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BigBang

Nucleosynthesis

Theemergenceofelementsintheuniverse

BenjaminTopper

Abstract.

Inthispaper,Iwillfirstgiveabriefoverviewofwhatgeneralrelativityhastosayabout

cosmology,gettinganexpandinguniverseasasolutiontoEinstein'sequation,i.e.auniverse

witha[thermal]history.Wewillgothroughthedifferentstepsofthebigbangnucleosynthesis,

brieflyjustifyingtheparticleantiparticleasymmetry(otherwisenonucleosynthesiswould

happen)andthenevaluatinganddiscussingindetailstheabundancesofthefirstelements.

IwillthendiscusstheconsequencesoftheBigBangBnucleosynthesisonmodernphysics:the

constraintsitgivesonthestandardmodel,ondarkmatter...


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Contents

1. Introductiontostandardcosmology.TheFriedmannLemaitreRobertsonWalkermodels.Thermalhistoryoftheuniverse

2. BigBangNucleosynthesis.Nuclearequilibrium

.Weakinteractionfreezeout

.Elementsynthesis:Primordialabundancesof , ,Deuterium, .

.Uncertainties

3. Constraintsonnewphysics.Numberofneutrinogenerations

.Neutrinomasses

.Darkmatter

4. Conclusion

Disclaimer.

ThislectureismainlybasedonJeanPhilippeUzanandPatrickPetersbookCosmologie

Primordiale(chapter4),MarkTroddenandSeanCarrollsTASILectures:Introductionto

Cosmology,andthereviewarticleBigBangnucleosynthesisandphysicsbeyondthe

StandardModel.

Thecompletelistofreferencesusedcanbefoundonthelastpageofthisdocument.


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Introductiontostandardcosmology

ThefirstcosmologicalsolutionstoEinsteinsequationsweregivenbyEinsteinhimselfinthe

early1917.HoweverthegeneralsolutionswereindependentlyfoundbyAlexandreFriedmann

andGeorgesLemaitreonlyin1922and1927.

ThestandardBigBangcosmologicalmodelisbasedonwhatisnowcalledtheCosmological

Principle,whichassumesthattheuniverseisspatiallyhomogeneousandisotropic.This

principleenforcesthegeometryoftheuniversetobeonethatisdescribedbyFriedmannand

LemaitresolutionstoEinsteinsequations.

a) TheFriedmannLemaitreRobertsonWalkermodelsThesymmetriesinducedbyhomogeneityandisotropyofspaceallowustowritethemetricina

verysimpleandelegantform:

1 sin

HereR(t)isthecosmicscalefactorwhichevolvesintimeanddescribestheexpansionor

contraction oftheuniverseandkisthescaled3spacecurvaturesignature(+1=elliptic,

0=euclidean,1=hyperbolicspace;itisaninformationonthelocalgeometryoftheuniverse).

AnotherusefulquantitytodefinefromthecosmicscalefactoristheHubbleparametergivenby:

TheHubbleparameteristhemeasureoftheexpansionrateoftheuniverse(itisanexpansion

ratebecauseitishomogenoustoaninversetime: )whichlinkstherecessionspeedofagalaxyvtoitsdistancedthroughthefollowinglaw,knownastheHubblelaw, .ComingbacktoEinsteinsfullfieldequation 8firstneedtoaskourselvesiswhatkindofenergymomentumtensorcanbeconsistentwith

Withtheenergydensityintherestframeofthefluidanditspressureinthesameframe,

,thequestionwe

observationsandthecosmologicalprinciple.Itturnsoutthatitisoftenuseful(andsimpler)to

considerthematteroftheuniverseasaperfectfluid:

beingthespatialmetric(includingthe ).


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PluggingitintoEinsteinsequations,wegetthetwofollowingequations,thefirstonebeingan

evolutionequationandthesecondoneaconstrainttoit:

12

4

2

6

83 3Wethereforegetadynamicaluniverse,towhichwecangiveanage:

~ 14.2

,

andwecannowturnourselvestounderstandingits[thermal]evolution,orinotherwordsits

thermodynamics.

b) ThermalhistoryoftheuniverseFromverybasicarguments,wecanunderstandthatthecompositionoftheuniversehasgreatly

evolved;letscall reactionrateforagivenparticleinteraction.Ifthatreactionrateismuch

higherthantheexpansionrateH,thentheinvolvedinteractioncanmaintainthoseparticlesin

athermodynamicequilibriumatatemperatureT;theycanthenbetreatedasFermiDiracor

BoseEinsteingases,obeyingthefollowingdistributionfunction:

1

With thedegeneracyfactor, isthechemicalpotential, ,andistheofthephoton call

However,if theparticleissaidtobedecoupled;theinteractioncanmaintainthemiceq

interactionisnot

Theparticledensityequationcanbecomputedfromthedistributionfunction:

temperatureofthekindofparticlestudied.Thetemperature s,T,is ed

temperatureoftheuniverse.

thermodyna uilibriumbetweentheparticleandtheotherconstituents.

