Thermal History of the Universe

Rewind the expansion of the universe and every distance shrinks, every density climbs, and the temperature of the cosmic plasma rises without bound. Because the expansion is very nearly adiabatic, the contents cool as they dilute — for a bath of relativistic particles the temperature tracks the inverse scale factor, $T\propto a^{-1}$ — so the present cold, structure-filled cosmos traces back to a state that was hot, dense, and almost perfectly homogeneous. The thermal history is the chronological catalogue of that cooling: the temperatures, times, and microphysical processes marking each transition from one regime to the next. A single inequality organises the whole story. Any reaction that keeps a species in equilibrium runs at a rate $\Gamma=n\langle\sigma v\rangle$ , competing with the expansion rate $H=\dot a/a$ . While $\Gamma\gg H$ a particle collides many times per Hubble time and stays thermalised; once $\Gamma$ falls below $H$ the reaction switches off, the species decouples, and its comoving abundance freezes. Neutrino decoupling, nucleosynthesis, recombination — each is the same freeze-out logic applied to a different interaction, and each has its own article. Here we thread them together. The connective tissue is thermodynamics: the effective relativistic degrees of freedom $g_*(T)$ set the energy density that drives the Friedmann equation, and comoving entropy conservation, $s\,a^3=\text{const}$ , ties the temperatures of decoupled species to those still coupled. Those two tools, plus the freeze-out criterion, order every landmark from the electroweak epoch to the release of the microwave background — and predict relics we can still measure today.

Key takeaways

The thermal history is the ordered sequence of epochs the hot early universe passed through as it expanded and cooled, its radiation temperature scaling as $T\propto a^{-1}$ while radiation dominates the energy budget.
Each epoch is fixed by a race between an interaction rate $\Gamma=n\langle\sigma v\rangle$ and the Hubble rate $H$ : a species stays in equilibrium while $\Gamma\gg H$ and freezes out once $\Gamma\lesssim H$ .
Landmark events in temporal order run electroweak transition ( $T\sim100\,\mathrm{GeV}$ ), QCD confinement ( $T\sim150\,\mathrm{MeV}$ ), neutrino decoupling ( $T\sim1\,\mathrm{MeV}$ ), $e^+e^-$ annihilation, nucleosynthesis ( $T\sim0.1\,\mathrm{MeV}$ ), and recombination ( $T\sim0.26\,\mathrm{eV}$ , $z\approx1100$ ).
Comoving entropy conservation links decoupled temperatures; it fixes $T_\nu=(4/11)^{1/3}T_\gamma$ after $e^+e^-$ annihilation reheats the photons, giving a predicted relic neutrino bath at $1.95\,\mathrm{K}$ today.
The effective degrees of freedom $g_*(T)$ drop in steps — from $106.75$ in the full Standard Model to $3.36$ after $e^+e^-$ annihilation — setting the expansion rate through the Friedmann equation, and leaving testable relics: light-element abundances, the CMB, and the cosmic neutrino background.
Key idea: whether a species is coupled to the cosmic plasma is decided by comparing its interaction rate $\Gamma=n\langle\sigma v\rangle$ with the Hubble rate $H$ — equilibrium while $\Gamma\gg H$ , freeze-out when $\Gamma\lesssim H$ — and that one criterion, together with $T\propto a^{-1}$ and comoving entropy conservation, orders the entire thermal history.

The core insight

Whether a species is coupled to the cosmic plasma is decided by comparing its interaction rate $\Gamma=n\langle\sigma v\rangle$ with the Hubble rate $H$ : equilibrium while $\Gamma\gg H$ , freeze-out when $\Gamma\lesssim H$ . Combined with $T\propto a^{-1}$ and comoving entropy conservation, this single criterion orders the entire thermal history.

