Critical Density and the Density Parameters

Cosmology answers an audacious question with a single number: how much stuff is there in the universe, and is it enough to bend space shut? The Friedmann equation supplies the yardstick. Written as a balance between the expansion rate and the self-gravity of the cosmic fluid, it names one particular density — the critical density $\rho_c=3H^2/8\pi G$ — at which the two exactly cancel and space is left perfectly flat. It is the cosmic analogue of escape velocity: below it the expansion is bound and the geometry closes; above it the geometry stays open. Crucially, $\rho_c$ is not a constant of nature: built from the instantaneous Hubble rate $H(t)$ , it was enormous in the early universe and is a mere five protons per cubic metre today. Dividing every actual density by this yardstick turns an equation about kilograms per cubic metre into one about pure fractions. Those fractions are the density parameters $\Omega_i=\rho_i/\rho_c$ , the working vocabulary of modern cosmology: matter, radiation and dark energy each report as a share of unity, and a single curvature term $\Omega_k$ absorbs whatever is left over. The Friedmann equation collapses to one sum rule, $\Omega_m+\Omega_r+\Omega_\Lambda+\Omega_k=1$ , valid at every epoch. Because the components dilute at different rates as the universe swells — matter as $a^{-3}$ , radiation as $a^{-4}$ , the cosmological constant not at all — the shares reshuffle over time, carving cosmic history into a radiation era, a matter era, and the dark-energy era we have just entered. This article builds $\rho_c$ and the $\Omega_i$ from the Friedmann equation, proves the sum rule, assembles the dimensionless expansion history $H(z)$ , and reads the fate of the cosmos straight off the numbers.

Key takeaways

The critical density $\rho_c=3H^2/(8\pi G)$ is the energy density that makes an FLRW universe spatially flat ( $k=0$ ); it tracks $H(t)$ and so is not a fixed constant.
Each component's density in units of $\rho_c$ defines a dimensionless density parameter $\Omega_i=\rho_i/\rho_c$ , the fraction of the closure density that species contributes.
Dividing the Friedmann equation by $H^2$ turns it into the exact sum rule $\Omega_m+\Omega_r+\Omega_\Lambda+\Omega_k=1$ , where $\Omega_k=-kc^2/(a^2H^2)$ carries spatial curvature with the opposite sign to $k$ .
Observation gives a nearly flat universe: $\Omega_{m,0}\approx0.31$ , $\Omega_{\Lambda,0}\approx0.69$ , $\Omega_{r,0}\approx9\times10^{-5}$ , $|\Omega_{k,0}|<0.005$ .
Different dilution laws pack into the dimensionless Friedmann equation $E(z)^2=\Omega_{r,0}(1+z)^4+\Omega_{m,0}(1+z)^3+\Omega_{k,0}(1+z)^2+\Omega_{\Lambda,0}$ , which fixes the whole expansion history and the eras within it.
Today $\rho_{c,0}\approx8.5\times10^{-27}\ \mathrm{kg\,m^{-3}}$ — about five hydrogen atoms per cubic metre, dilute beyond any laboratory vacuum yet decisive for the geometry and fate of everything.
Key idea: the Friedmann equation says the total density measured in units of the critical density, plus a curvature term, must equal one — $\sum_i\Omega_i+\Omega_k=1$ — and spatial flatness is just the special case $\Omega_{\mathrm{tot}}=1$ .

The core insight

The Friedmann equation is the statement that the total density measured in units of the critical density, plus a curvature term, must equal one: $\sum_i\Omega_i+\Omega_k=1$ . Flatness is the special case $\Omega_{\mathrm{tot}}=1$ .

Where this lives today

The density parameters are the output variables of every precision-cosmology campaign. The Planck satellite, folding CMB temperature and polarization with baryon acoustic oscillations, delivers the canonical $\Lambda$ CDM inventory: $\Omega_{m,0}\approx0.315$ (of which baryons are only $\Omega_{b,0}\approx0.049$ , the rest cold dark matter), $\Omega_{\Lambda,0}\approx0.685$ , $\Omega_{r,0}\approx9\times10^{-5}$ , and a curvature share consistent with zero, $|\Omega_{k,0}|<0.005$ — the empirical statement that space is flat to better than one percent. That flatness is exactly what cosmic inflation predicts, which is why $\Omega_{\mathrm{tot}}=1$ is read as a signature of the inflationary epoch rather than a coincidence. The same $E(z)$ built from these numbers fixes the luminosity distances of type Ia supernovae, the CMB acoustic scale, and the growth of large-scale structure. And the parameters host cosmology's sharpest tension: the Hubble tension, a roughly $5\sigma$ disagreement between the $H_0$ inferred from the early-universe $\Omega_i$ and the value measured locally from the distance ladder, makes the critical density — through its $H_0^2$ dependence — the focus of the field's most active open problem.

