Thermodynamics of the early universe, v.4

Transcription

1 Thermodynmics of the erly universe, v.4 A physicl description of the universe is possible when it is ssumed to be filled with mtter nd rdition which follows the known lws of physics. So fr there is no observtionl evidence ginst this ssumption. There re unknown components clled drk mtter nd energy which re being explored these dys, but they don t mtter much when the universe ws very young. And tht is the epoch we will consider in somewht more detil here. 1 Astrophysicl units The mthemticl formuls we use in cosmology tke simpler form when they re written in more pproprite units more suitble thn the more common SI units we usully use. Time is mesured in seconds nd lengths is meters. Since one yer contins = seconds, we cn lso use 1 yr = s (1) s time unit. Cosmologicl time scles re usully of the order of billion of yers nd thus it will lter be convenient to use the time unit 1 Gyr = s. We will lter discuss the ge of the universe which tody is known to be 13.7 Gyr. Insted of using the meter, we will be bit old-fshoned nd use 1 cm = 10 2 m s the unit length. Since the velocity of light is defined to be c = m/s = cm/s, we cn lso mesure distnces in units of time. For instnce, in one second lightpulse moves cm. This distnce is clled one lightsecond. A more common length unit is lightyer which we will denote by the unit lyr. It hs the mgnitude cm/s s, i.e. 1 lyr = cm (2) Astronomers re fond of using their own length unit clled the prsec. For resons we will not go into here, it is defined to be 1 pc = 3.26 lyr (3) Cosmologicl length scles re then typiclly of the order of millions of prsecs so tht one then uses 1 Mpc = 10 6 pc s suitble unit of length. Formlly, this unifiction of units comes bout by tking the velocity of light to be c = 1. Times nd lengths re then mesured in the sme units so tht 1 s = cm (4) 1

2 We re then using so-clled reltivistic units suitble for physicl problems involving Einstein s theory of reltivity. The symbol c for the velocity of light will be bsent from ll formuls. Energy nd mss re then expressed in terms of the sme units. From the definition of the energy unit joule J or the equivlent electronvolt ev given by 1 J = ev (5) it then follows tht mss of one grm corresponds to n mount of energy given by 1 g = ev (6) For instnce, elementry prticle msses cn be expressed in units of 1 MeV = 10 6 ev. The proton mss is then m p = MeV while the electron mss is m e = MeV. In strophysics mny processes re governed by quntum mechnics. The formuls will involve the Plnck-Dirc constnt h = Js = evs. This dditionl constnt of nture cn lso be mde to dispper in the mthemtics by introducing quntum mechnicl units defined by setting h = 1. Then we will hve ev = 1 s 1 (7) Needless to sy, this enbles us to mesure other physicl quntities in new units. For exmple, the ction of system is usully mesured in units of the Plnck constnt. It will now be dimensionless. Energy cn be expressed in units of inverse seconds s follows from (7). At very high energies the physicl processes resulting from quntum mechnics will lso be reltivistic. We cn then combine these quntum mechnicl units defined by h = 1 with the bove reltivistic units resulting from hving c = 1 into new, soclled HEP units. They re used in high energy prticle physics nd strophysics. In stndrd units we hve hc = MeV fm (8) where 1 fm = m is clled one femtometer or fermi. Setting now hc = 1, we find tht MeV equls one inverse fermi. A very useful reltion indeed for those working in this field of physics! Obviously, it is equivlent to (7) when time nd lengths re mesured in the sme units. Similrly, mss density mesured in g/cm 3 cn then first be converted to MeV/cm 3 using (6) nd then further on to MeV 4 with the help of (8). In fct, we hve the useful reltion GeV 4 = GeV/fm 3 (9) 2

