The Radiation-Dominated Era

For the first fifty thousand years of cosmic history the universe ran on light. Not on the stars and galaxies that dominate the night sky today, but on a seething plasma of relativistic quanta — photons, neutrinos, and, at the highest temperatures, every particle of the Standard Model moving so fast that its rest mass was an afterthought. This is the radiation-dominated era, and it is governed by a single dynamical fact: a gas of relativistic particles has a pressure equal to one third of its energy density, $p=\rho/3$ . That equation of state is not an accident of the early universe but a theorem about massless fields — it follows from the tracelessness of their stress-energy tensor — and once it is fed into the Friedmann equations it fixes everything else. Two consequences organize this article. First, energy conservation in an expanding spacetime turns $p=\rho/3$ into $\rho\propto a^{-4}$ , one power of the scale factor steeper than pressureless matter's $a^{-3}$ ; the extra power is the cosmological redshift of each quantum draining energy on top of the ordinary geometric dilution. Second, the flat Friedmann equation integrates that scaling into the clean power law $a(t)\propto t^{1/2}$ , so the Hubble rate is simply $H=1/(2t)$ . Because radiation dilutes faster than everything else, running the clock backward it always overtakes matter — there is a unique matter–radiation equality at $z_{\rm eq}\approx 3400$ — and the temperature climbs as $T\propto a^{-1}$ through neutrino decoupling and Big Bang nucleosynthesis. The single power law ties a cosmic clock to a thermometer, which is what makes the young universe a precision laboratory.

Key takeaways

The radiation-dominated era is the earliest well-understood epoch of the hot Big Bang, when relativistic species — photons, neutrinos, and at high enough temperature every light particle — dominate the energy budget.
The radiation equation of state $p=\rho/3$ (a consequence of the traceless stress tensor of massless fields) feeds the Friedmann equations to give $\rho\propto a^{-4}$ and, in a flat universe, $a(t)\propto t^{1/2}$ , so $H=1/(2t)$ .
The extra power over matter's $\rho_m\propto a^{-3}$ is the cosmological redshift stretching each quantum's wavelength, on top of the shared $a^{-3}$ dilution of number density.
Because radiation dilutes fastest, it must precede matter; the two are equal at matter–radiation equality, $z_{\rm eq}\approx 3400$ , $T_{\rm eq}\approx 0.8\ \mathrm{eV}$ .
The temperature scales as $T\propto a^{-1}\propto t^{-1/2}$ , the thermal density is $\rho=(\pi^2/30)\,g_*\,T^4$ , and the age–temperature clock $t\approx 2.4\,g_*^{-1/2}(T/\mathrm{MeV})^{-2}\,\mathrm{s}$ dates Big Bang nucleosynthesis to the first few minutes.
Key idea: Radiation's equation of state $p=\rho/3$ forces $\rho\propto a^{-4}$ — one power steeper than matter because of redshift — which in a flat universe drives the unique expansion law $a\propto t^{1/2}$ , $H=1/(2t)$ .

Where this lives today

The radiation era is not a closed chapter of textbook history — it is an active precision frontier. Big Bang nucleosynthesis remains one of the tightest tests of the standard cosmological model: measured primordial abundances of deuterium (now known to better than one percent from quasar absorption spectra) and helium-4 agree with predictions that depend directly on the expansion rate $H\propto\sqrt{g_*}\,T^2$ , which lets BBN weigh the relativistic content of the universe and bound the effective number of neutrino species $N_{\rm eff}=3.046$ — any extra light particle would speed up the clock and overproduce helium. The same $g_*$ physics is cross-checked by the cosmic microwave background: Planck measurements of the acoustic-peak heights pin the equality redshift to $z_{\rm eq}=3402\pm26$ and independently constrain $N_{\rm eff}$ , and the two very different probes agree. The predicted $1.95\ \mathrm{K}$ cosmic neutrino background is the great outstanding relic, with experiments such as PTOLEMY aiming at its first direct capture. And the era's small particle horizon is precisely the horizon problem that motivates a preceding epoch of inflation — making this era the hinge between the earliest instants and the observed thermal universe.

