXanthucrossesthey-

axis,andatthatinstantitemits

aradiosignalalongthey-axis,di-

rectedattheorigin.

Theradi-

ationisreceivedsometimelater

bytheEmmerac.Inthiscase,is

theradiationreceivedredshiftedorblueshifted?Whatistheredshiftz(where

againnegativevaluesofzcanbeusedtodescribeblueshifts)?

(c)(5points)Isthesequenceofeventsdescribedin(b)physicallydistinctfromthe

sequencedescribedin(a),orisitreallythesamesequenceofeventsdescribed

8.286QUIZ1REVIEW

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inareferenceframethatismovingrelativetothereferenceframeusedinpart

(a)?Explainyourreasoninginasentenceortwo.(Hint:notethatthereare

threeobjectsintheproblem:Xanthu,Emmerac,andthephotonsoftheradio

signal.)

�

PROBLEM

19:

A

TWO-LEVEL

HIGH-SPEED

MERRY-GO-

ROUND

(15points)

ThisproblemwasProblem3onQuiz1,2007.

Considerahigh-speedmerry-go-roundwhichissimilartotheonediscussedin

Problem3ofProblemSet1,butwhichhastwolevels.Thatis,therearefourevenly

spacedcarswhichtravelaroundacentralhubatspeedvatadistanceRfroma

centralhub,andalsoanotherfourcarsthatareattachedtoextensionsofthefour

radialarms,eachmovingataspeed2vatadistance2Rfromthecenter.Inthis

problemwewillconsideronlylightwaves,notsoundwaves,andwewillassume

thatvisnotnegligiblecomparedtoc,butthat2v<c.

WelearnedinProblemSet1thatthereisnoredshiftwhenlightfromonecarat

radiusRisreceivedbyanobserveronanothercaratradiusR.

(a)(5points)Supposethatcars5{8areallemittinglightwavesinalldirections.If

anobserverincar1receiveslightwavesfromeachofthesecars,whatredshift

zdoessheobserveforeachofthefoursignals?

(b)(10points)Supposethataspaceshipisrecedingtotherightatarelativistic

speedualongalinethroughthehub,asshowninthediagram.Supposethat

anobserverincar6receivesaradiosignalfromthespaceship,atthetimewhen

thecarisinthepositionshowninthediagram.Whatredshiftzisobserved?

8.286QUIZ1REVIEW

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PROBLEM

20:

SIGNAL

PROPAGATION

IN

A

FLAT

MATTER-

DOMINATED

UNIVERSE(55points)

ThefollowingproblemwasonQuiz1,2009.

Considera at,matter-dominateduniverse,withscalefactor

a(t)=bt2=3;

wherebisanarbitraryconstant.Forthefollowingquestions,theanswertoany

partmaycontainsymbolsrepresentingtheanswerstopreviousparts,whetheror

notthepreviouspartwasansweredcorrectly.

(a)(10points)Attimet=t1 ,alightsignalissentfromgalaxyA.Let`p;sA(t)

denotethephysicaldistanceofthesignalfromAattimet.(Notethatt=0

correspondstotheoriginoftheuniverse,nottotheemissionofthesignal.)

(i)FindthespeedofseparationofthelightsignalfromA,de�nedasd`p;sA=dt.

Whatisthevalueofthisspeed(ii)atthetimeofemission,t1 ,and(iii)what

isitslimitingvalueatarbitrarilylatetimes?

(b)(5points)Supposethatthereisasecondgalaxy,galaxyB,thatislocatedat

aphysicaldistancecH�1

fromAattimet1 ,whereH(t)denotestheHubble

expansionrateandcisthespeedoflight.(cH�1iscalledtheHubblelength.)

Supposethatthelightsignaldescribedabove,whichisemittedfromgalaxy

Aattimet1 ,isdirectedtowardgalaxyB.Atwhattimet2

doesitarriveat

galaxyB?

(c)(10points)Let`p;sB(t)denotethephysicaldistanceofthelightsignalfrom

galaxyBattimet.(i)Findthespeedofapproachofthelightsignaltowards

B,de�nedas�d`p;sB=dt.Whatisthevalueofthisspeed(ii)atthetimeof

emission,t1 ,and(iii)atthetimeofreception,t2 ?

(d)(10points)IfanastronomerongalaxyAobservesthelightarrivingfromgalaxy

Battimet1 ,whatisitsredshiftzBA?

(e)(10points)Supposethatthereisanothergalaxy,galaxy

C,alsolocatedataphysicaldistancecH�1

from

Aat

timet1 ,butinadirectionorthogonaltothatofB.If

galaxyBisobservedfromgalaxyCattimet1 ,whatis

theobservedredshiftzBC?Recallthatthisuniverseis

at,soEuclideangeometryapplies.

(f)(10points)SupposethatgalaxyA,attimet1 ,emitselectromagneticradiation

sphericallysymmetrically,withpoweroutputP.(Pmightbemeasured,for

example,inwatts,where1watt=1joule/second.)Whatistheradiation

energy uxJthatisreceivedbygalaxyB

attimet2 ,whentheradiation

reachesgalaxyB?(Jmightbemeasured,forexample,inwattspermeter2.

Unitsarementionedhereonlytohelpclarifythemeaningofthesequantities|

youranswershouldhavenoexplicitunits,butshouldbeexpressedintermsof

anyorallofthegivenquantitiest1 ,P,andc,plusperhapssymbolsrepresenting

theanswerstopreviousparts.)

8.286QUIZ1REVIEW

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PROBLEM

21:DID

YOU

DO

THEREADING?(25points)

ThefollowingproblemappearedonQuiz1of2011.

(a)(10points)Hubble'slawrelatesthedistanceofgalaxiestotheirvelocity.The

Dopplere�ectprovidesanaccuratetooltomeasurevelocity,whilethemea-

sureofcosmicdistancesismoreproblematic.Explainbrie ythemethodthat

Hubbleusedtoestimatethedistanceofgalaxiesinderivinghislaw.

(b)(5points)OneexpectsHubble'slawtoholdasaconsequenceoftheCosmo-

logicalPrinciple.WhatdoestheCosmologicalPrinciplestate?

(c)(10points)Giveabriefde�nitionforthewordshomogeneityandisotropy.

Thensayforeachofthefollowingtwostatementswhetheritistrueorfalse.

Iftrueexplainbrie ywhy.Iffalsegiveacounter-example.Youshouldassume

Euclideangeometry(whichWeinbergimplicitlyassumedinhisdiscussion).

(i)Iftheuniverseisisotropicaroundonepointthenithastobehomogeneous.

(ii)Iftheuniverseisisotropicaroundtwoormoredistinctpointsthenithas

tobehomogeneous.

(d)Bonusquestion:(2pointsextracredit)Ifweallowcurved(i.e.,non-Euclidean)

spaces,isittruethatauniversewhichisisotropicaroundtwodistinctpoints

hastobehomogeneous?

Iftrueexplainbrie ywhy,andotherwisegivea

counter-example.

�

PROBLEM

22:THETRAJECTORY

OFA

PHOTON

ORIGINAT-

ING

ATTHEHORIZON

(25points)

ThefollowingproblemappearedonQuiz1of2011.

Consideragaina atmatter-dominateduniverse,withascalefactorgivenby

a(t)=bt2=3;

wherebisaconstant.Lett0denotethecurrenttime.

(a)(5points)Whatisthecurrentvalueofthephysicalhorizon

distance

`p;horizon (t0 )?Thatis,whatisthepresentdistanceofthemostdistantmatter

thatcanbeseen,limitedonlybythespeedoflight.

(b)(5points)Consideraphotonthatisarrivingnowfromanobjectthatisjustat

thehorizon.Ourgoalistotracethetrajectoryofthisobject.Supposethatwe

setupacoordinatesystemwithusattheorigin,andthesourceofthephoton

alongthepositivex-axis.Whatisthecoordinatex0ofthephotonatt=0?

(c)(5points)Asthephotontravelsfromthesourcetous,whatisitscoordinate

x(t)asafunctionoftime?

(d)(5points)Whatisthephysicaldistance`p (t)betweenthephotonandusasa

functionoftime?

(e)(5points)Whatisthemaximumphysicaldistance`p;max (t)betweenthephoton

andus,andatwhattimetmaxdoesitoccur?

8.286QUIZ1REVIEW

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SOLUTIONS,FALL2011

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SOLUTIONS

PROBLEM

1:DID

YOU

DO

THEREADING?(35points)

a)DopplerpredictedtheDopplere�ectin1842.

b)Mostofthestarsofourgalaxy,includingoursun,lieina atdisk.Wetherefore

seemuchmorelightwhenwelookoutfromearthalongtheplaneofthedisk

thanwhenwelookinanyotherdirection.

c)Hubble'soriginalpaperontheexpansionoftheuniversewasbasedonastudy

ofonly18galaxies.Well,atleastWeinberg'sbooksays18galaxies.Formy

ownbookImadeacopyofHubble'soriginalgraph,whichseemstoshow

24blackdots,eachofwhichrepresentsagalaxy,asreproducedbelow.The

verticalaxisshowstherecessionvelocity,inkilometerspersecond.Thesolid

lineshowsthebest�ttotheblackdots,eachofwhichrepresentsagalaxy.Each

opencirclerepresentsagroupofthegalaxiesshownasblackdots,selectedby

theirproximityindirectionanddistance;thebrokenlineisthebest�ttothese

points.Thecrossshowsastatisticalanalysisof22galaxiesforwhichindividual

distancemeasurementswerenotavailable.IamnotsurewhyWeinbergrefers

to18galaxies,butitispossiblethatthetextofHubble'sarticleindicatedthat

18ofthesegalaxiesweremeasuredwithmorereliabilitythantherest.

8.286QUIZ1REVIEW

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SOLUTIONS,FALL2011

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d)e)

Duringatimeintervalinwhichthelinearsizeoftheuniversegrowsby1%,the

horizondistancegrowsbymorethan1%.Toseewhy,notethatthehorizon

distanceisequaltothescalefactortimesthecomovinghorizondistance.The

scalefactorgrowsby1%duringthistimeinterval,butthecomovinghorizon

distancealsogrows,sincelightfromthedistantgalaxieshashadmoretimeto

reachus.

f)ArnoA.PenziasandRobertW.Wilson,BellTelephoneLaboratories.

g)

(i)theaveragedistance

between

photons:

proportionaltothesizeof

theuniverse

(Photonsareneithercreatednordestroyed,sotheonly

e�ectisthattheaveragedistancebetweenthem

isstretchedwiththe

expansion.Sincetheuniverseexpandsuniformly,alldistancesgrowby

thesamefactor.)

(ii)thetypicalwavelengthoftheradiation:

proportionaltothesizeof

theuniverse(SeeLectureNotes3.)

(iii)thenumberdensityofphotonsintheradiation:

inverselypropor-

tionaltothecubeofthesizeoftheuniverse(From(i),theaveragedis-

tancebetweenphotonsgrowsinproportiontothesizeoftheuniverse.

Sincethevolumeofacubeisproportionaltothecubeofthelengthof

aside,theaveragevolumeoccupiedbyaphotongrowsasthecubeof

thesizeoftheuniverse.Thenumberdensityistheinverseoftheaverage

volumeoccupiedbyaphoton.)

(iv)theenergy

density

oftheradiation:

inverselyproportionaltothe

fourthpowerofthesizeoftheuniverse

(Theenergyofeachphotonis

proportionaltoitsfrequency,andhenceinverselyproportionaltoitswave-

length.Sofrom(ii)theenergyofeachphotonisinverselyproportional

tothesizeoftheuniverse,andfrom(iii)thenumberdensityisinversely

proportionaltothecubeofthesize.)