Where this lives today

The thermal history is the calibration standard of precision cosmology. The Planck satellite measured the effective number of neutrino species from the microwave background as $N_{\rm eff}=2.99\pm0.17$ , in striking agreement with the standard-model prediction $N_{\rm eff}=3.044$ (the small excess encodes the non-instantaneous nature of neutrino decoupling). Big Bang nucleosynthesis, run through modern codes such as PArthENoPE and PRIMAT, predicts a primordial helium mass fraction $Y_p\approx0.247$ and a deuterium abundance now measured to better than one percent in metal-poor gas along quasar sightlines — a two-parameter test of physics at one second that the data pass. Any new light species — a sterile neutrino, an axion, dark radiation — would raise $g_*$ , speed the expansion, and shift both $Y_p$ and the microwave-background damping tail, so the Simons Observatory and CMB-S4 aim to measure $N_{\rm eff}$ to $\pm0.03$ , turning the first second into a laboratory for particle physics far beyond any accelerator. The last frontier is the relic neutrino sea itself: the PTOLEMY project is attempting the fearsomely difficult direct capture of $1.95\,\mathrm{K}$ neutrinos on tritium — a measurement that would confirm the coldest, oldest prediction of the thermal history.

The physical picture

Picture the universe as a gas in a piston being drawn out. As the volume grows the gas cools, and each species only "feels" the cooling through the collisions that keep it thermalised. When those collisions become rarer than the rate at which the piston moves — when $\Gamma$ falls below $H$ — that species can no longer keep up: it thermally disconnects and coasts, its momentum distribution frozen into the shape it wore at decoupling and thereafter merely redshifting. Temperature is simultaneously a clock and a menu. At high $T$ every particle lighter than $T$ is present in comparable abundance and richly interconverting; as $T$ drops past each rest mass, that species annihilates or decays and dumps its entropy into the lighter species still in the bath. The sequence of these dropouts — top quark, electroweak bosons, the confinement of quarks and gluons into hadrons, then muons, then electrons — is exactly the descending staircase in the effective degrees of freedom $g_*(T)$ . The first figure is that staircase, with real numbers on every tread.

Figure

The staircase of relativistic degrees of freedom g∗(T): as the universe cools from left to right, each species annihilates once T drops below its mass and steps g∗ down — from 106.75 in the full Standard Model, through the QCD transition, to 10.75 at neutrino decoupling and 3.36 after e⁺e⁻ annihilation

Two rates govern everything, and they scale differently with temperature. The interaction rate typically runs as a high power of $T$ (for weak processes $\Gamma\propto T^5$ ), while the Hubble rate in radiation domination goes only as $H\propto T^2$ . So $\Gamma/H\propto T^3$ falls steeply as the universe cools: any reaction blisteringly fast early on eventually, and rather sharply, loses the race. Decoupling is therefore generic — only its timing varies. Read those freeze-outs and transitions off in order and you get the timeline of the cosmos, each rung tagged with a temperature and a time.

Figure

The thermal-history timeline from the electroweak epoch to the release of the microwave background: each rung is the freeze-out or transition of one interaction, tagged with its temperature and cosmic time; the plasma cools downward as T ∝ a⁻¹ and time runs forward

The core equations

The thermal history is built from a small, tightly linked set of relations: how temperature tracks expansion, how expansion is driven by the relativistic energy density, how interaction rates compete with expansion, and how entropy bookkeeping connects epochs. We collect them here, then derive the three results that do the most work. The temperature of a relativistic bath falls with the expansion because its entropy is fixed in a comoving volume. When the effective entropic degrees of freedom hold constant, this is a clean inverse-scale-factor law — the master clock of the radiation era.

T \propto a^{-1}\qquad(\text{relativistic bath, } g_{*s}\ \text{constant}) \qquad (1)

where: $T$ — temperature of the coupled plasma; $a$ — cosmic scale factor, normalised to $a_0=1$ today. Reading: double the scale factor and you halve the temperature, provided no species annihilates in between to spoil the count. The expansion rate itself is set by the total energy density through the Friedmann equation, and in the radiation era that density is dominated by relativistic species and fixed by $g_*(T)$ . This is what turns "temperature" into "time".

H^2 = \frac{8\pi G}{3}\,\rho\;,\qquad \rho = \frac{\pi^2}{30}\,g_*(T)\,T^4 \qquad (2)

where: $H$ — Hubble rate $\dot a/a$ ; $G$ — Newton's constant; $\rho$ — energy density of relativistic species; $g_*(T)$ — effective relativistic degrees of freedom for energy density. Reading: larger $g_*$ means a denser, faster-expanding universe at fixed temperature. Whether a given species keeps up with the plasma is decided by its reaction rate — the number of interactions each particle undergoes per unit time.

