The Thermal History of the Early Universe

The thermal history of the early universe is the story of how one number — the temperature $T$ of a hot, nearly homogeneous plasma — fell as space expanded, and of the fossils that each stage of cooling left behind. Because the young universe was radiation dominated and its reaction rates were, for long stretches, enormously faster than the expansion rate, the cosmic fluid sat in local thermodynamic equilibrium: at every instant a single temperature described it, driven steadily downward by the Friedmann expansion. The whole narrative is organised by one competition — between the reaction rate $\Gamma$ that keeps a species in equilibrium and the Hubble rate $H$ . While $\Gamma\gg H$ a species tracks equilibrium; when $\Gamma$ falls below $H$ the reaction can no longer keep pace, the species decouples, and whatever abundance it carried is frozen in as a relic. Two functions do the bookkeeping: the effective numbers of energy and entropy degrees of freedom, $g_*(T)$ and $g_{*s}(T)$ , which set $\rho=(\pi^2/30)g_*T^4$ and $s=(2\pi^2/45)g_{*s}T^3$ . As $T$ drops below the mass of each species, that species annihilates and dumps its entropy into the lighter particles still in contact, so $g_*$ and $g_{*s}$ fall in a staircase, and comoving entropy conservation $sa^3=\text{const}$ ties temperature to the scale factor as $T\propto g_{*s}^{-1/3}a^{-1}$ . Reading the thermometer downward gives the milestones — the electroweak and QCD transitions, neutrino decoupling near $1\,\mathrm{MeV}$ , nucleosynthesis at $\sim0.1\,\mathrm{MeV}$ , and recombination at $\sim0.26\,\mathrm{eV}$ — each of which we still observe today as the cosmic microwave background, the light-element abundances, and the predicted neutrino background at $1.95\,\mathrm{K}$ .

Key takeaways

As the FLRW universe expands and cools, its contents pass through equilibrium epochs interrupted by decouplings (freeze-out) and phase transitions, governed by the competition between the reaction rate $\Gamma$ and the Hubble rate $H$ .
In equilibrium the plasma temperature scales as $T\propto 1/a$ (up to entropy injections), so temperature is a clock: $T\sim 1\,\mathrm{MeV}$ at $t\sim 0.7\,\mathrm{s}$ , $T\sim 0.26\,\mathrm{eV}$ at recombination, $z\approx 1090$ .
The energy and entropy budgets are carried by $g_*(T)$ and $g_{*s}(T)$ , which fall in steps as heavy species annihilate — why $T\propto g_{*s}^{-1/3}a^{-1}$ rather than exactly $1/a$ .
Entropy conservation $sa^3=\text{const}$ makes $e^+e^-$ annihilation heat the photons but not the already-decoupled neutrinos, fixing $T_\nu/T_\gamma=(4/11)^{1/3}\approx 0.714$ and $T_{\nu,0}\approx 1.95\,\mathrm{K}$ .
Every freeze-out leaves a fossil: the CMB blackbody at $2.7255\,\mathrm{K}$ , the neutrino background, and the primordial abundances (about 75% H, 25% He by mass).
Key idea: A process keeps a species in equilibrium while $\Gamma\gtrsim H$ and freezes out a relic when $\Gamma\lesssim H$ ; meanwhile entropy conservation $sa^3=\text{const}$ links temperature to expansion, giving $T\propto g_{*s}^{-1/3}/a$ .

Where this lives today

The thermal-history framework is not a historical curiosity — it is a precision tool. Big-bang nucleosynthesis, run from the freeze-out physics above, predicts the primordial deuterium and helium abundances to the percent level, and matching them to observation pins the baryon-to-photon ratio at $\eta\approx 6\times 10^{-10}$ ( $\Omega_b h^2\approx 0.022$ ) — a number the CMB later confirmed independently, a great consistency check. The effective number of relativistic species, $N_{\rm eff}\approx 3.044$ , enters $g_*$ and hence the expansion rate; Planck's measurement of it from the CMB damping tail now bounds extra light degrees of freedom — sterile neutrinos, dark radiation, thermalised axions. The same freeze-out logic underlies the "WIMP miracle," in which a weak-scale particle freezes out with relic density $\Omega_\chi h^2\sim 3\times 10^{-27}\,\mathrm{cm^3\,s^{-1}}/\langle\sigma v\rangle$ , naturally of order the observed dark-matter abundance — the target of experiments like LZ and XENONnT. And the predicted neutrino background at $1.95\,\mathrm{K}$ is the quarry of the PTOLEMY project.

