The Matter-Dominated Era

For most of cosmic history the universe has been ruled by stuff that does not push back. After the radiation fireball cools and thins, the energy budget passes to pressureless matter — cold dark matter and, after recombination, ordinary baryons — and stays there for the longest dynamically distinct chapter of the expansion. The handover is pure bookkeeping: radiation energy dilutes as $a^{-4}$ because each photon both spreads with the volume and loses energy to the redshift, while matter dilutes only as $a^{-3}$ , the simple thinning of a fixed particle number across a growing box. Whatever head start radiation held, the steeper curve must eventually fall below the shallower one. The crossover — matter–radiation equality — lands at $z_{\mathrm{eq}}\approx 3400$ , and below it the Friedmann source term is matter. Once matter is in charge the dynamics collapse into something solvable in a line. A single perfect fluid with $w=0$ , fed through the conservation law and the flat Friedmann equation, yields the power law $a(t)\propto t^{2/3}$ : an expansion that grows without bound yet always decelerates, with $q=+\tfrac12$ . This same era is the workshop of cosmic structure. Because pressureless matter has nothing to resist its own gravity, every faint overdensity grows — the density contrast tracks the scale factor, $\delta\propto a$ , far more efficiently than the logarithmic crawl radiation allowed, turning a nearly smooth early universe into galaxies, clusters, and the cosmic web. The chapter does not last forever: dark energy, whose density never dilutes, overtakes matter near $z\approx 0.3$ and flips deceleration into acceleration, so the tidy $t^{2/3}$ law is excellent for $1\lesssim z\lesssim 1000$ but must yield to full $\Lambda$ CDM near today.

Key takeaways

The matter-dominated era is the epoch in which pressureless matter — cold dark matter plus baryons — governs the expansion, from matter–radiation equality at $z_{\mathrm{eq}}\approx 3400$ to dark-energy onset near $z\approx 0.3$ .
With $w=0$ the density dilutes as $\rho_m\propto a^{-3}$ , tracking a conserved particle number spread through a growing proper volume.
The flat Friedmann equation with that source gives $a(t)\propto t^{2/3}$ , so $H=2/(3t)$ and the expansion always decelerates, $q=+\tfrac12$ .
With no pressure to fight gravity the era grows structure: sub-horizon overdensities amplify as $\delta\propto a$ , turning CMB-level fluctuations into galaxies.
A flat matter-only model is only $t_0=2/(3H_0)\approx 9.3$ Gyr old — younger than the oldest stars — a classic pointer to dark energy and the failure of pure $t^{2/3}$ today.
Benchmarks: $\Omega_{m,0}\approx 0.31$ , $\Omega_{\Lambda,0}\approx 0.69$ , $z_{\mathrm{eq}}\approx 3400$ , recombination $z\approx 1090$ , matter– $\Lambda$ equality $z\approx 0.30$ .
Key idea: Pressureless matter ( $w=0$ ) dilutes as $\rho\propto a^{-3}$ ; substituting into the flat Friedmann equation gives $a\propto t^{2/3}$ , a decelerating expansion in which gravitational instability grows density perturbations linearly as $\delta\propto a$ .

Where this lives today

The matter era is the regime precision cosmology now measures to per-cent accuracy. Its key observable is the scale of matter–radiation equality: the matter power spectrum turns over at the comoving horizon size at $z_{\mathrm{eq}}$ , and that turnover — with the CMB acoustic peaks, whose heights depend on $\Omega_m h^2$ — is how Planck pins the matter density to better than a percent. The growing mode $\delta\propto a$ is a working tool too: the growth rate $f=\mathrm{d}\ln\delta/\mathrm{d}\ln a$ , measured through redshift-space distortions in surveys like DESI and Euclid, would flag modified gravity if it deviated. The end of the era is now mapped directly — baryon acoustic oscillations act as a standard ruler tracing the expansion through the deceleration-to-acceleration transition near $z\approx 0.6$ that the 1998 supernovae first glimpsed. And the era's tensions drive open problems: the " $S_8$ tension" between structure amplitudes from the CMB and from weak lensing may be a fluke, a systematic, or the first crack in the standard growth history.