\Gamma = n\,\langle\sigma v\rangle \qquad (3)

where: $\Gamma$ — reaction rate per particle; $n$ — number density of targets; $\langle\sigma v\rangle$ — thermally averaged cross section times relative velocity. Reading: many interactions per Hubble time ( $\Gamma\gg H$ ) means equilibrium; too few ( $\Gamma\lesssim H$ ) means freeze-out. The quantity conserved across the expansion is not temperature but the entropy inside a comoving volume, and it is the tool that carries information from one epoch to the next.

s = \frac{2\pi^2}{45}\,g_{*s}(T)\,T^3\;,\qquad S = s\,a^3 = \text{const} \qquad (4)

where: $s$ — entropy density; $g_{*s}$ — effective degrees of freedom for entropy; $S$ — total comoving entropy, conserved under adiabatic expansion. Reading: when a species drops out it heats the bath, so $g_{*s}$ falls while $sa^3$ stays fixed. Both $g_*$ and $g_{*s}$ are weighted head-counts of the light species, with fermions carrying a factor $7/8$ from Fermi–Dirac statistics and each decoupled species entering at its own temperature.

g_* = \sum_{i\,\in\,\text{bosons}} g_i\!\left(\frac{T_i}{T}\right)^{4} + \frac{7}{8}\sum_{i\,\in\,\text{fermions}} g_i\!\left(\frac{T_i}{T}\right)^{4} \qquad (5)

where: $g_i$ — internal (spin, colour) degrees of freedom of species $i$ ; $T_i$ — its temperature, which may differ from $T$ after decoupling; $7/8$ — the Fermi–Dirac versus Bose–Einstein statistical weight. Reading: photons contribute $2$ , each relativistic fermion its spin count times $7/8$ , and cold species drop out. The single most-quoted consequence of entropy bookkeeping is the offset between the neutrino and photon temperatures, opened up when $e^+e^-$ annihilation heats the photons but not the already-decoupled neutrinos.

T_\nu = \left(\frac{4}{11}\right)^{1/3} T_\gamma \approx 0.714\,T_\gamma \qquad (6)

where: $T_\nu$ — neutrino temperature after $e^+e^-$ annihilation; $T_\gamma$ — photon (CMB) temperature. Reading: neutrinos run about $29\%$ cooler than photons ever after — a prediction that reaches the relic backgrounds today. Integrating the radiation-era Friedmann equation converts any temperature into an age, dating each landmark on the timeline.

t \approx \frac{1}{2H} = \left(\frac{45}{16\pi^3 G\,g_*}\right)^{1/2}\frac{1}{T^2} \approx \frac{2.4}{\sqrt{g_*}}\left(\frac{1\,\mathrm{MeV}}{T}\right)^{2}\ \mathrm{s} \qquad (7)

where: $t$ — cosmic time since the Big Bang; $g_*$ — effective degrees of freedom at $T$ . Reading: a MeV is roughly a second, and the age scales as the inverse square of the temperature. Finally, the two present-day relic temperatures — one measured to five figures, one predicted and still unmeasured directly — are the observational anchors of the whole framework.

T_0 = 2.7255\ \mathrm{K}\;,\qquad T_{\nu,0} = 1.95\ \mathrm{K} \qquad (8)

where: $T_0$ — measured CMB temperature today (COBE/FIRAS); $T_{\nu,0}$ — predicted cosmic neutrino background temperature today. Reading: $T_{\nu,0}=(4/11)^{1/3}\times2.7255\,\mathrm{K}$ is a parameter-free prediction of the thermal history.

Derivation AWeak freeze-out and the one-MeV neutrino decoupling temperature

The goal is to show that the machinery of freeze-out collapses, for weak interactions, into a single order-of-magnitude estimate placing neutrino decoupling near $1\,\mathrm{MeV}$ — crucially above the electron mass, which is why neutrinos escape the reheating that $e^+e^-$ annihilation soon delivers to the photons. We build the weak rate from dimensional analysis, set it against the radiation-era Hubble rate, and solve $\Gamma=H$ for the decoupling temperature.

Step 1. For a relativistic species $n\sim T^3$ and the weak cross section grows as $\sigma\sim G_F^2 T^2$ , so $\Gamma\sim n\langle\sigma v\rangle\sim G_F^2 T^5$ , with $v\sim c$ and $G_F\approx1.17\times10^{-5}\,\mathrm{GeV}^{-2}$ . Why: the weak cross section carries two powers of $G_F$ and rises steeply with temperature — the reason weak processes are fast early and hopeless late.