3 which only involves units typicl of high energy nucler physics. Finlly, when the system is t finite temperture T the kinetic energies of prticles will be set by the mgnitude of k B T. The inverse Boltzmnn s constnt hs the vlue kb 1 = K/eV. Setting then k B = 1, we cn mesure temperture in electronvolts ev insted of Kelvin degrees K, i.e. 1 ev = K (10) This thermodynmic choice of units lso clens up the formuls lot. For instnce, entropy will now be dimensionless. 2 The Friedmn equtions The expnsion of the universe follows from Einsteins generl theory of reltivity. Smering ll the mtter nd rdition found in strs, glxies nd elsewhere out in uniform wy throughout the universe, the mthemtics becomes reltively simple nd cn be summed in two differentil equtions. They re usully nmed fter the Russin mthemticin Alexnder Friedmn who derived them in We will need them in the following. Insted of n exct derivtion, we will here give some non-reltivistic rguments which reproduce the essentil prts of of the equtions. Consider prticle with mss m distnce from some point in the universe we will cll the center. This is some rbitrry point since there is no rel center in the universe. In the rdil direction it hs the kinetic energy mȧ 2 /2 while it is pulled bck by grvittionl force due to the mss inside the rdius. If ρ is the uniform mss density of the universe, this mss is M = (4π/3) 3 ρ. Since it corresponds to potentil energy of GMm/, the totl energy of the prticle is E = mȧ 2 /2 GMm/. We cn now rewrite this s ȧ 2 + k = 8π 3 Gρ2 (11) when we introduce k = 2E/m. And this is exctly Friedmns first eqution! The only new ingredient we would hve hd if we hd derived it from Einstein s generl theory, is the vlue of the constnt k. It could then tke only three vlues depending on the curvture of the universe. When k = +1 the universe hs positive curvture nd finite size, while for k = 0 it is Eucliden flt nd infinite. Also in the cse of negtive curvture k = 1 it is infinite in extent but now with hyperbolic geometry. If we think of the mss m s prticle lunched out into spce from the mss M, it hs negtive energy when k = 1 nd cnnot escpe. It just reches mximl 3

4 m Figure 1: Prticle with mss m represents glxy nd is ttrcted by the mss inside sphere with rdius. height nd flls bck gin. This corresponds to finite universe. On the other hnd, when k = 1, it hs positive energy nd cn escpe to infinity. Then we hve n infinite universe. In the limiting cse k = 0 the prticle brely reches infinity nd the universe gin hs n infinite extent. Insted of describing this mechnicl problem using the energy principle, we could just s well do it with the help of Newtons second lw. It reltes the prticle ccelertion to the grvittionl force cting on it, i.e. mä = GMm/ 2. With the bove expression for the mss M, we then find ä/ = (4πG/3)ρ. And gin, this is Friedmns second eqution s long s we cn ignore the pressure p in the mtter. If the mtter moves non-reltivisticlly, we expect this to be negligible - s we will lter see. But for rdition we found the pressure p = ρ/3 nd thus hs size similr to the mss density ρ. A correct derivtion including both of these contributions led to the sme eqution s just derived but with the single replcement ρ ρ+3p. The correct form of Friedmns second eqution is therefore ä = 4πG ( ) ρ + 3p (12) 3 Notice tht in our units both the mss density ρ nd the pressure p hve the sme dimensions. Given n eqution of stte p = p(ρ) we now hve two differentil equtions for the two unknowns nd ρ. But they re not relly completely independent of ech 4

5 other. To show tht we tke the time derivtive of the first eqution (11) which gives 2ȧä = 8π 3 G d d (ρ2 )ȧ or ä = (4πG/3)d(ρ 2 )/d. Compring this with the second eqution (12) we thus must hve d d (ρ2 ) = (ρ + 3p) for consistency. Multiplying both sides with nd combining terms, it then follows tht d d (ρ3 ) = 3p 2 (13) But this simple result following directly from the Friedmn equtions is nothing else thn the second lw of thermodynmics du + pdv = T ds for n dibtic process defined by ds = 0. In fct, the energy in sphere of volume V = (4π/3) 3 is U = ρv nd dv = 4π 2 d which then gives (13). Hd we insted bsed our derivtion on Einsteins reltivistic theory of grvittion, we would hve found the sme result s consequence of locl conservtion of energy nd momentum during the expnsion. Thus the expnsion (or contrction) of the universe following from these equtions will be such tht the totl entropy of the rdition nd mtter in the universe remins constnt. Relly n mzing nd beutiful result! But perhps not so surprising. Since there is nothing else outside the universe, it is relly n isolted system for which the entropy should remin constnt. But only s long s the expnsion cn be considered to be thermodynmiclly reversible. But is it? And could there not be other sources of entropy besides the one considered here? We will come bck to these questions lter. In the bove derivtion the quntity strted out s rdius of some imginry sphere in the universe. We must now drop this nive, mentl picture. In more relistic description it is scle fctor which determines the distnce between points in the expnding universe. A simple illustrtion would be two-dimensionl universe corresponding to the surfce of blloon being blown up. The surfce hs positive curvture nd is finite in size. In this cse the scle fctor is the rdius of the blloon so tht the distnce D between two points in this universe is proportionl to it. Thus D = θ where θ is the ngulr seprtion between the two points seen from the center of the blloon. The points re fixed to the surfce during the expnsion nd the ngle θ therefore stys fixed during the expnsion of the blloon. 5