The physical picture

Picture a fixed number of dust grains sprinkled on an expanding rubber sheet. Their count never changes, but the proper volume swells as $a^3$ , so the number density — and, because each grain's rest energy is constant, the energy density — falls as $a^{-3}$ . There is no pressure in this dust: the grains just coast. That single fact organizes the era. Gravity has free rein both to slow the global expansion, dragging the scale factor onto its decelerating $t^{2/3}$ track, and to pull overdense patches into structure. Contrast the radiation it replaced: a photon gas loses energy twice — volume dilution $a^{-3}$ plus per-photon redshift $a^{-1}$ — for a total $a^{-4}$ , so the steeper curve always loses the race given enough expansion. That is why matter's takeover is guaranteed, not accidental. The first figure is that race drawn out: three lines on a log plot, and the shaded gap between two crossings is the matter era itself.

Figure

Energy density of each cosmic component versus scale factor: radiation ρ ∝ a⁻⁴ falls fastest and matter ρ ∝ a⁻³ overtakes it at matter–radiation equality z_eq ≈ 3400, then matter meets the constant dark-energy density ρ_Λ at z_Λ ≈ 0.30 — the shaded band between the two crossings is the matter-dominated era

Because matter carries no pressure, an overdense region has nothing to hold it up: it feels only its own excess gravity and collapses, with no restoring force to set off the acoustic oscillations that ring through the photon–baryon plasma. This is why structure grows so readily once matter dominates — the amplification $\delta\propto a$ is free gravitational growth, checked only by the expanding background (Hubble drag). The deceleration parameter $q=+\tfrac12$ is the signature stamped on the era: gravity is winning, the expansion is slowing, and the day $q$ flips negative is the fingerprint that matter has handed the universe to dark energy. The second figure shows that growth law directly and reads off the age it predicts.

Figure

The matter-era scale factor a(t) = (t/t₀)^(2/3) climbs but bends over — a decelerating expansion with H = 2/(3t) and q = +½; the dashed line is how it would coast at today's rate, and the curve falling below it is the deceleration. A flat matter-only universe is only t₀ = 2/(3H₀) ≈ 9.3 Gyr old, too young for the oldest stars

The core equations

The microphysical starting point is the equation of state. A perfect fluid is specified by the ratio of its pressure to its energy density, and cosmological "matter" is the fluid for which that ratio vanishes — cold, slow particles whose kinetic pressure is negligible against their rest energy.

w=\frac{p}{\rho c^2}=0 \qquad (1)

where: $p$ — isotropic pressure of the cosmic fluid; $\rho$ — mass-energy density; $w$ — equation-of-state parameter, $w=0$ for pressureless matter (dust). Reading: matter gravitates through its density alone, never through pressure. Feed $w=0$ into covariant conservation of the stress-energy tensor — the FLRW continuity equation — and the density law follows immediately: a conserved number of particles diluted through a proper volume growing as $a^3$ .

\dot{\rho}_m+3H\rho_m=0\;\Longrightarrow\;\rho_m=\rho_{m,0}\,a^{-3} \qquad (2)

where: $H\equiv\dot a/a$ — Hubble parameter; $a$ — scale factor, normalized so $a_0=1$ today; $\rho_{m,0}$ — present-day matter density. Reading: with no source or sink of particles, density scales inversely with the growing volume. With the source known, the flat first Friedmann equation closes the dynamics. In the interior of the era it is dominated by the single term $\rho_m$ ; curvature and cosmological-constant terms are negligible.

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

where: $G$ — Newton's gravitational constant; $\dot a$ — time derivative of the scale factor. Reading: the square of the expansion rate tracks the instantaneous matter density. Combining (2) and (3) and integrating gives the signature result of the era — a power-law expansion — with the age of a flat matter-only model in terms of the present expansion rate.

a(t)=\left(\frac{t}{t_0}\right)^{2/3},\qquad t_0=\frac{2}{3H_0} \qquad (4)

where: $t_0$ — present age in a flat matter-only ("Einstein–de Sitter") model; $H_0$ — present Hubble parameter. Reading: the universe grows forever but ever more slowly, its age a fixed fraction of the Hubble time $1/H_0$ . Differentiating the power law gives the expansion rate and the deceleration parameter, a pure number in this era.