Step 2. The Friedmann equation with $\rho\sim T^4$ gives $H\sim T^2/M_{\rm Pl}$ , where $M_{\rm Pl}=G^{-1/2}\approx1.2\times10^{19}\,\mathrm{GeV}$ . Why: in radiation domination the only scales are $T$ and the Planck mass, so dimensional analysis fixes $H$ .

Step 3. Setting $\Gamma=H$ gives $G_F^2 T^5\sim T^2/M_{\rm Pl}$ , hence $T_{\rm dec}^3\sim 1/(G_F^2 M_{\rm Pl})$ . Why: because $\Gamma/H\propto T^3$ the crossing is steep — a factor of two in $T$ changes $\Gamma/H$ eightfold, so decoupling is a genuine event, not a slow drift.

Step 4. Evaluating yields $T_{\rm dec}\sim1\,\mathrm{MeV}$ , comfortably above the electron rest mass $m_ec^2=0.511\,\mathrm{MeV}$ . Why: since neutrinos have already decoupled when $e^\pm$ annihilate just below $0.5\,\mathrm{MeV}$ , they miss the entropy dumped into the photons — the origin of $T_\nu<T_\gamma$ .

Go deeper · Derivation B · 1 minThe time–temperature relation of the radiation era

Derivation BThe time–temperature relation of the radiation era

The aim is to turn the Friedmann equation into a working stopwatch: a formula that reads the age of the universe off its temperature, so that "neutrino decoupling at an MeV" acquires the concrete meaning "about one second after the Big Bang". We use the radiation-era scaling $a\propto t^{1/2}$ to fix the Hubble rate in terms of time, insert the relativistic energy density, and solve for $t$ .

Step 1. In radiation domination $a\propto t^{1/2}$ , so $H=\dot a/a=1/(2t)$ . Why: with $\rho\propto a^{-4}$ the Friedmann equation integrates to a power law, and the exponent $1/2$ is what gives the clean factor of two relating $H$ and $t$ .

Step 2. Insert $\rho=\tfrac{\pi^2}{30}g_*T^4$ into $H^2=\tfrac{8\pi G}{3}\rho$ to get $\tfrac{1}{4t^2}=\tfrac{8\pi G}{3}\cdot\tfrac{\pi^2}{30}g_*T^4$ . Why: this couples the geometric clock $H=1/(2t)$ to the thermodynamic content $g_*T^4$ , which is exactly the bridge from temperature to time.

Step 3. Solve for $t$ to find $t=\big(\tfrac{45}{16\pi^3 G g_*}\big)^{1/2}T^{-2}$ , and restoring $\hbar$ and $c$ gives the practical form $t\approx 2.4\,g_*^{-1/2}(T/\mathrm{MeV})^{-2}\,\mathrm{s}$ . Why: the inverse-square temperature dependence means the universe spends far longer at low temperatures than high, so the late epochs stretch out over enormous times while the early ones flash by.

Step 4. At $T=1\,\mathrm{MeV}$ with $g_*=10.75$ this gives $t\approx0.7\,\mathrm{s}$ — the canonical "one second" — and at $T\approx0.1\,\mathrm{MeV}$ with $g_*=3.36$ it gives a few minutes, the nucleosynthesis window. Why: these two anchors, $1\,\mathrm{s}$ at an MeV and minutes at a tenth of an MeV, are the benchmarks every account of the early universe is calibrated against.

Go deeper · Derivation C · 1 minThe photon reheating factor at electron–positron annihilation

Derivation CThe photon reheating factor at electron–positron annihilation

The purpose is to pin down the exact number $(11/4)^{1/3}$ by which the photon temperature is boosted above the neutrino temperature when electrons and positrons annihilate, and thereby predict the present neutrino background temperature from the measured CMB temperature. The engine is comoving entropy conservation applied to the electromagnetic sector alone, since the neutrinos have already decoupled — counting degrees of freedom before and after annihilation is the whole calculation.