8.286QUIZ1REVIEW

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SOLUTIONS,FALL2011

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(v)thetemperatureoftheradiation:

inverselyproportionaltothesize

oftheuniverse(Thetemperatureisdirectlyproportionaltotheaver-

ageenergyofaphoton,whichaccordingto(iv)isinverselyproportional

tothesizeoftheuniverse.)

PROBLEM

2:THESTEADY-STATEUNIVERSETHEORY(25points)

a)(10points)AccordingtoEq.(3.7),

H(t)=

1a(t)

dad

t:

Sointhiscase

1a(t)

dad

t=H0;

whichcanberewrittenas

daa

=H0dt:

Integrating,

lna=H0t+c;

wherecisaconstantofintegration.Exponentiating,

a=beH0

t;

whereb=ecisanarbitraryconstant.

b)(15points)Consideracubeofside`cdrawnonthecomovingcoordinatesystem

diagram.Thephysicallengthofeachsideisthena(t)`c ,sothephysicalvolume

is

V(t)=a3(t)`3c:

Sincethemassdensityis�xedat�=�0 ,thetotalmassinsidethiscubeatany

giventimeisgivenby

M(t)=a3(t)`3c�0:

Intheabsenceofmattercreationthetotalmasswithinacomovingvolume

wouldnotchange,sotheincreaseinmassdescribedbytheaboveequation

8.286QUIZ1REVIEW

PROBLEM

SOLUTIONS,FALL2011

p.26

mustbeattributedtomattercreation.Therateofmattercreationperunit

timeperunitvolumeisthengivenby

Rate=

1V(t)

dMd

t

=

1

a3(t)`3c3a2(t)dad

t`3c�0

=3adad

t�0

=

3H0�0:

Youwerenotaskedtoinsertnumbers,butitisworthwhiletoconsiderthe

numericalvalueaftertheexam,toseewhatthisansweristellingus.Suppose

wetakeH0

=

70km-sec �1-Mpc �1,andtake�0

tobethecriticaldensity,

�c=3H20 =8�G.Then

Toputthisnumberintomoremeaningfulterms,notethatthemassofahy-

drogenatom

is1:67�10 �27

kg,andthat1year=3:156�107

s.Therate

ofmatterproductionrequiredforthesteady-stateuniversetheorycanthen

beexpressedasroughlyonehydrogenatompercubicmeterperbillionyears!

Needlesstosay,sucharateofmatterproductionistotallyundetectable,so

thesteady-statetheorycannotberuledoutbythefailuretodetectmatter

production.

8.286QUIZ1REVIEW

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SOLUTIONS,FALL2011

p.27

PROBLEM

3:DID

YOU

DO

THEREADING?(25points)

Thefollowing5questionsareeachworth5points:

(a)Inthe1940's,threeastrophysicistsproposeda\steadystate"theoryofcos-

mology,inwhichtheuniversehasalwayslookedaboutthesameasitdoes

now.Statethelastnameofatleastoneoftheseauthors.(Bonuspoints:you

canearn1pointeachfornamingtheothertwoauthors,andhenceupto2

additionalpoints,but1pointwillbetakeno�foreachincorrectanswer.)

Ans:(Weinberg,page8,orRyden,page16):HermannBondi,ThomasGold,

andFredHoyle.

(b)In1917,aDutchastronomernamedWillem

deSitterdidwhichoneofthe

followingaccomplishments:

(i)measuredthesizeoftheMilkyWaygalaxy,�ndingittobeaboutone

billionlight-yearsindiameter.

(ii)resolvedCepheidvariablestarsinAndromedaandtherebyobtainedper-

suasiveevidencethatAndromedaisnotwithinourowngalaxy,butis

apparentlyanothergalaxylikeourown.

(iii)publishedacatalog,NebulaeandStarClusters,listing103objectsthat

astronomersshouldavoidwhenlookingforcomets.

(iv)publishedamodelfortheuniverse,basedongeneralrelativity,which

appearedtobestaticbutwhichproducedaredshiftproportionaltothe

distance.

(v)discoveredthattheorbitalperiodsoftheplanetsareproportionaltothe

3/2powerofthesemi-majoraxisoftheirellipticalorbits.

Discussion:(i)isfalseinpartbecausedeSitterwasnotinvolvedinthemea-

surementofthesizeoftheMilkyWay,butthemostobviouserrorisinthesize

oftheMilkyWay.ItsactualdiameterisreportedbyWeinberg(p.16)tobe

about100,000light-years,althoughnowitisbelievedtobeabouttwicethat

large.(ii)isanaccuratedescriptionofanobservationbyEdwinHubblein

1923(Weinberg,pp.19-20).(iii)describestheworkofCharlesMessierin1781

(Weinberg,p.17).(v)isofcourseoneofKepler'slawsofplanetarymotion.

(c)In1964{65,ArnoA.PenziasandRobertW.Wilsonobserveda uxofmi-

crowaveradiationcomingfromalldirectionsinthesky,whichwasinterpreted

byagroupofphysicistsataneighboringinstitutionasthecosmicbackground

radiationleftoverfromthebigbang.Circlethetwoitemsonthefollowinglist

thatwerenotpartofthestorybehindthisspectaculardiscovery:

8.286QUIZ1REVIEW

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(i)BellTelephoneLaboratory

(ii)MIT

(iii)PrincetonUniversity

(iv)pigeons

(v)groundhogs

(vi)Hubble'sconstant

(vii)liquidhelium

(viii)7.35cm

(Grading:3ptsfor1correctanswer,5for2correctanswers,and-2foreach

incorrectanswer,buttheminimumscoreiszero.)

Discussion:Thediscoveryofthecosmicbackgroundradiationwasdescribed

insomedetailbyWeinberginChapter3.TheobservationwasdoneatBell

TelephoneLaboratories,inHolmdel,NewJersey.Thedetectorwascooledwith

liquidheliumtominimizeelectricalnoise,andthemeasurementsweremadeat

awavelengthof7.35cm.Duringthecourseoftheexperimenttheastronomers

hadtoejectapairofpigeonswhowereroostingintheantenna.Penziasand

Wilsonwerenotinitiallyawarethattheradiationtheydiscoveredmighthave

comefromthebigbang,butBernardBurkeofMITputthemintouchwith

agroupatPrincetonUniversity(RobertDicke,JamesPeebles,P.G.Roll,and

DavidWilkinson)whowereactivelyworkingonthishypothesis.

(d)ImportantpredictionsoftheCopernicantheorywerecon�rmedbythediscov-

eryoftheaberrationofstarlight(whichshowedthatthevelocityoftheEarth

hasthetime-dependenceexpectedforrotationabouttheSun)andbythebe-

havioroftheFoucaultpendulum(whichshowedthattheEarthrotates).These

discoveriesweremade

(i)duringCopernicus'lifetime.

(ii)approximatelytwoandthreedecadesafterCopernicus'death,respectively.

(iii)aboutonehundredyearsafterCopernicus'death.

(iv)approximatelytwoandthreecenturiesafterCopernicus'death,respec-

tively.

Rydendiscussesthisonp.5.Theaberrationofstarlightwasdiscoveredin

1728,whiletheFoucaultpendulumwasinventedin1851.

(e)IfoneaveragesoversuÆcientlylargescales,theuniverseappearstobeho-

mogeneousandisotropic.Howlargemusttheaveragingscalebebeforethis

homogeneityandisotropysetin?

(i)1AU(1AU=1:496�1011m).

(ii)100kpc(1kpc=1000pc,1pc=3:086�1016m=3.262light-year).

(iii)1Mpc(1Mpc=106pc).

(iv)10Mpc.

(v)100Mpc.

(vi)1000Mpc.

ThisissueisdiscussedinRyden'sbookonp.11.

8.286QUIZ1REVIEW

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p.29

PROBLEM

4:AN

EXPONENTIALLY

EXPANDING

UNIVERSE

(a)AccordingtoEq.(3.7),theHubbleconstantisrelatedtothescalefactorby

H=_a=a:

So

H=�a0 e�t

a0 e�t

=

�:

(b)AccordingtoEq.(3.8),thecoordinatevelocityoflightisgivenby

dxd

t=

ca(t)=

ca0e ��t:

Integrating,

x(t)=

ca0 Z

t0

e ��t0d

t 0

=

ca0 ��

1�e ��t0 �t0

=

c�a0 �1�e ��t �:

(c)FromEq.(3.11),orfromthefrontofthequiz,onehas

1+z=a(tr )

a(te ):

Herete=0,so

1+z=a0 e�tr

a0

=)

e�tr

=1+z

=)

tr=1�

ln(1+z):

(d)Thecoordinatedistanceisx(tr ),wherex(t)isthefunctionfoundinpart(b),

andtristhetimefoundinpart(c).So

e�tr

=1+z;

8.286QUIZ1REVIEW

PROBLEM

SOLUTIONS,FALL2011

p.30

and

x(tr )=

c�a0 �1�e ��tr �

=

c�a0 �1�1

1+z �

=

cZ

�a0 (1+z):

Thephysicaldistanceatthetimeofreceptionisfoundbymultiplyingbythe

scalefactoratthetimeofreception,so

`p (tr )=a(tr )x(tr )=

cze�tr

�(1+z)=

cz�:

PROBLEM

5:\DID

YOU

DO

THEREADING?"

(a)Thedistinguishingquantityis��=�c .Theuniverseisopenif<1, atif

=1,orclosedif>1.

(b)Thetemperatureofthemicrowavebackgroundtodayisabout3Kelvin.(The

bestdeterminationtodate*wasmadebytheCOBEsatellite,whichmeasured

thetemperatureas2:728�0:004Kelvin.Theerrorhereisquotedwitha

95%

con�dencelimit,whichmeansthattheexperimentersbelievethatthe

probabilitythatthetruevalueliesoutsidethisrangeisonly5%.)

(c)Thecosmicmicrowavebackgroundisobservedtobehighlyisotropic.

(d)ThedistancetotheAndromedanebulaisroughly2millionlightyears.

(e)1929.

(f)2billionyears.Hubble'svalueforHubble'sconstantwashighbymodern

standards,byafactorof5to10.

(g)Theabsoluteluminosity(i.e.,thetotallightoutput)ofaCepheidvariable

starappearstobehighlycorrelatedwiththeperiodofitspulsations.This

correlationcanbeusedtoestimatethedistancetotheCepheid,bymeasuring

theperiodandtheapparentluminosity.Fromtheperiodonecanestimatethe

absoluteluminosityofthestar,andthenoneusestheapparentluminosityand

the1=r2

lawfortheintensityofapointsourcetodeterminethedistancer.

(h)107light-years.

(i)ArnoA.PenziasandRobertW.Wilson,BellTelephoneLaboratories.

(j)PrincetonUniversity.

*AstrophysicalJournal,vol.473,p.576(1996):TheCosmicMicrowaveBack-

groundSpectrumfromtheFullCOBEFIRASDataSets,D.J.Fixsen,E.S.Cheng,

J.M.Gales,J.C.Mather,R.A.Shafer,andE.L.Wright.

8.286QUIZ1REVIEW

PROBLEM

SOLUTIONS,FALL2011

p.31

PROBLEM

6:AFLATUNIVERSEWITH

UNUSUALTIMEEVOLU-

TION

Thekeytothisproblemistoworkincomovingcoordinates.