6 For our three-dimensionl universe we hve the sme distnce lw D = θ between two points which follow the expnsion. The points we tlk bout here re just mthemticl ideliztions, in relity they cn be considered to be glxies. They re therefore seen to move wy from ech other with velocity V = Ḋ = θȧ. This we cn write s V = HD where H = ȧ is the Hubble prmeter. We hve thus rrived t one of the bsic properties of the expnding universe, nmely Hubbles lw sying tht every glxy moves wy from us with velocity proportionl to the distnce to the glxy. The best vlue for the bsic prmeter H observed tody when distnces re mesured in units of Mpc, is (14) H 0 = 71 ± 1 km s Mpc (15) In reltivistic units when distnces re mesured in seconds, this is equivlent to H 0 = 2.30± s 1 obtined using the conversion formuls t the beginning of this chpter. 3 Equtions of stte The Friedmn equtions cn only be solved when we know the eqution of stte p = p(ρ). Using it in (13) we cn then solve for ρ = ρ() which tells us how the mss density vries with the scle fctor. Combined with the first Friedmn eqution (11) we cn then solve for = (t) which tells us how the universe expnds with time. Energy in the form of rdition hs the eqution of stte p = ρ/3. We derived it first for photons, but it pplies to ll mssless prticles. It is lso vlid for mssive prticles when they re moving with moment much lrger thn their msses. This is know s the extreme reltivistic or ER limit opposite to the non-reltivistic or NR limit where the moment re much smller thn the mss of the prticles. Non-reltivistic prticles t temperture T is described by the idel gs eqution of stte p = nk B T when we cn ignore the interctions between them. Here n = ρ/m is the prticle number density when m is their common mss. Their verge squred velocity follows from the equiprtition principle s v 2 = 3k B T/m resulting in p = ρ v 2 /3. But for NR prticles v 2 1 nd we therefore cn tke p = 0 for such mtter. This simple eqution of stte is lso ssumed to hold for the unknown, 6

7 drk mtter even if we don t know nything bout the prticles it consists of. But this conclusion should be ok s long s the known lws of physics pplies to them. Until ten yers go it ws generlly believed tht the universe contined only rdition nd mtter. But since then it hs become cler from stronomicl observtions tht there is lso third component usully clled drk energy. Even if energy nd mtter re equivlent in reltivistic theories, drk mtter nd drk energy should not be confused with ech other. While drk mtter is ssumed to hve the bove NR eqution of stte p = 0, the eqution of stte for drk energy is found to be p = ρ. As we now will see, tht hs drmtic consequences. All these three equtions of stte cn be summrized in the simple eqution p = wρ (16) where the prmeter w = 1/3 for rdition, w = 0 for NR mtter nd w = 1 for drk energy. Inserted into the consistency eqution (13) it gives We esily rewrite it to obtin the form d d (ρ3 ) + 3wρ 2 = 0 dρ ρ = 3(1 + w)d fter expnding the derivtive. Now it is strightforwrd to integrte from n epoch with scle prmeter until tody when it hs the vlue 0, giving where ρ 0 is the mss or energy density tody. ( ) 0 3(1+w) ρ() = ρ 0 (17) For drk energy with w = 1 the dependency on the scle fctor is seen to go wy nd the energy density ρ v () = ρ v0 is constnt throughout the history of the universe. This is specil cse nd nd is equivlent to the cosmologicl constnt introduced by Einstein. He needed it in his very first cosmologicl model when he thought the universe ws sttic, i.e. forever the sme. When it ws few yers lter insted discovered by Hubble to expnd, Einstein did not consider this constnt ny longer physiclly relevnt. But tody it is needed gin to explin tht this expnsion is ctully observed to be ccelerting, i.e. ä > 0. In fct, from (12) we see tht ä < 0 s long s the density ρ nd pressure p re positive s they re for rdition nd mtter. But for drk energy p = ρ nd the resulting ccelertion is positive. 7