The physical picture

Picture the universe as a sealed box of uniformly expanding gas. As the box grows, every wavelength stretches and every particle loses energy, so the gas cools. While collisions are frequent it stays in equilibrium at a single temperature; once the box has expanded so far that a reaction can no longer fire fast enough, that reaction shuts off and the particles it was shuffling freeze at their last-equilibrium abundance. The cooling is not featureless — it is punctuated by the mileposts at which one species after another goes non-relativistic, annihilates, or decouples. The first figure lays the whole run out as one descending track, temperature against cosmic time.

Figure

The cosmic thermometer: expansion cools the plasma so its temperature falls as T ∝ 1/a, passing the electroweak transition (100 GeV, 10 ps), the QCD quark–hadron transition (150 MeV, 10 µs), neutrino decoupling (1 MeV, 0.7 s), nucleosynthesis (0.1 MeV, 3 min) and recombination (0.26 eV, 380 kyr) where the CMB is released

The single most important comparison on that track is $\Gamma$ against $H$ . Both are inverse times: $1/\Gamma$ is the typical interval between interactions, $1/H$ the expansion timescale, essentially the age of the universe. When interactions beat expansion, $\Gamma\gg H$ , a species keeps sampling equilibrium; when expansion wins, $\Gamma\ll H$ , it decouples and coasts, its momenta redshifting with the scale factor. What makes each decoupling sharp is that the rates carry different powers of $T$ : a weak rate falls as $T^5$ while $H$ falls only as $T^2$ , so once the steeper curve dives under the shallower one it stays there. The second figure makes this crossing — the freeze-out condition itself — into a picture.

Figure

The freeze-out criterion made visual: the weak rate Γ ∝ T⁵ falls far faster than the Hubble rate H ∝ T² as the universe cools, so above about 1 MeV interactions win and neutrinos track equilibrium, while below 1 MeV Γ drops under H and the neutrinos decouple, freezing out the cosmic neutrino background at T_ν = (4/11)^1/3 T_γ ≈ 1.95 K today

Entropy is the conserved quantity that keeps this bookkeeping honest. The total comoving entropy $S=sa^3$ is fixed under adiabatic expansion, so when a heavy species annihilates its entropy is handed to the lighter particles still in thermal contact, momentarily slowing their cooling. That is exactly why the photons end up warmer than the neutrinos: the neutrinos had already decoupled when $e^+e^-$ annihilated, so only the photons collected the released entropy. Every decoupling is a fossilization event — the early universe is an accelerator and a calorimeter whose run-record we still read today.

The core equations

The whole edifice rests on the equilibrium thermodynamics of a relativistic gas, packaged into two degrees-of-freedom counts. The energy and entropy densities sum over all species still relativistic and coupled, weighted by spin and by the Bose–Einstein versus Fermi–Dirac factor of $7/8$ .

\rho=\frac{\pi^2}{30}\,g_*(T)\,T^4,\qquad s=\frac{2\pi^2}{45}\,g_{*s}(T)\,T^3 \qquad (1)

where: $\rho$ — energy density; $s$ — entropy density; $g_*(T)$ — effective energy-density degrees of freedom; $g_{*s}(T)$ — effective entropy degrees of freedom; $T$ — photon temperature. Reading it: both densities are fixed by counting species, fermions contributing $7/8$ of an equal boson, with $g_*=g_{*s}$ whenever every species shares one temperature. The central dynamical statement is not an equation but an inequality: a process's fate is decided by whether it outruns the expansion. This one criterion sorts the thermal history into equilibrium epochs and freeze-outs.

\frac{\Gamma}{H}\gtrsim 1\ \Rightarrow\ \text{equilibrium};\qquad \frac{\Gamma}{H}\lesssim 1\ \Rightarrow\ \text{freeze-out} \qquad (2)

where: $\Gamma=n\langle\sigma v\rangle$ — interaction rate per particle; $H$ — Hubble rate; $n$ — target number density; $\langle\sigma v\rangle$ — thermally averaged cross section times velocity. Reading it: a species stays glued to equilibrium while collisions beat expansion, and coasts as a frozen relic once expansion wins. To evaluate that criterion we need $H(T)$ . In the radiation era curvature and the cosmological constant are negligible, and the flat Friedmann equation with the relativistic energy density collapses to a clean power law.

