Thermal History and Relic Abundances

The expansion of the universe changes temperature, density, and reaction timescales together. This chapter builds one continuous account: equilibrium thermodynamics sets the target state, collision terms decide whether species track it, entropy transfers establish relative temperatures, and frozen number or momentum distributions become later observables. The aim is not a catalog of early-universe events but a reusable method for deciding what is conserved, what falls out of equilibrium, and which approximation controls a quoted abundance.

Helpful background. States, Processes, and the Laws of Thermodynamics supplies Thermodynamics and Statistical Mechanics for equilibrium, entropy, chemical work, and detailed balance. The Einstein Equation supplies Einstein Equations and the convention underlying $H(T)$. Relativistic Matter and Compact Objects supplies Relativistic Matter and Compact Objects thermodynamics. Classical Field Modes and Canonical Variables supplies the Flat-Spacetime Quantum Field Theory mode convention, while Free Fields, Creation Operators, and Fock Space supplies occupation-state counting.

Enter the thermal history

Before starting, check that you can distinguish physical from comoving number density, explain why pressure performs work in an expanding comoving volume, and interpret $H^{-1}$ as a background timescale rather than a universal collision time. If any of these is unfamiliar, review the linked thermodynamics page and Continuity Equation and Adiabatic Expansion. The Friedmann Equations provide the expansion law used throughout.

The first five pages follow the main thermal sequence. The sixth applies the same methods to stable relics, whose decoupling epoch depends on their mass and interactions. Two shorter problem-solving routes are given below.

1. Thermodynamics in an Expanding Universe derives phase-space densities, $g_*$ and $g_{*s}$, sector entropy, and temperature evolution through thresholds.
2. Reaction Rates, Decoupling, and Freeze-Out derives the momentum-integrated Boltzmann equation, separates chemical from kinetic decoupling, and replaces the slogan $\Gamma=H$ by a tracking and residual-interaction test.
3. Neutrino Decoupling and Relic Radiation applies separate entropy conservation to the photon and neutrino sectors and explains the precision meaning of $N_{\rm eff}$.
4. Big-Bang Nucleosynthesis follows weak freeze-out and a nuclear network to helium and deuterium, then states the observational selections, likelihood, and versioned response calculation.
5. Recombination and Photon Decoupling separates the Saha curve from Peebles kinetics and turns $x_e(z)$ into a normalized, finite-width visibility function.
6. Thermal Relics and Abundance Calculations rewrites the number equation for $Y=n/s$, solves a cold-relic benchmark, and contrasts inverse-cross-section freeze-out with hot relics and entropy dilution.

For particle relic calculations, read pages 1, 2, and 6. For the thermal inputs to CMB and structure formation, read pages 1–5 in order. The full route shows how the same entropy and rate logic links MeV neutrino decoupling, nuclear abundances, atomic recombination, and more general stable relics.

One convention package

Thermal integrals use $\hbar=c=k_B=1$. Cosmic time is $t$ and $H=\dot a/a$. The radiation and entropy densities are

$$\rho_r=\frac{\pi^2}{30}g_*(T)T^4, \qquad s=\frac{2\pi^2}{45}g_{*s}(T)T^3,$$

where every page declares which bath defines $T$. A decoupled sector may have $T_X\ne T$, so its contribution cannot be inserted into $g_*(T)$ without the appropriate temperature ratio. During radiation domination,

$$H\simeq1.66\,g_*^{1/2}\frac{T^2}{M_{\rm Pl}}, \qquad M_{\rm Pl}=G^{-1/2}.$$

The coefficient $1.66$ uses the nonreduced Planck mass. Pages that use redshift set $a_0=1$ and distinguish physical momentum $p$ from comoving momentum $q=ap$. Relic calculations define $x=m/T$ and $Y=n/s$; observational BBN work separately declares its abundance ratios, data vector, covariance, and priors.

The chain from entropy to an observable

For a separately adiabatic bath,

$$g_{*s}(T)T^3a^3=\text{constant}.$$

That relation sets both the cooling law and the relative reheating of a sector that remains coupled while another has decoupled. Microscopic tracking then depends on the appropriate relaxation eigenvalue, schematically

$$\frac{\Gamma_{\rm rel}}{H} \quad\hbox{compared with}\quad \left|\frac{d\ln X_{\rm eq}}{d\ln a}\right|.$$

Here $X$ has collisionless dilution factored out, as for the yield $n/s$, a temperature ratio, or a spectral distortion amplitude. Once number-changing reactions fail, a stable species is conveniently evolved through

$$\frac{dY}{dx} =-\frac{s\langle\sigma v\rangle}{Hx} \left(1+\frac13\frac{d\ln g_{*s}}{d\ln T}\right) (Y^2-Y_{\rm eq}^2).$$

