Matter-Radiation Equality

Two density lines that had to cross

Take any expanding universe filled with both matter and radiation, and the two energy budgets will, in time, behave very differently. Non-relativistic matter — atoms, dark-matter particles, anything whose kinetic energy is negligible compared with its rest mass — dilutes as the volume of space. Double the linear size, and the density falls by a factor of eight. So ρ_m ∝ a⁻³, where a is the cosmological scale factor.

Radiation, by contrast, suffers a double penalty. The photon number density also dilutes as a⁻³, but each individual photon also has its wavelength stretched by the expansion — exactly the cosmological redshift — and so its energy E = hc/λ falls as a⁻¹. The product is ρ_r ∝ a⁻⁴, a steeper decline by one power of a. Massless neutrinos behave the same way until they become non-relativistic at extremely low temperatures, so the early-universe radiation budget is essentially photons plus three flavours of relativistic neutrinos.

The two lines have different slopes on a log-log plot of density versus scale factor, and they had to cross somewhere. The crossing point — defined by ρ_m(a_eq) = ρ_r(a_eq) — is matter-radiation equality. Plugging in present-day values Ω_m ≈ 0.315, Ω_r ≈ 9.24 × 10⁻⁵ gives the equality scale factor a_eq = Ω_r/Ω_m ≈ 2.94 × 10⁻⁴, or redshift z_eq ≈ 3402.

How well Planck pins it

The 2018 final Planck cosmological-parameters paper quotes z_eq = 3402 ± 26 (TT,TE,EE+lowE+lensing), a relative uncertainty under 1%. That extraordinary precision is not a luxury: z_eq enters the CMB power spectrum in several ways simultaneously, and the data lever-arm is enormous. Equality fixes the ratio of early to late-time gravitational potentials, which controls (1) the magnitude of the early integrated Sachs-Wolfe (ISW) effect at large angular scales, and (2) the gravitational driving of acoustic oscillations entering the horizon during radiation domination, which sets the heights of the high-l acoustic peaks. Combining the two anchors z_eq with extraordinary leverage.

Why structure could not grow before equality

Before equality, the universe expands under the gravitational impulse of a smooth, dominant radiation component. Cold dark-matter perturbations on sub-horizon scales feel three competing effects: their own self-gravity (which would amplify them), Hubble friction (which damps them), and — critically — no comparable density of pressureless matter to help them along. The result, worked out by Mészáros in 1974, is that the growing mode of the matter density contrast δ_m ≡ δρ_m/ρ_m evolves as

δ_m(a) ∝ 1 + (3/2)(a / a_eq)   during radiation domination

which means that deep in the radiation era (a ≪ a_eq), the perturbations are effectively frozen at their primordial value. Once a approaches a_eq, the (3/2)(a/a_eq) term kicks in and δ_m starts growing roughly linearly with the scale factor. Throughout matter domination,

δ_m(a) ∝ a   during matter domination

The takeaway is that cosmic structure formation, in the linear regime, only really gets going at z_eq. Before that, the radiation drives the expansion too fast for gravitational instability to amplify density contrasts. After that, gravity wins and the seeds of every galaxy and cluster start to grow.

The turnover of the matter power spectrum

The Mészáros suppression has a sharp scale-dependent signature in the matter power spectrum P(k). Modes with comoving wavenumber k that entered the horizon during the radiation era (k > k_eq) were suppressed; modes that entered during matter domination (k < k_eq) were not. The boundary between the two regimes is the wavenumber k_eq of perturbations that entered the horizon exactly at equality.

k_eq = a_eq H(a_eq) ≈ 0.01045 h Mpc⁻¹    (Planck)

This corresponds to a comoving wavelength of about 100 Mpc/h. On larger scales (smaller k), P(k) follows the primordial spectrum P(k) ∝ k^n_s, with n_s ≈ 0.965. On smaller scales (larger k), it falls as P(k) ∝ k^(n_s − 4) because of the Mészáros suppression of small-scale modes during the radiation era. The turnover sits precisely at k_eq and is a generic prediction of every cold-dark-matter cosmology. Measurements of galaxy clustering — SDSS, BOSS, eBOSS, DESI — recover this turnover and use it as one of several handles on cosmological parameters.

The early-universe timeline

Epoch	Redshift	Cosmic time	Dominant content	Key event
Inflation ends	—	≈ 10⁻³⁴ s	Inflaton	Reheating, primordial spectrum imprinted
BBN	~ 4 × 10⁸	~ 3 min	Radiation	D, ³He, ⁴He, ⁷Li synthesis
Neutrino decoupling	~ 6 × 10⁹	~ 1 s	Radiation	ν free-streaming
Matter-radiation equality	3402	~ 50,000 yr	matter ≈ radiation	linear growth begins, k_eq imprinted
Recombination / CMB	1090	~ 380,000 yr	Matter	Photons decouple, last scattering
Dark ages	1090 → ~30	0.38 → ~100 Myr	Matter	No stars yet, neutral H ° 21 cm
First stars / reionisation	~ 20 → 6	~ 200 Myr → 1 Gyr	Matter	Pop III, reionisation of IGM
Matter-Λ equality	~ 0.30	~ 10 Gyr	Matter ≈ Λ	Acceleration begins

Equality sits between Big Bang nucleosynthesis and recombination, separated from the CMB by about a factor of three in scale factor or roughly 330,000 cosmic years. That gap is what lets dark-matter perturbations begin amplifying before the photons even decouple, so the gravitational potentials that the CMB photons last-scatter through already carry the imprint of the growing dark-matter structure.

