---
title: The Thermal History of the Universe
draft: false
module: The Hot Big Bang
moduleNumber: 12
lessonNumber: 1
order: 1201
summary: >
  Running the expansion backward compresses and heats the universe, so its past
  is a sequence of thermal epochs set by temperature. Temperature scales as the
  inverse scale factor; species stay in equilibrium while their interaction rate
  exceeds the expansion rate and freeze out when it drops below. The effective
  degrees of freedom count the relativistic species and step down through mass
  thresholds, and neutrino decoupling just before electron-positron annihilation
  leaves a relic neutrino background slightly cooler than the photons.
topics: [The Hot Big Bang]
sources:
  - book: Ryden
    ref: "Ch. 9 — Nucleosynthesis and the Early Universe §9.1–9.2; Ch. 10"
  - book: Carroll & Ostlie
    ref: "Ch. 30 — The Early Universe; §30.1"
  - book: PDG
    ref: "Review of Particle Physics — Big-Bang Cosmology (Cosmological Parameters, Thermal History)"
---

The observed expansion, run backward, forces every proper volume to shrink and
every relativistic wavelength to blueshift, so the early universe was denser and
hotter than the present one. The history of the first fractions of a second is
therefore organized by temperature rather than time: each species of particle
participates in the thermal bath while its interactions are fast, and drops out
when the expansion outruns them. This lesson fixes the three quantities that
structure that history — the temperature-scale-factor law $T \propto a^{-1}$, the
freeze-out criterion $\Gamma < H$, and the effective degrees of freedom
$g_\ast(T)$ — and applies them to the milestone epochs and to the decoupling of
the cosmic neutrino background.

## Temperature and the scale factor

A gas of photons in thermal equilibrium at temperature $T$ has a blackbody
spectrum. As the universe expands by a factor $a$, every photon wavelength
stretches as $\lambda \propto a$, so the whole spectrum shifts self-similarly. A
blackbody stays a blackbody under this rescaling provided its temperature falls
as

$$
T \propto a^{-1},
$$

which follows because the peak wavelength $\lambda_{\text{peak}} \propto 1/T$
must track the redshift $\lambda_{\text{peak}} \propto a$. Equivalently, the
present photon temperature $T_0 = 2.725\ \text{K}$ and the temperature at
redshift $z$ are related by $T(z) = T_0(1+z)$, since $1+z = a_0/a$.

The energy density of a relativistic gas is set by its temperature. For a single
bosonic species with $g$ internal (spin) states in equilibrium,[^zeta]

$$
\varepsilon = g\,\frac{\pi^2}{30}\,\frac{(k_B T)^4}{(\hbar c)^3},
\qquad
n = g\,\frac{\zeta(3)}{\pi^2}\,\frac{(k_B T)^3}{(\hbar c)^3},
$$

with $\varepsilon$ the energy density and $n$ the number density; a fermionic
species carries a factor $7/8$ in $\varepsilon$ and $3/4$ in $n$ relative to a
boson with the same $g$, from the difference between Bose-Einstein and
Fermi-Dirac statistics. The photon has $g=2$ polarizations, giving the present
photon number density $n_{\gamma,0} \approx 411\ \text{cm}^{-3}$ and radiation
density parameter $\Omega_{\gamma}h^2 \approx 2.47 \times 10^{-5}$.[^ryden-thermal]

Because $\varepsilon \propto T^4 \propto a^{-4}$, the radiation energy density
dilutes one power of $a$ faster than the $a^{-3}$ dilution of a fixed number of
massive particles. That single power is the reason the universe, matter-poor in
energy today, was radiation-dominated in its past.

### Entropy conservation and the exact scaling

The expansion is adiabatic to excellent approximation: no heat flows across a
comoving surface because the universe is homogeneous. The entropy in a comoving
volume is therefore conserved. The entropy density of the relativistic bath is

$$
s = \frac{2\pi^2}{45}\,g_{\ast s}(T)\,k_B\,\frac{(k_B T)^3}{(\hbar c)^3},
$$

where $g_{\ast s}$ counts the entropic degrees of freedom (defined below).
Conservation of $s\,a^3$ gives the exact law

$$
g_{\ast s}(T)\,T^3\,a^3 = \text{const}
\quad\Longrightarrow\quad
T \propto g_{\ast s}^{-1/3}\,a^{-1}.
$$

Between mass thresholds $g_{\ast s}$ is constant and $T \propto a^{-1}$ holds
exactly. When a species becomes non-relativistic and annihilates, it dumps its
entropy into the remaining bath, $g_{\ast s}$ drops, and $T$ falls more slowly
than $a^{-1}$ for a short interval. This is the mechanism that later warms the
photons relative to the decoupled neutrinos.

