1 Lecture-05 Thermodynamics in the Expanding Universe Ping He ITP.CAS.CN

2 5.1 The Boltzmann Equation The early universe, mostly is in thermal equilibrium, however, there are notable departures from thermal equilibrium. neutrino decoupling, CMB decoupling, BBN, inflation, baryogenesis, relic WIMP decoupling, etc When decoupled The evolution of distribution function (DF) is simple for LTE or decoupling Difficult for the epoch just around decoupling, and we have a rough criterion for decoupling  is interaction rate per particle, H is the expansion rate

3 Eq-5.1 : can only be qualitative and semi-quantitative Boltzmann equation: deal with DF evolution precisely. e.g. BBN calculation Collision operator Liouville operator Non-relativistic for f(t, v, x) with external force F Generalized to covariant relativistic form, we have Gravitational force

4 For FRW model, With Robertson-Walker metric, Liouville operator is Using the definition of number density To integrate Boltzmann equation, we have

5 Interaction For example, just consider the number evolution for , the interaction is The collision term is +: bosons -: fermions

6 Invariant phase-space volume element

7 In the most general cases, the Boltzmann equations are a coupled set of integral-partial differential equations for all the species. Only one or two need to be considered, the others are in equilibrium, which is denoted as  Two simplifications: (1) T (or CP) invariance (2) Use Maxwell-Boltzmann statistics instead of Fermi-Dirac and Bose-Einstein. For any non-relativistic species, (m i -  i )/T >>1

8 Introduce comoving number density to scale out expansion effect Using the conservation of entropy per comoving volume (sa**3=constant) Introduce a time (or temperature) variable

9 During the radiation-dominated epoch Here, So, the Boltzmann equation can be rewritten as

Freeze Out: Origin of Species A species has an abundance of when in equilibrium. If the species freeze out, i.e., , at a temperature so that m/T ~ 1, the species can have a significant relic abundance left today. Assuming  is stable, that is, no spontaneous decay, only annihilation and inverse annihilation processes take place. Further more, assuming: 1. Symmetry between 2. in thermal equilibrium with zero chemical potential.

11 For example: weak weak & electromagnetic So, e - and e+ are in good thermal equilibrium Consider the factor in the collision term in the Bolt-Equ. Since are in thermal equilibrium, with, we have: Energy conservation, so we have

12 Now, the interaction term can be written in terms of, If define, Eq-5.21 can be:

13 The thermal-average annihilation cross section times velocity is: If, there are other annihilation channels, then we have: Then,

14 In the non-relativistic (x>>3) and in the relativistic (x<<3), the comoving number density: where Since

15 Y Y EQ x decouple:  Y=const In the following, we use Boltzmann equation to study hot and cold relics In both relativistic and non-relativistic cases, with decreasing T than H. So when decreases more rapid

Hot Relics relativistic, x f <<3, and Y EQ ~ const If the expansion is isentropic, then A species that decouples when it is relativistic is often called a hot relic. The present-day value:

17 Since the mass of  For light neutrinos (m<MeV), decouple when T ~ MeV For 2-component neutrino

18 For a massless species, its temperature is So when it decouples very early on, that is is very large, then T  is much less than photon temperature. They are called warm relic. From Eq-5.31, we know that depends upon If a species decouples very early on, say at T>300GeV, when 

Cold Relics Freeze out (or decoupling) occurs when the species is non-relativistic Easy: or (x f >> 3), the species is called cold relics. See Eq-5.25Eq phase-I phase-II

20 How to determine x f, and the detail around x ~ x f Parameterizing the cross section as (usually takes power-law dependence) n=0 s-wave annihilation, n=1 p-wave annihilation The Boltzmann Equation takes the form: Where

21 Define At early times (1 < x << x f ), Y tracks Y EQ very closely, are small, and an approximate solution is obtained by setting At late times ( x >> x f ), Y tracks Y EQ very poorly, can be safely neglected, so phase-I

22 Integrate from phase-II See Fig-5.1 Now, determine  define From Eq-5.40, we get, so where Choosing c(c+2)=n+1 gives the best fit, within 5%

23 By we can also get x f, and set Differ with just the pre-factor, for example, for The present-day number and mass density of relic 

24 Note that the relic density of   is inversely prop to cross section and mass of the particle So we find that and That is, the present-day mass density only depends upon the annihilation cross section at freeze out.

Two Examples of Cold Relics (1) A baryon-asymmetric Universe Taking nucleon -- antinucleon cross section Today’s value,, indicating that we live in a baryon-asymmetric universe. We will consider baryogenesis in the next lecture.

26 (2) A heavy stable neutrino (m >> MeV), decouples as non-relativistic The annihilation cross section is:

27 In Dirac case, where c2 ~ 5, g=2, g* ~ 60 Similar results hold for Majorana case

28 Since, we have: and

Recombination Revisited Calculate the residual ionization more precisely with Boltzmann equation number density for free electron, free proton, and hydrogen atoms The equilibrium ionization fraction: is the baryon-to-photon ratio, B=13.6eV is the binding energy of hydrogen. In the post-recombination era, where

30 Note that, following the evolution of the ionization fraction X e is analogous to following the evolution of the abundance of a stable, massive particle species. The Boltzmann equation for n e : where is the thermally-averaged recombination cross section:

31 Taking the universe to be matter-dominated, define Consider ， the deviation from ionization equilibrium. The equation governing the evolution of is: For an approximate solution is obtained by setting

32 On the other hand, for, so that can be neglected, and Integrating from, we obtain When free-out at x=x f, so that And

33 Then And the residual ionization fraction is If using the criterion for freeze out,, then

Concluding Remarks For, We have (3) And Boltzmann Equation Can be extended to (4) Results see Fig-5.1Fig-5.1 (1) (2)

35 References E.W. Kolb & M.S. Turner, The Early Universe, Addison-Wesley Publishing Company, 1993