(Radiation Dominated) Last Update: 21 June 2006

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1 Chaper Rik s Cosmology uorial: he ime-emperaure Relaionship in he Early Universe Chaper he ime-emperaure Relaionship in he Early Universe (Radiaion Dominaed) Las Updae: 1 June Inroduion n In Chaper we made use of he proporionaliy 1 / R (n = or for maer and respeively). We were able o derive explii expressions for he mean densiy of he early universe as a funion of ime and G. he emperaure of he universe an be found in erms of he densiy. Hene, he emperaure an also be found as a funion of ime. his is he subje of his Chaper. However, firs we say a lile abou he onens of he universe in he firs seonds and minues. We shall shorly see ha he emperaure of he universe a 1 seond is abou K. he universe will herefore onain a high densiy of gamma phoons. he ypial energy of phoons a his emperaure (~k) is around.5 MeV. his means ha any parile whose res mass is less han 1.5 MeV an readily be reaed ou of he ambien hermal energy, in onjunion wih is aniparile. he only pariles whose res masses are less han his are he eleron (and is aniparile, he posiron) and he neurinos. Consequenly, a imes before abou 1 seond we an expe he universe o onain a high densiy of elerons, posirons and all he speies of neurinos. In fa, here will be roughly similar numbers of hese pariles as phoons. he only oher pariles presen will be hose whih have been produed in he omplex proesses ourring even earlier and ha are sable 1. his inludes proons and neurons. However, proons and neurons have masses of abou 98 MeV. his is hree orders of magniude greaer han he ambien hermal energy. Consequenly, proons and neurons anno be reaed sponaneously as parile-aniparile pairs. In fa here are essenially no aniproons and anineurons. he universe has already deided o be omposed of maer raher han animaer. he proons and neurons are no presen as a onsequene of being reaed ou of he ambien hermal energy, in onras o he elerons and posirons. Hene, here is no reason o suppose ha he numbers of proons and neurons would be omparable wih he numbers of phoons, elerons and posirons. In fa here is jus one nuleon for abou every x 10 9 phoons. Amazingly, his was rue a 1 seond and remains rue oday. We give no derivaion of his huge raio here, bu raher regard i as one of he freely hosen universal parameers. Is value is probably deermined by he omplex ineraions ourring a muh earlier imes and mediaed by he srong nulear fore. A abou one milliseond he ypial hermal energy has dropped below ha required o reae parile-aniparile pairs for any oher pariles. Before his ime he nex lighes pariles, he muons, would be around in large numbers and, a lile earlier sill, he pions. For his reason we sar our sory a abou a milliseond. he 1 In his onex sable means having a half life longer han a few seonds. hus we inlude neurons, alhough hey are aually unsable wih a half life of abou 15 minues. I may also inlude pariles of dark maer, if hese are disin from boh ordinary maer and neurinos, as seems likely a presen. We exlude hem from onsideraion only beause we urrenly have no idea of heir properies. How an we alk abou muh earlier han 1 seond? ime is bes regarded on a logarihmi sale. hus, we mean a a value of log() muh less han zero. he Big Bang is pushed bak o minus infiniy on his sale. Page 1 of