The physical picture

Treat a large comoving ball of the cosmic fluid as a Newtonian system and the Friedmann equation reads like conservation of energy for a shell on its surface. The Hubble expansion supplies an outward "kinetic" term; the enclosed mass-energy pulls back with a "potential" term; and the spatial curvature constant $k$ is exactly the total energy left over. There is one density at which the two terms balance so precisely that the leftover vanishes: the critical density. It is the boundary between a bound orbit (the ball reaches a maximum size and falls back — a closed universe) and an unbound one (it coasts to infinity — an open universe), the cosmic twin of the escape-velocity threshold. The first figure draws those three fates.

Figure

Three fates of a matter-only universe set by the critical density: above ρ_c the geometry is closed (Ω_tot greater than 1) and the expansion reverses into a big crunch, at exactly ρ_c ≈ 8.5×10⁻²⁷ kg m⁻³ the geometry is flat (Ω_tot = 1) and coasts to zero rate, below ρ_c it is open (Ω_tot less than 1) and expands forever

Now divide the whole balance by $H^2$ . Densities become fractions of the critical density, and the shares of matter, radiation and dark energy, plus a curvature share, must sum to exactly one. This is more than bookkeeping: each share is measured by a different observation — galaxy clustering and lensing for $\Omega_m$ , the CMB acoustic peaks for $\Omega_b$ and $\Omega_k$ , distant supernovae for $\Omega_\Lambda$ — so their agreement, and their summing to unity, is a stringent test that the whole $\Lambda$ CDM picture hangs together. Because the components thin out at different rates, the shares are not fixed: run the clock backward and radiation, diluting fastest, wins; run it forward and the undiluting cosmological constant takes over. The second figure makes those crossovers quantitative.

Figure

How the three densities dilute with expansion in units of today's critical density: radiation ∝ (1+z)⁴ in red, matter ∝ (1+z)³ in amber, and the cosmological constant stays flat in green, so radiation ruled before matter–radiation equality at z ≈ 3400, matter ruled until dark energy overtook it near z ≈ 0.3, and today Ω_m,0 ≈ 0.315, Ω_Λ,0 ≈ 0.685, Ω_r,0 ≈ 9×10⁻⁵

The core equations

The starting point is the first Friedmann equation. Demanding spatial flatness — setting the curvature constant $k$ to zero — isolates one density, and that density is the critical density. It is the density that a flat universe with expansion rate $H$ must have; nothing more, nothing fundamental.

\rho_c(t)=\frac{3H^2(t)}{8\pi G} \qquad (1)

where: $\rho_c$ — critical energy density (usually quoted as an equivalent mass density); $H=\dot a/a$ — Hubble parameter; $G$ — Newton's constant. Reading: because $\rho_c\propto H^2$ and $H$ falls over cosmic history, the closure density was vast in the early universe and is tiny now — it is a moving target, not a fixed benchmark. Rather than carry densities in SI units, cosmology measures every component against $\rho_c$ . The ratio is the density parameter, a pure number that says what fraction of the closure density a given species supplies.

\Omega_i\equiv\frac{\rho_i}{\rho_c}=\frac{8\pi G\,\rho_i}{3H^2} \qquad (2)

where: $\Omega_i$ — dimensionless density parameter of component $i$ ; $\rho_i$ — its energy (mass) density. Reading: $\Omega_i>1$ for a species means it alone would over-close the universe; the observed shares are all fractions that must be added together. Spatial curvature is folded into the same language by defining a curvature parameter. It is not a physical fluid but a bookkeeping term that measures how far the geometry departs from flat. Note the sign: it is defined with the opposite sign to the curvature constant $k$ .