## Thermal equilibrium and the freeze-out criterion

A species remains in thermal equilibrium with the bath only while the reactions
that create and destroy it, or exchange energy with it, proceed faster than the
universe changes. The relevant comparison is between the interaction rate per
particle,

$$
\Gamma = n\,\langle \sigma v \rangle,
$$

with $n$ the number density of targets, $\sigma$ the cross section, and $v$ the
relative velocity, and the expansion rate, the Hubble parameter $H = \dot a / a$.
The ratio $\Gamma/H$ measures the number of interactions per Hubble time. The
governing rule is:

- **Coupled ($\Gamma \gg H$):** interactions are many per expansion time; the
  species tracks the equilibrium distribution at the common temperature $T$.
- **Decoupled / frozen out ($\Gamma \ll H$):** interactions are too slow to keep
  up; the species falls out of equilibrium, and its comoving number is fixed
  (apart from later decays).

Decoupling occurs near the crossover $\Gamma \simeq H$. Because both rates are
steep functions of temperature, the transition is fast, and treating it as an
instantaneous freeze-out at $\Gamma(T_{\text{dec}}) = H(T_{\text{dec}})$ is
accurate enough for the epoch temperatures.

$$
% caption: The per-particle interaction rate and the expansion rate both fall as
% the universe cools; a species freezes out where the falling interaction rate
% drops below the expansion rate.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->, black] (0,0) -- (9.4,0) node[right, black!70] {cooling, time increasing};
\draw[->, black] (0,0) -- (0,4.6) node[above, black!70] {rate (log)};
% interaction rate: steep decline
\draw[acc, very thick] (0.5,4.3) .. controls (3.0,2.6) and (4.5,1.2) .. (8.8,0.25);
\node[acc, anchor=west] at (6.2,0.7) {interaction rate};
% expansion rate: shallower decline
\draw[black, very thick, dashed] (0.5,3.4) .. controls (3.5,2.2) and (5.5,1.5) .. (8.8,0.9);
\node[black, anchor=west] at (6.4,1.35) {expansion rate};
% crossover
\fill[black!70] (5.05,1.62) circle (2pt);
\draw[black, densely dotted] (5.05,1.62) -- (5.05,0) node[below, black!70] {freeze-out};
\node[black!70, anchor=south west] at (5.15,1.7) {coupled to decoupled};
\end{tikzpicture}
$$

For the radiation era the expansion rate follows from the Friedmann equation with
$\varepsilon = (\pi^2/30)\,g_\ast (k_B T)^4/(\hbar c)^3$:

$$
H = \sqrt{\frac{8\pi G}{3}\,\frac{\varepsilon}{c^2}}
  = \left(\frac{4\pi^3 G}{45}\right)^{1/2}
    \frac{\sqrt{g_\ast}\,(k_B T)^2}{\hbar^{3/2} c^{5/2}}
\;\propto\; \sqrt{g_\ast}\; T^2 .
$$

Combining $a \propto t^{1/2}$ with this gives the time-temperature relation of the
radiation era, conveniently written for the standard model degrees of freedom as

$$
t \approx 2.4\,g_\ast^{-1/2}
\left(\frac{k_B T}{1\ \text{MeV}}\right)^{-2}\ \text{s}.
$$

At $k_B T = 1\ \text{MeV}$ the universe is about one second old; at the electroweak
scale $k_B T \sim 100\ \text{GeV}$ it is of order $10^{-11}\ \text{s}$.

## Effective degrees of freedom

The energy density of the whole relativistic bath sums over every species light
enough to be relativistic, $k_B T \gg m c^2$, each weighted by its statistics:

$$
\varepsilon_{\text{tot}} = \frac{\pi^2}{30}\,g_\ast(T)\,\frac{(k_B T)^4}{(\hbar c)^3},
\qquad
g_\ast = \sum_{\text{bosons}} g_i
       + \frac{7}{8}\sum_{\text{fermions}} g_i .
$$

The entropic count $g_{\ast s}$ has the same form but weights each species by
$T_i^3/T^3$ when a decoupled species (such as the neutrinos after their
decoupling) carries a different temperature than the photons. While everything
shares one temperature, $g_{\ast s} = g_\ast$.