[Somestudentshaveaskedmewhyonecannotuse\physical"coordinates,for

whichthecoordinatesreallymeasurethephysicaldistances.Inprincipleonecan

useanycoordinatesystemonlikes,butthecomovingcoordinatesarethesimplest.

InanyothersystemitisdiÆculttowritedownthetrajectoryofeitheraparticle

oralight-beam.Incomovingcoordinatesitiseasytowritethetrajectoryofeither

alightbeam,oraparticlewhichismovingwiththeexpansionoftheuniverse(and

hencestandingstillinthecomovingcoordinates).Note,bytheway,thatwhenone

saysthataparticleisstandingstillincomovingcoordinates,onehasnotreallysaid

verymuchaboutit'strajectory.Onehassaidthatitismovingwiththematter

which�llstheuniverse,butonehasnotsaid,forexample,howthedistancebetween

theparticleandoriginvarieswithtime.Theanswertothislatterquestionisthen

determinedbytheevolutionofthescalefactor,a(t).]

(a)Thephysicalseparationatto

isgivenbythescalefactortimesthecoordi-

natedistance.Thecoordinatedistanceisfoundbyintegratingthecoordinate

velocity,so

`p (to )=a(to ) Z

to

te

cdt 0

a(t 0)=bt1=3

o Zto

te

cdt 0

bt 01=3

=32

ct1=3

o ht2=3

o

�t2=3

e i

=32

cto h1�(te =to )2=3 i:

(b)Fromthefrontoftheexam,1

+z=a(to )

a(te )= �to

te �

1=3

=)

z= �to

te �

1=3�

1:

(c)Bycombiningtheanswersto(a)and(b),onehas

`p (to )=32

cto �1�

1

(1+z)2 �:

8.286QUIZ1REVIEW

PROBLEM

SOLUTIONS,FALL2011

p.32

(d)Thephysicaldistanceofthelightpulseattimetisequaltoa(t)timesthe

coordinatedistance.Thecoordinatedistanceattimetisequaltothestarting

coordinatedistance,`c (te ),minusthecoordinatedistancethatthelightpulse

travelsbetweentimeteandtimet.Thus,

`p (t)=a(t) �`c (te )� Z

tte

cdt 0

a(t 0) �

=a(t) �Z

to

te

cdt 0

a(t 0) � Z

tte

cdt 0

a(t 0) �

=a(t) Z

to

t

cdt 0

a(t 0)

=bt1=3 Zto

t

cdt 0

bt 01=3

=32

ct1=3 ht2=3

o

�t2=3 i

=

32ct "�tot �2=3�

1 #:

PROBLEM

7:ANOTHER

FLAT

UNIVERSEWITH

AN

UNUSUAL

TIMEEVOLUTION

(40points)

a)(5points)Thecosmologicalredshiftisgivenbytheusualform,

1+z=a(t0 )

a(te ):

Forlightemittedbyanobjectattimete ,theredshiftofthereceivedlightis

1+z=a(t0 )

a(te )= �t0

te �

:

So,

z= �t0

te �

�1:

b)(5points)Thecoordinatest0

andtearecosmictimecoordinates.The\look-

back"timeasde�nedintheexamisthentheintervalt0 �te .Wecanwrite

thisas

t0 �te=t0 �1�te

t0 �:

8.286QUIZ1REVIEW

PROBLEM

SOLUTIONS,FALL2011

p.33

Wecanusetheresultofpart(a)toeliminatete =t0infavorofz.From(a),

te

t0

=(1+z) �1=

:

Therefore,

t0 �te=t0 h1�(1+z) �1= i:

c)(10points)Thepresentvalueofthephysicaldistancetotheobject,`p (t0 ),is

foundfrom

`p (t0 )=a(t0 ) Z

t0

te

ca(t)dt:

Calculatingthisintegralgives

`p (t0 )=

ct 0

1� "

1t �1

0

�1

t �1

e

#:

Factoringt �1

0

outoftheparenthesesgives

`p (t0 )=

ct0

1� "1� �t0

te �

�1 #

:

ThiscanberewrittenintermsofzandH0usingtheresultofpart(a)aswell

as,

H0=

_a(t0 )

a(t0 )=

t0

:

Finallythen,

`p (t0 )=cH�1

0

1� h1�(1+z) �

1 i:

d)(10points)Anearlyidenticalproblem

wasworkedthroughinProblem8of

ProblemSet1.

Theenergyoftheobservedphotonswillberedshiftedbyafactorof(1+z).In

additiontherateofarrivalofphotonswillberedshiftedrelativetotherateof

photonemmission,reducingthe uxbyanotherfactorof(1+z).Consequently,

theobservedpowerwillberedshiftedbytwofactorsof(1+z)toP=(1+z)2.

8.286QUIZ1REVIEW

PROBLEM

SOLUTIONS,FALL2011

p.34

Imagineahypotheticalsphereincomovingcoordinatesasdrawnabove,cen-

teredontheradiatingobject,withradiusequaltothecomovingdistance`c .

Nowconsiderthephotonspassingthroughapatchofthespherewithphysical

areaA.IncomovingcoordinatesthepresentareaofthepatchisA=a(t0 )2.

Sincetheobjectradiatesuniformlyinalldirections,thepatchwillintercept

afraction(A=a(t0 )2)=(4�`2c )ofthephotonspassingthroughthesphere.Thus

thepowerhittingtheareaAis

(A=a(t0 )2)

4�`2c

P

(1+z)2

:

Theradiationenergy uxJ,whichisthereceivedpowerperarea,reachingthe

earthisthengivenby

J=

1

4�`p (t0 )2

P

(1+z)2

whereweused`p (t0 )=a(t0 )`c .Usingtheresultofpart(c)towriteJinterms

ofP;H0 ;z;and gives,

J=

H20

4�c2 �1�

�2

P

(1+z)2 h1�(1+z) �

1 i2

:

e)(10points)FollowingthesolutionofProblem

1ofProblem

Set1,wecan

introducea�ctitiousrelaystationthatisatrestrelativetothegalaxy,but

8.286QUIZ1REVIEW

PROBLEM

SOLUTIONS,FALL2011

p.35

locatedjustnexttothejet,betweenthejetandEarth.Asintheprevious

solution,therelaystationsimplyrebroadcaststhesignalitreceivesfromthe

source,atexactlytheinstantthatitreceivesit.Therelaystationtherefore

hasnoe�ectonthesignalreceivedbytheobserver,butallowsustodividethe

problemintotwosimpleparts.

Thedistancebetweenthejetandtherelaystationisveryshortcomparedto

cosmologicalscales,sothee�ectoftheexpansionoftheuniverseisnegligible.

Forthispartoftheproblemwecanusespecialrelativity,whichsaysthatthe

periodwithwhichtherelaystationmeasuresthereceivedradiationisgivenby

�trelaystation= s1�vc

1+vc

��tsource:

NotethatIhaveusedtheformulafrom

thefrontoftheexam,butIhave

changedthesizeofv,sincethesourceinthiscaseismovingtowardtherelay

station,sothelightisblue-shifted.ToobserversonEarth,therelaystationis

justasourceatrestinthecomovingcoordinatesystem,so

�tobserved=(1+z)�trelaystation

:

Thus,

1+zJ ��tobserved

�tsource

=

�tobserved

�trelaystation

�trelaystation

�tsource

=(1+z)jcosmological �(1+z)jspecialrelativity

=(1+z) s1�vc

1+vc

:

Thus,

zJ=(1+z) s1�vc

1+vc

�1:

Noteadded:Inlookingoverthesolutionstothisproblem,Ifoundthatasub-

stantialnumberofstudentswrotesolutionsbasedontheincorrectassumption

thattheDopplershiftcouldbetreatedasifitwereentirelyduetomotion.

ThesestudentsusedthespecialrelativityDopplershiftformulatoconvert

theredshiftzofthegalaxytoavelocityofrecession,thensubtractedfrom

thisthespeedvofthejet,andthenagainusedthespecialrelativityDoppler

shiftformulato�ndtheDopplershiftcorrespondingtothiscompositevelocity.

8.286QUIZ1REVIEW

PROBLEM

SOLUTIONS,FALL2011

p.36

However,asdiscussedattheendofLectureNotes3,thecosmologicalDoppler

shiftisgivenby

1+z��to

�te=a(to )

a(te );

(3.11)

andisnotpurelyane�ectcausedbymotion.Itisreallythecombinede�ect

ofthemotionofthedistantgalaxiesandthegravitational�eldthatexists

betweenthegalaxies,sothespecialrelativityformularelatingztovdoesnot

apply.

PROBLEM

8:DID

YOU

DO

THEREADING?

a)Thelinesweredark,causedbyabsorptionoftheradiationinthecooler,outer

layersofthesun.

b)IndividualstarsintheAndromedaNebulawereresolvedbyHubblein1923.

[Theothernamesanddatesarenotwithoutsigni�cance.In1609Galileo

builthis�rsttelescope;during1609-10heresolvedtheindividualstarsofthe

MilkyWay,andalsodiscoveredthatthesurfaceofthemoonisirregular,that

Jupiterhasmoonsofitsown,thatSaturnhashandles(laterrecognizedas

rings),thatthesunhasspots,andthatVenushasphases.In1755Immanuel

KantpublishedhisUniversalNaturalHistoryandTheoryoftheHeavens,in

whichhesuggestedthatatleastsomeofthenebulaearegalaxieslikeourown.

In1912HenriettaLeavittdiscoveredtherelationshipbetweentheperiodand

luminosityofCepheidvariablestars.Inthe1950sWalterBaadeandAllan

Sandagerecalibratedtheextra-galacticdistancescale,reducingtheaccepted

valueoftheHubbleconstantbyaboutafactorof10.]

c)

(i)True.[In1941,A.McKellardiscoveredthatcyanogencloudsbehaveasif

theyarebathedinmicrowaveradiationatatemperatureofabout2.3 ÆK,

butnoconnectionwasmadewithcosmology.]

(ii)False.[Anyradiationre ectedbythecloudsisfartooweaktobedetected.

Itisthebrightstarlightshiningthroughthecloudthatisdetectable.]

(iii)True.[Electromagneticwavesatthesewavelengthsaremostlyblocked

bytheEarth'satmosphere,sotheycouldnotbedetecteddirectlyuntil

highaltitudeballoonsandrocketswereintroducedintocosmicbackground

radiationresearchinthe1970s.Precisedatawasnotobtaineduntilthe

COBEsatellite,in1990.]

(iv)True.[ThemicrowaveradiationcanboosttheCNmoleculefromitsground

statetoalow-lyingexcitedstate,astateinwhichtheCandNatoms

rotateabouteachother.Thepopulationofthislow-lyingstateistherefore

8.286QUIZ1REVIEW

PROBLEM

SOLUTIONS,FALL2011

p.37

determinedbytheintensityofthemicrowaveradiation.Thispopulation

ismeasuredbyobservingtheabsorptionofstarlightpassingthroughthe

clouds,sincethereareabsorptionlinesinthevisiblespectrumcausedby

transitionsbetweenthelow-lyingstateandhigherenergyexcitedstates.]

(v)False.[Nochemicalreactionsareseen.]

d)Aristarchus.[TheheliocentricpicturewasneveracceptedbyotherGreek

philosophers,however,andwasnotreviveduntilthepublicationofDeRevo-

lutionibusOrbiumCoelestium(OntheRevolutionsoftheCelestialSpheres)by

Copernicusin1543.]

e)(ii)Anypatchofthenightskywouldlookasbrightasthesurfaceofthesun.