\Omega_k\equiv-\frac{k c^2}{a^2 H^2} \qquad (3)

where: $\Omega_k$ — effective curvature density parameter; $k$ — FLRW curvature constant ( $-1,0,+1$ ); $c$ — speed of light. Reading: positive spatial curvature ( $k>0$ , a closed universe) gives $\Omega_k<0$ , so a deficit of curvature share means an excess of matter-energy share. With curvature in the same units, the entire first Friedmann equation collapses to a single constraint that holds at every instant of cosmic history. This is the master relation of the article.

\Omega_m+\Omega_r+\Omega_\Lambda+\Omega_k=1 \qquad (4)

where: $\Omega_m$ — matter (baryonic plus cold dark) share; $\Omega_r$ — radiation (photons plus relativistic neutrinos) share; $\Omega_\Lambda$ — dark-energy or cosmological-constant share. Reading: the four shares are not independent — measure any three and the fourth is fixed. Collecting the genuine matter-energy contributions (everything but curvature) gives the total density parameter, whose distance from one is a direct readout of the curvature.

\Omega_{\mathrm{tot}}\equiv\Omega_m+\Omega_r+\Omega_\Lambda=1-\Omega_k \qquad (5)

where: $\Omega_{\mathrm{tot}}$ — sum of all non-curvature shares. Reading: $\Omega_{\mathrm{tot}}=1$ is flat, $>1$ is closed, $<1$ is open — the geometry is decided by whether the real density exceeds the critical density. When dark energy is modelled as a cosmological constant $\Lambda$ , its share can be written directly from $\Lambda$ , showing that $\Omega_\Lambda$ inherits the same $H^{-2}$ dependence as the critical density itself.

\Omega_\Lambda=\frac{\Lambda c^2}{3H^2} \qquad (6)

where: $\Lambda$ — cosmological constant (dimensions of inverse length squared). Reading: because $\rho_\Lambda=\Lambda c^2/8\pi G$ is fixed while $\rho_c$ falls, $\Omega_\Lambda$ climbs toward one as the universe ages. Feeding the dilution laws of each component into the Friedmann equation and normalising by the present rate $H_0$ produces the dimensionless expansion history — the single most-used equation in observational cosmology, the function fits everything from supernovae to the CMB.

\left(\frac{H}{H_0}\right)^2=\Omega_{r,0}(1+z)^4+\Omega_{m,0}(1+z)^3+\Omega_{k,0}(1+z)^2+\Omega_{\Lambda,0} \qquad (7)

where: $H_0$ — present Hubble constant; $z$ — redshift with $1+z=a_0/a$ ; $\Omega_{i,0}$ — present-day value of each share. Reading: each power of $(1+z)$ is a fingerprint — the term with the steepest power dominates at high redshift, which is why radiation ruled the early universe. Finally, evaluating equation (1) today with the measured Hubble constant gives the famous number that anchors the whole scheme.

\rho_{c,0}=\frac{3H_0^2}{8\pi G}\approx1.878\times10^{-26}\,h^2\ \mathrm{kg\,m^{-3}} \qquad (8)

where: $h$ — reduced Hubble constant, $H_0=100\,h\ \mathrm{km\,s^{-1}\,Mpc^{-1}}$ . Reading: with $h\approx0.67$ this is about $8.5\times10^{-27}\ \mathrm{kg\,m^{-3}}$ , some five protons per cubic metre — the density that decides the geometry of the cosmos.

Derivation ACritical density from the cosmic energy balance

The goal is to show that the critical density is not an arbitrary definition but the unique density at which a spatially flat universe is self-consistent, and to expose the escape-velocity analogy that gives it physical meaning. We reach the same $\rho_c=3H^2/8\pi G$ by two independent routes — a Newtonian energy argument on a comoving sphere and the flatness condition on the relativistic Friedmann equation — and confirm they agree, which is why the Newtonian picture is a trustworthy guide to the background dynamics.

Step 1. Take a sphere of comoving radius carrying physical radius $R=a\,x$ and uniform density $\rho$ ; a test shell on its surface recedes at $v=HR$ and encloses mass $M=\tfrac43\pi R^3\rho$ . Newtonian energy conservation for the shell reads $\tfrac12 v^2-\dfrac{GM}{R}=E$ , a constant. Why: by Birkhoff's theorem only the enclosed mass acts, so a local Newtonian ball faithfully mimics the global dynamics of a homogeneous universe.