H^2=\frac{8\pi G}{3}\rho=\frac{8\pi^3 G}{90}\,g_*(T)\,T^4,\qquad H\simeq 1.66\,g_*^{1/2}\,\frac{T^2}{M_{\rm Pl}} \qquad (3)

where: $G$ — Newton's constant; $M_{\rm Pl}=G^{-1/2}\approx 1.22\times 10^{19}\,\mathrm{GeV}$ — the (non-reduced) Planck mass; $g_*$ — energy degrees of freedom at $T$ . Reading it: $H$ rises as $T^2$ , so the early universe expands furiously fast, with the coefficient $\sqrt{4\pi^3/45}\approx 1.66$ carrying the constants. Comoving entropy conservation turns temperature into a clock and ties the plasma to the FLRW scale factor. Because $S=sa^3$ is fixed, temperature and scale factor are locked together up to the slowly stepping $g_{*s}$ .

s\,a^3=\text{const}\ \Rightarrow\ g_{*s}(T)\,T^3 a^3=\text{const}\ \Rightarrow\ T\propto g_{*s}^{-1/3}\,a^{-1} \qquad (4)

where: $a$ — FLRW scale factor; $sa^3$ — total comoving entropy, conserved for adiabatic expansion. Reading it: $T\propto 1/a$ holds exactly only when $g_{*s}$ is constant; across each annihilation threshold the temperature falls more slowly, as entropy is dumped into the surviving species. The most celebrated consequence of that entropy dump is the offset between photon and neutrino temperatures, a firm prediction that fixes the neutrino-background temperature.

\frac{T_\nu}{T_\gamma}=\left(\frac{4}{11}\right)^{1/3}\approx 0.714,\qquad T_{\nu,0}\approx 1.95\,\mathrm{K} \qquad (5)

where: $T_\nu$ — neutrino temperature (decoupled before annihilation); $T_\gamma$ — photon temperature (heated by $e^+e^-$ annihilation); $T_{\nu,0}$ — present-day value. Reading it: neutrinos miss the entropy injection, ending up cooler by a fixed factor — $1.95\,\mathrm{K}$ today against the CMB's $2.7255\,\mathrm{K}$ . Two more relations close the loop to observation: the neutron-to-proton ratio at weak decoupling sets the helium yield, and the time–temperature relation converts a temperature into a cosmic age.

\left.\frac{n_n}{n_p}\right|_{\rm freeze}=\exp\!\left(-\frac{Q}{T_f}\right),\quad Q=m_n-m_p=1.293\,\mathrm{MeV} \qquad (6)

where: $n_n,\,n_p$ — neutron and proton densities; $Q$ — neutron–proton mass difference; $T_f\approx 0.8\,\mathrm{MeV}$ — weak freeze-out temperature. Reading it: the neutron fraction is set by a Boltzmann factor at freeze-out, then whittled down by neutron decay until nucleosynthesis locks the survivors into helium.

t\simeq \frac{0.301\,g_*^{-1/2}\,M_{\rm Pl}}{T^2}\ \Rightarrow\ \left(\frac{t}{1\,\mathrm{s}}\right)\approx 2.42\,g_*^{-1/2}\left(\frac{T}{1\,\mathrm{MeV}}\right)^{-2} \qquad (7)

where: $t$ — cosmic time; $g_*$ — effective degrees of freedom at $T$ . Reading it: with $g_*\approx 10.75$ near $1\,\mathrm{MeV}$ the coefficient is $2.42/\sqrt{10.75}\approx 0.74$ , so $t\approx 0.74\,(T/\mathrm{MeV})^{-2}\,\mathrm{s}$ — the canonical "about one second."

Derivation AThe photon-to-neutrino temperature ratio from entropy conservation

The goal is to derive, from nothing but comoving entropy conservation, the exact factor by which today's neutrino background is cooler than the microwave background. Neutrinos decouple at $T\sim 1\,\mathrm{MeV}$ , just above the temperature at which electrons and positrons annihilate. Because the neutrinos have already left thermal contact, the entropy released by $e^+e^-\to\gamma\gamma$ flows only into the photons, heating them relative to the neutrinos by a factor we compute with a degrees-of-freedom count and the constancy of $sa^3$ in each decoupled sector.

Step 1. Count entropy degrees of freedom in the photon-coupled bath just before annihilation: photons give $g=2$ , and $e^+,e^-$ each give $g=2$ as fermions, so $g_{*s}^{\rm before}=2+\tfrac{7}{8}(2+2)=\tfrac{11}{2}$ . Why: only species still exchanging entropy with the photons count; the decoupled neutrinos form a separate, sealed sector.