[Explanation:Thecruxoftheargumentisthatthebrightnessofanobject,

measuredforexamplebythepowerperarea(i.e., ux)hittingtheretinaof

youreye,doesnotchangeastheobjectismovedfurtheraway.Thepower

fallso�withthesquareofthedistance,butsodoestheareaoftheimageon

yourretina|

sothepowerperareaisindependentofdistance.Underthe

assumptionsstated,yourlineofsightwilleventuallyhitastarnomatterwhat

directionyouarelooking.Theenergy uxonyourretinawillthereforebethe

sameasintheimageofthesun,sotheentireskywillappearasbrightasthe

surfaceofthesun.]

PROBLEM

9:A

FLATUNIVERSEWITH

a(t)/

t3=5

a)Ingeneral,theHubbleconstantisgivenbyH=_a=a,wheretheoverdotdenotes

aderivativewithrespecttocosmictimet.Inthiscase

H=

1bt3=5

35bt �2=5=

35t:

b)Ingeneral,the(physical)horizondistanceisgivenby

`p;horizon (t)=a(t) Z

t0

ca(t 0)dt 0:

Inthiscaseonehas

`p;horizon (t)=bt3=5 Zt

0

cbt 03=5dt 0=ct3=552 ht2=5�02=5 i=

52ct:

8.286QUIZ1REVIEW

PROBLEM

SOLUTIONS,FALL2011

p.38

c)Thecoordinatespeedoflightisc=a(t),sothecoordinatedistancethatlight

travelsbetweentA

andtB

isgivenby

`c= Z

tB

tA

ca(t 0)dt 0= Z

tB

tA

cbt 03=5dt 0=

5c

2b �t2=5

B

�t2=5

A �:

d)Thephysicalseparationisjustthescalefactortimesthecoordinateseparation,

so

`p (tA)=a(tA)`c=

52ctA "�tBt

A �2=5�

1 #:

`p (tB)=a(tB)`c=

52ctB "1� �tA

tB �

2=5 #

:

e)Letteqbethetimeatwhichthelightpulseisequidistantfromthetwogalaxies.

Atthistimeitwillhavetraveledacoordinatedistance`c =2,where`cisthe

answertopart(c).Sincethecoordinatespeedisc=a(t),thetimeteq

canbe

foundfrom:

Zteq

tA

ca(t 0)dt 0=12

`c

5c

2b �t2=5

eq

�t2=5

A �=5c

4b �t2=5

B

�t2=5

A �

Solvingforteq ,

teq= "t2=5

A

+t2=5

B

2

#5=2

:

f)AccordingtoHubble'slaw,thespeedisequaltoHubble'sconstanttimesthe

physicaldistance.Bycombiningtheanswerstoparts(a)and(d),onehas

v=H(tA)`p (tA)

=

35tA

52ctA "�tBt

A �2=5�

1 #=

32c "�tBt

A �2=5�

1 #:

8.286QUIZ1REVIEW

PROBLEM

SOLUTIONS,FALL2011

p.39

g)Theredshiftforradiationobservedattimetcanbewrittenas

1+z=

a(t)

a(te );

whereteisthetimethattheradiationwasemitted.Solvingforte ,

te=

t

(1+z)5=3

:

Asfoundinpart(d),thephysicaldistancethatthelighttravelsbetweente

andt,asmeasuredattimet,isgivenby

`p (t)=a(t) Z

tte

ca(t 0)dt 0=52

ct "1� �tet �2=5 #

:

Substitutingtheexpressionforte ,onehas

`p (t)=52

ct �1�

1

(1+z)2=3 �:

Asz!1,thisexpressionapproaches

limz!1`p (t)=52

ct;

whichisexactlyequaltothehorizondistance.Itisageneralrulethatthe

horizondistancecorrespondstoin�niteredshiftz.

h)Againwewillviewtheproblem

incomovingcoordinates.PutgalaxyBat

theorigin,andgalaxyAatacoordinatedistance`calongthex-axis.Drawa

sphereofradius`c ,centeredgalaxyA.AlsodrawadetectorongalaxyB,with

8.286QUIZ1REVIEW

PROBLEM

SOLUTIONS,FALL2011

p.40

physicalareaA(measuredatthepresenttime).

Theenergyfromthequasarwillradiateuniformlyonthesphere.Thedetector

hasaphysicalareaA,sointhecomovingcoordinatepictureitsareainsquare

notcheswouldbeA=a(tB)2.Thedetectorthereforeoccupiesafractionofthe

spheregivenby

[A=a(tB)2]

4�`2c

=

A

4�`p (tB)2

;

sothisfractionoftheemittedphotonswillstrikethedetector.

Nextconsidertherateofarrivalofthephotonsatthesphere.Inlecturewe

�guredoutthatifaperiodicwaveisemittedattimetA

andobservedattime

tB,thentherateofarrivalofthewavecrestswillbeslowerthantherateof

emissionbyaredshiftfactor1+z=a(tB)=a(tA).Thesameargumentwill

applytotherateofarrivalofphotons,sotherateofphotonarrivalatthe

spherewillbeslowerthantherateofemissionbythefactor1+z,reducingthe

energy uxbythisfactor.Inaddition,eachphotonisredshiftedinfrequency

by1+z.Sincetheenergyofeachphotonisproportionaltoitsfrequency,the

energy uxisreducedbyanadditionalfactorof1+z.Thus,therateatwhich

energyreachesthedetectoris

Powerhittingdetector=

A

4�`p (tB)2

P

(1+z)2

:

TheredshiftzofthelightpulsereceivedatgalaxyBisgivenby

1+z=a(tB)

a(tA)= �tBt

A �3=5

:

8.286QUIZ1REVIEW

PROBLEM

SOLUTIONS,FALL2011

p.41

Usingoncemoretheexpressionfor`P(tB)frompart(d),onehas

J=Powerhittingdetector

A

=

P(tA=tB)6=5

25�c2t2B �1� �tA

tB �

2=5 �2

:

TheproblemiswordedsothattA,andnotz,isthegivenvariablethatdeter-

mineshowfargalaxyAisfromgalaxyB.Inpractice,however,itisusually

moreusefultoexpresstheanswerintermsoftheredshiftzofthereceived

radiation.Onecandothisbyusingtheaboveexpressionfor1+ztoeliminate

tA

infavorofz,�nding

J=

P

25�c2t2B(1+z)2=3 �(1+z)2=3�1 �2

:

i)Lett 0AbethetimeatwhichthelightpulsearrivesbackatgalaxyA.Thepulse

mustthereforetravelacoordinatedistance`c(theanswertopart(c))between

timetB

andt 0A,so

Zt0A

tB

ca(t 0)dt 0=`c:

Usingtheanswerfrom(c)andintegratingtheleft-handside,

5c

2b �t 02=5

A

�t2=5

B �=5c

2b �t2=5

B

�t2=5

A �:

Solvingfort 0A;

t 0A= �2t2=5

B

�t2=5

A �5=2

:

PROBLEM

10:DID

YOU

DO

THEREADING?

a)Einsteinbelievedthattheuniversewasstatic,andthecosmologicaltermwas

necessarytopreventastaticuniversefromcollapsingundertheattractiveforce

ofnormalgravity.[Therepulsivee�ectofacosmologicalconstantgrowslin-

earlywithdistance,soifthecoeÆcientissmallitisimportantonlywhenthe

separationsareverylarge.Suchatermcanbeimportantcosmologicallywhile

stillbeingtoosmalltobedetectedbyobservationsofthesolarsystemoreven

thegalaxy.Recentmeasurementsofdistantsupernovas(z�1),whichyou

8.286QUIZ1REVIEW

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mayhavereadaboutinthenewspapers,makeitlooklikemaybethereisa

cosmologicalconstantafterall!Sincethecosmologicalconstantisthehotissue

incosmologythisseason,wewillwanttolookatitmorecarefully.Thebest

timewillbeafterLectureNotes7.]

b)Atthetimeofitsdiscovery,deSitter'smodelwasthoughttobestatic[although

itwasknownthatthemodelpredictedaredshiftwhich,atleastfornearby

galaxies,wasproportionaltothedistance].Fromamodernperspectivethe

modelisthoughttobeexpanding.

[Itseemsstrangethatphysicistsin1917couldnotcorrectlydetermineif

thetheorydescribedauniversethatwasstaticorexpanding,butthemath-

ematicalformalism

ofgeneralrelativitycanberatherconfusing.Thebasic

problemisthatwhenspaceisnotEuclideanthereisnosimplewaytoassign

coordinatestoit.Themathematicsofgeneralrelativityisdesignedtobevalid

foranycoordinatesystem,buttheunderlyingphysicscansometimesbeob-

scuredbyapeculiarchoiceofcoordinates.Achangeofcoordinatescannot

onlydistorttheapparentgeometryofspace,butitcanalsomixupspaceand

time.ThedeSittermodelwas�rstwrittendownincoordinatesthatmadeit

lookstatic,soeveryonebelieveditwas.LaterArthurEddingtonandHermann

Weyl(independently)calculatedthetrajectoriesoftestparticles,discovering

thatthey ewapart.]

c)n1=3,andn2=4.

d)Above3,000Ktheuniversewassohotthattheatomswereionized,dissociated

intonucleiandfreeelectrons.Ataboutthistemperature,however,theuniverse

wascoolenoughsothatthenucleiandelectronscombinedtoform

neutral

atoms.

[Thisprocessisusuallycalled\recombination,"althoughthepre�x\re-

"istotallyinaccurate,sinceinthebigbangtheorytheseconstituentshad

neverbeenpreviouslycombined.AsfarasIknowthewordwas�rstusedin

thiscontextbyP.J.E.Peebles,soIonceaskedhimwhythepre�xwasused.

Herepliedthatthiswordisstandardterminologyinplasmaphysics,andwas

carriedoverintocosmology.]

[Regardlessofitsname,recombinationwascrucialfortheclumpingof

matterintogalaxiesandstars,becausethepressureofthephotonsintheearly

universewasenormous.Whenthematterwasionized,thefreeelectronsinter-

actedstronglywiththephotons,sothepressureofthesephotonspreventedthe

matterfromclumping.Afterrecombination,however,thematterbecamevery

transparenttoradiation,andthepressureoftheradiationbecameine�ective.]

[Incidentally,atroughlythesametimeasrecombination(withbiguncer-

tainties),themassdensityoftheuniversechangedfrombeingdominatedby

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radiation(photonsandneutrinos)tobeingdominatedbynonrelativisticmat-

ter.Thereisnoknownunderlyingconnectionbetweenthesetwoevents,andit

seemstobesomethingofacoincidencethattheyoccurredataboutthesame

time.Thetransitionfrom

radiation-dominationtomatter-dominationalso

helpedtopromotetheclumpingofmatter,butthee�ectwasmuchweaker

thanthee�ectofrecombination|

becauseoftheveryhighvelocityofphotons

andneutrinos,theirpressureremainedasigni�cantforceevenaftertheirmass

densitybecamemuchsmallerthanthatofmatter.]