Fromthisweunderstandthattherealwaysexistsatemperatureforwhichthe

effectiveanymore;itissaidtobefrozen.

1


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Thisequation,appliedtobosonsandfermionsatdifferenttemperatures,givesthefollowing

table:

Limit Typeofparticle n

,

Bosons g3 TFermions

3g34 T Bosons g mT2

eE

g mT2

Fermions

eE

Table1:Thermodynamics

Thephotonsareknowntohavezerochemicalpotential(asthenumberofphotonsisnot

conserved),thereforetheparticleantiparticleannihilationprocess: mustsatisfythefollowingconservationlaw: .Pluggingthisintheparticledensityequation,wefindanasymmetryinthenumberofparticles

andantiparticles:

0

Note:atlowtemperature( )weget,asexpected,anexponentialsuppressionofthisasymmetrythatgoes

with .

like

.

Thisasymmetryexplainsthedominationofmatteroverantimatterandthereforeallowsthe

Theearlyuniversebeingdominatedbyradiation,wecanrewrite,using[table1]theexpansionrateof

83

formationofnuclei.

expansionoftheuniverse:

30


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Hbeinghomogenoustoaninversetime,wecanrewriteitas:

2.42 withTinMeV, beingthenumberofrelativisticdegreesoffreedomatagiventemperature.ThesynopsisoftheBigBangnucleosynthesiscanbesplitintothreemajorphases:

T>>1MeV Thermodynamicequilibriumbetweenallthecomponentsoftheuniverse;

Universesdominatedbyradiation;## ~

Photodissociationpreventsanycomplexnucleitoform.

1>T>0.7Mev Weakinteractioncannotmaintainequilibriumbetweenallparticles;neutrons

decouple:neutronfreezeout:## ~ .Freeneutronsdecayinto

protons;atomicnucleistayatthermodynamicequilibrium.Freezeout

temperature

.

0.7>T>0.05Mev Nuclearthermodynamicequilibriumcannotbemaintained.Electronpositronannihilationhasheatedthephotonbath.Atomicnucleiformthrough2body

collisions: (onlyonewaybecauseradiationdensityislowenough).

Table2:Thermalevolutionoftheuniverse


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BigBangNucleosynthesis

a) NuclearequilibriumFortemperaturesgreateroroftheorderof100Mev,theuniverseisdominatedbyrelativistic

particlesinequilibrium:electrons,positrons,neutrinosandphotons.Thecontributionfrom

nonrelativisticparticlecanbeneglected;theweakinteractionsbetweenneutrons,protonsand

leptons:

keepalltheparticles(aswellasthenonrelativisticbaryons)inthermodynamicequilibrium.

Attemperatureslargerthan1Mev,thenuclearinteractionsstillmaintaintheveryfirstnucleiinforegivenby:

n g mT2

andtheelectromagneticinteractionbetweenelectronsandpositrons:

thermodynamicequilibriumtheirfractioncanthereforebecomputedonlyusing

thermodynamicconsiderations,asshownbelow.

Thedensityofthosenonrelativisticnucleiisthere

e

gmT2

e

e e

with , thechemicalpotentialofprotonsandneutrons.gast echemicalequilibriumimposes

A Z eglectingthedifferenceinmassbetweenprotonsandneutronsintheprefactor,wecan

/ 2 2

Aslon hereactionratesarehighertheexpansionrate,th

thechemicalpotentialtobe:

N

extractfromthisformulathedensityratioofprotonsandneutrons:

2 ///

Rewritingtheexponentialinthenucleardensity,weget:

n g mT2 e gmT2

2nn mNT/ eB

ith isthebindingenergyofthenucleon.W


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Defining , , abundancesofucleiin ermodynamicequilibrium:

/

,wefinallyobtaintheatomicn th 3

2

/

with thebaryontophotonratio.Wecanusethisformulato ebindi ndthetemperatureat

getth ngenergyofthelightnucleia

whichtheirabundancewillbeatamaximum: 2.22 6.92 7.72 28.3 0.066 0.1 0.11 0.28Inparticular,itgivestheexactgoodprimordialevolutionfortheDeuteriumabundanceuptoits

leweunderstandthatevenatthermalequilibrium,nucleosynthesiscannotstart

b) Weakinteractionfreezeoutandneutronabundancetsomestage ,Xcanbesaidtofreezeoutifitsabundancestopsevolvinggreatly.

onrea rium:

760

peak,andfortheotherlightuntilelementsuntiltheydepartfromequilibrium(welldiscussthis

pointlater).