Step 2. Substitute $v=HR$ and $M=\tfrac43\pi R^3\rho$ to get $\tfrac12 H^2R^2-\tfrac43\pi G\rho R^2=E$ , then divide by $\tfrac12 R^2$ to obtain $H^2=\tfrac{8\pi G}{3}\rho+\dfrac{2E}{R^2}$ . Why: the $2E/R^2$ term is precisely the relativistic curvature term $-kc^2/a^2$ in disguise — a bound shell ( $E<0$ ) matches a closed universe ( $k>0$ ).

Step 3. Demand the leftover energy vanish, $E=0$ , the marginal case dividing bound from unbound. Then $H^2=\tfrac{8\pi G}{3}\rho$ , and the density that satisfies it is $\rho_c=\dfrac{3H^2}{8\pi G}$ . Why: $E=0$ is exactly the escape-velocity condition — the density at which the expansion coasts to a halt at infinity with nothing to spare, which is the flat case.

Step 4. Confirm from relativity: the first Friedmann equation is $H^2=\tfrac{8\pi G}{3}\rho-\dfrac{kc^2}{a^2}$ ; setting $k=0$ gives $H^2=\tfrac{8\pi G}{3}\rho_c$ , identical to Step 3. Why: the Newtonian energy constant $E$ and the relativistic curvature $k$ play the same role, so both derivations name the same $\rho_c$ — reassurance that the intuitive picture is exact for the background dynamics.

Go deeper · Derivation B · 1 minThe flatness sum rule from the Friedmann equation

Derivation BThe flatness sum rule from the Friedmann equation

The aim is to turn the dimensionful Friedmann equation into a dimensionless statement that the density shares plus a curvature share sum to one, and to make clear this holds at every epoch rather than just today. We divide through by $H^2$ , recognise each resulting term as a density parameter, and read off the constraint, which is the single most economical way to state the contents of the universe and the observational meaning of spatial flatness.

Step 1. Start from the first Friedmann equation with every fluid included, $H^2=\dfrac{8\pi G}{3}\rho_{\mathrm{tot}}-\dfrac{kc^2}{a^2}$ , where $\rho_{\mathrm{tot}}=\rho_m+\rho_r+\rho_\Lambda$ and the cosmological constant is folded in as $\rho_\Lambda=\Lambda c^2/8\pi G$ . Why: writing $\Lambda$ as a density lets it enter the same ledger as matter and radiation, so a single sum rule can carry all of them.

Step 2. Divide every term by $H^2$ : $1=\dfrac{8\pi G}{3H^2}\rho_{\mathrm{tot}}-\dfrac{kc^2}{a^2H^2}$ . Why: dividing by $H^2$ is the same as dividing by $\rho_c$ up to constants — it is the operation that converts kilograms per cubic metre into fractions of the closure density.

Step 3. Identify the pieces using the definitions $\Omega_i=\rho_i/\rho_c$ and $\Omega_k=-kc^2/(a^2H^2)$ , so that $\dfrac{8\pi G}{3H^2}\rho_{\mathrm{tot}}=\Omega_m+\Omega_r+\Omega_\Lambda$ and the curvature term is $\Omega_k$ . Why: each term is now a measurable, dimensionless share, and the equation is a constraint among observables rather than a differential equation.

Step 4. Collect the pieces to reach $\Omega_m+\Omega_r+\Omega_\Lambda+\Omega_k=1$ , valid at any time because $H$ , $a$ and every $\rho_i$ were evaluated at the same arbitrary epoch. Why: the sum rule is an identity, not a fit — flatness, $\Omega_k=0$ , is equivalent to $\Omega_{\mathrm{tot}}=1$ , i.e. the real density equalling the critical density.

Go deeper · Derivation C · 1 minThe dimensionless Friedmann equation and the eras of expansion

Derivation CThe dimensionless Friedmann equation and the eras of expansion

The purpose is to build the function $E(z)=H(z)/H_0$ that encodes the entire expansion history from the present-day density parameters, and to see why its terms scale with distinct powers of $(1+z)$ so that different components rule at different times. We insert the dilution laws into the Friedmann equation, normalise by $H_0^2$ , and read the powers — the calculation that organises cosmic history into radiation, matter and dark-energy eras.

Step 1. Write the Friedmann equation listing each component, $H^2=\dfrac{8\pi G}{3}(\rho_r+\rho_m+\rho_\Lambda)-\dfrac{kc^2}{a^2}$ . Why: keeping the species separate is essential because each dilutes differently, and it is that difference that drives the succession of eras.

Step 2. Insert the dilution laws from the continuity equation with equations of state $w_r=\tfrac13,\ w_m=0,\ w_\Lambda=-1$ , giving $\rho_r=\rho_{r,0}a^{-4}$ , $\rho_m=\rho_{m,0}a^{-3}$ , $\rho_\Lambda=\rho_{\Lambda,0}$ , and set $a_0=1$ so $a=1/(1+z)$ . Why: radiation carries an extra factor of $a^{-1}$ beyond simple volume dilution because expansion also redshifts each photon's energy, which is what makes its power steeper than matter's.

Step 3. Divide by $H_0^2$ and use $\rho_{i,0}/\rho_{c,0}=\Omega_{i,0}$ together with $-kc^2/(a_0^2H_0^2)=\Omega_{k,0}$ , where the curvature term inherits an $a^{-2}$ scaling; this yields $E(z)^2=\Omega_{r,0}(1+z)^4+\Omega_{m,0}(1+z)^3+\Omega_{k,0}(1+z)^2+\Omega_{\Lambda,0}$ . Why: the present-day shares are the only inputs, so a handful of measured numbers determines $H$ at every redshift — the reason the $\Omega_{i,0}$ are cosmology's central fitting parameters.

Step 4. Read the crossovers: radiation ( $\propto(1+z)^4$ ) overtakes matter ( $\propto(1+z)^3$ ) at high $z$ , while the constant $\Omega_{\Lambda,0}$ overtakes matter at low $z$ . Why: the steepest power always wins at large redshift and the flat term always wins in the far future, so the dominant term marches from radiation to matter to $\Lambda$ as the universe ages.

Limiting cases

How the density parameters behave in the characteristic regimes of cosmic history:

Case	Result	Meaning
Radiation-dominated early universe	$\Omega_r$ term $\propto(1+z)^4$ , $\ \rho_r\gg\rho_m,\rho_\Lambda$	Steepest dilution wins going back; radiation ruled until matter–radiation equality at $z_{eq}\approx3400$ .
Matter-dominated middle epoch	$\rho_m\propto(1+z)^3$ , $\ \Omega_m\to1$	Between equality and dark energy the geometry behaves as Einstein–de Sitter; structure grows in this era.
Dark-energy future	$\Omega_\Lambda\to1$ , $\ H\to H_0\sqrt{\Omega_{\Lambda,0}}$	As $a\to\infty$ the constant $\rho_\Lambda$ dominates; expansion approaches de Sitter with constant $H\approx56\ \mathrm{km\,s^{-1}\,Mpc^{-1}}$ .
Empty (Milne) universe	$\Omega_m=\Omega_r=\Omega_\Lambda=0$ , $\ \Omega_k=1$	Curvature alone balances expansion: $a\propto t$ , the coasting boundary case with $k<0$ .
Exactly flat, matter-only (Einstein–de Sitter)	$\Omega_m=1$ , all others zero, $a\propto t^{2/3}$	The critical universe of textbooks; expands forever but decelerates to zero rate — the knife-edge fate without $\Lambda$ .

Common trapGeometry alone does not fix the fate of the universe

It is tempting to read destiny straight off the curvature — closed recollapses, flat and open expand forever. That folklore is true only in a universe with no cosmological constant. Once $\Lambda>0$ is present, geometry and fate decouple: our universe is spatially flat ( $\Omega_{\mathrm{tot}}=1$ , $\Omega_k\approx0$ ) yet expands forever and in fact accelerates, because the decisive quantity is $\Omega_\Lambda$ , not $\Omega_k$ . Two more traps ride alongside. First, the critical density is not a constant of nature — built from $H(t)$ , it changes with time, and quoting " $\Omega_m\approx0.3$ " without the subscript zero (meaning today) hides that the shares evolve. Second, $\Omega_k$ carries the opposite sign to $k$ : a closed universe ( $k>0$ ) has $\Omega_k<0$ and $\Omega_{\mathrm{tot}}>1$ . Read fate from the dynamics — the full $E(z)$ — not from the label "flat".

Worked examples

1 · The present critical density: five protons per cubic metre (fully worked)

Evaluate $\rho_{c,0}=3H_0^2/(8\pi G)$ for $H_0=67.4\ \mathrm{km\,s^{-1}\,Mpc^{-1}}$ (Planck 2018) and express it in protons per cubic metre.

Step 1. Convert $H_0$ to SI using $1\ \mathrm{Mpc}=3.086\times10^{22}\ \mathrm{m}$ : $H_0=\dfrac{67.4\times10^3\ \mathrm{m\,s^{-1}}}{3.086\times10^{22}\ \mathrm{m}}=\mathbf{2.18\times10^{-18}\ s^{-1}}$ , the inverse Hubble time.

Step 2. Square it: $H_0^2=(2.18\times10^{-18})^2=\mathbf{4.77\times10^{-36}\ s^{-2}}$ .

Step 3. Insert into equation (1) with $G=6.674\times10^{-11}\ \mathrm{m^3\,kg^{-1}\,s^{-2}}$ : $\rho_{c,0}=\dfrac{3\times4.77\times10^{-36}}{8\pi\times6.674\times10^{-11}}=\dfrac{1.431\times10^{-35}}{1.677\times10^{-9}}=\mathbf{8.5\times10^{-27}\ kg\,m^{-3}}$ .

Step 4. Divide by the proton mass $m_p=1.673\times10^{-27}\ \mathrm{kg}$ : $\rho_{c,0}/m_p=8.5\times10^{-27}/1.673\times10^{-27}=\mathbf{5.1\ protons\ per\ m^3}$ , matching equation (8), $1.878\times10^{-26}\times0.674^2$ . The universe is critically dense at only about five hydrogen atoms per cubic metre — emptier than the best laboratory vacuum, yet enough to make space flat.

2 · Matter–radiation equality (fully worked)

Find the redshift at which matter and radiation contributed equally, given $\Omega_{m,0}=0.31$ and $\Omega_{r,0}=9.0\times10^{-5}$ .

Step 1. In the $E(z)^2$ expansion the two contributions are $\Omega_{m,0}(1+z)^3$ and $\Omega_{r,0}(1+z)^4$ ; the common $\rho_{c,0}$ cancels. Set them equal: $\Omega_{m,0}(1+z_{eq})^3=\Omega_{r,0}(1+z_{eq})^4$ .

Step 2. Cancel three powers of $(1+z)$ : $1+z_{eq}=\dfrac{\Omega_{m,0}}{\Omega_{r,0}}$ . Radiation carries one extra power, so exactly one factor survives.

Step 3. Plug in: $1+z_{eq}=\dfrac{0.31}{9.0\times10^{-5}}=\mathbf{3.4\times10^{3}}$ , so $z_{eq}\approx3400$ . The Planck 2018 value is $z_{eq}\approx3402$ , in agreement; note $\Omega_{r,0}$ must include relativistic neutrinos ( $\Omega_r=1.68\,\Omega_\gamma$ ) on top of the photon share $\Omega_\gamma\approx5.4\times10^{-5}$ , or $z_{eq}$ comes out too high. Radiation ruled the universe until $z\approx3400$ , roughly 50,000 years after the big bang.

3 · The onset of cosmic acceleration (fully worked)

A universe of matter plus a cosmological constant accelerates when $\rho_m<2\rho_\Lambda$ . Using $\Omega_{m,0}=0.315$ , $\Omega_{\Lambda,0}=0.685$ , find the redshift at which acceleration began, and contrast it with matter– $\Lambda$ equality.

Step 1. The acceleration equation $\ddot a/a=-\tfrac{4\pi G}{3}(\rho+3p)$ turns positive when $\rho+3p<0$ . With matter ( $p=0$ ) and $\Lambda$ ( $p=-\rho_\Lambda$ ), $\rho+3p=\rho_m-2\rho_\Lambda$ , so acceleration starts at $\rho_m=2\rho_\Lambda$ .

