Cosmology and particle physics

Transcription

1 Cosmology and particl physics Lctur nots Timm Wras Lctur 8 Th thrmal univrs - part IV In this lctur w discuss th Boltzmann quation that allows on to dscrib th volution of procsss in our univrs that ar not in quilibrium. Thn w discuss th formation of light lmnts during big bang nuclosynthsis and th rcombination of lctrons and protons into nutral hydrogn. 1 Th Boltzmann Equation Th numbr dnsity in th absnc of intractions (or in quilibrium) scals lik th invrs volum, i.. lik a 3, sinc it is a dnsity. This mans that it satisfis th quation dn dt + 3ȧ a n = dn dt + 3Hn = 1 d(na 3 ) a 3 dt = 0. (1) As w discussd in th prvious lcturs, thr ar ampl intractions in which two particl intract and bcom two nw particls. Ths can b schmatically writtn as , () which mans that particl 1 and annihilat and bcom particls 3 and 4 (and vic vrsa). Such intractions togthr with dcays of singl particls ar th most rlvant procsss in th arly univrs sinc th intraction of thr or mor particls is much mor unlikly bcaus ths thr or mor particls would hav to b all vry clos at th sam tim. Th Boltzmann quation dscribs th volution of th numbr dnsity n 1 of for xampl particl 1 in th prsnc of intractions. Hr w focus on th intraction (), in which cas th Boltzmann quation is givn by 1 d(n 1 a 3 ) a 3 dt = < σv > n 1 n + c n 3 n 4, (3) whr th first trm dscribs th rduction of n 1 du to annihilation of particls 1 with, whil th scond trm dscribs th production of 1 particls (and particls) du to th annihilation of 3 and 4 particls. Th fr paramtr c can b rlatd to th thrmally avragd cross-sction < σv >: W know from quation 1 that th right-hand-sid of quation 3 has to vanish in thrmal quilibrium, i.. for n i = n q i. This givs c = nq 1 n q n q 3 n q 4 < σv >. (4) 1

2 Th Boltzmann quation thn bcoms 1 d(n 1 a 3 ) = < σv > (n a 3 1 n nq 1 n q ) dt n q 3 n q n 3 n 4. (5) 4 This can b rwrittn as d log(n 1 a 3 ) d log(a) = Γ 1 H (1 nq 1 n q n q 3 n q 4 ) n 3 n 4, (6) n 1 n whr Γ 1 = n < σv >. Th abov quation dtrmins th volution of th numbr dnsity for particls spcis 1 as a function of a(t). Sinc a(t) grows with t in our univrs w can ssntially think of th abov Boltzmann quation as dtrmining th volution of spcis 1 with tim. W s that Γ 1 /H plays a crucial rol in dtrmining th volution of n 1 a 3. If th intraction rat Γ 1 bcoms small compard to th Hubbl rat H, w hav a frz out and th numbr dnsity of n 1 scals lik a constant tims a 3. To dscrib th volution of all th particls in our arly univrs on has to solv simultanously all th corrsponding coupld Boltzmann quations. This is of cours only possibl numrically and gos byond what w will discuss in class. Hr w will focus on a fw simpl intrsting cass that w can discuss mor or lss analytically and using th quilibrium rsults from th prvious lcturs. W will hncforth drop th suprscript q and just writ n i for th numbr dnsitis in quilibrium. Chmical potntials Bfor w discuss big bang nuclosynthsis it is usful to rviw th ffct of a non-zro chmical potntial. In th phas spac distribution function (s for xampl quation (1) in th lctur 7 nots) a non-zro chmical potntial lads to f ± (p) = 1 (E(p) µ)/t ± 1. (7) Whil again ach particl can hav a diffrnt chmical potntial, chmical quilibrium, which is rachd via intractions, lads to rlations btwn th chmical potntials. For xampl intraction lik th ons in quation () lad to µ 1 + µ = µ 3 + µ 4. (8) Non-zro chmical potntials will modify th xprssion for, for xampl, th numbr dnsity, so that for non-rlativistic particls in quilibrium it is givn by mt µ m n = g T. (9) π Howvr, if w tak ratios of numbr dnsitis in which th chmical potntial cancls du to quation (8), thn w don t rally nd th valus of th chmical potntials. Not, that photons can intract with lctrons via a doubl Compton scattring which lads to µ γ = 0. + γ + γ + γ, (10)