Fromthistab

beforeT=0.3MeV.Thermalphotonspreventformationoflargequantityofdeuteriumuntil

T>0.3MeV(photodissociation)eventhoughthecrosssectionfor ishigh.

A

Weakinteracti ctionratefor

keepsprotonsandneutronsinequilib

1 Futwhenthisreactionratebecomesoftheorderoftheexpansionrate,thethermal

H

3g G TB

equilibriumisbroken: T0.8 T 0.8MeV or t 1.15s

Therefore,whentheweakinteraction(i.e.theneutronabundance)freezesout,theneutronto

T 11

protonrationis:

1/6Justconsideringtheexpansiontoget eneutronabundanceatthebeginningof

icleitwill

e

6

th

nucleosynthesisisabittoonavethough,),asthefreeneutronisnotastablepart

decaywithalifetimeof887sfromthetimeweakinteractionfreezesoutto,temperaturthatallowsDeuteriumtoform(T=0.086MeV,t=180sec).Wethereforeget:

1 / 0.136


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c) Elementsynthesis:Primordialabundancesof , ,Deuterium,

utronsarelockedupandnoheavier

anyore

e

synthesisisthatno elementsexistwith

Letsstartouranalysiswithqualitativearguments:

TheformationofdeuteriumisacrucialfactorforthecontinuationoftheBigBang

nucleosynthesis:iftoomuch

isformed,thenne

elementscanbecreated.Howeveriftoolittleisformed,thenanimportantfactorinfurtherfusionismissing;especially,whendeuteriumcanform,helium4canformmuchm

easilybecausethefollowingreactions and enablethcreationoflargeamountsforHeliumthrough and .AveryimportantfactorintheBigBangnucleo stable

A=5andA=8(figure4).Thereforewewouldexpect togreatlythisanomaly decreaseany

lium

ndancethanHelium4(tritiumis

t

Figure1:Nuclearreactions http://physicsworld.com/cws/article/print/30680/1/PWfea4_0807

nucleationprocessatA=4andA=7(itisalotlesslikelyforA=4togotoA=6thantoA=5).

Fromthis,weunderstandthat:

Theabundanceof

shouldatfirstincreasethendecreasewhenHelium4canbe

onsumedinthereactions).produced(asitisbeingc Theabundancesof and shouldincreasethenlevelupordecreasewhenheisgettingcreated.Theyshouldhavealowerfinalabuesh decradioactivetherefor ould reaseoverlongperiodslongerthantheonestudied

here)

Helium4,whichisstable,andattheendofthefirstchain(beforeA=5)shouldbepresentinlargequantitiesattheendoftheBigBangnucleosynthesis.Alltheprevious

elementscanfusetoformHelium4,andHelium4isveryunlikely(butnot0)toreac

withprotonsorneutronstojumptoA=6

AbundancesofLithiumandBerylliumshouldbeverylowcomparedtotheother.Wecanexpect

and

tobetheendoftheBigBangnucleosynthesisbecauseofthe

A=8barrier(alsohelpedbythefactthatBe7isradioactiveandthereforedecaysto

lighterelements).

Lithiumvalley: destroyedbyprotonsbutBe7contribution.


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Nowletsgetintosomelittlecomputations:

TheeasiestestimationisforHelium4:wenowthatallfreeneutronsleftafterfreezeoutwill

getbounduptoHe4becauseofnucleistability.EstimationsonHe4abundancesdepend

mostlyontheneutrontoprotonrate,thusmostoftheuncertaintiesarebasedonthe

uncertaintiesofthefreeneutronlifetime(=887seconds)

~ 2 1 0.24 ForDeuterium,wecangetitscreationslopewiththethermalequilibriumequationgiven

previously(simplyusingMathematica,itworkperfectly);howevergettingitsfinalabundanceis

muchmoredifficultaswehavetocomputeallitssinkterms.Thesameproblemappearsfor

alltheotherelements.