Step 2. Count the same quantity just after annihilation, when only photons remain: $g_{*s}^{\rm after}=2$ . Why: the $e^+e^-$ pairs have dumped their entropy into radiation and gone, so the photon sector drops from $11/2$ to $2$ .

Step 3. Impose entropy conservation on the photon sector in a fixed comoving volume, $\tfrac{11}{2}(aT_\gamma)^3_{\rm before}=2\,(aT_\gamma)^3_{\rm after}$ . Why: the comoving entropy of a sector in internal equilibrium is separately conserved, so lowering its degree-of-freedom count must raise $aT_\gamma$ to compensate.

Step 4. Track the decoupled neutrinos, which free-stream so that $aT_\nu$ is constant and which shared the photon temperature before annihilation, giving $(aT_\nu)_{\rm after}=(aT_\gamma)_{\rm before}$ . Why: a decoupled relativistic species keeps $T\propto 1/a$ , so $aT_\nu$ is frozen at its pre-annihilation value.

Step 5. Take the ratio after annihilation: $\big(aT_\gamma/aT_\nu\big)^3=g_{*s}^{\rm before}/g_{*s}^{\rm after}=11/4$ , hence $T_\nu/T_\gamma=(4/11)^{1/3}\approx 0.714$ . Why: this fixes a permanent offset, so with $T_0=2.7255\,\mathrm{K}$ today we predict $T_{\nu,0}\approx 1.95\,\mathrm{K}$ . *This result gives $T_{\nu,0}\approx 1.95\,\mathrm{K}$ ; once the small residual heating of neutrinos from $e^+e^-$ annihilation is folded in, the effective number of neutrino species is $N_{\rm eff}\approx 3.04$ .*

Go deeper · Derivation B · 1 minNeutrino decoupling from the rate comparison

Derivation BNeutrino decoupling from the rate comparison

The goal is to show that the weak interactions holding neutrinos in equilibrium switch off near $T\sim 1\,\mathrm{MeV}$ , and to see why the switch is sharp. Neutrinos are kept thermalised by weak processes such as $\nu\bar\nu\leftrightarrow e^+e^-$ , whose cross section grows as $\sigma\sim G_F^2 T^2$ on dimensional grounds, acting on a relativistic target density $n\sim T^3$ . Setting the reaction rate equal to the Hubble rate gives the decoupling temperature, and comparing the temperature powers of the two rates shows why decoupling happens over a narrow window.

Step 1. Write the interaction rate per neutrino as $\Gamma=n\langle\sigma v\rangle\sim T^3\cdot G_F^2 T^2=G_F^2 T^5$ , with $G_F\approx 1.17\times 10^{-5}\,\mathrm{GeV}^{-2}$ . Why: $G_F$ has dimensions of inverse mass squared, so for relativistic particles the only rate one can build from $G_F$ and $T$ is $G_F^2 T^5$ — a very steep function of temperature.

Step 2. Write the Hubble rate from equation (3), $H\simeq 1.66\,g_*^{1/2}T^2/M_{\rm Pl}$ with $g_*\approx 10.75$ for the plasma of photons, $e^\pm$ and three neutrinos. Why: near $1\,\mathrm{MeV}$ these are the relativistic species, and $H\propto T^2$ — much gentler than $\Gamma$ .

Step 3. Equate the two, $G_F^2 T^5=1.66\,g_*^{1/2}T^2/M_{\rm Pl}$ , and solve: $T_{\rm dec}^3=1.66\,g_*^{1/2}/(G_F^2 M_{\rm Pl})$ . Why: the crossing of $\Gamma$ and $H$ is by definition where equilibrium can no longer be maintained.