How equality leaves a CMB fingerprint

The Planck constraint on z_eq does not come from observing equality directly — we cannot — but from two indirect signatures in the temperature and polarisation power spectra.

• Early ISW. Photons cross gravitational potentials between equality and recombination. In the radiation era those potentials decay; after equality they stabilise. A photon falling into a potential that subsequently decays gains net energy. The result is an enhancement of the temperature power spectrum on intermediate scales (around the first acoustic peak), with an amplitude that depends sensitively on how much of the photon's journey occurred in the radiation era versus the matter era — i.e., on z_eq.
• Driving of acoustic peaks. Modes that enter the horizon during radiation domination experience time-varying potential wells that drive the photon-baryon oscillation with a resonant boost. The boost amplitude depends on z_eq because z_eq sets the maximum scale that experienced this driving. Concretely, the heights of the high-l acoustic peaks (l ≳ 200) are sensitive to z_eq. Planck measures these peak ratios precisely, and inverts them to z_eq.

Both effects bolt z_eq to sub-percent precision in standard ΛCDM. Allowing for non-standard physics (extra radiation, modified gravity, time-varying dark energy) relaxes the constraint but also creates new directions in parameter space where z_eq is degenerate with the deviation — which is itself a powerful diagnostic.

What can shift z_eq

• Dark radiation (N_eff > 3.044). Adding light species — sterile neutrinos, thermal axions, asymmetric dark sectors — raises the radiation density at fixed photon temperature. Higher ρ_r pushes z_eq lower (later equality). Planck's bound is N_eff = 2.99 ± 0.17, consistent with the Standard Model prediction of 3.044.
• Different Ω_m or h. Since z_eq ≈ Ω_m h² × constant, increasing Ω_m h² pushes equality earlier. This is one route by which CMB observations constrain Ω_m h² to roughly half a percent.
• Modified gravity. Models that alter the expansion rate during radiation domination (some f(R) variants, some early-dark-energy proposals) move z_eq and, more importantly, change the relation between z_eq and the observed CMB peak structure.
• Early dark energy. An exotic energy component that turns on briefly near equality and then dissipates can shrink the sound horizon at recombination without modifying late-time observations — one current candidate for resolving the Hubble tension. These models are tightly constrained by their effect on z_eq and the high-l peak heights.

Worked example: where does z_eq come from?

Start with present-day matter and radiation density parameters. For Planck 2018 ΛCDM:

Ω_m = 0.315
Ω_r = Ω_γ + Ω_ν
Ω_γ = 2.473 × 10⁻⁵ h⁻²    (photons, fixed by T_CMB = 2.725 K)
Ω_ν = 1.692 × 10⁻⁵ h⁻²    (3 ultrarelativistic ν, includes 7/8 × (4/11)^(4/3) factor)
Ω_r ≈ 4.165 × 10⁻⁵ h⁻²
with h ≈ 0.674:
Ω_r ≈ 9.17 × 10⁻⁵

Then

1 + z_eq = Ω_m / Ω_r ≈ 0.315 / 9.17 × 10⁻⁵ ≈ 3434
z_eq ≈ 3433

The slight discrepancy with the headline Planck number (3402) reflects which species are still relativistic at equality (the heaviest neutrino is borderline, and the analysis convention matters), the exact T_CMB used, and the parameter-fit covariance. The order of magnitude is robust.

For the cosmic age, integrate the Friedmann equation with matter + radiation (Λ is utterly negligible at z = 3400):

t_eq = ∫₀^a_eq da / [a H(a)]
     ≈ (2/3) × (1/H_0) × Ω_m⁻¹/² × a_eq^(3/2) × [(2 − √2)]   (matter + radiation)
     ≈ 50,000 years

That 50,000-year number is what gets quoted as "when equality happened" — counted from the Big Bang singularity in the standard ΛCDM expansion history.

Common pitfalls

• Confusing equality with recombination. Equality (z ≈ 3400) and recombination (z ≈ 1100) are separated by a factor of three in scale factor and about 330,000 years. The CMB photons we see were last scattered well after equality, so by the time the universe became transparent it was already matter-dominated.
• Forgetting neutrinos in Ω_r. Computing z_eq with photons only — ignoring the relativistic neutrino contribution — gives z_eq ≈ 6000, way too high. The factor 1.692 × 10⁻⁵ h⁻² for relic neutrinos adds about 70% to the radiation density and pushes equality later by a similar amount.
• Assuming structure does not grow at all during radiation. The Mészáros result is "logarithmic, effectively frozen", not "exactly zero". Sub-horizon dark-matter perturbations grow as 1 + (3/2)(a/a_eq) — slow but non-trivial near equality.
• Forgetting horizon entry. The Mészáros suppression applies only to sub-horizon modes during radiation domination. Super-horizon modes grow as a² regardless. The k_eq turnover marks the boundary between these two regimes, not a sharp on/off switch.
• Treating z_eq as fixed by the photon temperature alone. z_eq depends on Ω_m h² as well, so different cosmologies with the same T_CMB can have different equality redshifts. This is one of the parameter degeneracies that CMB peak-height ratios break.