The same logic has different observational endpoints:

Process	Frozen or rapidly changing object	Later diagnostic
neutrino decoupling	comoving momentum distribution and sector entropy	$T_\nu/T_\gamma$, $N_{\rm eff}$, free streaming
weak and nuclear freeze-out	neutron fraction and nuclear yields	$Y_p$, D/H, lithium tension
hydrogen recombination	free-electron fraction $x_e(z)$	Thomson optical depth and visibility
stable massive relic	yield $Y=n/s$	$\Omega_\chi h^2$ after any dilution

No row is determined by $\Gamma=H$ alone. The relevant collision moment, entropy history, background $H(T)$, and integration through the transition must all be specified. The standard radiation-era thermodynamic relations and rate estimates are collected in Olive and Peacock 2026, §§22.3.1–22.3.3, printed pp. 10–13 (Official PDF), while the hot- and cold-relic limits are summarized in Baudis and Profumo 2026, §27.3, printed pp. 2–4 (Official PDF).

Precision boundaries

The analytic pages deliberately expose their approximations. State-counting formulas use the ultrarelativistic ideal-gas limit; interacting thresholds need an equation of state. Instantaneous neutrino decoupling gives the ideal $(4/11)^{1/3}$ ratio, while precision work solves quantum kinetic equations. Saha equilibrium locates atomic recombination but not the visibility peak; precision ionization histories require multilevel atoms and radiative transfer. A one-species relic equation excludes coannihilation, asymmetry, resonances, separate-sector kinetics, entropy production, and nonstandard expansion unless these are added explicitly.

The qualitative event sequence is developed in Tong 2019, §2.5.2, p. 117 (Official PDF), and the reaction-network and abundance-measurement boundaries are reviewed in Fields, Molaro, and Sarkar 2026, §§24.2–24.3, pp. 1–5 (Official PDF).

BBN is the chapter’s observational inference page. Its numerical statements are dated, tied to exact abundance samples and network versions, and do not turn a conditional two-observable fit into a model-independent measurement of new radiation. A complete particle-physics model or a theory of baryogenesis lies outside this chapter.

Review the chapter

A real relativistic boson $X$ decouples while the coupled bath has $g_{*s}=80$. The bath later reaches $g_{*s}=20$ without entropy production. A student claims that $T_X=T$ must remain true, that a massive relic’s final abundance equals its equilibrium abundance at the instant $\Gamma=H$, and that the CMB was emitted on the Saha recombination slice. Correct all three claims. Then state how a later bath-entropy increase by a factor $\Delta_S$ changes an already frozen massive-relic yield.

Solution

The decoupled boson has $T_Xa=\text{constant}$, while the coupled bath obeys $g_{*s}T^3a^3=\text{constant}$. Therefore

$$\frac{T_X}{T} =\left(\frac{20}{80}\right)^{1/3} =4^{-1/3}.$$

For one real bosonic state its conserved relativistic yield is

$$Y_X=0.278\frac{1}{80},$$

provided the two sectors shared a temperature at decoupling.

For a massive relic, $\Gamma\sim H$ only locates gradual departure. Residual annihilations after that crossing change the abundance, so one must integrate the yield equation and include $g_{*s}$ evolution. If the bath’s comoving entropy later grows by $\Delta_S$ while $n_\chi a^3$ stays fixed, then

$$Y_{\chi,\rm after} =\frac{Y_{\chi,\rm before}}{\Delta_S}.$$

Finally, Saha equilibrium does not capture the Lyman-$\alpha$/two-photon bottleneck and does not define a single emission surface. A kinetic ionization history supplies $\tau$, and the last-scattering probability is the finite-width distribution $g(\eta)=-\tau'e^{-\tau}$.

References

• Baudis, L. and S. Profumo, “Dark Matter,” in F. Takahashi et al. (Particle Data Group), Review of Particle Physics, International Journal of Modern Physics A 41, 2630011 (2026), revised August 2025, Official PDF.
• Fields, B. D., P. Molaro, and S. Sarkar, “Big Bang Nucleosynthesis,” in F. Takahashi et al. (Particle Data Group), Review of Particle Physics, International Journal of Modern Physics A 41, 2630011 (2026), revised August 2025, Official PDF.
• Olive, K. A. and J. A. Peacock, “Big-Bang Cosmology,” in F. Takahashi et al. (Particle Data Group), Review of Particle Physics, International Journal of Modern Physics A 41, 2630011 (2026), revised August 2025, Official PDF.
• Tong, D., Cosmology, University of Cambridge Part II Mathematical Tripos lecture notes, Michaelmas 2019, Official PDF.