8 Mtter hs w = 0 nd from (17) the density is seen to vry s ( ) 0 3 ρ m () = ρ m0 (18) This cn be understood from the simple fct the number of non-reltivistic prticles in comoving volume is constnt while the volume itself increses like 3. Similrly, for rdition with w = 1/3 we find ( ) 0 4 ρ r () = ρ r0 (19) If the rdition is mde up of photons with wvelength λ this will be stretched out by the expnsion nd therefore increse. The corresponding frequency ω thus decreses like 1/. Since the energy of ech photon is hω, it will therefore lso decrese like 1/. Combined with the volume increse 3, this explins the 1/ 4 decrese of the rditive energy density (19). If it is in thermodynmicl equilibrium with temperture T, it will be proportionl to T 4 ccording to the Stefn-Boltzmnn lw. Compring this with (19), one finds tht it vries like 0 T () = T 0 (20) during the expnsion. Just fter Big Bng the universe is therefore very hot. The photons from tht epoch re now observed s cosmic microwve bckground rdition (CMBR) nd hve the temperture T γ0 = ± K. It is extremely isotropic in ll directions, but there re tiny fluctutions in this temperture which re fctor ten thousnds smller. Their distribution cross the sky observed by the stellites COBE nd WMAP tells us very much bout the properties of the erly universe. In the next section we will find tht the scle fctor decreses when we go bck in the history of the universe. This is just nother wy of sying tht it is expnding with incresing time. Thus, when we look bck nd consider the universe just fter the Big Bng when 0, the rdition density (19) will be dominnt. In this erly stge of the evolution we cn therefore neglect the vcuum energy nd even the NR mtter energy. Only ER mtter will contribute s prt of the generl rdition energy density. The first Friedmn eqution (11) we cn write s k = 8π Gρ 2 H2 3 when we reorder it nd introduce the Hubble prmeter (14). It cn be rewritten s k/ 2 = H 2 (Ω 1) when we define Ω = ρ/ρ c where the so-clled criticl density is ρ c = 3H2 8πG (21) 8

9 For its numericl vlue we need the grvittionl constnt 8 cm3 G = (22) g s 2 which in reltivistic units is equivlent to G = cm/g. With the present vlue (15) for the Hubble constnt, the criticl density is then found to be ρ c0 = g/cm 3 = 5200 ev/cm 3 tody. It ws lrger in the pst. Mesurements by the WMAP stellite combined with other stronomicl observtions hve mde it cler during the lst few yers tht the universe is sptilly flt. The curvture prmeter is therefore k = 0 nd the reltive mtter density Ω = 1. Since we hve three kinds of mtter in the universe, we must therefore lwys hve Ω r + Ω m + Ω v = 1 (23) where Ω i = ρ i /ρ c for ech component. Different observtions give the present vlues Ω v0 = 0.73 nd Ω m0 = Thus the dominnt energy component of the universe tody is the vcuum energy. Mtter ppers s drk mtter nd ordinry, bryonic mtter with densities ρ d nd ρ b respectively so tht ρ m = ρ d + ρ b. The reltive mtter energy density Ω m = Ω d + Ω b is tody dominted by the drk component with Ω d0 = Therefore Ω b0 = 0.04 or only 4% of the energy content of the universe is due to the sme kind of mtter s we know here on Erth! The contributions from rdition is tody negligble. This follows from the Stefn- Boltzmnn lw which is ρ γ = π2 15 T 4 γ (24) in HEP units. With the present temperture T γ0 = ± K of the CMB rdition, this gives ρ γ0 = 0.26 ev/cm 3. Dividing by the criticl density (21), it follows tht Ω γ0 = One cn show tht the contribution from strlight in the universe is even smller. Lter we will see tht mssless neutrinos will give contribution of the sme, smll mgnitude. But tody we know tht the neutrinos re not exctly mssless nd thus cn give significntly bigger contribution. We will come bck to this prt lter. 4 Expnsion of the universe Our universe is flt universe with curvture prmeter k = 0 nd the first Friedmn eqution (11) simplifies to ȧ 2 = 8πGρ 2 /3 or H 2 = 8πG 3 ρ (25) 9

