Cosmological Preliminaries

FLRW Metric: $d{s}^2=d{t}^2-a^2(t)(\frac{d{r}^2}{1-k{r}^2}+r^2(d{\theta }^2+si{n}^2\theta d{\varphi }^2))=g_{\mu \nu }d{x}^{\mu }d{x}^{\nu }$

The constant $k=\{-1,0,+1\}$ corresponds to the spatial curvature (k=0 ⇒ flat universe).

The affine connection (akaChristoffel-symbols ) are: ${\Gamma }_{\nu \lambda }^{\mu }=\frac{1}{2}{g}^{\mu \sigma }({\partial }_{\nu }{g}_{\lambda \sigma }+{\partial }_{\lambda }{g}_{\sigma \nu }-{\partial }_{\sigma }{g}_{\nu \lambda })$

Hubble Parameter: $H\equiv \frac{\dot{a}(t)}{a(t)}$

Dark matter freeze-out occurs before Big Bang Nucleosynthesis. Thus, for a radiation dominated universe, $H=1.66g_{\ast }^{\frac{1}{2}}\frac{T^2}{M_P}$ where $M_P=1.22\times 10^{19}GeV$.

We define a dimensionless parameter $x=\frac{m}{T}$ and define $H(m)$ as:

$H(m)=1.66g_{\ast }^{\frac{1}{2}}\frac{m^2}{M_P}=H{x}^2$

The phase space distribution function f describes the occupancy number in phase space for a given particle in kinetic equilibrium and distinguishes between fermions and bosons.

$f=\frac{1}{e^{\frac{(E-\mu )}{T}}\pm 1}$, where the (-) sign corresponds to bosons and the (+) corresponds to fermions. E is the energy and µ is the chemical potential.

For species in chemical equilibrium, the chemical potential is conserved in the interactions. Thus, for processes such as $i+j\leftrightarrow c+d$, we have ${\mu }_i+{\mu }_j={\mu }_c+{\mu }_d$. Since the number of photons is not conserved in the interactions, ${\mu }_{\gamma }=0$.

Now, using the expression for the phase space distribution function f and integrating in phase space, we can compute a series of observables in the Universe. In particular, the number density of particles, n, the energy density, ρ, and pressure, p, for a dilute and weakly-interacting gas of particles with g internal degrees of freedom read

$n=\frac{g}{(2\pi {)}^3}\int f(\text{p})d^{3}p$

$\rho =\frac{g}{(2\pi {)}^3}\int E(\text{p})f(\text{p})d^{3}p$

$p=\frac{g}{(2\pi {)}^3}\int \frac{|{\text{p}}^2|}{3E(\text{p})}f(\text{p})d^{3}p$

Let us now define densities normalised by the time dependent volume $V(t)=a(t{)}^3$, the reason being in the absence of number changing processes, the comoving number density remains constant with time evolution. An expanding Universe is a closed system and in thermal equilibrium, the total entropy is conserved.

$TdS=d(\rho V)+pdV=d((\rho +p)V)-Vdp=0$

The entropy density is therefore $s=\frac{S}{V}=\frac{(\rho +p)}{V}$.

We define theyield as a fraction of the number density and entropy density as $Y=\frac{n}{s}$

The evolution of entropy density as a function of the temperature is given by

$s=\frac{2{\pi }^2}{45}{g}_{\ast s}{T}^3$, where the effective number of relativistic degrees of freedom for entropy is $g_{\ast s}={\sum }_{bosons}g(\frac{T_i}{T}{)}^3+\frac{7}{8}{\sum }_{fermions}g(\frac{T_i}{T}{)}^3$.

The energy density can be expressed as

$\rho =\frac{{\pi }^2}{30}{g}_{\ast }{T}^4$ in terms of relativistic number of degrees of freedom $g_{\ast }={\sum }_{bosons}g(\frac{T_i}{T}{)}^4+\frac{7}{8}{\sum }_{fermions}g(\frac{T_i}{T}{)}^4$.

In these two equations, T is the temperature of the plasma and $T_i$ is the effective temperature of each species.

The yield can be written as:

1. Relativistic Species

$n=\frac{g_{eff}}{{\pi }^2}\zeta (3)T^3$, where $g_{eff}=g$ for bosons and $g_{eff}=\frac{3}{4}g$ for fermions. Remember, $\zeta (3)\approx 1.202$.

Hence, $Y_{eq}=\frac{45}{2{\pi }^4}\zeta (3)\frac{g_{eff}}{g_{\ast s}}\approx 0.278\frac{g_{eff}}{g_{\ast s}}$

2. Non-relativistic Species

$n=g(\frac{mT}{2\pi }{)}^{3/2}{e}^{-m/T}$. Then at equilibrium, $Y_{eq}=\frac{45}{2{\pi }^4}(\frac{\pi }{8}{)}^{1/2}\frac{g}{g_{\ast s}}(\frac{m}{T}{)}^{3/2}{e}^{-m/T}$.