Itishoweverdoablewithoutacomputer,startingwiththeverygeneralstatement:

whereJ(t)and aretimedependantsourceandsinkterms.Thesolutiontothisequationis

Thisrequires,asexpected,carefulexaminationofallthedifferentreactionnetworksand

keepingtrackoftheirreactionrateastemperaturedecreases,thereforeitisaverytedious

computation(figure2)1

.

Generallyfreezeoutoccurswhen ~.ItispossibletoshowthatXapproachesthisequilibriumvaluefor

Figure2:Analyticcomputation

http://arxiv.org/PS_cache/hepph/pdf/9602/9602260v2.pdf

1Esmailzadethetal(1991);Dimopoulosetal(1988)


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TheanalyticalresultsshownheregivethegoodpredictionsforD,Helium3andLithium7

withinafactorof3,andHelium4within5%!

Moreover,thisanalyticcalculationhelpsinunderstandingthedepartofHelium4fromits

nuclearstatisticalequilibrium(thedashedlinearound0.6Mev).

Thisbehaviorisexplainedbythedeuteriumbottleneck(figure1):thecreationofHelium4is

delayeduntilenoughtritiumandhelium3areformed(itthenmimicstheevolutionofTandHelium3).

Thentheytoodepartfromnuclearstatisticalequilibriumatabout0.2MeVduetothe

deuteriumbottleneck.AtthispointHelium4,Helium3andTritiumallmimictheevolutionof

Deuteriumuntilitfinallydeviatesfromequilibriumat0.07MeV.

Wecannowcomparetheoryandtheobservation(figure4)onthefinalabundancesofthelight

elements:

Theory Observation

3.610. 5.510.

2.780.4410

1.5 0.1 10 6.7 10 1.210. 5.510 10. 1.50.5 0.2450.014 3 0.0002 887 0.009ln.

0.00150.23910.0020040.24430.2490.0 1.210. 5.510. 1.23.. 10

Figure3:Abundances Figure4:Elementgap

http://www.astro.ucla.edu/~wright/BBNS.html BigBangNucleosynthesisC.Ciemniak14.05.2004


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d) UncertaintiesAsnucleosynthesisinvolvesmanydifferentreactions,therearemanysourcesofuncertainty.

Themajoruncertaintiesin aredueto Theexperimentaluncertaintyontheneutronlifetime.Thecurrentvalueisconsidered

tobe2

887 2 .Achangeof2intheneutronlifetimecausesachangeinHelium4predictionofabout0.4%. Thenumberofneutrinogenerations;ifthenumberofgenerationsismorethan3,say

3.3(aswewillthisthatseemstobethecurrentboundonthenumberofneutrino

generations)thenwegetachangeofabout1.5%intheabundanceprediction.

Fortheotherelements,uncertaintiesinthenuclearcrosssectionscandramaticallymodifythe

abundancepredictions.TheycanalterDand byupto15%and byabout50%!Andallofthisbeingachainreactionachangeinthereactionrateofoneelementwillaffect

alltheothers.

TheseeffectscanbeneglectedfortheHelium4abundance( 0.3%)becauseaswehaveseenitispossibletocalculateitsabundanceonlyusingtheneutronsabundanceatnucleationtime;alreadygotveryprecis

Big an nucleosynthesisalsoallowstocalculatethebaryontophotonratio .1.7510

e,

cedHubble

toputconstraintsonotherobservableslikethe

wedidnothavetoconsideranyofthenuclearreactionstodothiscalculationsand

a eresult.

B g

Fromspectroscopicmeasurements,andtaking 3,onegetsthefollowingvalue: 5.15Whichcorrespondsto Bh 0.018 0.006where Bisthebaryoniccontentoftheuniversandtheredu constant.Aswewillseelaterthoseparametersallowus

numberofneutrinosgenerations,theirmasses,butalsototestthevalueofthefundamental

constantslikeGandthefinestructureconstant (constraintgivenbyBBNis

5%.)BigBangnucleosynthesisisthereforeatestforthehotBigBangmodel,nuclearphysics,and

astrophysicsingeneral.

2ParticleDataGroup


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Constraints

to nrateof e

niverse,thereforemoreneutronswillsurviveuntilnucleosynthesiswhichleadstoanincrease

NumberofNeutrino

Flavors

Modelpredictionsfor

He4

onnewphysics

a) NumberofneutrinogenerationsAnincreaseinthenumberofneutrino

leads anincreaseintheexpansio th

u

intheHelium4abundance.

2 ~ 0.227 3 ~ 0.242 4 ~ 0.254ConstraintsfromBigBangnucleosynthesisarestillverydifficulttoestimate(manydifferent

aluescanbefoundintheliterature).Itseemslikethebestfit ~30.3v .