H=\frac{2}{3t},\qquad q\equiv-\frac{\ddot a\,a}{\dot a^2}=+\frac{1}{2} \qquad (5)

where: $q$ — deceleration parameter; $q>0$ signals a slowing expansion. Reading: $\ddot a<0$ throughout — the gravitational drag of pressureless matter. To locate where the era begins and ends we need the full expansion law with every component present. Each term dilutes at its own rate, and the largest at a given $a$ names the reigning era.

H^2(a)=H_0^2\left[\Omega_{r,0}\,a^{-4}+\Omega_{m,0}\,a^{-3}+\Omega_{\Lambda,0}\right] \qquad (6)

where: $\Omega_{r,0}\sim 9\times10^{-5}$ — present radiation density parameter; $\Omega_{m,0}\approx 0.31$ — present matter density parameter; $\Omega_{\Lambda,0}\approx 0.69$ — present dark-energy density parameter. Reading: the three terms cross over as $a$ grows, defining the radiation, matter, and dark-energy eras in turn. Setting the radiation and matter terms equal fixes the opening boundary of the era, the redshift of matter–radiation equality.

1+z_{\mathrm{eq}}=\frac{\Omega_{m,0}}{\Omega_{r,0}}\approx 3400 \qquad (7)

where: $z_{\mathrm{eq}}$ — redshift where matter and radiation densities are equal. Reading: below it the budget is matter, and the $t^{2/3}$ law takes over from the radiation era's $t^{1/2}$ . Finally, the era's defining activity — the growth of structure — obeys the linearized perturbation equation, whose growing solution is the whole point.

\ddot{\delta}+2H\dot{\delta}=4\pi G\rho_m\,\delta,\qquad \delta\propto a\propto t^{2/3} \qquad (8)

where: $\delta\equiv\delta\rho/\rho$ — fractional density contrast; $2H\dot\delta$ — Hubble-drag term opposing collapse; $4\pi G\rho_m\delta$ — self-gravity driving growth. Reading: gravity beats drag, and the growing mode rises in lockstep with the scale factor. (The particle horizon here evaluates to $d_{\mathrm{p}}=a\int_0^t c\,\mathrm{d}t'/a=3ct$ .)

Derivation AFrom pressureless dust to the two-thirds power law

The goal is to travel from the microphysics to the observable expansion history: begin with a single perfect fluid of vanishing pressure, use covariant conservation to obtain its dilution law $\rho_m\propto a^{-3}$ , substitute that into the flat Friedmann equation, and integrate to recover the celebrated $a(t)\propto t^{2/3}$ together with the Hubble rate $H=2/(3t)$ , the deceleration parameter $q=+\tfrac12$ , and the age relation $t_0=2/(3H_0)$ . Everything downstream rests on this one chain, so each link is checked against a limit or a textbook value.

Step 1. Start from the FLRW continuity equation $\dot\rho+3H(\rho+p/c^2)=0$ — the first law of thermodynamics for an adiabatically expanding comoving volume — and set $p=0$ to get $\dot\rho_m+3H\rho_m=0$ . Separating variables gives $\mathrm{d}\rho_m/\rho_m=-3\,\mathrm{d}a/a$ , hence $\rho_m=\rho_{m,0}\,a^{-3}$ . Why: with no pressure doing work, the only way energy leaves a comoving cell is geometric dilution, so density scales as inverse proper volume.

Step 2. Insert the dilution law into the flat Friedmann equation and multiply by $a^2$ to obtain $\dot a^2\,a=(8\pi G/3)\rho_{m,0}\equiv C^2$ , a constant. Why: with a time-independent right-hand side the equation is a clean separable ODE whose only smooth solution is a power law.

Step 3. Try the ansatz $a=Bt^{n}$ . Then $\dot a^2 a=B^3 n^2 t^{3n-2}$ , and for this to equal the constant $C^2$ at all times we need the time exponent to vanish, $3n-2=0$ , so $n=\tfrac23$ . Why: matching powers of $t$ is the whole content of the solution — the exponent is forced by the geometry of dilution, not chosen, which is why every flat matter universe grows the same way.

Step 4. Fix the normalization by demanding $a(t_0)=1$ , giving $a(t)=(t/t_0)^{2/3}$ . Differentiating, $H=\dot a/a=2/(3t)$ , so today $H_0=2/(3t_0)$ , i.e. $t_0=2/(3H_0)$ . Why: the same power law ties the present age rigidly to the present expansion rate — no free parameter is left once the model is flat and matter-only.

Step 5. Compute the deceleration parameter: from $a\propto t^{2/3}$ , $\dot a\propto t^{-1/3}$ and $\ddot a\propto -\tfrac29 t^{-4/3}<0$ , so $q=-\ddot a a/\dot a^2=+\tfrac12$ . Why: $\ddot a<0$ says gravity decelerates the expansion, and its era-independent value $\tfrac12$ is the fingerprint later supernova data went looking for.

Step 6. Put in numbers: with $H_0\approx 70\ \mathrm{km\,s^{-1}\,Mpc^{-1}}$ , $1/H_0\approx 13.97$ Gyr, so $t_0=2/(3H_0)\approx 9.3$ Gyr. Why: memorably too small — younger than the oldest globular clusters — a historical fingerprint of dark energy.

Result.

a(t)=\left(\frac{t}{t_0}\right)^{2/3},\qquad H(t)=\frac{2}{3t},\qquad q=+\frac12,\qquad t_0=\frac{2}{3H_0}

Go deeper · Derivation B · 1 minThe linear growth of density perturbations

Derivation BThe linear growth of density perturbations

The goal is to show that the matter era is the epoch in which structure actually grows, by solving the linearized density-contrast equation and exhibiting its growing mode. We work in the Newtonian, sub-horizon regime — scales well inside the Hubble radius, where relativistic subtleties do not enter — reduce the equation to Euler form using the matter-era background, and read off the two modes. The punchline, $\delta\propto a$ , is what distinguishes efficient matter-era growth from the stalled growth of the radiation era.

Step 1. The linearized continuity, Euler, and Poisson equations for a self-gravitating pressureless fluid combine into $\ddot\delta+2H\dot\delta-4\pi G\rho_m\,\delta=0$ : a driven oscillator with friction. Why: the middle term $2H\dot\delta$ is Hubble drag and the last is self-gravity — structure formation is the contest between these two.

Step 2. Substitute the matter-era background: $H^2=(8\pi G/3)\rho_m$ gives $4\pi G\rho_m=\tfrac32 H^2$ , and with $H=2/(3t)$ this is $4\pi G\rho_m=2/(3t^2)$ , $2H=4/(3t)$ . Why: the $t^{2/3}$ background from the previous derivation makes the coefficients scale as $1/t$ and $1/t^2$ , turning an abstract equation into a solvable one.

Step 3. The equation becomes $\ddot\delta+\dfrac{4}{3t}\dot\delta-\dfrac{2}{3t^2}\delta=0$ , a second-order Euler equation. Trying $\delta\propto t^{p}$ gives the indicial equation $p(p-1)+\tfrac43 p-\tfrac23=0$ , i.e. $3p^2+p-2=0$ . Why: Euler equations are solved by power laws, so the problem reduces to a quadratic whose roots are the two independent modes.

Step 4. Solve: $3p^2+p-2=(3p-2)(p+1)=0$ , so $p=\tfrac23$ or $p=-1$ . The growing mode is $\delta_+\propto t^{2/3}\propto a$ ; the decaying mode $\delta_-\propto t^{-1}$ is soon irrelevant. Why: after any transient the growing mode dominates, and because it equals the scale factor, an overdensity at equality is amplified by the same factor the universe expands.

Step 5. Interpret $\delta\propto a$ . In the radiation era the same fluid barely grows — the Mészáros effect leaves at most logarithmic creep — because radiation does not cluster and its rapid expansion swamps self-gravity. Why: structure "switches on" at equality; the matter era is the window in which the seeds of galaxies are actually grown.

Result.

\ddot{\delta}+2H\dot{\delta}-4\pi G\rho_m\,\delta=0\;\Longrightarrow\;\delta_+\propto a\propto t^{2/3},\quad \delta_-\propto t^{-1}

Go deeper · Derivation C · 1 minLocating the boundaries — equality, acceleration, and the end

Derivation CLocating the boundaries — equality, acceleration, and the end

The goal is to pin the two ends of the matter era to explicit redshifts, so the era is a bracketed interval, not a vague adjective. We equate pairs of density terms in the full Friedmann sum: radiation with matter for the opening boundary, matter with the cosmological constant for where dark energy catches up, and — because acceleration begins before densities are even equal — the condition $\ddot a=0$ for where the expansion stops slowing. Each redshift is evaluated with Planck density parameters.

Step 1. Radiation and matter dilute as $a^{-4}$ and $a^{-3}$ , so $\rho_r/\rho_m=(\Omega_{r,0}/\Omega_{m,0})a^{-1}$ . Setting this to unity gives $1+z_{\mathrm{eq}}=\Omega_{m,0}/\Omega_{r,0}\approx 0.31/9\times10^{-5}\approx 3400$ . Why: the opening whistle of the era — below $z_{\mathrm{eq}}$ matter outweighs radiation and $t^{2/3}$ replaces the radiation era's $t^{1/2}$ .

Step 2. For the closing boundary, $\rho_\Lambda$ is constant while $\rho_m\propto a^{-3}$ , so $\rho_\Lambda/\rho_m=(\Omega_{\Lambda,0}/\Omega_{m,0})a^{3}$ . The densities match at $1+z_\Lambda=(\Omega_{\Lambda,0}/\Omega_{m,0})^{1/3}\approx(0.69/0.31)^{1/3}\approx 1.30$ , i.e. $z_\Lambda\approx 0.30$ . Why: note the exponent is $+\tfrac13$ , not $-\tfrac13$ , which would nonsensically give a negative redshift.

Step 3. Acceleration begins earlier than density equality. The acceleration equation changes sign when $\rho+3p=0$ ; with $p_\Lambda=-\rho_\Lambda c^2$ this is $\rho_m=2\rho_\Lambda$ , giving $1+z_{\mathrm{acc}}=(2\Omega_{\Lambda,0}/\Omega_{m,0})^{1/3}\approx 1.63$ , so $z_{\mathrm{acc}}\approx 0.6$ . Why: dark energy's negative pressure counts three times in the source, so the universe accelerates while matter still outweighs $\Lambda$ two-to-one — the transition supernovae detected near $z\approx 0.6$ .

Result.

1+z_{\mathrm{eq}}=\frac{\Omega_{m,0}}{\Omega_{r,0}}\approx 3400,\qquad 1+z_{\mathrm{acc}}=\left(\frac{2\Omega_{\Lambda,0}}{\Omega_{m,0}}\right)^{1/3}\approx 1.6,\qquad 1+z_\Lambda=\left(\frac{\Omega_{\Lambda,0}}{\Omega_{m,0}}\right)^{1/3}\approx 1.3

With Planck 2018 parameters $\Omega_{m,0}=0.315$ , $\Omega_{r,0}=9.2\times10^{-5}$ , $\Omega_{\Lambda,0}=0.685$ , the three boundary redshifts evaluate to $z_{\mathrm{eq}}\approx 3.4\times10^3$ , $z_{\mathrm{acc}}\approx 0.63$ , and $z_\Lambda\approx 0.30$ .

Limiting cases

How the matter-era results connect to the regimes on either side, and to the sub-horizon world of structure:

Case	Result	Meaning
Early limit (toward equality)	$\rho_r/\rho_m=a_{\mathrm{eq}}/a=(1+z)/(1+z_{\mathrm{eq}})$	As $z\to z_{\mathrm{eq}}\approx 3400$ radiation becomes comparable; the pure $t^{2/3}$ law is corrected and matches smoothly onto the radiation era's $a\propto t^{1/2}$ .
Late limit (toward dark energy)	$\rho_\Lambda/\rho_m=(a/a_\Lambda)^3,\ \ 1+z_\Lambda=(\Omega_\Lambda/\Omega_m)^{1/3}\approx 1.30$	As $a$ grows, the constant $\rho_\Lambda$ overtakes $\rho_m\propto a^{-3}$ near $z\approx 0.30$ ; thereafter $a\propto e^{Ht}$ replaces $t^{2/3}$ and the expansion accelerates.
Curvature limit	$\lvert\Omega_k\rvert/\Omega_m\propto a$	A spatial-curvature term scales as $a^{-2}$ , diluting more slowly than matter ( $a^{-3}$ ); in an open or closed matter universe curvature would eventually dominate — but observations give $\Omega_k\approx 0$ .
Newtonian / sub-horizon limit	$\ddot\delta+2H\dot\delta-4\pi G\rho_m\,\delta=0$	For perturbations well inside the horizon the relativistic growth equation reduces to its Newtonian form, with growing solution $\delta\propto a\propto t^{2/3}$ .