The count is a step function of temperature. Each time $k_B T$ falls below a
particle's rest energy, that species annihilates and stops contributing. Reading
down from high temperature in the Standard Model:

- **$k_B T \gtrsim 300\ \text{GeV}$:** all Standard Model particles are
  relativistic, $g_\ast = 106.75$.
- **below the top, Higgs, $W$, $Z$ thresholds:** $g_\ast$ falls in steps to
  $\approx 86.25$ through the quark-hadron transition region.
- **quark-hadron transition ($k_B T \sim 150\ \text{MeV}$):** quarks and gluons
  confine into hadrons; almost all hadrons are heavy and drop out, leaving
  photons, electrons, positrons, neutrinos, and pions; $g_\ast \approx 17.25$
  just below, then $10.75$ once the pions and muons annihilate.
- **$k_B T \lesssim 0.5\ \text{MeV}$ (after electron-positron annihilation):**
  photons and the three decoupled neutrino species remain; $g_\ast \approx 3.36$
  and $g_{\ast s} \approx 3.91$.

$$
% caption: The effective relativistic degrees of freedom step down as the
% temperature falls through each particle rest-mass threshold, from about 107 in
% the Standard Model to roughly 3.4 after electron-positron annihilation.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->, black] (0,0) -- (9.6,0) node[right, black!70] {cooling to the right};
\draw[->, black] (0,0) -- (0,4.4) node[above, black!70] {degrees of freedom};
% descending staircase (log-height schematic)
\draw[acc, very thick]
  (0.3,4.0) -- (2.2,4.0) -- (2.2,3.3) -- (3.6,3.3) -- (3.6,2.4)
  -- (5.4,2.4) -- (5.4,1.5) -- (7.0,1.5) -- (7.0,0.6) -- (9.2,0.6);
% threshold annotations along the axis
\node[black!70, anchor=north] at (2.2,-0.05) {W, Z, top};
\node[black!70, anchor=north] at (3.9,-0.05) {quark-hadron};
\node[black!70, anchor=north] at (5.6,-0.05) {muons, pions};
\node[black!70, anchor=north] at (7.2,-0.05) {electrons};
% level labels
\node[acc, anchor=south] at (1.1,4.05) {about 107};
\node[acc, anchor=south] at (4.4,2.45) {about 17};
\node[acc, anchor=south] at (8.0,0.65) {about 3};
\end{tikzpicture}
$$

The height of $g_\ast$ enters $H \propto \sqrt{g_\ast}\,T^2$ and the
time-temperature relation, so knowing the staircase is what turns a temperature
into a cosmic age.

## The sequence of epochs

With the scaling laws in hand, the early history is a march down in temperature.
The named epochs, from earliest and hottest to the recombination that ends the
plasma era, are set by the physics that becomes relevant at each temperature.

| Epoch | $k_B T$ | approximate time | defining physics |
| --- | --- | --- | --- |
| Planck | $10^{19}\ \text{GeV}$ | $10^{-43}\ \text{s}$ | quantum gravity; classical spacetime breaks down |
| Grand unification | $10^{16}\ \text{GeV}$ | $10^{-38}\ \text{s}$ | GUT symmetry breaking; possible monopole, baryogenesis era |
| Electroweak | $100\ \text{GeV}$ | $10^{-11}\ \text{s}$ | electroweak symmetry breaking; $W$, $Z$ acquire mass |
| Quark-hadron | $150\ \text{MeV}$ | $10^{-5}\ \text{s}$ | quarks and gluons confine into hadrons |
| Neutrino decoupling | $1\ \text{MeV}$ | $1\ \text{s}$ | weak rates fall below $H$; neutrinos free-stream |
| Nucleosynthesis | $0.1\ \text{MeV}$ | $3\ \text{min}$ | light nuclei form once photodissociation stops |
| Matter-radiation equality | $0.8\ \text{eV}$ | $50{,}000\ \text{yr}$ | $\varepsilon_m = \varepsilon_r$; growth of structure turns on |
| Recombination | $0.3\ \text{eV}$ | $380{,}000\ \text{yr}$ | electrons bind to protons; photons decouple, CMB released |

The upper rows rest on particle physics not yet tested at those energies and are
correspondingly uncertain; from neutrino decoupling downward the physics is
laboratory-calibrated, and the predictions — the light-element abundances and the
CMB — are the quantitative successes of the hot Big Bang.