Step 4. Evaluate: $T_{\rm dec}\approx (3.3\times 10^{-9}\,\mathrm{GeV}^3)^{1/3}\approx 1.5\,\mathrm{MeV}$ , of order $1\,\mathrm{MeV}$ ; and since $\Gamma/H\propto T^3$ , halving $T$ shifts the balance by nearly an order of magnitude. Why: the $T^5$ -versus- $T^2$ mismatch makes the transition abrupt — weak reactions dominate above $\sim1\,\mathrm{MeV}$ and are irrelevant below, so the neutrino spectrum freezes at that instant. *This order-of-magnitude estimate gives $T_{\rm dec}\approx 1\,\mathrm{MeV}$ , consistent with the more precise result.*

Go deeper · Derivation C · 1 minThe recombination redshift from the Saha equation

Derivation CThe recombination redshift from the Saha equation

The goal is to explain why the universe stayed ionised until it had cooled to $T\approx 0.26\,\mathrm{eV}$ — nearly fifty times below the hydrogen binding energy — and to locate the last-scattering surface at $z\approx 1090$ . The reaction $e^-+p\leftrightarrow H+\gamma$ sits in Saha equilibrium while reactions are fast, but the huge photon-to-baryon ratio means enough high-energy photons in the Wien tail re-ionise any hydrogen that forms, delaying recombination far below $T\sim B=13.6\,\mathrm{eV}$ . Solving the Saha equation for the ionisation fraction pins down where the plasma finally becomes neutral.

Step 1. Define the ionisation fraction $X_e=n_e/(n_e+n_H)=n_p/n_B$ with $n_B=n_p+n_H$ , using charge neutrality $n_e=n_p$ . Why: $X_e$ runs from $1$ (fully ionised) to $0$ (fully neutral) and is the natural variable in which to state the Saha balance.

Step 2. Rewrite the Saha relation in terms of $X_e$ : $X_e^2/(1-X_e)=n_B^{-1}(m_e T/2\pi)^{3/2}e^{-B/T}$ , with $n_B=\eta\,n_\gamma\propto\eta T^3$ and $\eta\approx 6\times 10^{-10}$ . Why: dividing by the baryon density exposes the enormous prefactor set by the tiny baryon-to-photon ratio $\eta$ .

Step 3. The prefactor $\sim\eta^{-1}(m_e/T)^{3/2}$ is of order $10^{9}$ or larger, so $X_e$ stays near $1$ until $e^{-B/T}$ has fallen to about $10^{-9}$ . Why: it is not $T\sim B$ but $e^{-B/T}\sim\eta$ that triggers recombination, requiring $T\approx B/40\approx 0.3\,\mathrm{eV}$ — the delay is entirely due to photons outnumbering baryons a billion to one.

Step 4. Solve $X_e=\tfrac12$ to find $T_{\rm rec}\approx 0.26\,\mathrm{eV}\approx 3000\,\mathrm{K}$ , then convert with $T\propto(1+z)$ and $T_0=2.7255\,\mathrm{K}$ : $1+z_{\rm rec}\approx 3000/2.7255\approx 1100$ . Why: temperature redshifts as $1/a$ , so a temperature ratio is directly a redshift.

Step 5. Photon decoupling follows recombination: the Thomson rate $\Gamma=n_e\sigma_T$ drops below $H$ once free electrons are mopped up, at $z_{\rm dec}\approx 1090$ , $t_{\rm dec}\approx 380{,}000\,\mathrm{yr}$ , releasing the CMB. Why: the disappearance of free electrons lets photons stream freely, so the last-scattering surface sits just after the ionisation fraction plunges. *This gives the last-scattering redshift $z_*\approx 1090$ and recombination temperature $\approx 0.26\,\mathrm{eV}$ , in agreement with observation.*

Limiting cases

The behaviour of any species is set by two comparisons: its temperature against its mass, and its interaction rate against the expansion. The four regimes below cover the whole thermal history.

Case	Result	Meaning
Ultra-relativistic, coupled ( $T\gg m$ )	$\rho_i=\frac{\pi^2}{30}g_i T^4,\ n_i\propto T^3$	Bosons and fermions act as radiation; fermions carry a factor $7/8$ in $\rho$ relative to a boson of equal $g_i$
Non-relativistic, coupled ( $T\ll m$ )	$n_i=g_i\!\left(\frac{m_iT}{2\pi}\right)^{3/2}e^{-m_i/T}$	Number density is Boltzmann-suppressed; the species annihilates away and hands its entropy to lighter particles
Decoupled / free-streaming ( $\Gamma\ll H$ )	$f(p,t)=f_{\rm eq}\!\left(\tfrac{p\,a}{a_{\rm dec}}\right),\ p\propto 1/a$	The phase-space distribution is frozen in comoving momentum; the effective temperature redshifts as $T\propto 1/a$ even for massless relics
Radiation-era clock	$t\simeq 0.301\,g_^{-1/2}\tfrac{M_{\rm Pl}}{T^2}\Rightarrow t/\mathrm{s}\approx 2.42\,g_^{-1/2}(T/\mathrm{MeV})^{-2}$	With $g_*\approx 10.75$ near $1\,\mathrm{MeV}$ this gives $t\approx 0.74\,(T/\mathrm{MeV})^{-2}\,\mathrm{s}$ , the standard nucleosynthesis clock