Credits:arXiv:astroph/9706069


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b) NeutrinomassesAlthoughinthenaveversionoftheStandardModelneutrinosaremassless,recent

experimentshavetendedtoshowthatneutrinoswereactuallymassive(fromneutrino

oscillationexperimentsmainly3

).

Eventhoughtheirmassisthoughttobeextremelysmall,thelargequantityofneutrinosimplies

thatanynonzeromasshaveanobservableimpactoncosmology.

Fromnucleosynthesis,wecangetanupperlimittothemassofalltheneutrinos,bycombining

therelicneutrinoabundancewithobservationalboundsonpresentenergydensity(thanksto

WMAP):

2 94

summingoverallneutrinospeciesthatarerelativisticatdecoupling(i.e. 1).Ifweconsiderthatneutrinosaremoremassiveandthereforewherenotrelativistic(!),then

ticdecoupling,thewegetalowerlimit

ontheirmass: 2.Thereforenostableneutrinocanhaveamassinthe100eVto2GeVrange.

urrentneutrinomassestimationsare:

,

160 ,

24

nregion.

theyfalloutofchemicalequilibriumbeforetherelativis

C

2.1Thatisfornow,onlytheelectronneutrinoisknowntohaveamassoutoftheforbiddec) DarkmatterDirectobservationsofluminousmattergivethefollowingresulttowardthecontentofthe

universe:

h ~ 0.005ComparingthistothepreviousvalueofBh,wequicklyunderstandthatthereisaproblem:

n30%!Wherearetheother70%?

Baryonic=>no nucleonicdarkmatter,planetarymassblackholes,strangequarknuggets(shouldhaveenhancedproductionofheavyelementsduringBBN).

Nonbaryonic=>Relicparticles?Newparticles?3SuperKamiokande,1998


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Conclusion

Foralongtime,primordialnucleosynthesisandspectroscopicmeasurementswheretheonlywayto

predictandtestthelightelementsabundances,andtogetavalueforthebaryoniccontentofthe

B)haveallowedto

ccontentoftheuniverse:

h 0.02240.0009whichcorrespondstoabaryontophotonratio:

universe.Howeverrecentcalculations4oftheCosmicMicrowaveBackground(CM

measurewithextremeaccuracythebaryoni

6.14 0.25 10Ifweredoallthepreviouscalculationswiththosevalues,wegetthefollowingabundances:

predictionusing

CMBvalues

Element Theoretical

2.600.170.19 10 1.040.04 1050.04 0.2470.0004 4.150.450.49 1010

photonratio.

TheverticallinerepresentsWMAP

ThecurvesaretheBBNpredictions.

Uzan&Peter

Onthisgraphareshownthe

theoreticalpredictionsfor

Deuterium,Helium3,Helium4and

Lithium7withrespecttothebaryon

valuefor,thehorizontalonesindicatethespectroscopic

measurementsforeachelement,the

widthrepresentingtheuncertainties.

Credits:CosmologiePirmordiale

Tho arein theoreticalpredictions.Nuclearreactionratesarebeing

reanalyzedbecause, ,aslightchangeinoneofthemcouldgreatlyaffectallthe

predictions.

seresults conflictwiththe

aswehaveseen

4WMAPresultsontheCMB


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References

Books

CosmologiePrimordialeJeanPhilippeUzan&PatrickPeter

Princ P.J.EPeeblesiplesofphysicalcosmology

ArXiV

Xiv:astroph/9706069:BigbangNucle

astroph/0009506:LightElem

arXiv:astroph/9905211:PrimordialLithiu

rXiv:astroph/0008495:WhatIsTheBBN

rXiv:astroph/9905320:PrimordialNucl

arXiv:hepph/9602260:BigBangnucleosynthesisandphysicsbeyondtheStandardModel

ar osynthesisEntersthePrecisionEra

arXiv: entNucleosynthesis

mandBigBangNucleosynthesis

a PredictionfortheBaryonDensityandHowReliableIsIt?

a eosynthesis:TheoryandObservations

OtherInternetresources

http://www.astro.ucla.edu/~wright/BBNS.html

http://wwwthphys.physics.ox.ac.uk/users/SubirSarkar/mytalks/groningen05.pdf

http://www.astro.uu.se/~bg/cosmology

http://www.mpia.de/homes/rix/BBN_Lect.pdf

roddenandSeanCarrollTASILectures:IntroductiontoCosmology

.pdf

MarkT

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