PROBLEM

11:ANOTHERFLATUNIVERSEWITH

a(t)/

t3=5

a)AccordingtoEq.(3.7)oftheLectureNotes,

H(t)=

1a(t)

dad

t:

Forthespecialcaseofa(t)=bt3=5,thisgives

H(t)=

1bt3=535

bt �2=5=

35t:

b)AccordingtoEq.(3.8)oftheLectureNotes,thecoordinatevelocityoflight(in

comovingcoordinates)isgivenbyd

xdt=

ca(t):

SincegalaxiesAandBhavephysicalseparation`0attimet1 ,theircoordinate

separationisgivenby

`c=

`0

bt3=5

1

:

Theradiosignalmustcoverthiscoordinatedistanceinthetimeintervalfrom

t1tot2 ,whichimpliesthatZ

t2

t1

ca(t)dt=

`0

bt3=5

1

:

Usingtheexpressionfora(t)andintegrating,

5c

2b �t2=5

2

�t2=5

1 �=

`0

bt3=5

1

;

8.286QUIZ1REVIEW

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SOLUTIONS,FALL2011

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whichcanbesolvedfort2togive

t2= �1+

2`0

5ct1 �

5=2

t1:

c)Themethodisthesameasinpart(b).Thecoordinatedistancebetweenthe

twogalaxiesisunchanged,butthistimethedistancemustbetraversedinthe

timeintervalfromt2tot3 .So,

Zt3

t2

ca(t)dt=

`0

bt3=5

1

;

whichleadsto

5c

2b �t2=5

3

�t2=5

2 �=

`0

bt3=5

1

:

Solvingfort3gives

t3= "�t2

t1 �

2=5

+

2`0

5ct1 #

5=2

t1:

Theaboveanswerisperfectlyacceptable,butonecouldalsoreplacet2byusing

theanswertopart(b),whichgives

t3= �1+

4`0

5ct1 �

5=2

t1:

[Alternatively,onecouldhavebeguntheproblembyconsideringthefull

roundtripoftheradiosignal,whichtravelsacoordinatedistance2`cduring

thetimeintervalfromt1tot3 .Theproblemthenbecomesidenticaltopart(b),

exceptthatthecoordinatedistance`cisreplacedby2`c ,andt2isreplacedby

t3 .Oneisledimmediatelytotheanswerintheformofthepreviousequation.]

d)Cosmictimeisde�nedbythereadingofsuitablysynchronizedclockswhichare

eachatrestwithrespecttothematteroftheuniverseatthesamelocation.(For

thisproblemwewillnotneedtothinkaboutthemethodofsynchronization.)

Thus,thecosmictimeintervalbetweenthereceiptofthemessageandthe

8.286QUIZ1REVIEW

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SOLUTIONS,FALL2011

p.45

responseisthesameaswhatismeasuredonthegalaxyBclocks,whichis�t.

Theresponseisthereforesentatcosmictimet2+�t.Thecoordinatedistance

betweenthegalaxiesisstill`0 =a(t1 ),so

Zt4

t2+�t

ca(t)dt=

`0

bt3=5

1

:

Integrationgives

5c

2b ht

2=5

4

�(t2+�t)2=5 i=

`0

bt3=5

1

;

whichcanbesolvedfort4togive

t4= "�t2+�t

t1

�2=5

+

2`0

5ct1 #

5=2

t1:

e)Fromtheformulaatthefrontoftheexam,

1+z=a(tobserved )

a(temitted )=

a(t4 )

a(t2+�t)= �t4

t2+�t �

3=5

:

So,

z=a(tobserved )

a(temitted )=

a(t4 )

a(t2+�t)= �t4

t2+�t �

3=5�

1:

f)If�tissmallcomparedtothetimethatittakesa(t)tochangesigni�cantly,

thentheintervalbetweenasignalsentatt3andasignalsentatt3+�twillbe

receivedwitharedshiftidenticaltothatobservedbetweentwosuccessivecrests

ofawave.Thus,theseparationbetweenthereceiptoftheacknowledgement

andthereceiptoftheresponsewillbeafactor(1+z)timeslongerthanthe

timeintervalbetweenthesendingofthetwosignals,andtherefore

t4 �t3=(1+z)�t+O(�t2)

= �t4

t2+�t �

3=5

�t+O(�t2):

8.286QUIZ1REVIEW

PROBLEM

SOLUTIONS,FALL2011

p.46

Sincetheanswercontainsanexplicitfactorof�t,theotherfactorscanbe

evaluatedtozerothorderin�t:

t4 �t3= �t4

t2 �

3=5

�t+O(�t2);

whereto�rstorderin�tthet4inthenumeratorcouldequallywellhavebeen

replacedbyt3 .

Forthosewhopreferthebruteforceapproach,theanswertopart(d)can

beTaylorexpandedinpowersof�t.To�rstorderonehas

t4=t3+

@t4

@�t �����t=0�t+O(�t2):

Evaluatingthenecessaryderivativegives

@t4

@�t= "�t2+�t

t1

�2=5

+

2`0

5ct1 #

3=2�

t2+�t

t1

��3=5

;

whichwhenspecializedto�t=0becomes

@t4

@�t �����t=0= "�t2

t1 �

2=5

+

2`0

5ct1 #

3=2�

t2

t1 �

�3=5

:

Usingthe�rstboxedanswertopart(c),thiscanbesimpli�edto

@t4

@�t �����t=0= �t3

t1 �

3=5 �t2

t1 �

�3=5

= �t3

t2 �

3=5

:

PuttingthisbackintotheTaylorseriesgives

t4 �t3= �t3

t2 �

3=5

�t+O(�t2);

inagreementwiththepreviousanswer.

8.286QUIZ1REVIEW

PROBLEM

SOLUTIONS,FALL2011

p.47

PROBLEM

12:THEDECELERATION

PARAMETER

Fromthefrontoftheexam,weareremindedthat

�a=�4�3

G�a

and

�_aa �2

=8�3

G��kc2

a2

;

whereadotdenotesaderivativewithrespecttotimet.Thecriticalmassdensity

�cisde�nedtobethemassdensitythatcorrespondstoa at(k=0)universe,so

fromtheequationaboveitfollowsthat

�_aa �2

=8�3

G�c:

Substitutingintothede�nitionofq,we�nd

q=��a(t)a(t)

_a2(t)=��aa �a_a �2

= �4�3

G� ��3

8�G�c �=12��

c=

12:

PROBLEM

13:A

RADIATION-DOMINATED

FLATUNIVERSE

The atnessofthemodeluniversemeansthatk=0,so

�_aa �2

=8�3

G�:

Since

�(t)/1

a4(t);

itfollowsthat

dad

t=const

a

:

Rewritingthisas

ada=constdt;

8.286QUIZ1REVIEW

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SOLUTIONS,FALL2011

p.48

theinde�niteintegralbecomes

12a2=(const)t+c 0;

wherec 0isaconstantofintegration.Di�erentchoicesforc 0correspondtodi�erent

choicesforthede�nitionoft=0.Wewillfollowthestandardconventionofchoosing

c 0=0,whichsetst=0tobethetimewhena=0.Thustheaboveequationimplies

thata2/t,andtherefore

a(t)/t1=2

foraphoton-dominated atuniverse.

PROBLEM

14:DID

YOU

DO

THEREADING?(25points)

(a)In1826,theastronomerHeinrichOlberwroteapaperonaparadoxregarding

thenightsky.WhatisOlber'sparadox?Whatistheprimaryresolutionofit?

(Ryden,Chapter2,Pages6-8)

Ans:Olber'sparadoxisthatthenightskyappearstobedark,insteadofbeing

uniformlybright.Theprimaryresolutionisthattheuniversehasa�niteage,

andsothelightfrom

starsbeyondthehorizondistancehasnotreachedus

yet.(However,eveninthesteady-statemodeloftheuniverse,theparadox

isresolvedbecausethelightfromdistantstarswillbered-shiftedbeyondthe

visiblespectrum).

(b)WhatisthevalueoftheNewtoniangravitationalconstantGinPlanckunits?

ThePlancklengthisoftheorderof10 �35m,10 �15m,1015m,or1035m?

(Ryden,Chapter1,Page3)

Ans:G=1inPlanckunits,byde�nition.

ThePlancklengthisoftheorderof10 �35m.(Notethatthisanswercouldbe

obtainedbyaprocessofeliminationaslongasyourememberthatthePlanck

lengthismuchsmallerthan10 �15m,whichisthetypicalsizeofanucleus).

(c)WhatistheCosmologicalPrinciple?IstheHubbleexpansionoftheuniverse

consistentwithit?

(Weinberg,Chapter2,Pages21-23;Ryden,Chapter2,Page11)

Ans:TheCosmologicalPrinciplestatesthatthereisnothingspecialaboutour

locationintheuniverse,i.e.theuniverseishomogeneousandisotropic.

Yes,theHubbleexpansionisconsistentwithit(sincethereisnocenterof

expansion).

8.286QUIZ1REVIEW

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(d)Inthe\StandardModel"oftheuniverse,whentheuniversecooledtoabout

3�10aK,itbecametransparenttophotons,andtodayweobservetheseasthe

CosmicMicrowaveBackground(CMB)atatemperatureofabout3�10bK.

Whataretheintegersaandb?

(Weinberg,Chapter3;Ryden,Chapter2,Page22)

a=3,b=0.

(e)Whatdidtheuniverseprimarilyconsistofatabout1/100thofasecondafter

theBigBang?Includeanyconstituentthatisbelievedtohavemadeupmore

than1%ofthemassdensityoftheuniverse.

(Weinberg,Chapter1,Page5)

Ans:Electrons,positrons,neutrinos,andphotons.

PROBLEM

15:SPECIALRELATIVITYDOPPLER

SHIFT(20points)

(a)Theeasiestwaytosolvethisproblemisbyadoubleapplicationofthestandard

special-relativityDopplershiftformula,whichwasgivenonthefrontofthe

exam:

z= s1+�

1���1;

(18.1)

where�=v=c.Rememberingthatthewavelengthisstretchedbyafactor

1+z,we�ndimmediatelythatthewavelengthoftheradiowavereceivedat

Alpha-7isgivenby

�Alpha�7= s1+vs =c

1�vs =c�emitted

:

(18.2)

Thephotonsthatarereceivedbytheobserverareinfactneverreceivedby

Alpha-7,butthewavelengthfoundbytheobserverwillbethesameasif

Alpha-7actedasarelaystation,receivingthephotonsandretransmittingthem

atthereceivedwavelength.So,applyingEq.(18.1)again,thewavelengthseen

bytheobservercanbewrittenas

�observed= s1+vo =c

1�vo =c�Alpha�7

:

(18.3)

CombiningEqs.(18.2)and(18.3),

�observed= s1+vo =c

1�vo =c s

1+vs =c

1�vs =c�emitted;

(18.4)

8.286QUIZ1REVIEW

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SOLUTIONS,FALL2011

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so�nally

z= s1+vo =c

1�vo =c s

1+vs =c

1�vs =c �1:

(18.5)

(b)AlthoughweusedthepresenceofAlpha-7indeterminingtheredshiftzof

Eq.(18.5),theredshiftisnotactuallya�ectedbythespacestation.Sothe

special-relativityDopplershiftformula,Eq.(18.1),mustdirectlydescribethe

redshiftresultingfromtherelativemotionofthesourceandtheobserver.Thus

s1+vtot =c

1�vtot =c �1= s1+vo =c

1�vo =c s

1+vs =c

1�vs =c �1:

(18.6)

Theequationabovedeterminesvtotintermsofvoandvs ,sotherestisjust

algebra.Tosimplifythenotation,let�tot �vtot =c,�o �vo =c,and�s �vs =c.