3 Big bang nuclosynthsis Big bang nuclosynthsis rfrs to th formation of atomic nucli during th cooling of our arly univrs. Rcall that aftr th QCD phas transition around 150M V quarks form colorlss bound stats that includ protons and nutrons. During th continuous cooling of our univrs, th numbr dnsitis of ths non-rlativistic baryons is xponntially dcaying until, du to th initial antisymmtry btwn baryons and anti-baryons, w ar lft with a rsidual amount of baryonic mattr in th form of protons and nutrons and havir nucli. Th protons and nutrons can bind via th strong forc into atomic nucli and via th wak forc nutrons and protons can convrt into ach othr. All ths procsss ar initially in quilibrium and w want to undrstand with which rlic abundanc of nucli w ar lft, onc ths procsss drop out of quilibrium du to th cooling of our univrs. Th two rasons why w can actually do that without solving many coupld Boltzmann quations ar firstly that ssntially no lmnts havir than Hlium ar cratd during big bang nuclosynthsis, so that w can just focus our attntion on Hydrogn and Hlium and scondly that initially w hav only nutrons and protons in quilibrium without any rlvant amount of havir nucli..1 Protons and nutrons At tmpraturs abov 1M V protons and nutrons ar in quilibrium du to wak intractions of th form n + ν p + +. (11) On can argu that th chmical potntials for lctrons and nutrinos ar ngligibl small, so that quation (8) tlls us that µ p = µ n. Taking th ratio of th proton and nutron numbr dnsitis, th chmical potntial thn simply cancls (s quation (9)) and w find n n = ( mn m p mn mp T. (1) Rcalling th proton and nutron masss m p = 938.7MV and m n = MV, w s that thir ratio is vry clos to 1 and thir diffrnc is m n m p = 1.3MV. So at larg tmpraturs T 1MV w hav th sam numbr of nutrons and protons, whil at nrgis blow T 1MV, th ratio of nutron and proton numbr dnsitis is xponntially dcaying. Howvr, as w hav sn last tim whn w discussd nutrinos, procsss that involv th wak intractions lik th on in quation (11) will bcom irrlvant at nrgis blow roughly 1MV, sinc Γ/H 1 for T 1MV (s quation (14) in th lctur 7 nots). Actually a mor carful analysis rvals that th wak intractions bcom irrlvant at T.8MV which lads to n n = ( mn m p mn mp T 1.3MV.8MV.. (13) Onc th tmpratur drops furthr th finit liftim of th nutron bcoms important. In particular, a fr nutron can dcay via n p ν, (14) 3

4 which lads to an xponntial dcay of th nutron numbr dnsity n n n n t 886s. t 886s, (15) whr w usd that th man liftim of a fr nutron is 886s. Th dcay of th nutrons stops onc thy ar bound into nucli which happns around t 330s which lads to n n.14. (16) t 330s. Havir nucli Lt us study a procss that involvs th production of th lightst nuclus that is not just a proton, i.. dutrium. On nutron and on proton can form dutrium (and a photon): n + p + D + + γ. (17) As w argud abov, th photon s chmical potntial vanishs so that th chmical potntials cancl in th following ratios n D = 3 ( π n n 4 T m D m n m p m D mn mp T, (18) whr w usd g n = g p = and g D = 3. Th ratio btwn th masss is approximatly /m p, howvr, th diffrnc in th mass of th dutrium and its two constitutions is th binding nrgy m N + m p m D.MV. At nrgis wll blow th proton and nutron masss, i.. at T 1GV, th numbr dnsitis of protons and nutrons ar not xponntially dcaying anymor but ar dtrmind by th non-zro baryon numbr in our univrs, i.. by quation (1) in th lctur 5 nots: n n n b ζ(3) n γ = 10 T 3, (19) π whr w usd quation (11) in th lctur 6 nots for th photon numbr dnsity. Using this in quation (18), w gt n D T.MV 8 T (0) m p This implis that for T = 1MV, w hav n D / 10 1 and for roughly T.066MV w hav n D / 1. This mans that at tmpraturs abov T.1MV th dutrium abundanc is ngligibl and th sam is tru for vn havir nucli..3 Nuclosynthsis Now w hav all pics ilac and can discuss th cration of nucli that ar not just a proton. Our starting point ar protons and nutrons. As w mntiond bfor, procsss 4