Step 1. Just before annihilation the electromagnetic bath is photons plus $e^\pm$ pairs, giving $g_{*s}^{\rm before}=2+\tfrac{7}{8}(2+2)=\tfrac{11}{2}$ ; afterwards only photons remain, so $g_{*s}^{\rm after}=2$ . Why: the two photon polarisations count as $2$ , while the electron and positron each contribute two spin states weighted by $7/8$ — the annihilating pairs are precisely the entropy that must go somewhere.

Step 2. Comoving entropy in the electromagnetic sector is conserved, $g_{*s}^{\rm before}(aT)^3=g_{*s}^{\rm after}(aT_\gamma)^3$ , because the annihilation is adiabatic and happens while the $e^\pm$ are still coupled. Why: entropy has nowhere to leak, so the shrinking degree-of-freedom count must be compensated by a rise in $aT_\gamma$ — the photons are heated relative to where they would otherwise be.

Step 3. The neutrinos, already decoupled, simply keep $aT_\nu=\text{const}$ equal to the pre-annihilation $aT$ , so $(aT_\gamma)^3=\tfrac{11}{4}(aT_\nu)^3$ and $T_\gamma/T_\nu=(11/4)^{1/3}\approx1.401$ . Why: this is the crux — because the neutrinos are spectators to the entropy dump, the ratio of the two temperatures is frozen at exactly $(11/4)^{1/3}$ and never changes again.

Step 4. Inverting gives $T_\nu=(4/11)^{1/3}T_\gamma$ , so with $T_0=2.7255\,\mathrm{K}$ today the relic neutrinos sit at $T_{\nu,0}=(4/11)^{1/3}\times2.7255=1.95\,\mathrm{K}$ . Why: this is a parameter-free prediction the CMB temperature converts into a number — a cosmic neutrino sea at $1.95\,\mathrm{K}$ that experiments are now straining to detect directly.

Limiting cases

The thermal history collapses into a few recognisable regimes, each governed by how $\Gamma$ compares with $H$ and by whether $g_{*s}$ is changing.

Case	Result	Meaning
Deep radiation era	$H=\sqrt{\tfrac{8\pi G}{3}\rho},\ \rho=\tfrac{\pi^2}{30}g_*T^4$	Energy density set by relativistic species; $H\propto g_^{1/2}T^2$ and $t\propto g_^{-1/2}T^{-2}$ — the clock for every epoch before matter–radiation equality.
Tightly coupled	$\Gamma=n\langle\sigma v\rangle\gg H$	Species tracks the equilibrium distribution at the common temperature; abundances follow Bose, Fermi, or Boltzmann forms with no memory of the past.
Frozen out	$\Gamma\lesssim H$	Comoving abundance fixed; the species free-streams and its momenta redshift with an independently evolving effective temperature — relic neutrinos, the CMB, dark-matter freeze-out.
Entropy-releasing transition	$g_{s}\ \text{falling}\Rightarrow T_\gamma\propto g_{s}^{-1/3}a^{-1}$	An annihilating species heats the coupled bath, so the photon temperature falls more slowly than $a^{-1}$ — as at $e^+e^-$ annihilation.

Common trap“The universe simply cooled as T ∝ 1/a all the way down”

The clean law $T\propto a^{-1}$ holds only for a relativistic bath whose entropic degree-of-freedom count $g_{*s}$ is not changing. Every time a species annihilates, it dumps its entropy into the remaining coupled particles, and the correct relation is $T\propto g_{*s}^{-1/3}a^{-1}$ : the coupled temperature falls more slowly than $1/a$ across the transition. At $e^+e^-$ annihilation this is not a nicety — it is the entire origin of the $(4/11)^{1/3}$ offset between the neutrino and photon temperatures. Assume a strict $T\propto a^{-1}$ through that event and you predict $T_\nu=T_\gamma$ , missing the coldness of the relic neutrino sea and, with it, a testable pillar of the standard cosmology. The lesson generalises: $T\propto a^{-1}$ is the rule between thresholds, and the interesting physics lives precisely at the thresholds where it breaks.

Worked examples

1 · Dating the epochs from temperature (fully worked)

Use the radiation-era time–temperature relation to find the age of the universe at neutrino decoupling ( $T=1\,\mathrm{MeV}$ , $g_*=10.75$ ) and at the onset of nucleosynthesis ( $T=0.1\,\mathrm{MeV}$ , $g_*=3.36$ ).

