Thermodynamics in an Expanding Universe

A thermal history is not simply the rule $T\propto a^{-1}$. That scaling holds for the momentum-temperature scale of a decoupled ultrarelativistic thermal distribution, or for an adiabatic ultrarelativistic bath with fixed effective degrees of freedom. Mass thresholds instead redistribute entropy and change the temperature–scale-factor relation. This page derives the energy, pressure, and entropy of an early-universe bath, includes chemical potentials, and makes the assumptions behind comoving entropy conservation explicit.

Required background. States, Processes, and the Laws of Thermodynamics supplies Thermodynamics and Statistical Mechanics for entropy, chemical work, and equilibrium states.

Helpful background. The Einstein Equation supplies the Einstein-equation convention, while The Friedmann Equations and Continuity Equation and Adiabatic Expansion supply its FLRW reduction and conservation law; 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; and Free Fields, Creation Operators, and Fock Space supplies occupation-state counting.

Equilibrium densities and chemical potentials

Use $\hbar=c=k_{\rm B}=1$ in thermal integrals. In an ideal free-particle or well-defined quasiparticle gas, a species of degeneracy $g_i$, energy $E_i=(p^2+m_i^2)^{1/2}$, temperature $T_i$, and chemical potential $\mu_i$ has

$$f_i(p)= \frac{1}{\exp[(E_i-\mu_i)/T_i]\pm1},$$

with the upper sign for fermions. For bosons, $\mu_i\leq m_i$ when the lowest one-particle energy is $m_i$; equality is the condensation threshold, and the continuum formula below omits any condensate contribution. Isotropy gives

$$\begin{aligned} n_i&=\frac{g_i}{2\pi^2}\int_0^\infty p^2f_i\,dp,\\ \rho_i&=\frac{g_i}{2\pi^2}\int_0^\infty p^2E_i f_i\,dp,\\ p_i&=\frac{g_i}{6\pi^2}\int_0^\infty \frac{p^4}{E_i}f_i\,dp. \end{aligned}$$

For a noninteracting equilibrium component, the Euler relation gives

$$s_i=\frac{\rho_i+p_i-\mu_i n_i}{T_i}.$$

A reaction $\sum_i\nu_i A_i\leftrightarrow0$ in chemical equilibrium imposes $\sum_i\nu_i\mu_i=0$. Thus photons have $\mu_\gamma=0$ when their number can change, while a particle–antiparticle pair has $\mu_{\bar i}=-\mu_i$ for a conserved charge. Small baryon and lepton asymmetries barely affect the total radiation density, but their chemical potentials remain essential for net charge densities.

For the total equilibrated system, the entropy formula follows from extensivity, $E=TS-pV+\sum_i\mu_iN_i$, rather than from the special form of an ideal gas. That total identity remains useful when interacting species exchange energy and charge, even when a species-by-species split is not exact. Chemical equilibrium removes only the combinations of chemical potentials changed by active reactions; each exactly conserved charge leaves one independent potential. If a reaction freezes out, the associated constraint can cease to hold even while elastic scattering continues to maintain a common kinetic temperature.

Choose a reference bath temperature $T$ and define the exact effective functions from the total energy and entropy densities assigned to the radiation bath:

$$g_*(T)=\frac{30\rho_{\rm rad}}{\pi^2T^4}, \qquad g_{*s}(T)=\frac{45s_{\rm rad}}{2\pi^2T^3}.$$

For ideal species with negligible chemical potentials in the ultrarelativistic limit $m_i/T_i\to0$, these definitions reduce to

$$\begin{aligned} g_*&=\sum_{\rm bosons}g_i\left(\frac{T_i}{T}\right)^4 +\frac78\sum_{\rm fermions}g_i\left(\frac{T_i}{T}\right)^4,\\ g_{*s}&=\sum_{\rm bosons}g_i\left(\frac{T_i}{T}\right)^3 +\frac78\sum_{\rm fermions}g_i\left(\frac{T_i}{T}\right)^3. \end{aligned}$$

The two functions need not agree across a mass threshold or when sectors have different temperatures. A semi-relativistic species must be evaluated with the full integrals above; an interacting plasma, especially through the QCD crossover, requires a controlled equation of state rather than free-species step counting. The Particle Data Group derives the ideal integrals and definitions in Olive and Peacock 2025/2026, §§22.3.1–22.3.2, printed pp. 10–12, eqs. (22.35)–(22.44) (PDF).

For example, well above the electron mass but below the muon mass, ideal photons, electron–positron pairs, and three active neutrino–antineutrino pairs at one temperature give

$$g_*=g_{*s}=2+\frac78(4+6)=10.75.$$

The two photon polarizations contribute $2$, the electron and positron spin states contribute $4$, and the populated neutrino helicities contribute $6$. The factor $7/8$ comes from the relativistic Fermi energy integral relative to the Bose integral at the same temperature. This count explains the value used in the weak-decoupling estimate; it must change when the sectors separate or the electrons become nonrelativistic.