5 involving mor than two particls ar vry rar so that th initial procss must b th formation of dutrium from on proton and on nutron as shown in quation (17). Only onc dutrium is formd, which as w saw abov happns around T.66MV, can Hlium b producd via D + p + 3 H + γ, D + 3 H 4 H + p +. (1) Th binding nrgy of 4 H, B H, is largr than that of dutrium B D. This lads to an nhancmnt of th numbr dnsity of Hlium compard to that of dutrium n H n D BH BD T. () This is similar to quation (0), whr dutrium is favord at low tmpraturs, xcpt that hr w don t hav a supprssion factor. This mans that hlium is almost immdiatly producd aftr dutrium and that all nutrons nd up in 4 H nucli. Sinc ach 4 H atoms contains two nutrons, this allows us to asily dtrmin th fraction of hlium to hydrogn in our univrs n H n H = n H = 1 n n = 7%. (3) This answr is vry clos to a full numrical analysis that solvs all th coupld Boltzmann quations and which givs somthing lik 6.% 1. Sinc th mass of an Hlium nuclus is 16 roughly four tims as larg as th proton mass, w find that roughly on fourth of th mass of ordinary mattr in our arly univrs is in th form of Hlium and th rst in th form of Hydrogn. This prfctly agrs with obsrvations and is on of th grat succsss of big bang nuclosynthsis and shown in figur 1. Byond Hlium You probably wondr why havir atomic nucli don t form during big bang nuclosynthsis (and how thy appard in our univrs). Th rason that thy arn t formd from protons, nutrons, dutrium and hlium is th following: As w hav sn abov, bfor hlium can b formd, protons and nutrons nd to first combin to form a substantial amount of dutrium. During this tim th univrs kps cooling and th nucli loos part of thir kintic nrgy, which maks it hardr to ovrcom th Coloumb barrir (i.. to bring togthr two positivly charg nucli). Mor importantly, onc a larg amount of 4 H is formd, ths can only combin to form 8 B which is unstabl and dcays fastr than it can b formd. Vry small amounts of Tritium and 3 H that ar also cratd during big bang nuclosynthsis can combin with 4 H to form 7 Li of which w obsrv tiny amounts today. 1 So big bang nuclosynthsis producs only vry light lmnts. As brifly mntiond last tim, th havir lmnts that w s today in our univrs and that w ar mad of ar cratd in th first stars through nuclar fusions. Onc th havist of ths stars xplod in suprnova, ths havy lmnts ar rlasd into th stllar mdium and can bcom part of scond gnration stars and form plants. 1 Actually w obsrv slightly mor Lithium than thortical prdictd (s figur 1) which might rquir small modifications of big bang nuclosynthsis. 5