Common trapT ∝ 1/a is exact everywhere

It is not. The relation $T\propto 1/a$ holds only while $g_{*s}$ is constant — that is, between annihilation thresholds. Whenever a species becomes non-relativistic and annihilates (most importantly $e^+e^-\to\gamma\gamma$ ), its entropy is dumped into the photon bath, $g_{*s}$ drops, and equation (4) forces $T$ to fall more slowly than $1/a$ across that step. This is not a minor correction: it is precisely why the photon temperature ends up above the neutrino temperature by $(11/4)^{1/3}$ , and why the neutrino background is predicted at $1.95\,\mathrm{K}$ rather than $2.7255\,\mathrm{K}$ . Treating $T\propto 1/a$ as exact through $e^+e^-$ annihilation erases the entire photon–neutrino offset.

Worked examples

1 · The nucleosynthesis clock (fully worked)

Using the radiation-era time–temperature relation, find the cosmic age at neutrino decoupling ( $T=1\,\mathrm{MeV}$ , $g_*=10.75$ ) and at the deuterium-formation temperature ( $T=0.07\,\mathrm{MeV}$ , $g_*=3.36$ ).

Step 1. At $T=1\,\mathrm{MeV}$ the coefficient in equation (7) is $2.42/\sqrt{10.75}=\mathbf{0.738}$ , so $t\approx 0.74\,(T/\mathrm{MeV})^{-2}\,\mathrm{s}=\mathbf{0.74\,s}$ . This is the origin of the loose "about one second" for neutrino decoupling — the precise number is three-quarters of a second.

Step 2. At $T=0.07\,\mathrm{MeV}$ , after $e^+e^-$ annihilation, the relativistic content is photons plus three decoupled neutrinos, $g_*\approx 3.36$ , so the coefficient is $2.42/\sqrt{3.36}=1.32$ . Then $t\approx 1.32\,(0.07)^{-2}\,\mathrm{s}\approx\mathbf{2.7\times 10^{2}\,s}$ ; nucleosynthesis proper begins a little earlier, near $t\approx 3\,\mathrm{min}$ — the famous "first three minutes."

Step 3. Check the trend: dropping $T$ by $\sim 14$ lengthens $t$ by $\sim 14^2\approx 200$ , exactly as the inverse-square law demands. The clock reads t ≈ 0.74 s at 1 MeV and t ≈ 3 min at the onset of nucleosynthesis.

2 · The temperature of the cosmic neutrino background today

The CMB is a blackbody at $T_0=2.7255\,\mathrm{K}$ . Predict the present neutrino background temperature, as an energy and as a fraction of the CMB photon energy density (massless neutrinos).

Step 1. Apply the entropy result of Derivation 1: neutrinos decoupled before $e^+e^-$ annihilation, so $T_\nu/T_\gamma=(4/11)^{1/3}=0.7138$ . Hence $T_{\nu,0}=0.7138\times 2.7255\,\mathrm{K}=\mathbf{1.945\,K}$ , rounded to $1.95\,\mathrm{K}$ .

Step 2. In energy units $k_B T_{\nu,0}=(8.617\times 10^{-5}\,\mathrm{eV/K})\times 1.945\,\mathrm{K}=\mathbf{1.68\times 10^{-4}\,eV}$ — a fraction of a milli-electron-volt, which is why neutrinos heavier than this are non-relativistic today and the background is so hard to detect.

Step 3. For each relativistic neutrino species the energy density carries the fermion factor $7/8$ and the temperature ratio to the fourth power, $\rho_\nu/\rho_\gamma=\tfrac{7}{8}(4/11)^{4/3}$ per species; summing over three species gives $3\times\tfrac{7}{8}(4/11)^{4/3}=0.681$ . The neutrino background sits at 1.95 K today and carries about 68% of the CMB photon energy density while relativistic.

3 · The primordial helium mass fraction

From the weak freeze-out ratio, estimate the fraction of baryonic mass locked into helium-4.