Then

1+�tot=1+�o

1��o

1+�s

1��s(1��tot )

�tot �1+1+�o

1��o

1+�s

1��s �=1+�o

1��o

1+�s

1��s �1

�tot �(1��o ��s+�o �s )+(1+�o+�s+�o �s )

(1��o )(1��s )

�=

(1+�o+�s+�o �s )�(1��o ��s+�o �s )

(1��o )(1��s )

�tot [2(1+�o �s )]=2(�o+�s )

�tot=

�o+�s

1+�o �s

vtot=

vo+vs

1+vo vs

c2

:

(18.7)

The�nalformulaistherelativisticexpressionfortheadditionofvelocities.

Notethatitguaranteesthatjvtot j�caslongasjvo j�candjvs j�c.

8.286QUIZ1REVIEW

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PROBLEM

16:DID

YOU

DO

THEREADING?(25points)

(a)(4points)Whatwasthe�rstexternalgalaxythatwasshowntobeatadistance

signi�cantlygreaterthanthemostdistantknownobjectsinourgalaxy?How

wasthedistanceestimated?

Ans:(Weinberg,page20)The�rstgalaxyshowntobeatadistancebeyondthe

sizeofourgalaxywasAndromeda,alsoknownbyitsMessiernumber,M31.

Itisthenearestspiralgalaxytoourgalaxy.Thedistancewasdetermined

(byHubble)usingCepheidvariablestars,forwhichtheabsoluteluminosityis

proportionaltotheperiod.AmeasurementofaparticularCepheid'speriod

determinesthestar'sabsoluteluminosity,which,comparedtothemeasured

luminosity,determinesthedistancetothestar.(Hubble'sinitialmeasurement

ofthedistancetoAndromedausedabadly-calibratedversionofthisperiod-

luminosityrelationshipandconsequentlyunderestimatedthedistancebymore

thanafactoroftwo;nonetheless,theinitialmeasurementstillshowedthat

theAndromedaNebulawasanorderofmagnitudemoredistantthanthemost

distantknownobjectsinourowngalaxy.)

(b)(5points)Whatisrecombination?Didgalaxiesbegintoformbeforeorafter

recombination?Why?

Ans:(Weinberg,pages64and73)Recombinationreferstotheformationof

neutralatomsoutofchargednucleiandelectrons.Galaxiesbegantoform

afterrecombination.Priortorecombination,thestrongelectromagneticinter-

actionsbetweenphotonsandmatterproducedahighpressurewhiche�ectively

counteractedthegravitationalattractionbetweenparticles.Oncetheuniverse

becametransparenttoradiation,thematternolongerinteractedsigni�cantly

withthephotonsandconsequentlybegantoundergogravitationalcollapseinto

largeclumps.

(c)(4points)InChapterIVofhisbook,Weinbergdevelopsa\recipeforahot

universe,"inwhichthematteroftheuniverseisdescribedasagasinthermal

equilbriumataveryhightemperature,inthevicinityof109K(severalthou-

sandmilliondegreesKelvin).Suchathermalequilibrium

gasiscompletely

describedbyspecifyingitstemperatureandthedensityoftheconservedquan-

tities.Whichofthefollowingisonthislistofconservedquantities?Circleas

manyasapply.

(i)baryonnumber

(ii)energyperparticle

(iii)protonnumber

(iv)electriccharge

(v)pressure

Ans:(Weinberg,page91)Thecorrectanswersare(i)and(iv).Athirdcon-

servedquantity,leptonnumber,wasnotincludedinthemultiple-choiceoptions.

(d)(4points)ThewavelengthcorrespondingtothemeanenergyofaCMB(cosmic

microwavebackground)photontodayisapproximatelyequaltowhichofthe

8.286QUIZ1REVIEW

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SOLUTIONS,FALL2011

p.52

followingquantities?(Youmaywishtolookupthevaluesofvariousphysical

constantsattheendofthequiz.)

(i)2fm(2�10 �15m)

(ii)2microns(2�10 �6m)

(iii)2mm(2�10 �3m)

(iv)2m.

Ans:(Ryden,page23)Thecorrectansweris(iii).

Ifyoudidnotrememberthisnumber,youcouldestimatetheanswerbyremem-

beringthatthecharacteristictemperatureofthecosmicmicrowavebackground

isapproximately3Kelvin.Thetypicalphotonenergyisthenontheorderof

kT,fromwhichwecan�ndthefrequencyasE=h�.Thewavelengthofthe

photonisthen�=�=c.Thisapproximationgives�=5:3mm,whichisnot

equaltothecorrectanswer,butitismuchclosertothecorrectanswerthanto

anyoftheotherchoices.

(e)(4points)Whatistheequivalenceprinciple?

Ans:(Ryden,page27)Initssimplestform,theequivalenceprinciplesaysthat

thegravitationalmassofanobjectisidenticaltoitsinertialmass.Thisequality

impliestheequivalentstatementthatitisimpossibletodistinguish(without

additionalinformation)betweenanobserverinareferenceframeaccelerating

withacceleration~aandanobserverinaninertialreferenceframesubjecttoa

gravitationalforce�mobs ~a.

(Actually,whattheequivalenceprinciplereallysaysisthattheratioofthe

gravitationaltoinertialmassesmg =miisuniversal,thatis,independentofthe

materialpropertiesoftheobjectinquestion.Theratiodoesnotnecessarily

needtobe1.However,onceweknowthatthetwotypesofmassesarepro-

portional,wecansimplyde�nethegravitationalcouplingG

tomakethem

equal.Toseethis,consideratheoryofgravitywheremg =mi=q.Thenthe

gravitationalforcelawis

mi a=�GMmg

r2

;

or

a=�GqM

r2

:

Atthispoint,ifwede�neG0=Gq,wehaveagravitationaltheorywithgravi-

tationalcouplingG0andinertialmassequaltogravitationalmass.)

(f)(4points)WhyisitdiÆcultforEarth-basedexperimentstolookatthesmall

wavelengthportionofthegraphofCMBenergydensityperwavelengthvs.

wavelength?

8.286QUIZ1REVIEW

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Ans:(Weinberg,page67)TheEarth'satmosphereisincreasinglyopaquefor

wavelengthshorterthan.3cm.Therefore,radiationatthesewavelengthswill

beabsorbedandrescatteredbytheEarth'satmosphere;observationsofthe

cosmicmicrowavebackgroundatsmallwavelengthsmustbeperformedabove

theEarth'satmosphere.

PROBLEM

17:

TRACING

A

LIGHT

PULSE

THROUGH

A

RADIATION-DOMINATED

UNIVERSE

(a)Thephysicalhorizondistanceisgiveningeneralby

`p;horizon=a(t) Z

tf

0

ca(t)dt;

sointhiscase

`p;horizon=bt1=2 Ztf

0

cbt1=2dt=

2ctf:

(b)Ifthesourceisatthehorizondistance,itmeansthataphotonleavingthe

sourceatt=0wouldjustbereachingtheoriginattf .So,te=0.

(c)Thecoordinatedistancebetweenthesourceandtheoriginisthecoordinate

horizondistance,givenby

`c;horizon= Z

tf

0

cbt1=2dt=

2ct1=2

fb

:

(d)Thephotonstartsatcoordinatedistance2c ptf=b,andbytimetitwillhave

traveledacoordinatedistanceZ

t0

cbt 01=2dt 0=2c pt

b

towardtheorigin.Thusthephotonwillbeatcoordinatedistance

`c=2cb �p

tf �p

t �

fromtheorigin,andhenceaphysicaldistance

`p (t)=a(t)`c=

2c �pttf �t �:

8.286QUIZ1REVIEW

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SOLUTIONS,FALL2011

p.54

(e)To�ndthemaximumof`p (t),wedi�erentiateitandsetthederivativetozero:

d`p

dt= rtft�2 !c;

sothemaximumoccurswhen

rtf

tmax

=2;

or

tmax=14

tf:

PROBLEM

18:TRANSVERSEDOPPLER

SHIFTS

(a)Describingtheeventsinthecoordinatesystemshown,theXanthuisatrest,

soitsclocksrunatthesamespeedasthecoordinatesystemtimevariable,t.

Theemissionofthewavecrestsoftheradiosignalarethereforeseparatedbya

timeintervalequaltothetimeintervalasmeasuredbythesource,theXanthu:

�t=�ts:

SincetheEmmeracismovingperpendiculartothepathoftheradiowaves,

atthemomentofreceptionitsdistancefrom

theXanthuisataminimum,

andhenceitsrateofchangeiszero.Hencesuccessivewavecrestswilltravel

thesamedistance,aslongasc�t�a.Sincethewavecreststravelthesame

distance,thetimeseparationoftheirarrivalattheEmmeracis�t,thesame

asthetimeseparationoftheiremission.TheclocksontheEmmerac,however,

andrunningslowlybyafactorof

=

1

q1�v2

c2

:

Thetimeintervalbetweenwavecrestsasmeasuredbythereceiver,onthe

Emmerac,isthereforesmallerbyafactorof ,

�tr=�ts

:

Thus,thereisablueshift.Theredshiftparameterzisde�nedby

�tr

�ts=1+z;

8.286QUIZ1REVIEW

PROBLEM

SOLUTIONS,FALL2011

p.55

so

1 =1+z;

or

z=1�

:

Recallthat >1,sozisnegative.

(b)Describingthissituationinthecoordinatesystemshown,thistimethesource

ontheXanthuismoving,sotheclocksatthesourcearerunningslowly.The

timebetweenwavecrests,measuredincoordinatetimet,isthereforelargerby

afactorof than�ts ,thetimeasmeasuredbytheclockonthesource:

�t= �ts:

SincetheradiosignalisemittedwhentheXanthuisatitsminimumsepara-

tionfromtheEmmerac,therateofchangeoftheseparationiszero,soeach

wavecresttravelsthesamedistance(againassumingthatc�t�a).Sincethe

Emmeracisatrest,itsclocksrunatthesamespeedasthecoordinatetimet,

andhencethetimeintervalbetweencrests,asmeasuredbythereceiver,is

�tr=�t= �ts:

Thusthetimeintervalasmeasuredbythereceiverislongerthanthatmeasured

bythesource,andhenceitisaredshift.Theredshiftparameterzisgivenby

1+z=�tr

�ts= ;

so

z= �1:

(c)Theeventsdescribedin(a)canbemadetolookalotliketheeventsdescribed

in(b)bytransformingtoaframeofreferencethatismovingtotherightat

speedv0

|

i.e.,bytransformingtotherestframeoftheEmmerac.Inthis

frametheEmmeracisofcourseatrest,andtheXanthuistravelingonthe

trajectory

(x=�v0 t;y=a;z=0);

asinpart(b).However,justasthetransformationcausesthex-component

ofthevelocityoftheXanthutochangefromzerotoanegativevalue,sothe

x-componentofthevelocityoftheradiosignalwillbetransformedfromzeroto

anegativevalue.Thusinthisframetheradiosignalwillnotbetravelingalong

they-axis,sotheeventswillnotmatchthosedescribedin(b).Thesituations

describedin(a)and(b)arethereforephysicallydistinct(whichtheymustbe

iftheredshiftsaredi�erent,aswecalculatedabove).

8.286QUIZ1REVIEW

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p.56

PROBLEM

19:A

TWO-LEVELHIGH-SPEED

MERRY-GO-ROUND

(15points)

(a)Sincetherelativepositionsofallthecarsremain�xedasthemerry-go-round

rotates,eachsuccessivepulsefrom

anygivencartoanyothercartakesthe

sameamountoftimetocompleteitstrip.ThustherewillbenoDopplershift

causedbypulsestakingdi�erentamountsoftime;theonlyDopplershiftwill

comefromtimedilation.