Common trapThe era is not a single event, and t^(2/3) is not exact today

Two traps. First, do not equate the matter-dominated dynamical era with the post-recombination epoch. Matter domination begins at equality, $z_{\mathrm{eq}}\approx 3400$ , when matter first outweighs radiation; recombination — protons and electrons forming neutral hydrogen, releasing the CMB — happens much later at $z\approx 1090$ . Both sit inside the era, but they are distinct: the universe is matter-dominated for a factor of three in redshift before it turns transparent. Second, $a\propto t^{2/3}$ is exact only for a flat, single-component matter universe. In real $\Lambda$ CDM the cosmological constant overtakes matter near $z\approx 0.3$ and the expansion accelerates from $z\approx 0.6$ , so the pure power law fails near today even though it is excellent for $1\lesssim z\lesssim 1000$ .

Worked examples

1 · The age of a flat matter-only universe (confidence builder)

Take a flat matter-only ("Einstein–de Sitter") universe with $H_0=70\ \mathrm{km\,s^{-1}\,Mpc^{-1}}$ . Find its present age, and compare with the oldest stars and the full $\Lambda$ CDM age.

Step 1. Convert $H_0$ to inverse seconds: with $1\ \mathrm{Mpc}=3.086\times10^{19}\ \mathrm{km}$ , $H_0=2.27\times10^{-18}\ \mathrm{s^{-1}}$ , so $1/H_0=4.41\times10^{17}\ \mathrm{s}$ . Dividing by $3.156\times10^{16}\ \mathrm{s\,Gyr^{-1}}$ gives $1/H_0=\mathbf{13.97\ Gyr}$ .

Step 2. Apply the age relation (4): $t_0=\tfrac23(1/H_0)=\tfrac23\times 13.97=\mathbf{9.31\ Gyr}$ . The two-thirds factor is the whole story — a decelerating universe reaches a given size sooner than one coasting at the present rate, so it is younger than the naive Hubble time.

Step 3. Interpret. The oldest globular-cluster stars are dated at $\gtrsim 12$ –13 Gyr, so a $9.3$ -Gyr universe would be younger than the stars it contains. The full $\Lambda$ CDM integral $t_0=\int_0^1 \mathrm{d}a/(aH)$ with $\Omega_m=0.31$ , $\Omega_\Lambda=0.69$ instead gives $\approx 13.8$ Gyr. The pure matter model gives 9.3 Gyr; the deficit of about 4 Gyr is a classic quantitative pointer to dark energy.

2 · Equality redshift and how much structure grows

Using $\Omega_{m,0}=0.315$ and $\Omega_{r,0}=9.2\times10^{-5}$ , find the equality redshift. Then, treating growth as Einstein–de Sitter ( $\delta\propto a$ ), estimate the factor by which a linear overdensity at equality is amplified by today.

Step 1. Equality is where the two dilution laws cross, equation (7): $1+z_{\mathrm{eq}}=\Omega_{m,0}/\Omega_{r,0}=0.315/(9.2\times10^{-5})=3424$ . So $z_{\mathrm{eq}}\approx\mathbf{3400}$ and the corresponding scale factor is $a_{\mathrm{eq}}=1/3424=\mathbf{2.9\times10^{-4}}$ .

Step 2. The growing mode tracks the scale factor, $\delta\propto a$ , so the amplification from equality to now is the ratio of scale factors: $\delta_0/\delta_{\mathrm{eq}}=a_0/a_{\mathrm{eq}}=1+z_{\mathrm{eq}}\approx\mathbf{3400}$ . A seed of amplitude $\delta_{\mathrm{eq}}\sim10^{-5}$ (CMB level) reaches $\delta\sim3\times10^{-2}$ — still linear, but on its way to galaxy-forming collapse.

Step 3. A caveat. The pure- $\delta\propto a$ estimate overcounts: late-time dark energy stalls growth, so after $z\approx 0.3$ the growing mode falls behind $a$ and the actual growth is suppressed by roughly $20$ – $25\%$ . Growth over the era is about $\times 3400$ in Einstein–de Sitter, reduced to about $\times 2600$ once dark-energy suppression is included.

Physics of the portrait

The matter era carries not a discoverer's name but a model's name. The flat, matter-only universe that solves to $a\propto t^{2/3}$ is the Einstein–de Sitter universe — for six decades simply what most cosmologists meant by "the universe."

Origins · 1 minEinstein–de Sitter universe and how this was discovered

Einstein–de Sitter universe

the flat matter-only cosmology a ∝ t^(2/3), proposed 1932 · relativistic cosmology and structure formation

The Einstein–de Sitter universe is the simplest expanding cosmology Einstein's field equations allow: spatially flat, filled only with pressureless matter, no curvature and no cosmological constant. Its history runs through the founders of relativistic cosmology. Alexander Friedmann wrote down the expanding solutions in 1922, and Georges Lemaître rediscovered them in 1927 and tied them to the recession of galaxies — yet Einstein resisted a dynamical universe for years, having introduced the cosmological constant precisely to keep it static. When Hubble's 1929 data made expansion undeniable, Einstein abandoned $\Lambda$ , and in 1932 he and Willem de Sitter published a short, decisive note: with expansion established there was no need for curvature or a cosmological term, and the data fit the flat, matter-filled model that now bears both names. Its virtue is its poverty of parameters — fix $H_0$ and everything follows: $q=+\tfrac12$ , the age $t_0=2/(3H_0)$ , and the growth law $\delta\propto a$ that made it the default backdrop for structure-formation theory through the 1970s and 1980s. Its downfall was equally clean: it predicts an age younger than the oldest stars and a decelerating expansion, and by 1998 the Type Ia supernova surveys had measured $q_0<0$ instead. It was retired as the description of today, but survives as the exact description of the long matter era that shaped almost everything we see.

History

1922

— Friedmann finds the expanding-universe solutions of Einstein's field equations

1927

— Lemaître connects cosmic expansion to the recession of galaxies

1932

— Einstein and de Sitter publish the flat matter-only universe, a ∝ t^(2/3)

1970

— Peebles formalizes gravitational growth of structure, δ ∝ a in the matter era

1998

— supernova surveys measure q₀ < 0, ending matter-era deceleration

Going further

Texts. Misner, Thorne & Wheeler, Gravitation (1973), Ch. 27 · Wald, General Relativity (1984), Ch. 5 · Carroll, Spacetime and Geometry (2004), Ch. 8 · Hartle, Gravity (2003), Ch. 18 · Schutz, A First Course in General Relativity (2009), Ch. 12 · Dodelson, Modern Cosmology (2003), Ch. 2, 7 · Peebles, Principles of Physical Cosmology (1993). All results follow from standard flat-FLRW cosmology. The continuity equation with $w=0$ gives $\rho_m\propto a^{-3}$ ; the flat Friedmann equation integrates to $a\propto t^{2/3}$ , $H=2/(3t)$ , $q=+\tfrac12$ , $t_0=2/(3H_0)$ ; and the growth indicial equation $3p^2+p-2=0$ has roots $p=\tfrac23,-1$ . Density-parameter anchors $\Omega_{m,0}\approx 0.31$ , $\Omega_{\Lambda,0}\approx 0.69$ , $\Omega_{r,0}\approx 9\times10^{-5}$ , $z_{\mathrm{eq}}\approx 3400$ , $z_{\mathrm{rec}}\approx 1090$ are from Planck 2018 (A&A 641, A6, 2020). The late-time markers evaluate to $z_{\mathrm{acc}}\approx 0.63$ from $\rho_m=2\rho_\Lambda$ and $z_\Lambda\approx 0.30$ from $1+z_\Lambda=(\Omega_\Lambda/\Omega_m)^{1/3}\approx 1.30$ (note the exponent is $+\tfrac13$ ), and the matter-only age is $t_0\approx 9.3$ Gyr from $1/H_0\approx 14.0$ Gyr.

