The Hot Big Bang: Thermodynamics and Equilibrium
Statistical mechanics of relativistic species in the early universe: number densities, energy density, entropy, effective degrees of freedom, and the Boltzmann equation
Overview
At temperatures above \(\sim 1\) MeV, the universe was a hot, dense plasma in which particles interacted rapidly enough to maintain thermal equilibrium. The thermodynamic properties of this plasma — encoded in the number density, energy density, pressure, and entropy — determine the expansion rate and set the stage for all subsequent cosmological events: nucleosynthesis, recombination, and structure formation.
1. Equilibrium Distribution Functions
A species \(i\) with mass \(m_i\), chemical potential \(\mu_i\), and internal degrees of freedom \(g_i\)(spin, colour, etc.) has the equilibrium phase-space distribution:
$$f_i(p) = \frac{1}{e^{(E_i - \mu_i)/T} \pm 1}$$
where \(+\) is for fermions (Fermi-Dirac) and \(-\) for bosons (Bose-Einstein), and \(E_i = \sqrt{p^2 + m_i^2}\).
2. Number Density
The number density of species \(i\) in equilibrium is:
Equilibrium Number Density
$$n_i = \frac{g_i}{2\pi^2}\int_0^\infty \frac{p^2\,dp}{e^{(E_i - \mu_i)/T} \pm 1}$$
In the relativistic limit \(T \gg m_i\) with \(\mu_i = 0\):
$$n_i = \begin{cases} \dfrac{\zeta(3)}{\pi^2}\,g_i\,T^3 & \text{(bosons)} \\[8pt] \dfrac{3}{4}\dfrac{\zeta(3)}{\pi^2}\,g_i\,T^3 & \text{(fermions)} \end{cases}$$
where \(\zeta(3) \approx 1.202\) is the Riemann zeta function.
3. Energy Density and Effective Degrees of Freedom
The total energy density of all relativistic species is:
Radiation Energy Density
$$\boxed{\rho_R = \frac{\pi^2}{30}\,g_*(T)\,T^4}$$
where the effective number of relativistic degrees of freedom is:
$$g_*(T) = \sum_{\text{bosons}} g_i\left(\frac{T_i}{T}\right)^4 + \frac{7}{8}\sum_{\text{fermions}} g_i\left(\frac{T_i}{T}\right)^4$$
The factor \(7/8\) arises from the difference between the Bose-Einstein and Fermi-Dirac integrals. At temperatures above the QCD phase transition (\(T \gtrsim 200\) MeV) with all Standard Model species relativistic,\(g_* = 106.75\).
4. Entropy Conservation
The entropy density of the relativistic plasma is:
Entropy Density
$$s = \frac{\rho + p}{T} = \frac{2\pi^2}{45}\,g_{*S}(T)\,T^3$$
where \(g_{*S}\) counts entropy degrees of freedom (equal to \(g_*\)when all species share the same temperature). The total entropy in a comoving volume is conserved: \(S = sa^3 = \text{const}\). This implies:
$$T \propto g_{*S}^{-1/3}\,a^{-1}$$
When species annihilate (e.g., \(e^+e^-\) annihilation below \(T \sim 0.5\) MeV), their entropy is transferred to the remaining species, heating them relative to any decoupled species (such as neutrinos).
5. Hubble Rate in the Radiation Era
Combining the Friedmann equation with the radiation energy density gives:
Hubble Rate vs. Temperature
$$\boxed{H = \sqrt{\frac{\pi^2 g_*}{90}}\,\frac{T^2}{M_{\text{Pl}}}}$$
where \(M_{\text{Pl}} = 1/\sqrt{G} \approx 1.22 \times 10^{19}\) GeV is the Planck mass.
This relation is the master clock of the early universe: it tells us the age of the universe at any temperature, \(t \sim 1/(2H)\). At \(T = 1\) MeV,\(t \approx 1\) second.
6. The Boltzmann Equation
When interaction rates become comparable to the Hubble rate, species fall out of equilibrium. The evolution of the number density \(n_\chi\) of a species \(\chi\)undergoing \(\chi\chi \leftrightarrow \text{SM SM}\) annihilation is governed by the Boltzmann equation:
Boltzmann Equation for Annihilation
$$\boxed{\dot{n}_\chi + 3Hn_\chi = -\langle\sigma v\rangle\left(n_\chi^2 - n_{\text{eq}}^2\right)}$$
Here \(\langle\sigma v\rangle\) is the thermally averaged annihilation cross section. The \(3Hn_\chi\) term accounts for dilution from expansion. Equilibrium is maintained when \(\Gamma = n_{\text{eq}}\langle\sigma v\rangle \gg H\).
It is convenient to define the yield \(Y = n/s\) and\(x = m_\chi/T\). Then the Boltzmann equation becomes:
$$\frac{dY}{dx} = -\frac{s\langle\sigma v\rangle}{Hx}\left(Y^2 - Y_{\text{eq}}^2\right)$$
Freeze-out occurs at \(x_f \sim 20\text{--}25\) for weakly interacting massive particles (WIMPs), giving a relic abundance that depends inversely on the annihilation cross section — the famous “WIMP miracle.”
7. Neutrino Decoupling and \(e^+e^-\) Annihilation
Neutrinos decouple from the plasma when weak interaction rates fall below the Hubble rate at\(T_{\nu,\text{dec}} \approx 1\) MeV. Shortly after, at \(T \sim 0.5\) MeV, electrons and positrons annihilate, heating the photon bath but not the decoupled neutrinos. Entropy conservation gives:
$$\frac{T_\gamma}{T_\nu} = \left(\frac{11}{4}\right)^{1/3} \approx 1.401$$
This temperature ratio persists to the present day. The cosmic neutrino background has a current temperature \(T_\nu \approx 1.95\) K, compared to\(T_\gamma = 2.725\) K for the CMB. The total radiation energy density today includes both photons and neutrinos:
$$\rho_R = \left[1 + \frac{7}{8}\left(\frac{4}{11}\right)^{4/3} N_{\text{eff}}\right]\rho_\gamma$$
with \(N_{\text{eff}} = 3.044\) in the Standard Model, accounting for slight neutrino heating from \(e^+e^-\) annihilation before complete decoupling.