Step 1. Write the relation in its practical form, $t\approx 2.42\,g_*^{-1/2}(T/\mathrm{MeV})^{-2}\,\mathrm{s}$ , obtained by inserting $\rho=\tfrac{\pi^2}{30}g_*T^4$ into $H^2=\tfrac{8\pi G}{3}\rho$ and using $H=1/(2t)$ (Derivation 2). The prefactor $2.42$ already carries the restored factors of $\hbar$ and $c$ .

Step 2. At neutrino decoupling, $g_*=10.75$ so $\sqrt{g_*}=3.28$ , and $T=1\,\mathrm{MeV}$ gives $(T/\mathrm{MeV})^{-2}=1$ . Thus $t=2.42/3.28\approx\mathbf{0.74\ s}$ — the origin of the textbook statement that neutrinos freeze out "about one second" after the Big Bang. The corresponding Hubble rate is $H=1/(2t)\approx\mathbf{0.68\ s^{-1}}$ .

Step 3. At the nucleosynthesis threshold, $g_*=3.36$ (the $e^\pm$ have annihilated) so $\sqrt{g_*}=1.83$ and $(0.1)^{-2}=100$ , giving $t=(2.42/1.83)\times100\approx\mathbf{132\ s}\approx2.2\ \mathrm{min}$ . Deuterium survives past its photodissociation bottleneck slightly later, at $T\approx0.07\,\mathrm{MeV}$ , giving $t\approx4.5\,\mathrm{min}$ — hence the first few minutes.

Step 4. The inverse-square scaling is the headline: a tenfold drop in temperature stretches the age a hundredfold. Neutrino decoupling at t ≈ 0.7 s, nucleosynthesis over the first few minutes — a MeV is a second, a tenth of an MeV is minutes.

2 · The relic neutrino sea today (fully worked)

From the measured CMB temperature $T_0=2.7255\,\mathrm{K}$ , predict the present temperature and number density of the cosmic neutrino background.

Step 1. Neutrinos decouple just before $e^+e^-$ annihilation, so they miss the photon reheating and settle at $T_\nu=(4/11)^{1/3}T_\gamma$ (Derivation 3). Numerically $(4/11)^{1/3}=0.7138$ , so today $T_{\nu,0}=0.7138\times2.7255=\mathbf{1.95\ K}$ .

Step 2. The photon number density of a blackbody at $T_0$ is $n_\gamma=411\,\mathrm{cm^{-3}}$ . Fermi statistics and the temperature ratio give the neutrino density per flavour (particle plus antiparticle) as $n_\nu=\tfrac{3}{4}\cdot\tfrac{4}{11}\,n_\gamma=\tfrac{3}{11}n_\gamma$ .

Step 3. Evaluate: $n_\nu=\tfrac{3}{11}\times411=\mathbf{112\ cm^{-3}}$ per flavour, so all three flavours together give $\approx\mathbf{336\ cm^{-3}}$ of relic neutrinos in every cubic centimetre of space right now.

Step 4. Their temperature $1.95\,\mathrm{K}$ corresponds to a mean energy of order $1.7\times10^{-4}\,\mathrm{eV}$ , far below the $\gtrsim0.05\,\mathrm{eV}$ mass scale from oscillation experiments, so at least two species are non-relativistic today and act as hot dark matter. A sea of ~336 neutrinos per cm³ at 1.95 K pervades space — predicted exactly, not yet detected directly.

3 · Why neutrinos end up colder than photons — the ordering test

Confirm from the numbers that neutrino decoupling precedes $e^+e^-$ annihilation, which is the reason $T_\nu<T_\gamma$ .

Step 1. Neutrino decoupling occurs at $T_{\rm dec}\approx1\,\mathrm{MeV}$ (Derivation 1), where the weak rate $\Gamma\propto G_F^2T^5$ falls below $H\propto T^2$ .

Step 2. Electrons and positrons remain relativistic and abundant only while $T\gtrsim m_ec^2=0.511\,\mathrm{MeV}$ ; below that they annihilate into photons, dumping their entropy into the electromagnetic bath.

Step 3. Because $1\,\mathrm{MeV}>0.511\,\mathrm{MeV}$ , the neutrinos have already decoupled when the $e^\pm$ annihilate. They are spectators to the entropy dump, so only the photons are heated — by the factor $(11/4)^{1/3}\approx1.40$ . The ordering (decouple at 1 MeV, then reheat at 0.5 MeV) is exactly what leaves the neutrinos permanently ~29% cooler than the photons.