Step 1. At freeze-out $T_f\approx 0.8\,\mathrm{MeV}$ the ratio is set by the Boltzmann factor of equation (6): $n_n/n_p=e^{-Q/T_f}=e^{-1.293/0.8}=\mathbf{0.20}$ , about one neutron per five protons.

Step 2. Between freeze-out and deuterium formation at $t\approx 200\,\mathrm{s}$ , free neutrons decay ( $\tau_n\approx 879\,\mathrm{s}$ ), thinning the ratio to $n_n/n_p\approx \mathbf{1/7}$ . Almost every surviving neutron is then swept into helium-4.

Step 3. With two neutrons per helium nucleus, $Y_p=\dfrac{2(n_n/n_p)}{1+n_n/n_p}=\dfrac{2/7}{8/7}=\mathbf{0.25}$ . Y_p ≈ 0.25 matches the observed 0.247 — a fossil of the neutron-to-proton ratio frozen in at one second.

Physics of the portrait

The thermal history of the universe is the achievement of one audacious idea: that the present cosmos is the cooled ash of a hot, dense beginning whose temperature can be computed back through time. That idea, and the prediction of a relic radiation bath, belongs above all to George Gamow.

Origins · 1 minGeorge Gamow and how this was discovered

George Gamow

1904–1968, Russian-American physicist · the hot big bang and primordial nucleosynthesis

George Gamow took the expanding-universe geometry of Friedmann — whose student he had been in Leningrad — and asked what its matter was doing while it expanded. In the 1940s he argued that the early universe was not merely dense but hot, a furnace in which nuclear reactions could forge the elements. With his student Ralph Alpher he worked out primordial nucleosynthesis in the 1948 paper mischievously signed Alpher, Bethe, Gamow, and with Robert Herman, Alpher drew the epochal conclusion: a hot beginning must leave a relic radiation bath, which they estimated at a few kelvin. The prediction was forgotten for over a decade until Penzias and Wilson stumbled on the microwave background in 1965. Gamow's framework — equilibrium thermodynamics run backward through the Friedmann expansion, freeze-out setting relic abundances, cooling radiation as the fossil of a hot past — is exactly the machinery of this article. He got several numbers wrong, and the heavy elements come from stars, not the Big Bang; but the central claim, that the universe has a computable thermal history whose relics we can measure, made cosmology a quantitative science.

History

1946

— Gamow argues that the early universe was hot and dense, not merely compressed

1948

— Alpher and Herman predict a relic radiation background of a few kelvin

1965

— Penzias and Wilson detect the microwave background near 3 K

1977

— big-bang nucleosynthesis is first used to bound the number of neutrino species

1990

— the COBE satellite shows the CMB is a near-perfect blackbody at 2.735 K

Going further

Texts. Kolb & Turner, The Early Universe (1990), chs. 3–5 · Dodelson & Schmidt, Modern Cosmology, 2nd ed. (2020), chs. 2–3 · Weinberg, Cosmology (2008), ch. 3 · Carroll, Spacetime and Geometry (2004), ch. 8 · Mukhanov, Physical Foundations of Cosmology (2005), ch. 3 · Particle Data Group, Review of Particle Physics (2022), Astrophysics & Cosmology section. Numerical values used throughout: $T_0=2.7255\,\mathrm{K}$ , $T_{\nu,0}\approx 1.95\,\mathrm{K}$ , $(4/11)^{1/3}=0.7138$ , $Q=m_n-m_p=1.293\,\mathrm{MeV}$ , $B=13.6\,\mathrm{eV}$ , $z_{\rm rec}\approx 1090$ , $t_{\rm dec}\approx 3.8\times 10^5\,\mathrm{yr}$ , $\eta\approx 6\times 10^{-10}$ , $N_{\rm eff}\approx 3.044$ , $G_F\approx 1.17\times 10^{-5}\,\mathrm{GeV^{-2}}$ , $M_{\rm Pl}\approx 1.22\times 10^{19}\,\mathrm{GeV}$ . Key internal checks: $g_{*s}$ before and after $e^+e^-$ annihilation is $11/2$ versus $2$ , confirming $(4/11)^{1/3}$ ; the Hubble coefficient follows from $\sqrt{4\pi^3/45}=1.66$ ; the time–temperature coefficient satisfies $0.301\,M_{\rm Pl}/(1\,\mathrm{MeV})^2=2.42\,\mathrm{s}$ , giving $t\approx 0.74\,\mathrm{s}$ at $1\,\mathrm{MeV}$ with $g_*=10.75$ ; and $Y_p\approx 0.25$ follows from $n_n/n_p\approx 1/7$ .