2 Chaper Rik s Cosmology uorial: he ime-emperaure Relaionship in he Early Universe omposiion of he universe afer his ime was simpler han a earlier imes. his simpliiy of omposiion and sruure was o persis for some hundreds of housands of years, a whih ime he firs neural aoms formed. Bu arguably he universe only sars o beome more omplex in is sruure when graviaional lumping of maer leads o he formaion of he firs sars, afer he order of hundreds of millions of years. Bu we ge ahead of ourselves. Le us firs derive he emperaure of he early universe.. Radiaion Dominaed Era he energy densiy due o phoons is simply he well known blak body densiy. he energy densiy is relaed o he radiaed power flux, whih in urn is given by he Sefan-Bolzmann law, as follows, 1 energy J (..1) he mass densiy equivalen is obained by dividing by, hene we have, mass (phoons only) (..) he Sefan-Bolzmann onsan is = 5.67 x 10-8 Wm - K -. Oher forms of will onribue also. he only oher of relevane afer abou he firs few minues is ha due o neurinos and ani-neurinos. (Before ~1 seonds here are sill elerons and posirons in omparable numbers, bu 99% of he posirons have annihilaed by minues see Chaper 7). I an be shown ha he hree speies of neurinos plus anineurinos onribue 68.1% as muh energy as he phoons (see Chaper 5). Noe ha afer ~1 se he neurinos deouple from he res of he universe (see Chaper 6). Afer ~1 ses he neurinos are herefore a a lower emperaure han he phoons, sine he laer benefi from he addiional hermal energy arising from he eleron-posiron annihilaions whereas he former do no. he addiional 68.1% of energy akes aoun of his differene in emperaure. Hene, mass 6.7 (phoons plus neurinos) (..) Equaing his densiy o ha derived in Chaper, gives, 6.7 G G hene 0.00 (..) hene, using G = 6.67 x kg -1 m se - gives, Page of

3 Chaper Rik s Cosmology uorial: he ime-emperaure Relaionship in he Early Universe emperaure of Everyhing Exep Neurinos afer Posiron Annihilaion (i.e. afer a few minues): x 10 (K, wih in se.) (..5) [Noe ha he inlusion of he hird speies of neurino, he au neurino, raher han jus wo, hanges he oeffiien in Equ.(..5) only marginally, from 1.8 o 1.]. he derivaion of Equs.(.., 5) has used he expression in Chaper for he densiy 1 /, whih is derived from he power-law expression for he size sale variaion R n, and hese hold only a suffiienly early imes (or for fla spaeime). We will see in Chaper 5 ha he annihilaion of he eleron-posiron pairs inreases he phoon emperaure by a faor of 1.0. hus, prior o ~1 ses, boh he phoons and he neurinos - and he elerons and posirons would have been a a emperaure of, emperaure of Everyhing Before ~1 ses, and for Neurinos a all imes x 10 (K, wih in se.) (..6) We shall see in Chaper ha Equ.(..5) gives quie an impressive prediion of he emperaure of he osmi mirowave bakground. However, sine COBE/WMAP have provided a very aurae measuremen of his emperaure, we shall use his measuremen in Chaper o fine-une he values of he oeffiiens in Equs.(..5,6).. Relaion Beween emperaure and Size Radiaion Era Noe ha from (..) ogeher wih 1/R (from Chaper ) i follows ha, R 1/ ( dominaed era) (..6) Aually, his is merely a re-saemen of he fa ha he wavelengh of he varies in proporion o he size sale R. Hene he frequeny, energy and emperaure of he phoon, are inversely proporional o R.. Relaion Beween emperaure and Size - Maer Dominaed Era As far as he is onerned, ha is he phoons and he neurinos, Equ.(..6) oninues o hold irrespeive of he fa ha maer now aouns for more massenergy han he. Consequenly, we may wrie simply, R 1/ (, any era) (..7) As for he emperaure of he, now dominan, maer ha is anoher quesion. As i happens (and i s no obvious why) maer forms neural aoms, and hene sops ineraing wih he eleromagnei, a abou he same ime ha maer beomes dominan in erms of mass-energy. Consequenly, maer eases o be in hermal equilibrium one i beomes dominan. Prior o his ime we ould safely Page of

4 Chaper Rik s Cosmology uorial: he ime-emperaure Relaionship in he Early Universe assume ha he emperaure of he maer and he emperaure of he were he same. We an no longer make his assumpion in he era of maer dominane. And i is no rue. Alhough we will no prove his here, following he deoupling of maer and, he emperaure of maer iniially falls below ha of he. Ulimae he emperaure of maer a leas some of i is desined o beome far greaer han ha of he osmi mirowave bakground. his happens as soon as graviaional lumping sars o our. Page of

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