10 since H = ȧ/. Here ρ is the totl mss-energy density with the three principl prts ρ = ρ r + ρ m + ρ v. Ech vries with the scle prmeter ccording to (17). Dividing by H0 2 nd expressing the present mss densities by their reltive mgnitudes, we then get H 2 H 2 0 ( ) 0 4 = Ω r0 + Ωm0 ( ) Ωv0 This is now first-order differentil eqution for the scle prmeter when we tke the squre root on both sides. Writing it s d [ Ω r0 ( 0 ) 4 + Ωm0 ( ) ] 0 3 1/2 + Ωv0 = H 0 dt (26) we see tht it cn be solved by direct integrtion. The solution = (t) describes the evolution of the universe. Ten yers go it ws generlly believed tht the vcuum energy could be ignored so tht the universe ws mss dominted tody. In such cse we hve Ω v0 = 0 nd gin neglecting the smll contribution from rdition tody, it follows tht Ω m0 = 1 from (23) to keep the universe is flt. From the mster eqution (36) we then integrte from n initil time t = 0 with = 0 to lter time t with scle prmeter, i.e. 0 d ( ) 1/2 t = H0 dt We hve here lso neglected the rdition contribution t very erly times which is not completely correct. For better ccurcy it must be included. Ignoring this smll compliction, the bove integrls re now elementry nd give for the scle fctor (t) = 0 (3H 0 t/2) 2/3 (27) It increses with time s illustrted in Fig.2 nd becomes equl to the present vlue 0 t time when 3H 0 t 0 = 2 or t 0 = 2 3H 0 (28) With the vlue (15) for the Hubble constnt this gives t 0 = 9.4 Gyr or somewht less thn ten billion yers. It would be the ge of the universe nd would not chnge very much including the rdition prt. The problem with this simple cosmology ws tht the universe contins clusters of strs being nerly thirteen billion yers old. And they must be younger thn the 10

11 t Figure 2: In mss-dominted universe the scle fctor increses s t 2/3. universe! The simplest wy to obtin longer ge ws to include the vcuum energy. We must then repet the bove clcultion leding to the somewht more difficult integrl 0 d (Ω v0 2 + Ω m0 3 0/) 1/2 = H 0 Here gin Ω m0 + Ω v0 = 1. It cn be done using hyperbolic functions resulting in t 0 dt ( )1/3 ( Ωm0 3 ) (t) = 0 sinh 2/3 Ω v0 H 0 t Ω v0 2 (29) Agin the ge of the universe t 0 is determined by (t 0 ) = 0 which gives t 0 = ( ) 2 1/2 Ωv0 rcsinh (30) 3H 0 Ωv0 Ω m0 Inserting the mesured vlues for the different prmeters we find the more relistic ge t 0 = 13.7 Gyr. It is consistent with the ge of the oldest globulr str clusters nd is now the generlly ccepted ge of the universe. It is esy to see from (29) tht the expnsion of the universe is tody ctully ccelerting, i.e. ä > 0. It hs ctully been doing tht for the lst severl billion yers. 11

12 In the future when t t 0 the hyperbolic function becomes simple exponentil function resulting in ( ) 1/3 Ωm0 (t) = 0 e Ω v0 H 0 t 4Ω v0 This type of cosmology ws first studied by the Dutch stronomer de Sitter just fter Einstein hd pplied his generl theory of reltivity to the universe. On the other hnd, t erlier times the expnsion ws insted decelerting. This cn be seen directly from the second Friedmn eqution (12) giving ä < 0 since then p > 0 when only mtter nd rdition re present. (31) 5 The rdition-dominted erly universe At very erly times when 0 the rdition term domintes in the eqution of motion (36). It then simplifies to d = 0 2 Ωr0 H 0 dt which gives the solution ( (t) = 0 2 Ω r0 H 0 t ) 1/2 (32) fter integrtion. It is only vlid s long s 0. With incresing time the mtter component of the universe will eventully strt to mtter resulting in somewht fster expnsion of the type (27). The rdition energy density ρ will decrese s (19) s function of the scle fctor nd therefore s ρ = ρ r0 /4Ω r0 H 2 0t 2 for incresing times when we use the bove result for (t). Now Ω r0 = ρ r0 /ρ c0 where the criticl density tody is ρ c0 = 3H 2 0/8πG. Thus we find the importnt formul 3 ρ = (33) 32πGt 2 relting directly the density in the erly universe to its ge. Wht we cll rdition here, is not only mde up of photons, but ll prticles with msses much smller thn the temperture T in the universe. Furthermore, we will ssume tht ll chemicl potentils re zero so tht the number of prticles nd ntiprticles re the sme. We will lso ssume tht interctions between the prticles mking up the rdition cn be neglected. This is highly non-trivil ssumption nd is relted to the property tht ll prticles re described by soclled non-abelin guge theories hving this crucil property. The rdition energy density ρ is then simply given by the Stefn-Boltzmnn formul, djusted by the spin degrees of freedom for ech prticle type. Using HEP units with c = h = 1 we cn thus write ρ = g π2 30 T 4 (34) 12