Wewilldescribetheeventsfrom

thepointofviewofaninertialreference

frameatrestrelativetothehubofthemerry-go-round,whichwewillcallthe

laboratoryframe.Thisistheframeinwhichtheproblemisdescribed,inwhich

theinnercarsaremovingatspeedv,andtheoutercarsaremovingatspeed

2v.Inthelaboratoryframe,thetimeintervalbetweenthewavecrestsemitted

bythesource�tLab

S

willbeexactlyequaltothetimeinterval�tLab

O

between

twocrestsreachingtheobserver:

�tLab

O

=�tLab

S

:

Theclocksonthemerry-go-roundcarsaremovingrelativetothelaboratory

frame,sotheywillappeartoberunningslowlybythefactor

1=

1

p1�v2=c2

fortheinnercars,andbythefactor

2=

1

p1�4v2=c2

fortheoutercars.Thus,ifwelet�tS

denotethetimebetweencrestsas

measuredbyaclockonthesource,and�tO

asthetimebetweencrestsas

8.286QUIZ1REVIEW

PROBLEM

SOLUTIONS,FALL2011

p.57

measuredbyaclockmovingwiththeobserver,thenthesequantitiesarerelated

tothelaboratoryframetimesby

2 �tS=�tLab

S

and 1 �tO

=�tLab

O

:

Tomakesurethatthe -factorsareontherightsideoftheequation,you

shouldkeepinmindthatanytimeintervalshouldbemeasuredasshorteron

themovingclocksthanonthelabclocks,sincetheseclocksappeartorun

slowly.Puttingtogethertheequationsabove,onehasimmediatelythat

�tO

= 2

1�tS

:

Theredshiftzisde�nedby

�tO

�(1+z)�tS

;

so

z= 2

1 �1= s1�v2

c2

1�4v2

c2

�1:

(b)Forthispartoftheproblemisusefultoimaginearelaystationlocatedjustto

therightofcar6inthediagram,atrestinthelaboratoryframe.Therelay

stationrebroadcaststhewavesasitreceivesthem,andhencehasnoe�ecton

thefrequencyreceivedbytheobserver,butservesthepurposeofallowingus

toclearlyseparatetheproblemintotwoparts.

The�rstpartofthediscussionconcernstheredshiftofthesignalasmeasured

bytherelaystation.Thiscalculationwouldinvolveboththetimedilationand

8.286QUIZ1REVIEW

PROBLEM

SOLUTIONS,FALL2011

p.58

achangeinpathlengthsbetweensuccessivepulses,butwedonotneedtodo

it.Itisthestandardsituationofasourceandobservermovingdirectlyaway

fromeachother,asdiscussedattheendofLectureNotes1.TheDopplershift

isgivenbyEq.(1.33),whichwasincludedintheformulasheet.Writingthe

formulaforarecessionspeedu,itbecomes

(1+z)jrelay= s1+uc

1�uc

:

Ifweagainusethesymbol�tS

forthetimebetweenwavecrestsasmeasured

byaclockonthesource,thenthetimebetweenthereceiptofwavecrestsas

measuredbytherelaystationis

�tR

= s1+uc

1�uc

�tS

:

Thesecondpartofthediscussionconcernsthetransmissionfrom

therelay

stationtocar6.Thevelocityofcar6isperpendiculartothedirectionfrom

whichthepulseisbeingreceived,sothisisatransverseDopplershift.Any

changeinpathlengthbetweensuccessivepulsesissecondorderin�t,soitcan

beignored.Theonlye�ectisthereforethetimedilation.Asdescribedinthe

laboratoryframe,thetimeseparationbetweencrestsreachingtheobserveris

thesameasthetimeseparationmeasuredbytherelaystation:

�tLab

O

=�tR

:

Asinpart(a),thetimedilationimpliesthat

2 �tO

=�tLab

O

:

Combiningtheformulasabove,

�O

=

1 2 s

1+uc

1�uc

�tS

:

Again�tO

�(1+z)�tS,so

z=

1 2 s

1+uc

1�uc

�1= s�1�

4v2

c2 ��1+uc �

1�uc

�1:

8.286QUIZ1REVIEW

PROBLEM

SOLUTIONS,FALL2011

p.59

PROBLEM

20:

SIGNAL

PROPAGATION

IN

A

FLAT

MATTER-

DOMINATED

UNIVERSE(55points)

(a)-(i)Ifwelet`c (t)denotethecoordinatedistanceofthelightsignalfromA,then

wecanmakeuseofEq.(3.8)fromthelecturenotesforthecoordinatevelocity

oflight:

d`c

dt=

ca(t):

(20.1)

Integratingthevelocity,`

c (t)= Z

tt1

cdt 0

a(t 0)=cb Zt

t1

dt 0

t 02=3

=3cb ht1=3�t1=3

1 i:

(20.2)

Thephysicaldistanceisthen

`p;sA(t)=a(t)`c (t)=bt2=33cb ht1=3�t1=3

1 i

=3c �t�t2=3t1=3

1 �

=3ct "1� �t1t �1=3 #

:

(20.3)

Wenowneedtodi�erentiate,whichisdonemosteasilywiththemiddleline

oftheaboveequation:

d`p;sA

dt

=c "3�2 �t1t �1=3 #

:

(20.4)

(ii)Att=t1 ,thetimeofemission,theaboveformulagives

d`p;sA

dt

=c:

(20.5)

Thisiswhatshouldbeexpected,sincethespeedofseparationofthelight

signalatthetimeofemissionisreallyjustalocalmeasurementofthespeed

oflight,whichshouldalwaysgivethestandardvaluec.

8.286QUIZ1REVIEW

PROBLEM

SOLUTIONS,FALL2011

p.60

(iii)Atarbitrarilylatetimes,thesecondterminbracketsinEq.(20.4)becomes

negligible,so

d`p;sA

dt

!3c:

(20.6)

Althoughthisanswerislargerthanc,itdoesnotviolaterelativity.Oncethe

signalisfarfromitsoriginitiscarriedbytheexpansionoftheuniverse,and

relativityplacesnospeedlimitontheexpansionoftheuniverse.

(b)ThispartoftheprobleminvolvesH(t1 ),sowecanstartbyevaluatingit:

H(t)=

_a(t)

a(t)=

ddt (bt2=3)

bt2=3

=

23t:

(20.7)

Thus,thephysicaldistancefromAtoBattimet1is

`p;BA

=32

ct1:

(20.8)

Thecoordinatedistanceisthephysicaldistancedividedbythescalefactor,so

`c;BA

=cH�1(t

1 )

a(t1 )

=

32ct1

bt2=3

1

=3c

2bt1=3

1

:

(20.9)

Sincelighttravelsatacoordinatespeedc=a(t),thelightsignalwillreachgalaxy

Battimet2if

`c;BA

= Zt2

t1

cbt 02=3dt 0

=3cb ht1=3

2

�t1=3

1 i:

(20.10)

Settingtheexpressions(20.9)and(20.10)for`c;BA

equaltoeachother,one

�nds12

t1=3

1

=t1=3

2

�t1=3

1

=)

t1=3

2

=32

t1=3

1

=)

t2=278

t1:

(20.11)

(c)-(i)Physicaldistancesareadditive,soifoneaddsthedistancefromAandthelight

signaltothedistancefromthelightsignaltoB,onegetsthedistancefromA

toB:

`p;sA+`p;sB

=`p;BA

:

(20.12)

8.286QUIZ1REVIEW

PROBLEM

SOLUTIONS,FALL2011

p.61

But`p;BA(t)isjustthescalefactortimesthecoordinateseparation,a(t)`c;BA.

Usingthepreviousrelations(20.3)and(20.9)for`p;sA(t)and`c;BA,we�nd

3ct "1� �t1t �1=3 #

+`p;sB(t)=32

ct1=3

1

t2=3

;

(20.13)

so

`p;sB(t)=92

ct1=3

1

t2=3�3ct=3ct "32 �t1t �1=3�

1 #:

(20.14)

Asacheck,onecanverifythatthisexpressionvanishesfort=t2=(27=8)t1 ,

andthatitequals(3=2)ct1

att=t1 .Butweareaskedto�ndthespeedof

approach,thenegativeofthederivativeofEq.(20.14):

Speedofapproach=�d`p;sB

dt

=�3ct1=3

1

t �1=3+3c

=

3c "1� �t1t �1=3 #

:

(20.15)

(ii)Atthetimeofemission,t=t1 ,Eq.(20.15)gives

Speedofapproach=0:

(20.16)

Thismakessense,sinceatt=t1

galaxyBisoneHubblelengthfromgalaxy

A,whichmeansthatitsrecessionvelocityisexactlyc.Therecessionvelocity

ofthelightsignalleavingAisalsoc,sotherateofchangeofthedistancefrom

thelightsignaltoBisinitiallyzero.

(iii)Atthetimeofreception,t=t2=(27=8)t1 ,Eq.(20.15)gives

Speedofapproach=c;

(20.17)

whichisexactlywhatisexpected.Asinpart(a)-(ii),thisisalocalmeasure-

mentofthespeedoflight.

8.286QUIZ1REVIEW

PROBLEM

SOLUTIONS,FALL2011

p.62

(d)To�ndtheredshift,we�rst�ndthetimetBA

atwhichalightpulsemustbe

emittedfromgalaxyBsothatitarrivesatgalaxyAattimet1 .Usingthe

coordinatedistancegivenbyEq.(20.9),thetimeofemissionmustsatisfy

3c

2bt1=3

1

= Zt1

tBA

cbt 02=3dt 0=3cb �

t1=3

1

�t1=3

BA �;

(20.18)

whichcanbesolvedtogive

tBA

=18

t1:

(20.19)

Theredshiftisgivenby

1+zBA

=

a(t1 )

a(tBA)= �t1

tBA �

2=3

=4:

(20.20)

Thus,

zBA

=3:

(20.21)

(e)ApplyingEuclideangeometrytothetriangleC-A-Bshowsthatthephysical

distancefromCtoB,attimet1 ,is p2cH�1.Thecoordinatedistanceisalso

largerthantheA-Bseparationbyafactorof p2.Thus,

`c;BC

=3 p2c

2b

t1=3

1

:

(20.22)

IfwelettBC

bethetimeatwhichalightpulsemustbeemittedfromgalaxy

BsothatitarrivesatgalaxyCattimet1 ,we�nd

3 p2c

2b

t1=3

1

= Zt1

tBC

cbt 02=3dt 0=3cb �

t1=3

1

�t1=3

BC �;

(20.23)

whichcanbesolvedto�nd

tBC

= 1�p

22 !3

t1:

(20.24)

Then

1+zBC

=

a(t1 )

a(tBC)= �t1

tBC �

2=3

=

1

�1�p

22 �2

;

(20.25)

8.286QUIZ1REVIEW

PROBLEM

SOLUTIONS,FALL2011

p.63

and

zBC

=

1

�1�p

22 �2 �1:

(20.26)

Fullcreditwillbegivenfortheanswerintheformabove,butitcanbesimpli�ed

byrationalizingthefraction:

zBC

=

1

�1�p

22 �2 �

1+p

22 �

2

�1+p

22 �

2 �1

=1+p

2+12

14

�1

=

5+4 p2:

(20.27)

Numerically,zBC

=10:657.