The physical picture

Take a fixed comoving box that stretches with the universe and fill it two ways. Fill it with cold dust — massive particles at rest — and its energy is the particle count times a fixed rest mass, so as the volume swells the density falls as $\rho_m\propto a^{-3}$ : pure geometric dilution. Now fill the same box with photons. Their number also dilutes as $a^{-3}$ , but each photon is a wave whose wavelength is stretched by the expansion, and a stretched photon carries less energy, $E=hc/\lambda\propto a^{-1}$ . Multiply the two effects and radiation thins as $\rho_r\propto a^{-4}$ — one power faster than matter. That single extra power is the hinge on which the whole thermal history turns: because $a^{-4}$ and $a^{-3}$ diverge without bound as $a\to 0$ , radiation is guaranteed to win at early enough times, no matter how negligible it is today. The first figure makes the crossover concrete — three components, three slopes, one moment of equality.

Figure

Energy density of each component versus scale factor on log axes: radiation ∝ a⁻⁴ (steepest), matter ∝ a⁻³, dark energy constant. Running toward the past (left) radiation overtakes matter at matter–radiation equality, a_eq ≈ 2.9×10⁻⁴, i.e. 1 + z_eq ≈ 3400, with Ω_m0 ≈ 0.31 and Ω_r0 ≈ 9×10⁻⁵

The pressure of a relativistic gas is not a small correction to a dust universe — it doubles the gravitational deceleration. Relativistic particles carry momentum comparable to their energy, so they resist compression far more strongly than slow dust; the net effect is that the universe decelerates faster and the scale factor grows as $t^{1/2}$ rather than matter's $t^{2/3}$ . Run the film backward and the plasma heats without limit, $T\propto 1/a$ , through a sequence of thermal milestones set by particle physics. The second figure is the workhorse relation of the era — the cosmic clock read against the cosmic thermometer, with the two landmarks, neutrino decoupling and the assembly of the first nuclei, that turn it into a quantitative theory.

Figure

The age–temperature relation on log axes: cosmic time versus plasma temperature, T ∝ t⁻¹ᐟ² from t ≈ 2.4 g*⁻¹ᐟ² (T/MeV)⁻² s. Landmarks: neutrino decoupling and neutron–proton freeze-out at T ≈ 1 MeV, t ≈ 1 s; the deuterium bottleneck opening and helium-4 assembly at T ≈ 0.07 MeV, t ≈ 200 s

The core equations

Everything begins with the equation of state. For a gas whose quanta are ultrarelativistic — kinetic energy far above rest mass — the pressure is locked to one third of the energy density, and the dimensionless ratio $w$ takes the value that names the whole era.

p=\frac{1}{3}\rho,\qquad w\equiv\frac{p}{\rho}=\frac{1}{3} \qquad (1)

where: $p$ — isotropic pressure of the cosmological fluid; $\rho$ — energy density (with $c=1$ ); $w$ — dimensionless equation-of-state parameter. Reading: relativistic particles carry substantial pressure ( $w=1/3$ ), where cold dust carries none ( $w=0$ ). Energy conservation in an expanding FLRW spacetime is the continuity equation, the $\nu=0$ component of $\nabla_\mu T^{\mu\nu}=0$ . It says the energy in a comoving volume changes both by dilution and by the pressure doing work on the expansion.

\dot\rho+3H(\rho+p)=0 \qquad (2)

where: $\dot\rho$ — time derivative of energy density; $H=\dot a/a$ — Hubble parameter; $a$ — scale factor. Reading: the $3H\rho$ term is ordinary dilution; the extra $3Hp$ is the $pV$ work that makes radiation lose energy faster than dust. Integrating (2) with $w=1/3$ gives the scaling that defines radiation. The exponent is $3(1+w)=4$ , so the density falls as the fourth power of the scale factor.

\rho_r(a)=\rho_{r0}\left(\frac{a_0}{a}\right)^{4}\propto a^{-4} \qquad (3)

where: $\rho_{r0}$ — radiation energy density today; $a_0$ — present scale factor (conventionally $1$ ). Reading: three of the four powers are the shared geometric dilution of quanta; the fourth is the redshift of each quantum's energy, unique to radiation. The gravitational half of the story is the Friedmann equation. For a spatially flat universe it ties the expansion rate directly to the energy density, with no curvature or $\Lambda$ term surviving at early times.

H^2=\left(\frac{\dot a}{a}\right)^2=\frac{8\pi G}{3}\,\rho \qquad (4)

where: $G$ — Newton's constant; $\rho$ — total energy density, here $\rho\approx\rho_r$ . Reading: because $\rho_r\propto a^{-4}$ dominates the curvature term $a^{-2}$ and the constant $\Lambda$ as $a\to 0$ , the flat-radiation approximation is essentially exact in the early universe. Substituting (3) into (4) and integrating yields the signature of the era: a scale factor rising as the square root of cosmic time, and a Hubble rate that is simply the reciprocal of twice the age.

a(t)\propto t^{1/2},\qquad H=\frac{1}{2t} \qquad (5)

where: $t$ — cosmic (proper) time since the Big Bang. Reading: the expansion rate is fixed with no free parameters, so knowing the time fixes $H$ and vice versa — the property that lets one date events by their temperature. To connect the abstract $\rho$ to a physical thermometer, use the thermal energy density of a relativistic plasma — a Stefan–Boltzmann law summed over every species still relativistic, weighted by its internal degrees of freedom.

\rho=\frac{\pi^2}{30}\,g_*\,T^4\qquad(\hbar=c=k_B=1) \qquad (6)

where: $T$ — photon temperature; $g_*$ — effective number of relativistic degrees of freedom. Reading: $g_*$ counts spin and polarization states (fermions weighted by $7/8$ ), running from $106.75$ above the electroweak scale down to $3.36$ today, and it directly rescales the expansion rate through $H\propto\sqrt{g_*}\,T^2$ . Combining $\rho\propto T^4$ with $\rho\propto a^{-4}$ and the expansion law gives the two scaling relations that make the era a clock and a thermometer at once.

T\propto a^{-1}\propto t^{-1/2} \qquad (7)

where: $T$ — temperature of the relativistic plasma. Reading: for slowly varying $g_*$ the temperature is just the inverse scale factor; sudden drops in $g_*$ (species annihilating) reheat the surviving plasma slightly, the correction $T\propto g_*^{-1/3}a^{-1}$ . Finally, setting $\rho_r=\rho_m$ locates the boundary of the era. Since $\rho_r/\rho_m\propto a^{-1}$ , the ratio of today's density parameters fixes the redshift and temperature of equality.

1+z_{\rm eq}=\frac{\Omega_{m0}}{\Omega_{r0}}\approx 3400,\qquad T_{\rm eq}\approx 0.8\ \mathrm{eV} \qquad (8)

where: $\Omega_{m0}\approx 0.31$ — present matter density parameter; $\Omega_{r0}\approx 9\times10^{-5}$ — present radiation density parameter (photons plus neutrinos); $z_{\rm eq}$ — redshift of equality. Reading: before this redshift the universe was radiation dominated; after it, matter took over and structure began to grow.

Derivation AFrom the equation of state to the square-root expansion law

The goal is to carry the single input $p=\rho/3$ all the way to the era's fingerprint, $a(t)\propto t^{1/2}$ and $H=1/(2t)$ : deriving the equation of state from the masslessness of the fields, integrating the continuity equation for the density scaling, and solving the flat Friedmann equation in closed form. Each link is elementary, but stringing them together shows that one thermodynamic fact about relativistic matter dictates the entire dynamics of the young cosmos, with no adjustable constants left over.

Step 1. Start from the perfect-fluid stress-energy tensor $T^{\mu\nu}=(\rho+p)u^\mu u^\nu+p\,g^{\mu\nu}$ and take its trace with $u^\mu u_\mu=-1$ , giving $T^\mu{}_\mu=-\rho+3p$ . For a gas of massless quanta the electromagnetic (or any conformal) stress tensor is traceless, $T^\mu{}_\mu=0$ , so $-\rho+3p=0$ and therefore $p=\tfrac13\rho$ , i.e. $w=1/3$ . Why: masslessness is the physical origin of the equation of state — it is conformal invariance, not any property of the expansion, that fixes $w$ .

Step 2. Insert $p=\rho/3$ into the continuity equation (2), so $\dot\rho=-3H(\rho+p)=-3H\rho(1+\tfrac13)=-4H\rho$ , which rearranges to $\dot\rho/\rho=-4\,\dot a/a$ . Integrate both sides: $\ln\rho=-4\ln a+\text{const}$ , hence $\rho\propto a^{-4}$ . Why: the factor $1+w=4/3$ is exactly what promotes matter's three powers of dilution to radiation's four; the arithmetic is trivial but it is where the redshift power enters the dynamics.

Step 3. Feed the scaling into the flat Friedmann equation (4): $(\dot a/a)^2=(8\pi G/3)\,\rho_{r0}\,a^{-4}$ . Multiply through by $a^2$ to get $\dot a^2 = (8\pi G\rho_{r0}/3)\,a^{-2}$ , take the square root, and rearrange to $a\,\dot a=C$ with $C\equiv\sqrt{8\pi G\rho_{r0}/3}$ a positive constant. Note that $a\,\dot a=\tfrac12\,\mathrm{d}(a^2)/\mathrm{d}t$ . Why: the substitution collapses a nonlinear equation for $a$ into a linear one for $a^2$ , which is why the radiation era admits an exact elementary solution where general FLRW models do not.

Step 4. Integrate $\tfrac12\,\mathrm{d}(a^2)/\mathrm{d}t=C$ with the initial condition $a=0$ at $t=0$ (the singularity), giving $a^2=2Ct$ and therefore $a\propto t^{1/2}$ . Differentiate to obtain the Hubble rate: $H=\dot a/a=\tfrac12\,\mathrm{d}(\ln a^2)/\mathrm{d}t=\tfrac12\cdot(1/t)=1/(2t)$ . Why: the boundary condition $a(0)=0$ is what selects the physically expanding branch and pins the origin of cosmic time to the Big Bang.

Step 5. Check against the general single-fluid result $a\propto t^{2/[3(1+w)]}$ : with $w=1/3$ the exponent is $2/(3\cdot\tfrac43)=1/2$ , and the companion formula $H=2/[3(1+w)t]$ gives $H=1/(2t)$ , both matching. Matter ( $w=0$ ) would instead give $t^{2/3}$ and $H=2/(3t)$ . Why: reproducing the textbook general power law from the specific derivation is the internal consistency check that no factor was dropped.

Go deeper · Derivation B · 1 minThe age–temperature relation and the Big Bang nucleosynthesis clock

Derivation BThe age–temperature relation and the Big Bang nucleosynthesis clock

The goal is to convert the geometric relation $H=1/(2t)$ into an explicit stopwatch that reads cosmic age directly from plasma temperature — the formula on which the entire quantitative theory of primordial nucleosynthesis rests. We assemble the thermal density, the Friedmann equation and the expansion law into $t(T)$ , restore the constants that natural units hide, and evaluate at neutron–proton freeze-out to confirm the celebrated "first second" of the universe.

Step 1. Write the total density thermally, $\rho=(\pi^2/30)\,g_*\,T^4$ , summing every relativistic species with its degrees of freedom. Substitute into the flat Friedmann equation to get $H^2=(8\pi G/3)(\pi^2/30)\,g_*\,T^4=(4\pi^3 G/45)\,g_*\,T^4$ . Why: this is the step that injects particle physics into gravity — the value of $g_*$ , set by which species are still relativistic, controls the expansion rate at fixed temperature.

Step 2. Use the radiation-era identity $H=1/(2t)$ from the previous derivation, so $1/(4t^2)=(4\pi^3 G/45)\,g_*\,T^4$ . Solve for the age: $t=\big(45/(16\pi^3 G\,g_*)\big)^{1/2}\,T^{-2}$ , i.e. $t\propto g_*^{-1/2}\,T^{-2}$ . Why: the inverse-square dependence on temperature is the signature that the young universe cools rapidly at first and ever more slowly — most of the interesting nuclear physics is crammed into the first few minutes.

Step 3. Restore constants. In particle units the coefficient becomes $t\simeq 0.301\,g_*^{-1/2}\,M_{\rm Pl}/T^2$ with $M_{\rm Pl}=1.22\times10^{19}\,\mathrm{GeV}$ , which reduces to the practical form $t\approx 2.42\,g_*^{-1/2}\,(T/\mathrm{MeV})^{-2}\ \mathrm{s}$ . Why: expressing time in seconds and temperature in MeV is what turns an abstract scaling into a number an experimentalist can test against measured light-element abundances.

Step 4. Evaluate at neutron–proton freeze-out. There $T\approx 1\ \mathrm{MeV}$ and the relativistic content (photons, electrons and positrons, three neutrino species) gives $g_*\approx 10.75$ , so $t\approx 2.42/\sqrt{10.75}\approx 0.7\ \mathrm{s}$ . This is the moment that fixes the neutron-to-proton ratio and hence the primordial helium abundance. Why: this single evaluation is the origin of the famous statement that the universe was "one second old" when its helium was decided — and it is a direct output of the radiation-era dynamics. *The numerical check gives $t(1\ \mathrm{MeV},\,g_*=10.75)=0.74\ \mathrm{s}$ , confirming the coefficient $2.42$ in the age–temperature formula.*

Go deeper · Derivation C · 1 minEntropy conservation and the relic neutrino temperature

Derivation CEntropy conservation and the relic neutrino temperature

The goal is to derive the exact factor $T_\nu/T_\gamma=(4/11)^{1/3}$ relating today's cosmic neutrino background to the photon background — a clean prediction of the radiation era following purely from entropy conservation in an expanding plasma. The physics is that neutrinos decouple while still relativistic and thereafter cool independently, so when electron–positron pairs annihilate they dump their entropy into the photons alone, leaving the two baths at different temperatures forever after.

Step 1. In thermal equilibrium the comoving entropy is conserved, $s\,a^3=\text{const}$ with $s\propto g_{*s}T^3$ , where $g_{*s}$ is the entropy-weighted count of relativistic degrees of freedom. Neutrinos decouple at $T\approx 1\ \mathrm{MeV}$ ; below this they free-stream and their temperature simply redshifts as $T_\nu\propto a^{-1}$ . Why: decoupling is what allows the neutrino and photon temperatures to diverge — before it, collisions keep them locked together.

Step 2. Just above $T=m_e$ the interacting plasma contains photons ( $g=2$ ) and electron–positron pairs ( $g=\tfrac78\times4=3.5$ ), so $g_{*s}=2+3.5=5.5=11/2$ . Just below, the pairs have annihilated and only photons remain, $g_{*s}=2$ . Why: the annihilation removes the $e^\pm$ entropy from the coupled bath, and it has nowhere to go but the photons, heating them relative to the already-decoupled neutrinos.

Step 3. Impose entropy conservation of the coupled sector across annihilation: $g_{*s}^{\rm before}(a\,T_\gamma)^3\big|_{\rm before}=g_{*s}^{\rm after}(a\,T_\gamma)^3\big|_{\rm after}$ , so $(a T_\gamma)^3$ grows by the factor $11/4$ . The neutrinos, decoupled, keep $a T_\nu=\text{const}$ , so the ratio becomes $T_\nu/T_\gamma=(4/11)^{1/3}\approx 0.714$ . Why: the entire $11/4$ comes from counting degrees of freedom before and after — no dynamics beyond entropy bookkeeping is needed.

Step 4. Evaluate today with the measured photon temperature $T_\gamma=2.725\ \mathrm{K}$ : $T_\nu=(4/11)^{1/3}\times 2.725\ \mathrm{K}\approx 1.95\ \mathrm{K}$ . This relic sea, never yet detected directly, contributes to $\Omega_{r0}$ and hence to $z_{\rm eq}$ . Why: the $1.95\ \mathrm{K}$ neutrino background is a concrete, falsifiable relic of the radiation era, and its energy density feeds straight back into the equality redshift of equation (8). *The entropy-weighted degrees of freedom are $g_{*s}=11/2$ before annihilation and $2$ after, giving $(4/11)^{1/3}\times 2.725\ \mathrm{K}=1.945\ \mathrm{K}$ .*

Limiting cases

How the radiation-dominated solution sits among the neighbouring FLRW regimes:

Case	Result	Meaning
Deep radiation era ( $a\to 0$ )	$a\propto t^{1/2},\ \ H=\dfrac{1}{2t},\ \ \rho\propto a^{-4}$	The exact early-time behaviour of a flat universe; radiation swamps matter ( $a^{-3}$ ), curvature ( $a^{-2}$ ) and $\Lambda$ (constant).
Matter–radiation equality	$\rho_r=\rho_m\ \Rightarrow\ 1+z_{\rm eq}=\dfrac{\Omega_{m0}}{\Omega_{r0}}\approx 3400$	Handover to the matter era; in conformal time the exact two-fluid solution interpolates smoothly, $a\propto\eta^2$ deep in matter domination.
General single fluid	$a\propto t^{2/[3(1+w)]},\ \ w=\tfrac13\Rightarrow t^{1/2}$	Radiation is the $w=1/3$ member; matter ( $w=0$ ) gives $t^{2/3}$ , a stiff fluid ( $w=1$ ) gives $t^{1/3}$ .
Small spatial curvature	$H^2=\dfrac{8\pi G}{3}\rho-\dfrac{k}{a^2}$	Curvature $\propto a^{-2}$ is negligible when $a^{-4}\gg a^{-2}$ , so the flat $t^{1/2}$ law holds to extraordinary accuracy early on.
Constant $g_*$ vs. threshold crossings	$T\propto a^{-1}$ vs. $T\propto g_*^{-1/3}a^{-1}$	Between annihilation thresholds temperature tracks $1/a$ exactly; when a species annihilates, $g_*$ drops and the survivors are briefly reheated.

Common trapRadiation scales as a⁻⁴ simply because it has pressure

The steeper falloff is often mis-attributed to pressure "pushing" the density down. Keep the two effects distinct. The factor $a^{-3}$ is pure geometry: a fixed number of quanta spread through a growing comoving volume, identical for photons and cold matter. The extra factor $a^{-1}$ is the cosmological redshift of each quantum's energy, $E\propto 1/\lambda\propto 1/a$ — the piece unique to radiation. Pressure enters the bookkeeping through the $pV$ work term $3Hp$ in the continuity equation, the formal accountant for that redshift, but the physical content is stretched wavelengths, not a mechanical push. And the clean $a\propto t^{1/2}$ law is strictly the flat, radiation-only result: it holds only while radiation is the sole dominant component, breaking down near equality.

Worked examples

1 · Dating neutron freeze-out and the first nuclei (fully worked)

Using the age–temperature relation $t\approx 2.42\,g_*^{-1/2}(T/\mathrm{MeV})^{-2}\ \mathrm{s}$ , find the cosmic age at neutron–proton freeze-out ( $T\approx 1\ \mathrm{MeV}$ , $g_*\approx 10.75$ ) and at the deuterium bottleneck ( $T\approx 0.07\ \mathrm{MeV}$ , $g_*\approx 3.36$ ), and give the Hubble rate at freeze-out.

Step 1. Age at freeze-out. Substitute $T=1\ \mathrm{MeV}$ and $g_*=10.75$ : $t=2.42/\sqrt{10.75}\times(1)^{-2}=2.42/3.279\approx \mathbf{0.74\ s}$ . So the neutron-to-proton ratio is frozen in during the first second of cosmic history.

Step 2. Hubble rate there. From $H=1/(2t)$ , $H=1/(2\times0.74\ \mathrm{s})\approx \mathbf{0.68\ s^{-1}}$ . The expansion timescale $H^{-1}\approx 1.5\ \mathrm{s}$ is what the weak interactions must beat to keep neutrons and protons in equilibrium; once the weak rate falls below $H$ , the ratio freezes.

Step 3. Age at the deuterium bottleneck. Now $T=0.07\ \mathrm{MeV}$ and (after $e^\pm$ annihilation) $g_*=3.36$ : $t=2.42/\sqrt{3.36}\times(0.07)^{-2}=2.42/1.833\times 204=1.32\times204\approx \mathbf{2.7\times10^{2}\ s}$ , a few minutes. Deuterium only becomes stable against photodissociation once the temperature has fallen this far, and helium-4 assembles immediately after.

Step 4. Interpretation. Between roughly $1\ \mathrm{s}$ and $200\ \mathrm{s}$ the neutrons surviving freeze-out (net of decay) are swept into helium, fixing $Y_p\approx 0.247$ . All of Big Bang nucleosynthesis is squeezed into the first few minutes because $t\propto T^{-2}$ makes the clock run fast when it is hot. Freeze-out at t ≈ 0.7 s, H ≈ 0.7 s⁻¹; helium assembled by t ≈ few hundred seconds.

2 · The redshift, temperature, and timing of matter–radiation equality (fully worked)

Given $\Omega_{m0}=0.31$ and $\Omega_{r0}=9.0\times10^{-5}$ (photons plus three neutrino species) and today's CMB temperature $T_0=2.725\ \mathrm{K}$ , find the redshift, temperature, and rough age of the universe at matter–radiation equality.

Step 1. Equality redshift. The densities scale as $\rho_m=\Omega_{m0}(1+z)^3$ and $\rho_r=\Omega_{r0}(1+z)^4$ (in units of the critical density). Setting them equal, one power of $(1+z)$ survives: $1+z_{\rm eq}=\Omega_{m0}/\Omega_{r0}=0.31/(9.0\times10^{-5})\approx \mathbf{3.4\times10^{3}}$ .

Step 2. Equality temperature. Temperature tracks redshift as $T=T_0(1+z)$ , so $T_{\rm eq}=2.725\ \mathrm{K}\times3444\approx 9.4\times10^{3}\ \mathrm{K}$ . Converting, $k_B T_{\rm eq}=(9.4\times10^{3}\ \mathrm{K})\times(8.62\times10^{-5}\ \mathrm{eV/K})\approx \mathbf{0.81\ eV}$ — well below any atomic binding, so this is still deep in the ionized-plasma era, before recombination at $z\approx 1100$ .

Step 3. Rough age at equality. Equality sits at the radiation-to-matter transition, so a pure-radiation estimate is only indicative; scaling the clock (with $g_*=3.36$ ) to $T_{\rm eq}\approx8.1\times10^{-7}\ \mathrm{MeV}$ gives $t\sim 2.42/\sqrt{3.36}\times(8.1\times10^{-7})^{-2}\ \mathrm{s}\sim 2\times10^{12}\ \mathrm{s}$ , about $\mathbf{5\times10^{4}\ yr}$ — the full two-fluid calculation gives a comparable value.

Step 4. Interpretation. Everything before $z_{\rm eq}\approx 3400$ is radiation dominated; after it, matter perturbations begin to grow and the seeds of galaxies take hold. The value of $z_{\rm eq}$ is itself measured from the heights of the CMB acoustic peaks, closing the loop between this simple ratio and precision cosmology. z_eq ≈ 3400, T_eq ≈ 0.8 eV, t_eq ≈ 5×10⁴ yr.

Physics of the portrait

The radiation-dominated era is the intellectual home of the hot Big Bang, and no one did more to insist that the early universe was hot — and therefore radiation dominated — than George Gamow, who turned a cooling fireball into a quantitative nuclear-physics laboratory.

Origins · 1 minGeorge Gamow and how this was discovered

George Gamow

1904–1968 · hot Big Bang and primordial nucleosynthesis

George Gamow was a Ukrainian-born theoretical physicist whose restless imagination ranged from quantum tunnelling in alpha decay to the genetic code, but his enduring cosmological legacy is the hot Big Bang. In the 1940s Gamow argued that if the early universe was radiation dominated and extremely hot, the observed abundances of the light elements could be forged in its first minutes as it expanded and cooled — the physics captured by exactly the age–temperature relation of this article. With his student Ralph Alpher he wrote the famous 1948 "alpha-beta-gamma" paper (Bethe's name added for the pun) laying out primordial nucleosynthesis, and that same year Alpher and Robert Herman used the framework to predict that the universe should be bathed in a relic radiation background at a temperature of a few kelvin — the cosmic microwave background, discovered by Penzias and Wilson in 1965. Gamow understood, well before it was fashionable, that the equation of state $p=\rho/3$ and the resulting $a\propto t^{1/2}$ dynamics make the young universe a controlled thermodynamic system, tying a cosmic clock to a thermometer. His insistence on a hot, radiation-first beginning is why the era described here is not speculation but tested history.

History

1922

— Friedmann derives the expanding-universe solutions of general relativity

1927

— Lemaître links cosmic expansion to redshift in a dynamic universe

1948

— Alpher, Bethe and Gamow model light-element formation in a hot early universe

1948

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

1965

— Penzias and Wilson detect the cosmic microwave background at 2.7 K

Going further

Texts. Misner, Thorne & Wheeler, Gravitation, §27 · Wald, General Relativity, Ch. 5 · Carroll, Spacetime and Geometry, Ch. 8 · Hartle, Gravity, Ch. 18 · Schutz, A First Course in General Relativity, Ch. 12 · Kolb & Turner, The Early Universe, Ch. 3 · Dodelson, Modern Cosmology, Ch. 2 · Weinberg, Cosmology, Ch. 3. The key results of this article are: (1) $w=1/3$ from the traceless stress tensor; (2) $\rho\propto a^{-4}$ by integrating the continuity equation; (3) $a\propto t^{1/2}$ , $H=1/(2t)$ from the flat Friedmann equation; (4) the age–temperature coefficient $t\approx 2.42\,g_*^{-1/2}(T/\mathrm{MeV})^{-2}\,\mathrm{s}$ , evaluating to $0.74\ \mathrm{s}$ at $T=1\ \mathrm{MeV}$ , $g_*=10.75$ ; (5) $z_{\rm eq}=\Omega_{m0}/\Omega_{r0}\approx 3400$ and $T_{\rm eq}\approx 0.81\ \mathrm{eV}$ ; (6) $T_\nu/T_\gamma=(4/11)^{1/3}\Rightarrow 1.95\ \mathrm{K}$ from entropy conservation. Numerical inputs used throughout: $\Omega_{m0}\approx0.31$ , $\Omega_{r0}\approx9\times10^{-5}$ , $g_*=106.75$ at high $T$ , $g_{*s}=11/2$ before $e^\pm$ annihilation, $Y_p\approx0.247$ , $N_{\rm eff}=3.046$ , $z_{\rm eq}=3402\pm26$ .

Prerequisites. The Friedmann Equations · The Cosmological Fluid and Equation of State · The Scale Factor and Hubble Parameter · Cosmological Redshift · The FLRW Metric. Leads to. The Matter-Dominated Era · Critical Density and the Density Parameters · Cosmological Horizons · Cosmological Redshift.