13 when we lso set k B = 1, mesuring temperture in energy units ccording to the conversion formul (10). Here we hve lso introduced the spin degenercy fctor which is g = 1 for ech bosonic spin degree of freedom nd g = 7/8 for the corresponding fermion spin degrees of freedom. For exmple, photon hs two possible spin directions nd therefore g γ = 2. A mssless neutrino hs lwys the spin in the opposite direction of its momentum nd therefore g ν = 7/8 while n electron cn hve its spin both long nd opposite to the momentum giving g e = 7/4. Combining (33) nd (34) we cn now express the squred ge of the universe directly by its temperture, t 2 = π 3 g GT 4 In the sme units the grvittionl constnt (22) is found to hve the vlue 1 G = GeV (35) so tht 1 GT = 1.49 MeV (T/MeV) 4 MeV 4 Here T/MeV is the temperture mesured in units of MeV which we henceforth will denote by T MeV. Now from (8) we hve MeV = 1.0 fm 1 = s 1 nd therefore 1/GT 4 = 64.5 s 2 /TMeV 4. Using this in the bove formul for the squred ge of the universe, it results in t 2 = 5.85 s 2 /g TMeV 4. Tking the squre root nd mesuring the time in units of seconds, i.e. defining the dimensionless time t s = t/s, we finlly hve the very importnt formul t s = 2.42 (36) g T 2 MeV It gives directly the ge of the universe mesured in units of seconds, in terms of its temperture, mesured in MeV. We cn therefore lredy sy tht the universe must hve temperture round 1 MeV when it is pproximtely one second old. Notice tht there is no nucler or elementry prticle physics informtion used in getting this result! 6 Formtion of nucleons All the known elementry prticles re described within theoreticl frmework clled the stndrd model. It predicts tht when they re in therml equilibrium t 13

14 temperture T 100 GeV they re ll mssless nd will thus mke up the rdition content of the erly universe. Therml equilibrium will exist when the expnsion rte of the universe is slower thn the rection rte between the different prticles. These rtes cn be clculted nd we know when equilibrium thermodynmics cn be used to describe their properties. The bosonic prticles re 8 colored guons, 4 electrowek vector bosons nd 4 spinless Higgs prticles. Since the first two groups hve spin S = 1 like the photon, their contribution to the g-fctor becomes g B = (8 + 4) = 28 (37) In the fermion sector there re 6 qurks, ech in three different colors, 3 leptons like the electron nd 3 neutrinos. With the sme contribution from their ntiprticles, the fermionic g-fctor becomes g F = 7 8 ( ) 7 ( ) = 90 (38) 8 For the totl g-fctor g = g B + g F needed in the formul (36) we thus obtin g = 427/4 = A temperture of T = 500 GeV is then seen to correspond to n ge of t = s of the universe! For even erlier times nd therefore lso higher tempertures, we will need improvements of the stndrd model. So fr tht is unknown territory which will hopefully be explored in the not too distnt future. Let us now sketch the therml history s the universe evolves from these erly times nd cools down. When the temperture hs become T < 100 GeV most of the prticles hve become mssive except for the photon nd the gluons. But the temperture is still so high tht most of them cn still be considered to be mssless nd therefore described s rdition. Exceptions re the top qurk with mss of 174 GeV nd the wek bosons with msses round GeV. But these re ll unstble nd will decy into lighter prticles. When the temperture hs cooled below T = 1 GeV the universe consists of u, d, s colored qurks nd their ntiprticles, gluons, photons, neutrinos nd ntineutrinos plus the leptons e ± nd µ ±. They cn ll be considered to be mssless s long s T > 200 MeV. The totl g-fctor is therefore g = (8 + 1) ( ( ) ) 2 = 247/4 (39) or g = In this temperture intervl the universe is therefore round t = 10 6 s old. After the temperture hs fllen below T = 100 MeV the qurks, ntiqurks nd gluons hve combined into bryons with strong interctions. Erlier there must 14