(f)FollowingthesolutiontoProblem6ofProblemSet2,wedrawadiagramin

comovingcoordinates,puttingthesourceatthecenterofasphere:

TheenergyfromgalaxyAwillradiateuniformlyoverthesphere.Ifthedetector

hasphysicalareaAD,theninthecomovingcoordinatepictureithascoordinate

areaAD=a2(t

2 ),sincethedetectionoccursattimet2Thefullcoordinatearea

8.286QUIZ1REVIEW

PROBLEM

SOLUTIONS,FALL2011

p.64

ofthesphereis4�`2c

;BA,sothefractionofphotonsthathitthedetectoris

fraction= �A=a(t2 )2 �

4�`2c

;BA

:

(20.28)

AsinProblem6,thepowerhittingthedetectorisreducedbytwofactorsof

(1+z):onefactorbecausetheenergyofeachphotonisproportionaltothe

frequency,andhenceisreducedbytheredshift,andonemorefactorbecause

therateofarrivalofphotonsisalsoreducedbytheredshiftfactor(1+z).

Thus,

Powerhittingdetector=P �A=a(t2 )2 �

4�`2c

;BA

1

(1+z)2

=P �A=a(t2 )2 �

4�`2c

;BA

�a(t1 )

a(t2 ) �

2

=P

A

4�`2c

;BA

a2(t

1 )

a4(t

2 ):

(20.29)

Theenergy uxisgivenby

J=Powerhittingdetector

A

;

(20.30)

so

J=

P

4�`2c

;BA

a2(t

1 )

a4(t2 ):

(20.31)

Fromhereitisjustalgebra,usingEqs.(20.9)and(20.11),anda(t)=bt2=3:

J=

P

4� h3c

2b t

1=3

1 i2b2t4=3

1

b4t8=3

2

=

P

4� h3c

2b t

1=3

1 i2

b2t4=3

1

�278 �8=3

b4t8=3

1

=

P

4� h3c2t1=3

1 i2

t4=3

1

�32 �8

t8=3

1

=

28

310�

Pc2t21

=

256

59;049�

Pc2t21

:

(20.32)

8.286QUIZ1REVIEW

PROBLEM

SOLUTIONS,FALL2011

p.65

Itisdebatablewhichofthelasttwoexpressionsisthesimplest,soIhaveboxed

bothofthem.Onecouldalsowrite

J=1:380�10 �3

Pc2t21

:

(20.33)

PROBLEM

21:DID

YOU

DO

THEREADING?(25points) y

(a)(10points)TodeterminethedistanceofthegalaxieshewasobservingHubble

usedsocalledstandardcandles.Standardcandlesareastronomicalobjects

whoseintrinsicluminosityisknownandwhosedistanceisinferredbymeasuring

theirapparentluminosity.First,heusedasstandardcandlesvariablestars,

whoseintrinsicluminositycanberelatedtotheperiodofvariation.Quoting

Weinberg'sTheFirstThreeMinutes,chapter2,pages19-20:

In1923EdwinHubblewasforthe�rsttimeabletoresolvetheAndromeda

Nebulaintoseparatestars.Hefoundthatitsspiralarmsincludedafewbright

variablestars,withthesamesortofperiodicvariationofluminosityaswas

alreadyfamiliarforaclassofstarsinourgalaxyknownasCepheidvariables.

Thereasonthiswassoimportantwasthatintheprecedingdecadetheworkof

HenriettaSwanLeavittandHarlowShapleyoftheHarvardCollegeObserva-

toryhadprovidedatightrelationbetweentheobservedperiodsofvariationof

theCepheidsandtheirabsoluteluminosities.(Absoluteluminosityisthetotal

radiantpoweremittedbyanastronomicalobjectinalldirections.Apparent

luminosityistheradiantpowerreceivedbyusineachsquarecentimeterofour

telescopemirror.Itistheapparentratherthantheabsoluteluminositythatde-

terminesthesubjectivedegreeofbrightnessofastronomicalobjects.Ofcourse,

theapparentluminositydependsnotonlyontheabsoluteluminosity,butalso

onthedistance;thus,knowingboththeabsoluteandtheapparentluminosities

ofanastronomicalbody,wecaninferitsdistance.)Hubble,observingtheap-

parentluminosityoftheCepheidsintheAndromedaNebula,andestimating

theirabsoluteluminosityfromtheirperiods,couldimmediatelycalculatetheir

distance,andhencethedistanceoftheAndromedaNebula,usingthesimple

rulethatapparentluminosityisproportionaltotheabsoluteluminosityand

inverselyproportionaltothesquareofthedistance.

Healsousedparticularlybrightstarsasstandardcandles,aswededucefrom

page25:

Returningnowto1929:Hubbleestimatedthedistanceto18galaxiesfrom

theapparentluminosityoftheirbrigheststars,andcomparedthesedistances

withthegalaxies'respectivevelocities,determinedspectroscopicallyfromtheir

Dopplershifts.

8.286QUIZ1REVIEW

PROBLEM

SOLUTIONS,FALL2011

p.66

Note:sincefromreadingjustthe�rstpartofWeinberg'sdiscussiononecould

beinducedtothinkthatHubbleusedjustCepheidsasstandardcandles,stu-

dentswhomentionedonlyCepheidsgot9pointsoutof10.Infact,however,

HubblewasabletoidentifyCepheidvariablesinonlyafewgalaxies.The

Cepheidswerecrucial,becausetheyservedasacalibrationforthelargerdis-

tances,buttheywerenotinthemselvessuÆcient.

(b)(5points)QuotingWeinberg'sTheFirstThreeMinutes,chapter2,page21:

Wewouldexpectintuitivelythatatanygiventimetheuniverseoughttolook

thesametoobserversinalltypicalgalaxies,andinwhateverdirectionsthey

look.(Here,andbelow,Iwillusethelabel\typical"toindicategalaxiesthatdo

nothaveanylargepeculiarmotionoftheirown,butaresimplycarriedalong

withthegeneralcosmic owofgalaxies.)

Thishypothesisissonatural(at

leastsinceCopernicus)thatithasbeencalledtheCosmologicalPrincipleby

theEnglishastrophysicistEdwardArthurMilne.

SotheCosmologicalprinciplebasicallystatesthattheuniverseappearsasho-

mogeneousandisotropic(onscalesofdistancelargeenough)toanytypicalob-

server,wheretypicalisreferredtoobserverswithsmalllocalmotioncompared

totheexpansion ow.Rydengivesamoregeneralde�nitionofCosmological

Principle,whichisvalidaswell.QuotingRyden'sIntroductiontoCosmology,

chapter2,page11or14(dependingonwhichversion):

However,moderncosmologistshaveadoptedthecosmologicalprinciple,

whichstates:Thereisnothingspecialaboutourlocationintheuniverse.The

cosmologicalprincipleholdstrueonlyonlargescales(of100Mpcormore).

(c)(10points)QuotingagainRyden'sIntroductiontoCosmology,chapter2,page

9or11:

Sayingthattheuniverseisisotropicmeansthattherearenopreferreddirec-

tionsintheuniverse;itlooksthesamenomatterwhichwayyoupointyour

telescope.Sayingthattheuniverseishomogeneousmeansthatthereareno

preferredlocationsintheuniverse;itlooksthesamenomatterwhereyouset

upyourtelescope.

(i)False.Iftheuniverseisisotropicaroundonepointitdoesnotneedtobe

homogeneous.Acounter-exampleisadistributionofmatterwithspherical

symmetry,thatis,withadensitywhichisonlyafunctionoftheradius

butdoesnotdependonthedirection:�(r;�;�)��(r).Inthiscaseforan

observeratthecenterofthedistributiontheuniverselooksisotropicbut

itisnothomogeneous.

(ii)

True.ForthecaseofEuclideangeometryisotropyaroundtwoormore

distinctpointsdoesimplyhomogeneity.Weinbergshowsthisinchapter

2,page24.Considertwoobservers,andtwoarbitrarypointsAandB

8.286QUIZ1REVIEW

PROBLEM

SOLUTIONS,FALL2011

p.67

whichwewouldliketoproveequivalent.Consideracirclethroughpoint

A,centeredonobserver1,andanothercirclethroughpointB,centered

onobserver2.IfCisapointontheintersectionofthetwocircles,then

isotropyaboutthetwoobserversimpliesthatA=CandB

=C,and

henceA=B.(ThisargumentwasgoodenoughforWeinbergandhence

goodenoughtodeservefullcredit,butitisactuallyincomplete:onecan

�ndpointsAandBforwhichthetwocircleswillnotintersect.Onyour

nextproblemsetyouwillhaveachancetoinventabetterproof.)

(d)(2pointsextracredit)False.IfwerelaxthehypothesisofEuclideangeome-

try,thenisotropyaroundtwopointsdoesnotnecessarilyimplyhomogeneity.

Acounter-examplewementionedinclassisatwo-dimensionaluniversecon-

sistingofthesurfaceofasphere.ThinkofthesphereinthreeEuclidean

dimensions,butthemodel\universe"consistsonlyofitstwo-dimensionalsur-

face.Imaginelatitudeandlongitudelinestogivecoordinatestothesurface,

andimagineamatterdistributionthatdependsonlyonlatitude.Thiswould

notbehomogeneous,butitwouldlookisotropictoobserversatboththenorth

andsouthpoles.Whilethisexampledescribesatwo-dimensionaluniverse,

whichthereforecannotbeouruniverse,wewilllearnshortlyhowtoconstruct

athree-dimensionalnon-Euclideanuniversewiththesesameproperties.

ySolutionwrittenbyDanieleBertolini.

PROBLEM

22:THETRAJECTORYOFAPHOTON

ORIGINATING

ATTHEHORIZON

(25points)

(a)Theykeyideaisthatthecoordinatespeedoflightisgivenby

dxd

t=

ca(t);

sothecoordinatedistance(innotches)thatlightcantravelbetweent=0and

now(t=t0 )isgivenby

`c= Z

t0

0

cdt

a(t):

Thecorrespondingphysicaldistanceisthehorizondistance:

`p;horizon (t0 )=a(t0 ) Z

t0

0

cdt

a(t):

Evaluating,`

p;horizon (t0 )=bt2=3

0 Zt0

0

cdt

bt2=3

=t2=3

0 h3ct1=3

0 i=

3ct0:

8.286QUIZ1REVIEW

PROBLEM

SOLUTIONS,FALL2011

p.68

(b)Asstatedinpart(a),thecoordinatedistancethatlightcantravelbetween

t=0andt=t0isgivenby`

c= Z

t0

0

cdt

a(t)=3ct1=3

0b

:

Thus,ifweareattheorigin,att=0thephotonmusthavebeenat

x0=3ct1=3

0b

:

(c)Thephotonstartsatx=x0

att=0,andthentravelsinthenegativex-

directionatspeedc=a(t).Thus,it'spositionattimetisgivenby

x(t)=x0 � Z

t0

cdt 0

a(t 0)=3ct1=3

0b

�3ct1=3

b

=

3cb �

t1=3

0

�t1=3 �:

(d)Sincethecoordinatedistancebetweenusandthephotonisx(t),measuredin

notches,thephysicaldistance(in,forexample,meters)isjusta(t)timesx(t).

Thus.

`p (t)=a(t)x(t)=

3ct2=3 �t1=3

0

�t1=3 �:

(e)To�ndthemaximumof`p (t),wesetthederivativeequaltozero:

d`p (t)

dt

=

ddt h3c �t2=3t1=3

0

�t �i=3c "23 �t0t �1=3�

1 #=0;

so

�t0

tmax �

1=3

=32

=)

tmax= �23 �3

t0=

827t0:

Themaximumdistanceisthen

`p;max=`p (tmax )=3c �23 �2

t2=3

0 �t1=3

0

� �23 �

t1=3

0 �=3c �23 �2 �13 �

t0

=

49ct0: