The Thermal History of the Universe

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

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,¹

with $\epsilon$ the energy density and $n$ the number density; a fermionic species carries a factor $7/8$ in $\epsilon$ 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 ${\Omega }_{\gamma }{h}^2\approx 2.47\times 10^{-5}$.²

Because $\epsilon \propto T^4\propto a^{-4}$, 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

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

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,

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.

For the radiation era the expansion rate follows from the Friedmann equation with $\epsilon =({\pi }^2/30)g_{\ast }(k_{B}T{)}^4/(\hslash c{)}^3$:

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

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:

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 $g_{\ast s}\approx 3.91$.

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.

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.

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

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

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 $g_{\ast s}=2+\frac{7}{8}(4)=11/2$ 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

The prediction is a cosmic neutrino background at present temperature

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.³

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

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 ${\epsilon }_m={\epsilon }_r$ and using the present density parameters,

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.

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, and the release of the photon bath at recombination near $k_{B}T\sim 0.3\text{eV}$.