$$
% caption: Temperature falls as the inverse scale factor along the expansion, and
% the milestone epochs are fixed intervals of temperature strung along the
% timeline from the Planck era to recombination.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->, black] (0,0) -- (10.2,0) node[right, black!70] {cosmic time (log)};
% temperature falling curve above the axis
\draw[acc, very thick] (0.4,3.9) .. controls (3.0,1.9) and (6.0,0.9) .. (9.8,0.45);
\node[acc, anchor=west] at (7.2,1.2) {temperature falling};
% epoch tick marks
\foreach \x/\lab in {0.6/Planck, 2.0/electroweak, 3.6/quark-hadron, 5.4/nucleosynthesis, 7.4/equality, 9.2/recombination}{
  \draw[black] (\x,0.1) -- (\x,-0.1);
  \node[black!70, rotate=32, anchor=east] at (\x,-0.15) {\lab};
}
\end{tikzpicture}
$$

## Neutrino decoupling and the relic background

The neutrinos illustrate the freeze-out criterion sharply and leave an observable
relic. Neutrinos are kept in equilibrium by weak interactions such as
$\nu\bar\nu \leftrightarrow e^+ e^-$ and $\nu e \to \nu e$, whose cross section
scales as $\sigma \sim G_F^2 (k_B T)^2$ in the relativistic regime. With
$n \propto T^3$ the interaction rate is

$$
\Gamma_{\nu} = n\,\langle\sigma v\rangle \sim G_F^2 (k_B T)^5 .
$$

The expansion rate in the radiation era is $H \propto T^2$, so

$$
\frac{\Gamma_\nu}{H} \propto \frac{G_F^2 (k_B T)^5}{T^2}
\;\propto\; \left(\frac{k_B T}{1\ \text{MeV}}\right)^3 .
$$

The ratio drops below unity at $k_B T \approx 1\ \text{MeV}$. Below this the
neutrinos free-stream: they retain a relativistic Fermi-Dirac distribution, but
their temperature simply redshifts as $T_\nu \propto a^{-1}$, decoupled from the
photons.

Shortly afterward, at $k_B T \approx 0.5\ \text{MeV} \approx m_e c^2$, electrons
and positrons become non-relativistic and annihilate,
$e^+ e^- \to \gamma\gamma$. Their entropy is delivered to the photons but not to
the already-decoupled neutrinos, so the photon bath is reheated relative to the
neutrino bath. Entropy conservation fixes the ratio. Before annihilation the
coupled bath has $g_{\ast s} = 2 + \tfrac{7}{8}(4) = 11/2$ for photons plus
$e^\pm$; afterward only the photons remain with $g_{\ast s} = 2$. Conservation of
$g_{\ast s}(a T_\gamma)^3$ across the annihilation, applied to the photon sector
while the neutrino temperature keeps redshifting, gives

$$
\frac{T_\nu}{T_\gamma} = \left(\frac{4}{11}\right)^{1/3} \approx 0.714 .
$$

$$
% caption: Neutrinos decouple near one MeV; the subsequent electron-positron
% annihilation heats the photons but not the neutrinos, leaving the relic
% neutrino background cooler than the photons by the factor (4/11)^{1/3}.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->, black] (0,0) -- (9.6,0) node[right, black!70] {cooling to the right};
\draw[->, black] (0,0) -- (0,4.2) node[above, black!70] {temperature (log)};
% shared temperature before decoupling
\draw[black, very thick] (0.4,3.9) -- (3.4,2.2);
\node[anchor=south] at (1.4,3.2) {shared bath};
% split point
\fill[black!70] (3.4,2.2) circle (2pt);
\draw[black, densely dotted] (3.4,2.2) -- (3.4,0) node[below, black!70] {decoupling};
% photon branch reheated (kink upward relative to neutrino)
\draw[acc, very thick] (3.4,2.2) .. controls (4.4,2.05) and (4.9,2.0) .. (5.1,1.95) -- (9.0,0.95);
\node[acc, anchor=south] at (7.6,1.2) {photons};
% neutrino branch, cooler
\draw[black, very thick, dashed] (3.4,2.2) -- (9.0,0.55);
\node[black, anchor=north] at (7.6,0.62) {neutrinos};
% annihilation marker
\draw[black, densely dotted] (4.7,0) -- (4.7,2.0) node[midway, right, black, xshift=1pt] {annihilation};
\end{tikzpicture}
$$

The prediction is a **cosmic neutrino background** at present temperature

$$
T_{\nu,0} = \left(\frac{4}{11}\right)^{1/3} T_0 = 1.95\ \text{K},
$$

with number density $n_\nu \approx 112\ \text{cm}^{-3}$ per flavor. Its direct
detection is beyond current technology, but the same physics enters the radiation
energy budget through the effective neutrino number
$N_{\text{eff}} \approx 3.046$, which the CMB damping tail measures, providing an
indirect confirmation.[^pdg-nu]