Builds on: The FLRW Metric · The Friedmann Equations · The Scale Factor and Hubble Parameter · The Cosmological Fluid and Equation of State · The Radiation-Dominated Era Leads to: The Matter-Dominated Era · Cosmological Redshift · Critical Density and the Density Parameters

Ask the tutor: Ask me to walk you through any single milestone — the entropy count behind $T_\nu/T_\gamma=(4/11)^{1/3}$ , the rate crossing that sets neutrino decoupling, or the Saha delay that fixes the last-scattering surface — and we will rebuild the number together.

What actually is g_* and why does it change with temperature?

$g_*$ is the effective number of relativistic degrees of freedom — a weighted count of every species light enough to be relativistic and still in thermal contact, bosons at full weight and fermions at $7/8$ . It sets the energy density through $\rho=(\pi^2/30)g_*T^4$ . It changes because "relativistic" is a moving target: as the plasma cools below the mass of each species, that species goes non-relativistic, annihilates, and drops out. So $g_*$ starts above $100$ (all Standard Model particles), falls to $10.75$ around $1\,\mathrm{MeV}$ (photons, electrons, three neutrinos), and settles near $3.4$ after $e^+e^-$ annihilation. The companion $g_{*s}$ counts the same species with a cube rather than a fourth power and governs entropy.

Is cosmological cooling the same as a gas cooling when you let it expand?

Very nearly. A relativistic gas expanding adiabatically does cool, and comoving entropy conservation $sa^3=\text{const}$ is exactly the statement that the expansion is adiabatic — no heat flows in or out of a comoving volume because the universe is homogeneous, so there is nowhere for heat to flow from. The one twist absent in a laboratory gas is that species drop out of the mixture as it cools: when electrons and positrons annihilate they release their entropy into the surviving photons, so the photon "gas" is briefly reheated relative to a simple $1/a$ cooling. A gas of fixed composition has no such steps and cools smoothly.

Why do neutrinos decouple while photons stay coupled much longer?

Because they interact through different forces. Neutrinos are held in the plasma only by the weak interaction, whose rate falls as $G_F^2 T^5$ — extraordinarily steep — so below about $1\,\mathrm{MeV}$ the weak rate plunges under the Hubble rate and neutrinos decouple. Photons scatter off free electrons via the electromagnetic Thomson cross section, far larger, which does not shut off until the free electrons disappear at recombination near $0.26\,\mathrm{eV}$ . The near-millionfold temperature gap between the two decouplings is exactly why the neutrino and photon backgrounds carry different temperatures today.

If the universe was in equilibrium, how can anything interesting survive?

This is the deep point: perfect equilibrium erases history, because an equilibrium state depends only on temperature, not on how you got there. Everything observable is a departure from equilibrium — a freeze-out. Neutrino decoupling, the survival of a small neutron fraction, dark-matter freeze-out, and recombination all happen precisely because a reaction fell out of equilibrium and left an abundance stuck at a non-equilibrium value. The competition $\Gamma$ versus $H$ is the arbiter: while $\Gamma\gg H$ the universe forgets its past, and only when $\Gamma$ drops below $H$ does it start to remember.

Why is recombination not the same event as photon decoupling?

They are close in time but conceptually distinct. Recombination is the chemistry: free electrons and protons combine into neutral hydrogen as the ionisation fraction falls around $z\approx 1300$ – $1000$ . Photon decoupling is the optics: the moment the photon mean free path grows longer than the horizon, so photons stop scattering and stream freely to us. Decoupling lags recombination slightly, because photons keep scattering off the last free electrons until the ionisation fraction has dropped far enough; the surface of last scattering sits at $z\approx 1090$ . Recombination is the cause, decoupling the consequence, and the CMB is the light released at the latter.

Has the cosmic neutrino background actually been detected?

Not directly, though its existence is established indirectly beyond reasonable doubt. Its radiation effect is seen through $N_{\rm eff}\approx 3.044$ , which the CMB and nucleosynthesis both measure and which matches three thermalised neutrino species heated slightly by residual $e^+e^-$ annihilation. Catching a relic neutrino is enormously hard: at $1.95\,\mathrm{K}$ their energies are sub-milli-electron-volt and their cross sections minuscule. The PTOLEMY experiment aims to capture them by neutrino capture on tritium, which has no threshold — a direct image of the universe at one second of age.