Comoving entropy and the cooling law

Let $t$ be cosmic time, a dot denote $d/dt$, and $H=\dot a/a$. For a fixed comoving coordinate cell, its physical volume is proportional to $a^3$. Homogeneous perfect-fluid conservation is

$$\dot\rho+3H(\rho+p)=0.$$

The first law for that cell can be arranged as

$$T\,d(sa^3) =d(\rho a^3)+p\,d(a^3) -\sum_i\mu_i\,d(n_i a^3).$$

Continuity makes the first two terms cancel. For a reaction progress $d(n_i a^3)=\nu_i\,d\xi$, chemical equilibrium gives $\sum_i\nu_i\mu_i=0$. More generally, equilibrium chemical potentials are linear combinations of exactly conserved charges, so $\sum_i\mu_i\,d(n_i a^3)=\sum_A\mu_A\,dQ_A=0$. Therefore

$$\frac{d}{dt}(s a^3)=0$$

only for an adiabatic sector: local equilibrium must be adequate, dissipative entropy production must be negligible, and no entropy may enter from another sector. Under those hypotheses,

$$\boxed{g_{*s}(T)T^3a^3=\text{constant}}, \qquad \boxed{T\propto g_{*s}^{-1/3}a^{-1}}.$$

At fixed $g_{*s}$ this reduces to $T\propto a^{-1}$. When $g_{*s}$ varies smoothly,

$$\frac{d\ln T}{d\ln a} =-\left[ 1+\frac{T}{3g_{*s}}\frac{dg_{*s}}{dT} \right]^{-1}.$$

When a species becomes nonrelativistic and annihilates into particles still coupled to the bath, the remaining bath is reheated relative to any already decoupled sector. Across an idealized transfer interval with distinct endpoint scale factors,

$$g_{*s,-}T_-^3a_-^3 =g_{*s,+}T_+^3a_+^3.$$

If a decoupled relativistic species $X$ had $T_{X,-}=T_-$ and simply redshifts, $T_{X,+}=T_-a_-/a_+$. Its final temperature relative to the reheated bath is

$$\frac{T_{X,+}}{T_+} =\left(\frac{g_{*s,+}}{g_{*s,-}}\right)^{1/3}.$$

This is relative reheating across a finite expansion interval, not a literal temperature discontinuity at fixed volume. A genuinely out-of-equilibrium decay or dissipative process instead requires an entropy-production term and can dilute previously frozen comoving abundances.

The distinction between total and sector entropy matters at every threshold. Before decoupling, one common temperature and one total $g_{*s}$ describe the coupled bath. After decoupling, each isolated and separately adiabatic sector conserves its own $s_i a^3$; isolation alone does not forbid internal entropy production. Annihilation in one sector reheats only that sector. A table of particle masses is therefore not enough to reconstruct temperatures: one must also know which reactions still exchange energy.

Expansion rate and temperature

During radiation domination,

$$H^2=\frac{8\pi G}{3}\rho_{\rm rad}$$

gives

$$\boxed{ H(T)=1.66\,g_*^{1/2}\frac{T^2}{M_{\rm Pl}} }, \qquad M_{\rm Pl}=G^{-1/2}.$$

The numerical coefficient uses the nonreduced Planck mass. This relation connects microphysical reaction rates to the cosmic clock, but it must be revised if another component contributes appreciably to $H$ or if the radiation bath is not in the stated equilibrium state. The exact normalization appears in Olive and Peacock 2025/2026, §22.3.3, printed p. 13, eq. (22.47) (Official PDF). Tong develops the distributions, hot-big-bang scaling, and sector reheating in Tong 2019, §§2.4.1–2.4.2, printed pp. 107–111, eqs. (2.34)–(2.40), §2.5.1, printed pp. 112–114, eqs. (2.41)–(2.45), and §2.5.4, printed pp. 124–125 (Official PDF).

Checks and limitations

Dimensions provide a fast check: $\rho\sim T^4$, $s\sim T^3$, and $H\sim T^2/M_{\rm Pl}$. A collisionless relativistic distribution obeys $p\propto a^{-1}$ and retains its shape in comoving momentum even when it no longer shares the bath temperature. Once particles become nonrelativistic, calling the redshifted momentum scale a “temperature” does not by itself restore kinetic equilibrium.

Entropy conservation applies sector by sector only when interactions between sectors are negligible and each sector is separately adiabatic. Adequate local equilibrium justifies the thermodynamic derivation above; collisionless Liouville evolution can conserve a fine-grained kinetic entropy by a different argument. Applying the total $g_{*s}$ to a decoupled species is the common route to a wrong relic temperature.

Nor should an abrupt step in a plotted $g_{*s}(T)$ be mistaken for a physical temperature discontinuity in a smooth crossover. The instantaneous formula is an integrated limiting case. Precision evolution uses the continuous equation of state, and a first-order transition would additionally require latent heat, phase coexistence, and possible entropy production to be modeled explicitly.

Exercises

Photons, equilibrium electron–positron pairs, and neutrinos initially shared a common temperature. Assume neutrinos then decouple, no other entropy-transfer event intervenes, and electron–positron annihilation is idealized as a rapid but finite expansion interval from $a_-$ to $a_+$. Derive the final neutrino-to-photon temperature ratio.

Solution

Before annihilation, the entropy degrees of freedom still coupled to photons are

$$g_{*s,-}=2+\frac78(2+2)=\frac{11}{2}.$$

Afterward only the two photon polarizations remain, so $g_{*s,+}=2$. Entropy conservation in the electromagnetic bath gives

$$\frac{11}{2}T_-^3a_-^3 =2T_{\gamma,+}^3a_+^3.$$

The decoupled neutrino temperature redshifts as $T_{\nu,+}=T_-a_-/a_+$. Dividing this relation by the electromagnetic entropy equation cancels the otherwise unspecified expansion factor and gives

$$\boxed{ \frac{T_\nu}{T_\gamma} =\left(\frac{4}{11}\right)^{1/3} }.$$

Noninstantaneous decoupling and finite-temperature QED effects produce the small corrections treated in Neutrino Decoupling and Relic Radiation.

References

• Olive, K. A., and J. A. Peacock. “Big-Bang Cosmology.” In Particle Data Group, Review of Particle Physics 2026. Review revised August 2025; issue dated 1 June 2026. Official PDF.
• Tong, David. Cosmology. University of Cambridge Part II Mathematical Tripos lecture notes, Michaelmas 2019. Official PDF.