Step 2. In shares, $\Omega_{m,0}(1+z_{\mathrm{acc}})^3=2\,\Omega_{\Lambda,0}$ , hence $(1+z_{\mathrm{acc}})^3=\dfrac{2\times0.685}{0.315}=4.35$ and $1+z_{\mathrm{acc}}=4.35^{1/3}=1.63$ , giving $\mathbf{z_{\mathrm{acc}}\approx0.63}$ — acceleration switched on about 7 billion years ago.

Step 3. Contrast with matter– $\Lambda$ density equality, $\Omega_{m,0}(1+z)^3=\Omega_{\Lambda,0}$ : $(1+z)^3=0.685/0.315=2.17$ , giving $z\approx0.30$ . Dark energy became the dominant density near $z\approx0.3$ , but because its negative pressure counts three-fold in the source, it began accelerating the expansion earlier, at $z\approx0.6$ .

Physics of the portrait

Weighing the universe against $\rho_c$ turned "is space open or closed?" from metaphysics into an experiment — and the equation traces to one short-lived Russian mathematician.

Origins · 1 minAlexander Friedmann and how this was discovered

Alexander Friedmann

1888–1925 · the expanding-universe solutions and the density that divides cosmic fates

Alexander Friedmann was a St Petersburg mathematician who, in papers of 1922 and 1924, did what Einstein had refused to do: he took the field equations at face value and let the universe move. Dropping the static assumption, he found the full family of homogeneous, isotropic solutions — expanding, contracting and oscillating worlds — and with them the pivotal insight of this article: there is a special density, set by the expansion rate, that separates a universe destined to recollapse from one that expands forever. That threshold is the critical density, and the ratio of the real density to it is what we now call $\Omega$ . Einstein at first dismissed the work as an error, then printed a retraction admitting Friedmann was right. Friedmann never saw it vindicated; he died of typhoid in 1925 at thirty-seven, four years before Hubble measured the galactic recession his equations had anticipated. Georges Lemaître rediscovered the expanding solution in 1927 and tied it to observation, and in 1932 Einstein and de Sitter singled out the exactly-critical, spatially flat, matter-filled model that bears their names — the $\Omega=1$ universe that dominated theoretical cosmology for sixty years. Friedmann's equation, and the closure density buried inside it, remain the scaffolding on which every modern measurement of the cosmic inventory is hung.

History

1922

— Friedmann derives the expanding solutions and the density dividing eternal expansion from recollapse

1929

— Hubble's law fixes H, turning the critical density into a measurable quantity

1932

— Einstein and de Sitter model the flat, exactly-critical, matter-filled universe

1998

— supernova surveys find ΩΛ ≈ 0.7 and cosmic acceleration

2018

— Planck confirms spatial flatness to high precision, absolute Ωk below 0.005

Going further

Texts. Misner, Thorne & Wheeler, Gravitation (1973), §27 · Wald, General Relativity (1984), §5.2 · Carroll, Spacetime and Geometry (2004), §8.3 · Hartle, Gravity (2003), Ch. 18 · Schutz, A First Course in General Relativity, 2nd ed., Ch. 12 · Dodelson, Modern Cosmology, 2nd ed., Ch. 2 · Peebles, Principles of Physical Cosmology (1993), §5 · Planck 2018 results VI, A&A 641, A6 (2020). Key numerical checks: $\rho_{c,0}$ evaluated at $H_0=67.4\ \mathrm{km\,s^{-1}\,Mpc^{-1}}$ with $G=6.674\times10^{-11}$ gives $\approx8.5\times10^{-27}\ \mathrm{kg\,m^{-3}}$ , or $\approx5$ protons per cubic metre; matter–radiation equality follows from $z_{eq}=\Omega_{m,0}/\Omega_{r,0}-1\approx3400$ with $\Omega_{m,0}\approx0.31$ and $\Omega_{r,0}\approx9\times10^{-5}$ ; the onset of acceleration satisfies $(1+z_{\mathrm{acc}})^3=2\Omega_{\Lambda,0}/\Omega_{m,0}$ , giving $z_{\mathrm{acc}}\approx0.6$ ; and the sum rule is recovered by dividing the Friedmann equation through by $H^2$ . Numerical density parameters, $z_{eq}$ and the flatness bound are from Planck 2018 (A&A 641, A6).