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

Ask the tutor: Ask me to walk through why a decelerating $t^{2/3}$ universe grows structure as $\delta\propto a$ , or to trace exactly where and why the real universe departs from the Einstein–de Sitter law.

Why does matter dilute as a⁻³ but radiation as a⁻⁴?

Both thin out by the same geometric factor $a^{-3}$ : a fixed number of particles or photons spread through a proper volume growing as $a^3$ . Radiation then suffers one extra power of $a^{-1}$ because every photon's energy is itself redshifted, $E\propto 1/a$ . Matter's rest energy does not redshift — a proton weighs the same today as at equality — so matter keeps only the volume factor. That one extra power guarantees the steeper radiation curve eventually drops below the matter curve, making matter's takeover inevitable.

Why is the expansion decelerating during this era?

Because the only thing gravitating is pressureless matter, and ordinary matter attracts. The acceleration equation $\ddot a/a=-\tfrac{4\pi G}{3}(\rho+3p/c^2)$ has a strictly negative right-hand side when $p=0$ , so $\ddot a<0$ always; concretely $a\propto t^{2/3}$ gives $\ddot a\propto -t^{-4/3}<0$ , summarized by $q=+\tfrac12$ . The expansion is like a ball thrown upward: still rising, but ever more slowly. Only a component with sufficiently negative pressure — dark energy, $p<-\tfrac13\rho c^2$ — can flip the sign and accelerate the expansion, as $\Lambda$ does near $z\approx 0.6$ .

What is the difference between matter–radiation equality and recombination?

They are two separate events, both inside the matter era, and confusing them is common. Equality, at $z_{\mathrm{eq}}\approx 3400$ , is a dynamical milestone: matter density first exceeds radiation density, so the Friedmann source — and the expansion law — becomes matter-dominated. Recombination, at $z\approx 1090$ , is an atomic milestone: the plasma cools enough to form neutral hydrogen, the universe turns transparent, and the CMB is released. Recombination happens well after equality, when the universe is already firmly matter-dominated. Structure growth switches on at equality; the CMB photons we see were freed at recombination.

Why does structure grow so much faster in the matter era than the radiation era?

It comes down to what gravitates and how fast the background expands. In the radiation era the dominant component does not clump, and the rapid expansion drags growth to a near halt — matter perturbations grow at most logarithmically, the Mészáros effect. Once matter dominates, self-gravity $4\pi G\rho_m\delta$ finally beats Hubble drag, and the density contrast grows in lockstep with the scale factor, $\delta\propto a$ . Over the era that is amplification by a factor of thousands — enough to turn the $10^{-5}$ CMB ripples into the seeds of collapsed galaxies and clusters.

Why is a flat matter-only universe too young, and what does that tell us?

The age relation is $t_0=2/(3H_0)$ . With $H_0\approx 70\ \mathrm{km\,s^{-1}\,Mpc^{-1}}$ the Hubble time is about $14$ Gyr, so the two-thirds factor gives $t_0\approx 9.3$ Gyr. But the oldest globular-cluster stars are dated at $\gtrsim 12$ Gyr — older than the universe that contains them, which is impossible. The resolution is that the universe is not pure matter today: a cosmological constant makes the recent expansion accelerate, and the full $\Lambda$ CDM age integral gives $\approx 13.8$ Gyr. The "age crisis" of the 1990s was one of several arguments pointing toward dark energy before the supernovae clinched it.

Is the Einstein–de Sitter model wrong, or still useful?

Both, depending on what you ask. As a description of today it is wrong: it predicts deceleration and too small an age, whereas we observe acceleration since $z\approx 0.6$ and an age near $13.8$ Gyr. But as a description of the long matter era — roughly $1\lesssim z\lesssim 1000$ , most of cosmic time — it is an excellent approximation, because there dark energy and radiation are both negligible and the universe really is a flat bath of pressureless matter. Every structure-formation calculation still uses $\delta\propto a$ and $a\propto t^{2/3}$ as the backbone for that epoch, adding the dark-energy correction on top. It is retired as a cosmology, not as a tool.

The Matter-Dominated Era

Where this lives today

The physical picture

The core equations

Limiting cases

Worked examples

1 · The age of a flat matter-only universe (confidence builder)

2 · Equality redshift and how much structure grows

Physics of the portrait

Going further

Before this

Where this leads