Physics of the portrait

The thermal history is where nuclear physics, particle physics, and cosmology fused into one predictive story — and the person who first insisted that the early universe was hot, its heat leaving a measurable relic, was George Gamow.

Origins · 1 minGeorge Gamow and how this was discovered

George Gamow

1904–1968 · hot Big Bang and primordial nucleosynthesis

George Gamow was a Russian-American theorist whose career reads like a tour of twentieth-century physics: as a young man he explained alpha decay by quantum tunnelling, then turned to stellar energy, and finally — decisively — to cosmology. In the late 1940s, with his graduate student Ralph Alpher and collaborator Robert Herman, Gamow argued that if the universe began hot and dense, nuclear reactions in its first minutes would forge the light elements from a primordial neutron-proton soup. The 1948 paper on this idea (published under Alpher, Bethe, and Gamow — the "αβγ" pun was Gamow's, Bethe added in absentia) launched the theory of Big Bang nucleosynthesis. The deeper insight came the same year: Alpher and Herman realised that the same hot beginning must leave a bath of thermal radiation cooled by the expansion to a few kelvin today, and estimated about $5\,\mathrm{K}$ — the first prediction of the cosmic microwave background. The idea languished until Penzias and Wilson stumbled on a $3\,\mathrm{K}$ excess in 1965, which Dicke, Peebles, and colleagues recognised as Gamow's relic radiation. He lived just long enough to see it discovered, vindicating the central claim of his cosmology: that the universe has a thermal history at all, written in relics we can still read.

History

1931

— Lemaître proposes the primeval-atom origin of the expanding universe

1948

— the αβγ paper computes primordial element synthesis in a hot early universe

1948

— Alpher and Herman predict a relic radiation of about 5 K today

1965

— Penzias and Wilson detect the cosmic microwave background near 3 K

1990

— COBE/FIRAS confirms a near-perfect 2.7 K blackbody spectrum

Going further

Texts. Ryden, Introduction to Cosmology, 2nd ed., ch. 9–10 · Dodelson & Schmidt, Modern Cosmology, 2nd ed., ch. 2–3 · Kolb & Turner, The Early Universe, ch. 3, 5 · Mo, van den Bosch & White, Galaxy Formation and Evolution, ch. 3 · Weinberg, Cosmology, ch. 3 · Carroll & Ostlie, An Introduction to Modern Astrophysics, 2nd ed., ch. 30. The key numerical results threading through this article are: the $g_*$ staircase values $106.75$ , $61.75$ , $17.25$ , $10.75$ , $3.36$ from standard Standard-Model degree-of-freedom counting; $T_\nu/T_\gamma=(4/11)^{1/3}=0.714$ , giving $T_{\nu,0}=1.95\,\mathrm{K}$ today, from the entropy argument at $e^+e^-$ annihilation; the time–temperature prefactor yielding $t\approx0.74\,\mathrm{s}$ at $1\,\mathrm{MeV}$ with $g_*=10.75$ and a few minutes at $0.1\,\mathrm{MeV}$ with $g_*=3.36$ ; the weak freeze-out estimate $T_{\rm dec}\sim1\,\mathrm{MeV}$ from $G_F^2T^5\sim T^2/M_{\rm Pl}$ , which exceeds $m_ec^2=0.511\,\mathrm{MeV}$ ; the electromagnetic degree-of-freedom drop $11/2\to2$ across $e^+e^-$ annihilation; and the relic densities $n_\gamma=411\,\mathrm{cm^{-3}}$ and $n_\nu=112\,\mathrm{cm^{-3}}$ per flavour. The CMB temperature $T_0=2.7255\,\mathrm{K}$ is the COBE/FIRAS measurement; recombination at $z\approx1100$ is standard; $N_{\rm eff}=3.044$ is the current Standard-Model value.

Prerequisites. Equilibrium Thermodynamics of the Early Universe · Effective Number of Relativistic Degrees of Freedom · Entropy Conservation and Comoving Entropy. Leads to. Decoupling and Freeze-Out · Neutrino Decoupling and the Cosmic Neutrino Background · Big Bang Nucleosynthesis · Recombination and the Saha Equation.