Prerequisites. The Friedmann Equations · The FLRW Metric · The Scale Factor and Hubble Parameter · The Cosmological Fluid and Equation of State. Leads to. The Radiation-Dominated Era · The Matter-Dominated Era · The Dark-Energy-Dominated Era · The Cosmological Constant and Dark Energy · Cosmological Horizons.

Ask the tutor: Ask me to walk you through weighing the whole universe against $\rho_c$ — from the escape-velocity picture to why a flat cosmos still accelerates.

Is the critical density the density the universe actually has?

Only if space is exactly flat. The critical density is a reference value — the density a universe with the observed expansion rate would need in order to be geometrically flat. Whether the real density matches it is an experimental question, answered by the density parameter $\Omega_{\mathrm{tot}}=\rho/\rho_c$ . The remarkable finding is that $\Omega_{\mathrm{tot}}$ is one to within about a percent, so the real density does sit essentially at the critical value today — a measured result, not a definition. In a closed or open universe the actual density would be above or below $\rho_c$ respectively.

Why does the critical density change with time if it decides the geometry?

Because $\rho_c=3H^2/8\pi G$ is tied to the Hubble rate, which shrinks as the expansion decelerates (and, with dark energy, tends to a constant). In the early universe $H$ was huge, so $\rho_c$ was huge; today it is about five protons per cubic metre. What does not change is $\Omega_k$ : the sum rule $\Omega_{\mathrm{tot}}+\Omega_k=1$ holds at every epoch, so if space is flat now it was flat always, even though the real density and the critical density have fallen by many orders of magnitude in lockstep. Geometry is a ratio, and the ratio is conserved.

Does Ω = 1 mean the universe will expand forever or recollapse?

By itself it means neither — that is the crucial subtlety. In a universe made only of matter and radiation, $\Omega_{\mathrm{tot}}=1$ (flat) is the marginal case that expands forever but decelerates toward zero rate, while $\Omega_{\mathrm{tot}}>1$ recollapses. Add a cosmological constant and the link between geometry and destiny breaks: our universe is flat and accelerating, and will expand forever, because dark energy's negative pressure drives the expansion regardless of curvature. The fate is set by the full dynamics encoded in $E(z)$ — specifically by $\Omega_\Lambda$ — not by the geometric label alone.

What is the difference between Ωm and Ωb, and where is dark matter?

$\Omega_m$ is the total matter share — everything that dilutes as $a^{-3}$ and clusters gravitationally. It splits into ordinary baryonic matter, $\Omega_{b,0}\approx0.049$ , and cold dark matter, $\Omega_{c,0}\approx0.26$ . So of the $\Omega_{m,0}\approx0.31$ matter budget, only about a sixth is the atoms of stars, gas and planets; the rest is dark matter, seen through its gravity in galaxy rotation, lensing and the CMB peak heights but not through light. Dark energy, $\Omega_\Lambda\approx0.69$ , is not matter at all and sits outside $\Omega_m$ .

Why is radiation negligible today but dominant in the early universe?

Radiation dilutes faster than matter: its density falls as $a^{-4}$ against matter's $a^{-3}$ , because expansion not only spreads photons through a growing volume but also stretches each photon's wavelength, sapping its energy. Running the clock forward, that extra factor of $a$ drives $\Omega_r$ down to today's $\approx9\times10^{-5}$ , utterly subdominant. Running it backward, the same factor makes radiation climb fastest, so it overtook matter at $z_{eq}\approx3400$ and ruled everything earlier. The single extra power of $(1+z)$ in the $E(z)^2$ expansion is the whole story of why we live in a matter-and-dark-energy universe rather than a radiation one.

How do we actually measure Ωk to conclude the universe is flat?

The sharpest handle is the cosmic microwave background. The physical size of the sound horizon at recombination is known from early-universe physics, so its observed angular size on the sky depends on the geometry the light traversed: in a closed universe angles are magnified, in an open one shrunk, in a flat one preserved. The measured first acoustic peak sits almost exactly at the flat-universe prediction, giving $|\Omega_{k,0}|<0.005$ when combined with baryon acoustic oscillations. Independently, the sum rule infers $\Omega_k=1-\Omega_m-\Omega_\Lambda$ from supernovae and clustering; all routes agree that space is flat to sub-percent precision.