6 Figur 1: Th mass fraction of svral light nucli as thortical prdictd (colord band) and masurd by WMAP (gry band). 3 Rcombination and photon dcoupling Aftr nucli ar formd around T.06MV, which corrsponds roughly to a tim of 3 minuts aftr th big bang, our univrs contains a soup of positivly chargd nucli, fr lctrons and photons (as wll as dcoupld nutrinos). During th furthr xpansion and cooling of our univrs th nrgy dnsity of radiation dcays lik a 4, whil th nrgy dnsity in th non-rlativistic mattr dcays lik a 3. Roughly 60,000 yars aftr th big bang th nrgy dnsitis in radiation and mattr ar qual and our univrs ntrs its mattr dominatd ra. Anothr 00,000 yars latr lctrons and nucli start to form nutral atoms, during a priod that is usually calld rcombination. Onc th rcombination nds and th univrs ssntially consists of nutral atoms, th photons in th cosmic microwav background can stram frly until today and tll us about th univrs 380,000 yars aftr th big bang as wll as much arlir tims. A mor accurat nam would b combination sinc thy hav nvr bn combind bfor. 6

7 3.1 Th Saha quation Th procss that kps lctrons, protons and photons in quilibrium during th first 00,000 yars aftr th big bang is + p + H + γ. (4) At a tmpratur of T 1V all particls xcpt th photons ar non-rlativistic and in (chmical and thrmal) quilibrium du to th abov procss. Rcalling that th photon chmical potntial vanishs, w can look at th following ratio in which th chmical potntials cancl 3 n H mh π mp+m m H = T, (5) n m m p T whr w usd that g H = 4 = g g p. Using that m p m H and that th binding nrgy of hydrogn is m p + m m H = 13.6V and th fact that our univrs is lctrically nutral which implis n =, w find n H π 13.6V = n T, (6) m T Nxt w dfin th fr lctron fraction as th ratio of fr lctrons to baryons X n n b. (7) As w hav sn abov in quation (3), mor than 90% of th baryon numbr is du to th protons, that can b in th form of positivly chargd nucli or in th form of nutral hydrogn n H, so that 9 ζ(3) n b + n H = n + n H 10 T 3, (8) π whr w usd quation (19). From th dfinition in quation (7) w thn find 1 X = ( + n H ) n X n Using this in quation (6) w find th Saha quation 3. Rcombination 1 X X 9 ζ(3) = 10 π ( + n H ) = n H n n b. (9) ( πt m 13.6V T. (30) Th Saha quation allows us to gt an stimat for th nrgis at which rcombination happnd. Taking th onst of rcombination as th tmpratur T bginning at which X =.9 and th nd of rcombination as th tmpratur T nd at which X =.1 w find from quation (30) that T bginning.35v and T nd.30v. Th rason that ths rsults ar so much smallr than th 13.6V binding nrgy is that thr ar many, many mor photons than baryons and that th black body spctrum of th photons has a tail of high nrgy photons that kp th Hydrogn ionizd until th avrag tmpratur of th photon bath is wll blow th binding nrgy of Hydrogn. 3 Hr w us n H to dnot th nutral hydrogn only so that n H. 7

8 3.3 Photon dcoupling Th so calld tim of last scattring at which th lctrons and photons scattr for th last tim via Thompson scattring + γ + γ, (31) is actually happning vn latr around a tim whn X.01. W s this as follows: Th cross-sction for Thompson scattring is σ T 10 3 MV and th corrsponding intraction rat is givn by 9 ζ(3) Γ T n σ T = n b X σ T 10 T 3 X π σ T. (3) In ordr to dtrmin th tmpratur at dcoupling T dc w hav to chck whn th abov intraction rat is of th sam siz as th Hubbl xpansion rat. During mattr domination th Hubbl rat is givn by 4 This implis H = H 0 Ωm,0 ( T T 0. (33) Γ(T D ) = H(T D ) T 3 D X (T D ) = 10 9 π H 0 Ωm,0. (34) ζ(3) σ T T 3 0 W can numrically solv this quation and find T D.6V and X (T D ).003. Th tmpratur T D.6V corrsponds to a tim of 380,000 yars aftr th big bang and a rdshift of z This follows from th first Fridmann quation and th fact that a(t) 1/T. H = H 0 8πG 3H0 ρ m = H0 a0 Ω m,0 a(t) 8