Cosmological particle freeze-out

Decoupling of a particle abundance from chemical equilibrium when interactions become too slow compared with cosmic expansion. Pair annihilation can continue after equilibrium production is negligible, leaving a nonzero relic abundance.

Number-changing reactions involving a relic candidate and another nearly degenerate species. If reactions interchange the species rapidly enough, the effective annihilation rate depends on their joint abundances and channels. A single-species relic estimate can then fail even for fixed candidate mass.

In a supplied equilibrium-cutoff model with $P=m{c}^2/(k_B{T}_f)$, present number density can contain $P^{3/2}{e}^{-P}$. The logarithmic derivative is $3/(2P)-1$, making small changes in a large cosmological particle freeze-out parameter important for the inferred abundance.

The effective count appearing in thermal entropy density $s\propto g_{\ast s}{T}^3$. Conservation of comoving entropy implies $g_{\ast s}{T}^3{a}^3$ is constant, so stable relic dilution includes $g_{\ast s,0}/g_{\ast s,f}$ when expressed using temperatures.

In a stable-particle approximation with negligible later annihilation, comoving particle number is conserved and physical number density falls as $a^{-3}$. Replacing scale-factor ratios by photon-temperature ratios additionally assumes no change in the entropy degrees of freedom.

For an imposed equilibrium cutoff $m{c}^2=P{k}_B{T}_f$ and density $n_f=6[m{c}^2{k}_B{T}_f/(\hslash c{)}^2{]}^{3/2}{e}^{-P}$, dilution by $(T_0/T_f{)}^3$ cancels the explicit mass in the final number density. The rest-energy density still scales with particle mass. This result assumes constant entropy degrees of freedom and no post-cutoff annihilation; entropy conservation adds a factor $g_{\ast s,0}/g_{\ast s,f}$ when these assumptions are relaxed.

A thermal relic is a surviving particle population whose earlier abundance was set by chemical thermal equilibrium with the cosmic plasma. After number-changing reactions cease and cosmic entropy is conserved, its comoving abundance is described by the yield $Y=n/s$. A relativistically decoupled species can retain an appreciable yield, whereas a species remaining in thermal equilibrium after becoming nonrelativistic is subject to Boltzmann suppression. Its later mass density, lifetime and momentum distribution determine its cosmological effects.

A rate proportional to $T$ competes with radiation-era expansion proportional to $T^2$. The interaction-to-expansion ratio therefore grows as the universe cools, crossing unity near $T_{\mathrm{eq}}={\alpha }_Y^2{m}_{\mathrm{pl}}/(1.66\sqrt{g_{\ast }})$. If a phase transition occurs at $T_c\gg T_{\mathrm{eq}}$, this high-temperature interaction never establishes thermal equilibrium before the transition. Subsequent faster number-changing processes may still establish a thermal distribution; the initial state is not automatically Fermi-Dirac just because the particles are relativistic.

In a radiation-dominated toy model with $\Gamma ={\alpha }_Y^4{T}^5/m_Y^4$, setting the interaction rate equal to the Hubble parameter gives the displayed decoupling scale. The transition $T_D=m_Y$ defines $m_{\mathrm{nr}}$. For smaller masses, $x_D=m_Y/T_D=(m_{\mathrm{nr}}/m_Y{)}^{1/3}\gg 1$. An instantaneous chemically equilibrated cosmological particle freeze-out estimate gives $n_Y/s=[45g_Y/(2{\pi }^2{g}_{\ast s})(2\pi {)}^{-3/2}]x_D^{3/2}{e}^{-x_D}$. This estimate presupposes number-changing thermal equilibrium, radiation domination and the assumed rate law, rather than following from elastic thermalization alone.

For a stable nonrelativistic thermal relic, compare energy densities rather than number densities. If $m_Y=m_{\mathrm{nr}}/x^3$ and $Y_Y=C{x}^{3/2}{e}^{-x}$, its density relative to baryons is $(C{m}_{\mathrm{nr}}/m_B{Y}_B)x^{-3/2}{e}^{-x}$. Requiring this ratio below an allowed dark-matter-to-baryon ratio $R$ means $x+\frac{3}{2}\log x\ge \log [C{m}_{\mathrm{nr}}/(R{m}_B{Y}_B)]$. With fixed rate parameters this excludes the weakly interacting large-mass branch, while the sufficiently Boltzmann-suppressed smaller-mass branch can satisfy the density bound. Decay or a nonthermal initial abundance requires a different constraint.

For $dY/dx=-\lambda Y^2/x^2$ after freeze-out, the exact limit is $Y_{\infty }=[Y_f^{-1}+\lambda /x_f{]}^{-1}$. The frequently used estimate $x_f/\lambda$ additionally neglects the initial inverse abundance.