Ask the tutor: Ask me to walk through why a single fact — the pressure of light equals one third of its energy density — dictates the entire expansion, temperature, and nuclear history of the first few minutes of the universe.

Why does radiation dilute as a⁻⁴ when matter dilutes as a⁻³?

Both share the same geometric dilution: a fixed number of quanta spread through a comoving volume that grows as $a^3$ , giving a factor $a^{-3}$ common to photons and cold matter alike. Radiation acquires one extra power, $a^{-1}$ , because each photon is a wave whose wavelength is stretched by the expansion, and a longer wavelength means less energy, $E=hc/\lambda\propto a^{-1}$ . Multiply the two and $\rho_r\propto a^{-4}$ . A massive particle at rest has a fixed rest energy no matter how the box grows, so it gets no such redshift factor and stays at $a^{-3}$ . The extra power is entirely the cosmological redshift of individual quanta.

If the universe is expanding, why does a ∝ t¹ᐟ² mean it is decelerating?

Because the expansion speed $\dot a\propto t^{-1/2}$ decreases with time even though $a$ itself keeps growing. Differentiating twice, $\ddot a\propto -t^{-3/2}<0$ , so the expansion is slowing — gravity from the radiation is pulling the brake. This is exactly what you expect when the dominant component has positive energy and pressure: both $\rho$ and $p$ appear with a decelerating sign in the acceleration equation $\ddot a/a=-\tfrac{4\pi G}{3}(\rho+3p)$ , and for radiation $\rho+3p=2\rho$ , twice the matter value. Only a component with strongly negative pressure, like a cosmological constant, can reverse the sign and accelerate the expansion — which is what happens billions of years later.

What is g* and why does it change with time?

$g_*$ is the effective number of relativistic degrees of freedom — a weighted count of every particle species whose temperature exceeds its mass, with each bosonic spin/polarization state counting $1$ and each fermionic state counting $7/8$ (from Fermi versus Bose statistics). It sets the thermal energy density through $\rho=(\pi^2/30)g_*T^4$ and hence the expansion rate. It changes because, as the universe cools, species become non-relativistic and annihilate: above the electroweak scale essentially the whole Standard Model is relativistic and $g_*\approx 106.75$ ; below the electron mass only photons and the relic neutrinos remain, and $g_*$ has fallen to about $3.36$ . Each drop briefly reheats the surviving plasma.

The universe is matter and dark energy dominated now, so why care about the radiation era?

Because it left fossils we measure today, and because it was once the whole show. Even though radiation is only about $9\times10^{-5}$ of the present energy budget, its $a^{-4}$ scaling means that going back far enough it dominated everything — for the first roughly fifty thousand years. That era set the primordial abundances of the light elements, produced the relic neutrino background, and fixed the growth-suppression scale that shapes the matter power spectrum and the cosmic-microwave-background acoustic peaks. The redshift of matter–radiation equality, $z_{\rm eq}\approx 3400$ , is imprinted directly on the CMB and is one of the numbers precision cosmology measures most sharply. The radiation era is invisible today only in the sense that its light has cooled, not that its consequences have faded.

How do we know the equality redshift is about 3400 rather than just calculating it?

Both the calculation and the observation agree, which is the point. The estimate $1+z_{\rm eq}=\Omega_{m0}/\Omega_{r0}$ needs the two density parameters, and $\Omega_{r0}$ is known very precisely because it is fixed by the measured CMB temperature ( $2.725\ \mathrm{K}$ ) plus the calculable neutrino contribution. Independently, the physical scale of equality is stamped onto the CMB: modes entering the horizon before equality were driven differently from those entering after, so the relative heights of the acoustic peaks encode $z_{\rm eq}$ directly. Planck's peak-height measurements give $z_{\rm eq}=3402\pm26$ , in excellent agreement with the density-ratio estimate.

What came before the radiation era?

Standard cosmology is confident back to Big Bang nucleosynthesis at $T\sim 1\ \mathrm{MeV}$ ( $t\sim 1\ \mathrm{s}$ ), where the radiation-era predictions are directly tested; earlier, the same $a\propto t^{1/2}$ dynamics are believed to hold up through the electroweak scale, with less direct evidence. The era's small, slowly growing particle horizon creates the horizon problem — causally disconnected patches share the same temperature — which motivates an epoch of accelerated expansion, cosmic inflation, preceding radiation domination. Inflation would have ended by reheating the universe into the hot plasma described here, effectively setting the initial condition ( $a=0$ at $t=0$ becomes the end of reheating). At the Planck scale quantum gravity takes over and the classical FLRW picture breaks down.