The energy density in the relativistic bath after $e^\pm$ annihilation, counting
photons plus three neutrino species at their reduced temperature, is

$$
\varepsilon_r = \left[ 1 + \frac{7}{8}\,N_{\text{eff}}
\left(\frac{4}{11}\right)^{4/3} \right] \varepsilon_\gamma
\approx 1.68\,\varepsilon_\gamma ,
$$

so the neutrinos raise the total radiation density by about 68% over the photons
alone. This total is what sets the epoch of matter-radiation equality.

## Matter-radiation equality

Radiation dilutes as $a^{-4}$ and matter as $a^{-3}$, so their ratio grows
$\propto a$ and there is a crossover redshift where the two are equal. Setting
$\varepsilon_m = \varepsilon_r$ and using the present density parameters,

$$
1 + z_{\text{eq}} = \frac{\Omega_{m,0}}{\Omega_{r,0}}
\approx \frac{0.31}{9.0 \times 10^{-5}} \approx 3400 ,
$$

with $\Omega_{r,0}$ including the neutrino contribution above. Before equality the
universe expands as $a \propto t^{1/2}$ (radiation); after it, as
$a \propto t^{2/3}$ (matter). Equality matters for structure: density
perturbations in the dark matter can only grow appreciably once matter dominates
the expansion, so $z_{\text{eq}}$ sets the scale imprinted on the matter power
spectrum and the turnover discussed in the structure-formation lesson.

$$
% caption: Radiation density falls one power of the scale factor faster than
% matter, so the two curves cross at equality; radiation dominates before,
% matter after, until dark energy takes over at late times.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->, black] (0,0) -- (9.6,0) node[right, black!70] {scale factor (log)};
\draw[->, black] (0,0) -- (0,4.6) node[above, black!70] {density (log)};
% radiation: steeper slope (falls faster to the right)
\draw[acc, very thick] (0.4,4.3) -- (8.8,0.3);
\node[acc, anchor=south west] at (5.6,1.2) {radiation};
% matter: shallower slope
\draw[black, very thick, dashed] (0.4,3.2) -- (8.8,1.1);
\node[black, anchor=north] at (2.0,2.5) {matter};
% dark energy: flat line
\draw[black, very thick, densely dotted] (0.4,0.9) -- (8.8,0.9);
\node[black, anchor=south] at (7.3,0.92) {dark energy};
% equality crossing
\fill[black!70] (2.35,3.15) circle (2pt);
\draw[black, densely dotted] (2.35,3.15) -- (2.35,0) node[below, black!70] {equality};
\end{tikzpicture}
$$

The thermal-history framework of this lesson supplies the initial conditions for
the two epochs the following lessons treat in detail: the freeze-out of the
neutron-to-proton ratio near $k_B T \sim 1\ \text{MeV}$ and the assembly of light
nuclei at $k_B T \sim 0.1\ \text{MeV}$ in [Big Bang
nucleosynthesis](/astrophysics-cosmology/the-hot-big-bang/big-bang-nucleosynthesis),
and the release of the photon bath at [recombination](/astrophysics-cosmology/the-hot-big-bang/recombination-and-the-cosmic-microwave-background)
near $k_B T \sim 0.3\ \text{eV}$.

[^ryden-thermal]: Ryden, §9.1–9.2 — the temperature-scale-factor law, the relativistic energy density, and the sequence of thermal epochs; the photon number density and radiation density parameter follow from the blackbody spectrum at $T_0$.
[^pdg-nu]: Particle Data Group, Review of Particle Physics, Big-Bang Cosmology — the freeze-out condition $\Gamma \simeq H$, the effective degrees of freedom, the neutrino temperature ratio $(4/11)^{1/3}$, and $N_{\text{eff}} = 3.046$. https://pdg.lbl.gov
[^zeta]: The **Riemann zeta function** is $\zeta(s)=\sum_{n=1}^{\infty} n^{-s}=1+2^{-s}+3^{-s}+\cdots$. It enters these gas integrals through the standard Bose result $\int_0^\infty \frac{x^{s-1}}{e^x-1}\,\d x=\Gamma(s)\,\zeta(s)$. The value $\zeta(3)\approx 1.2021$ used here is Apéry's constant, with no closed form in terms of $\pi$.