Ask the tutor: Ask me to walk you from any temperature to its cosmic time and dominant physics, or to derive the $(4/11)^{1/3}$ neutrino offset step by step.

Why does the temperature of radiation fall as the inverse scale factor?

Because the expansion conserves the entropy inside a comoving volume. For a relativistic bath $s\propto g_{*s}T^3$ and $s\,a^3$ is fixed, so $g_{*s}T^3a^3=\text{const}$ ; when $g_{*s}$ is constant this reduces to $T\propto a^{-1}$ . A complementary picture: each photon's wavelength stretches as $\lambda\propto a$ , so its energy $hc/\lambda\propto a^{-1}$ , and a blackbody stays a blackbody with $T\propto a^{-1}$ . Both views agree, and both fail gently at annihilation thresholds where $g_{*s}$ drops and the coupled temperature falls a little more slowly.

What is the difference between g∗ and g∗s?

They are two weighted counts of the relativistic species, coinciding whenever every species shares one temperature. The energy-density count $g_*$ weights each species by $(T_i/T)^4$ (energy density scales as $T^4$ ); the entropy count $g_{*s}$ weights by $(T_i/T)^3$ (entropy scales as $T^3$ ). While all species share the photon temperature the two are equal. After neutrino decoupling they part company: the neutrinos sit at $T_\nu=(4/11)^{1/3}T_\gamma$ , and the different powers make $g_*\approx3.36$ while $g_{*s}\approx3.91$ today. Use $g_*$ in the Friedmann equation and $g_{*s}$ in entropy bookkeeping.

If we cannot see anything from before recombination, how do we know the thermal history is right?

We read it from relics. The microwave background is a near-perfect blackbody at $2.7255\,\mathrm{K}$ , exactly the thermalised radiation a hot early universe must leave, and its anisotropies encode the plasma's composition at recombination. Big Bang nucleosynthesis predicts the abundances of deuterium, helium-4, and lithium from the conditions at one second to a few minutes; measured D and He match to percent precision with a single free parameter, the baryon density. And the framework predicts a relic neutrino background whose gravitational effect ( $N_{\rm eff}\approx3$ ) is seen in both the microwave background and nucleosynthesis. Three probes of three epochs agree — that is the evidence.

What actually orders the epochs — why electroweak before QCD before nucleosynthesis?

Temperature orders them, because it falls monotonically with expansion and each event has a characteristic energy scale: the electroweak transition near the weak-boson scale $T\sim100\,\mathrm{GeV}$ ; the QCD transition near the strong scale $T\sim150\,\mathrm{MeV}$ ; neutrino decoupling where the weak rate loses to expansion, $T\sim1\,\mathrm{MeV}$ ; nucleosynthesis where photons stop dissociating deuterium, $T\sim0.1\,\mathrm{MeV}$ . Since these scales are widely separated and the universe cools through them in order, the sequence is fixed, and the time–temperature relation converts each scale into an age.

Are the electroweak and QCD "transitions" true phase transitions?

In the Standard Model, not sharp ones. Both are now understood as smooth crossovers rather than first-order transitions with latent heat and bubble nucleation. Lattice calculations show the QCD transition near $150\,\mathrm{MeV}$ is a rapid but continuous crossover from quark-gluon plasma to hadron gas, and the electroweak transition for the observed Higgs mass is likewise a crossover. This matters: a genuine first-order transition could have sourced gravitational waves or driven baryogenesis, so the crossover nature constrains those scenarios. For thermal-history bookkeeping the distinction is mild — $g_*$ still steps down — but for why there is more matter than antimatter, it is central.

Why is N_eff equal to 3.044 rather than exactly 3?

Because neutrino decoupling is not perfectly instantaneous. The idealised story has neutrinos fully decoupled before $e^+e^-$ annihilation, so they receive none of its entropy and $N_{\rm eff}=3$ exactly. In reality decoupling overlaps slightly with the onset of annihilation, so the highest-energy neutrinos are still weakly coupled and skim a little of the released energy. Including this partial reheating, plus finite-temperature QED corrections and neutrino oscillations, raises the count to $N_{\rm eff}=3.044$ . The small excess is a genuine, calculable prediction — and measuring it precisely is one way the microwave background tests the standard thermal history.