15 hve been slightly more qurks thn ntiqurks becuse we know tht the ordinry mtter in the universe is mde up of nucleons, i.e. protons p nd neutrons n, nd not ntinucleons. The nucleons re now stble prticles with msses m p = MeV nd m n = MeV. They will therefore not contribute to rdition energy, but insted mke up the non-reltivistic bryonic mtter density. The muon leptons µ ± hve mss m µ = MeV, but re unstble nd will be replced by the decy products e ±, photons nd the neutrinos with their ntiprticles. As long s T > 1 MeV these prticles cn be considered s rdition with g = ( ) 2 = 43/4 (40) When the temperture hs dropped down to T = 1 MeV, it follows then from (36) tht the expnsion of the universe hs lsted very close to t = 1 s. At these tempertures protons nd neutrons re still in therml equilibrium with the other prticles through rections like p + e n + ν e n + e + p + ν e medited by the wek force. Since these nucleons re non-reltivistic, their number densities re given by the Mxwell distribution n i = g i ( mi T 2π ) 3/2 e m i/t where the spin degenercy fctor g i = 2 for these spin-1/2 fermions. Since the proton hs slightly smller mss thn the neutron, there will be somewht more protons thn neutrons in the universe t these tempertures ccording to the formul n n n p = ( mn m p (41) ) 3/2 e m/t (42) where m = m n m p = 1.29 MeV. Here s before we ignore ny chemicl potentils. This rtio will therefore decrese when the temperture sinks below T = 1 MeV nd there will thus be fewer neutrons thn protons in the universe. More detiled clcultions show tht when the temperture hs fllen to T = 0.8 MeV, the bove rections goes out of equilibrium. The corresponding rtio (42) then hs the vlue n n /n p = exp( 1.29/0.8) = 0.2 = 1/5. One sys tht there is nucluon frezze-out. At this stge of the evolution there re therefore five protons for ech neutron, i.e. the nucleon content of the universe is pppppn. The number frction of neutrons is therefore X n = n n n n + n p = 1 = n p /n n = (43)

16 Becuse the universe is ssumed to be electriclly neutrl, there will be the sme number of electrons s protons. 7 Neutrino freeze-out During this epoch the three neutrinos types ν e, ν µ nd ν τ nd their ntiprticles remin in therml equilibrium with ech others through wek interctions like e + + e ν + ν e ± + ν e ± + ν, e ± + ν e ± + ν The corresponding rection rtes decrese with decresing tempertures nd when T flls below 1 MeV they become slower thn the expnsion rte of the universe. Thus the neutrinos will lso go out of therml equilibrium round this temperture. Their number densities will still be described by the Fermi-Dirc distribution, but will no longer be ffected by the presence of other prticles. Throughout this epoch there is lso smll contribution to the energy density from the bryons nd the cold drk mtter, but since these prticles re non-reltivistic nd few in number compred with the others, these contributions cn be ignored. When decresing temperture the positrons dispper round T = 0.5 MeV. Until then they re in equilibrium with the electrons nd photons through rections like e + + e γ + γ. For this rection to proceed, the photons need to hve n energy greter thn 0.5 MeV which is the mss of the electron/positron. For lower tempertures essentilly ll the positrons nnihilte with electrons nd re converted to photons ccording to e + + e γ + γ. Thus the density of photons increse, corresponding to n increse in the photon temperture. This we cn now clculte since we know tht the totl entropy S = sv hs to be conserved. For the volume we cn tke V = (4π/3) 3 while the entropy density is s = 4ρ/3T when there re no chemicl potentils. From the energy density (34) we then get from the equlity of entropy before nd fter nnihiltion ( ) (T γ ) 3 before (T ν) 3 before = 2 (T γ ) 3 fter (T ν) 3 fter Since the neutrinos re decoupled, we will hve (T ν ) before = (T ν ) fter. Thus it follows tht (T γ ) fter /(T γ ) before = (11/4) 1/3. But before the nnihiltion the neutrino nd photon tempertures were the sme nd therefore ( ) ( ) Tγ 11 1/3 = = (44) 4 T ν fter 16

17 This rtio will now remin the sme until the present epoch when T γ0 = 2.73 K. Thus the temperture of the cosmic neutrino bckground rdition tody is T ν0 = 1.95 K. So fr we hve ssumed the neutrinos to be mssless nd their properties given by stndrd Fermi-Dirc sttisticl mechnics. But we know tody tht they ctully hve non-zero msses of mgnitudes somewht less thn 1 ev. So the bove description is only vlid when T ν m ν. But for these tempertures their number density is given by n ν+ ν = (3/4)n γ which will be vlid down to tempertures T ν < m ν since the neutrinos re stble prticles. The number density of one neutrino flvor tody is therefore n ν+ ν (t 0 ) = 3 4 n 4 γ(t 0 ) 11 = cm 3 = 112 cm 3 (45) nd the corresponding non-reltivistic mss density ρ ν+ ν (t 0 ) = m ν n ν+ ν (t 0 ) For the energy density rtio resulting from the three neutrino flvors known tody we therefore hve 3 Ω ν0 = ρ ν+ ν (t 0 )/ρ c0 = M ν (46) 47 ev ν=1 where M ν = 3 ν=1 m ν. We hve here used the previous vlue ρ c0 = 5200 ev/cm 3 for the criticl mss density tody. We thus see tht if the neutrinos hd hd somewht lrger msses, round 5-10 ev, they could hve been very good cndidtes for drk mtter! 8 Helium production The lightest nd most stble nucleus is found in the helium He tom nd is usully clled n α prticle mde up of two protons nd two neutrons. Above we found tht the nucleon content of the universe ws pppppnnppppp when the temperture hd fllen below 0.8 MeV. If the He nucleus could form directly from this mixture, we see from (43) tht there would be one α prticle mong eight protons. The weight frction of helium nuclei would therefore be Y α = 4/(4 + 8) = 1/3 = 0.33 since ech weighs four times s much s proton. This is somewht higher thn the observed rtio The reson is tht n α- nucleus cnnot form directly from two protons nd two neutrons. Insted one must first form deuterium d from the rection p + n d + γ (47) 17

18 where the photon hs the energy E γ = m n + m p m d = 2.22 Mev. But this intermedite production will not tke plce right-wy for tempertures T < 0.8 MeV. The reson is tht this vlue represents the verge temperture of the blckbody photon rdition. There will remin smll til in the Plnck distribution of photons with energies E > 2.22 Mev even t these lower tempertures. And s long s tht is the cse, the rection (47) will lso go in the reverse direction, breking up the deuterium nuclei. More detiled clcultions show tht first when the temperture hs sunk to T = 0.08 MeV will the deuterium nuclei strt to survive. And then follows very fst rections like d + d α + γ nd others producing stble helium nuclei. A very smll mount of deuterium will void this formtion nd survive to be observed in the universe. The rdition will now consist only of photons nd the decoupled neutrinos with the temperture given by (44). For the effective g-fctor we then hve g = ( 4 ) 4/3 = 3.36 (48) 11 From the mster formul (36) follows then tht the corresponding ge of the universe is or three nd one hlf minutes. t = 2.42 s = 210 s 2 We cn now understnd why we bove got result for the frction of helium which ws bit too lrge. Since free neutrons re rdioctively unstble nd decy like n p + e + ν e with lifetime of τ = 890 s, the number of neutrons vilble for deuterium production will be reduced by the fctor exp 210/890 = Thus the finl weight frction of helium becomes Y α = = 0.26 which is very close to the observed vlue. After 210 seconds evolution the min ingredients of the universe is then photons, neutrinos, electrons, protons nd α-prticles forming plsm with complicted, electromgnetic interctions. It will continue to be rdition dominted for very long time nd cooling down ccording to the formul (36). Nothing much hppens nd it is very dull period. Eventully the temperture will be become few ev nd neutrl hydrogen toms H will form ccording to p + e H + γ (49) nd similrly for the formtion of neutrl helium toms He. Erlier the photon energies very so high tht these rections lso went just s likely in the opposite 18

19 direction. But from this time on there re no chrged prticles in the universe nd the photons cn move freely. This trnsition t time t dec to neutrl mtter is usully clled decoupling or time for lst scttering. The photons we see tody in the CMB rdition comes from this epoch. Assuming rdition dominnce ll the wy to this decoupling time, we cn then clculte it from the mster formul (36). We know tht when the temperture ws T 0.1 MeV the ge ws t 100 s. Thus it follows tht t dec = ( T T dec ) 2t ( ) s = s which roughly corresponds to t dec yers. From then on the lmost uniform mtter in the universe will slowly strt to clump together by the grvittionl interction, strs will form which lter cluster into glxies nd so on until we hve the universe we see tody. April 2010, Finn Rvndl 19