The Friedmann Equations

The Friedmann equations are where general relativity stops being a theory of local geometry and becomes a theory of the universe as a whole. Feed the FLRW metric — the unique spacetime allowed by the Cosmological Principle of homogeneity and isotropy — into Einstein's ten coupled field equations and symmetry strips away every degree of freedom except one: the entire cosmos, its geometry, history, and future, collapses onto a single function of time, the scale factor $a(t)$ , and the field equations reduce to just two independent ordinary differential equations for it. The first Friedmann equation (the $00$ -component) fixes the expansion rate $H=\dot a/a$ from the summed energy density plus a curvature term. The second, from the spatial components, carries a relativistic shock: pressure gravitates — not mass alone but the combination $\varepsilon+3P$ , so a component with sufficiently negative pressure pushes space apart. The third, the fluid equation, is local energy conservation $\nabla_\mu T^{\mu\nu}=0$ , not independent: the contracted Bianchi identity makes any two of the three imply the third. Close the system with an equation of state $P=w\varepsilon$ and each ingredient dilutes at its own rate as space stretches: radiation as $a^{-4}$ , matter as $a^{-3}$ , vacuum energy not at all. That single fact orders cosmic history into radiation, matter, and dark-energy eras and — recast in the density parameters $\Omega_i$ — turns the first equation into the master law $H^2(a)=H_0^2\sum_i\Omega_{i,0}\,a^{-3(1+w_i)}$ from which every distance, age, and horizon in $\Lambda$ CDM is computed.

Key takeaways

Impose the FLRW metric on Einstein's equations and the whole universe reduces to one function, the scale factor $a(t)$ ; two independent ODEs govern its evolution.
The first Friedmann equation is a cosmic energy budget setting the expansion rate: $H^2=\tfrac{8\pi G}{3c^2}\varepsilon-\tfrac{kc^2}{a^2}$ , with curvature $k$ playing the role of total energy.
The second (acceleration) equation, $\ddot a/a=-\tfrac{4\pi G}{3c^2}(\varepsilon+3P)$ , shows that pressure gravitates: negative pressure ( $P<-\varepsilon/3$ ) drives acceleration.
With $P=w\varepsilon$ each component scales as $\varepsilon\propto a^{-3(1+w)}$ , so radiation, matter, and $\Lambda$ dominate in turn and the flat solutions are $a\propto t^{1/2},\,t^{2/3},\,e^{Ht}$ .
The critical density $\varepsilon_c=3c^2H^2/8\pi G$ divides open, flat, and closed geometries and recasts the equations in the dimensionless $\Omega$ language of the $\Lambda$ CDM concordance model.
Key idea: impose the FLRW metric on Einstein's equations and the whole universe reduces to one degree of freedom, $a(t)$ ; the first Friedmann equation is an energy budget that sets the expansion rate, and the second reveals that energy density and pressure gravitate.

The core insight

Impose the FLRW metric on Einstein's equations and the whole universe reduces to one degree of freedom, $a(t)$ : the first Friedmann equation is an energy budget setting the expansion rate, and the second shows that energy density and pressure gravitate.

Where this lives today

The Friedmann equations run inside every modern cosmology pipeline. When the Planck team fits the cosmic microwave background, or the Dark Energy Survey and DESI fit millions of galaxy positions and supernova brightnesses, the model being fit is the master expansion law of equation (6): the data constrain $H_0$ and the $\Omega_i$ , and everything else — the sound horizon, the distance to last scattering, the growth of structure — is an integral over the $H(a)$ those equations define. The current frontier is a genuine tension in this fit. The $H_0$ inferred from the CMB by running the equations forward from the early universe ( $\approx67$ ) disagrees at $5\sigma$ with the local distance-ladder value ( $\approx73\ \mathrm{km\,s^{-1}\,Mpc^{-1}}$ ) — the Hubble tension, possibly a sign of new physics. And DESI's 2024–2025 baryon-acoustic-oscillation data hint that the dark-energy $w$ may not be exactly $-1$ but evolving; if so, the constant $\Omega_\Lambda$ term must become a time-dependent density. Either way, the questions are posed entirely in the language Friedmann wrote down in 1922.

The physical picture

Strip the universe down to what the Cosmological Principle allows and you are left with a uniform fluid filling space, expanding so that every proper distance scales with one number, $a(t)$ . The first Friedmann equation is then nothing more exotic than energy conservation for that fluid. A test galaxy on the edge of a large sphere of the medium recedes with velocity $v=Hr$ , carrying kinetic energy, while the enclosed mass pulls it back with potential energy; their sum is a conserved constant — precisely what general relativity reinterprets as the spatial curvature $k$ . A bound sphere (negative energy) is a closed, positively curved universe destined to recollapse; zero energy is spatially flat, coasting to a halt only after infinite time; positive energy is open and expands forever. The whole "fate of the universe" question is the sign of one number.

Figure

Three fates from one equation: solving H² = 8πG ρ/3 − kc²/a² for a single matter component gives a closed universe (k = +1) that recollapses, a flat one (k = 0) that coasts as t^(2/3), and an open one (k = −1) that expands forever — while adding a cosmological constant Λ bends the curve upward into acceleration

The second equation adds a twist with no Newtonian counterpart. In Newton's gravity only mass density attracts; in general relativity the source is the full stress–energy tensor, whose active gravitational density is $\varepsilon+3P$ — pressure gravitates on the same footing as energy. Ordinary matter and radiation have zero or positive pressure and therefore decelerate the expansion, as intuition demands. But a medium with pressure more negative than $-\varepsilon/3$ flips the sign of $\ddot a$ and accelerates it — the reason dark energy can blow the universe apart against its own gravity. The equations sort the cosmos into eras automatically: whichever component dilutes slowest must eventually win, and since radiation thins as $a^{-4}$ , matter as $a^{-3}$ , and vacuum energy not at all, the late universe inevitably becomes dark-energy dominated and accelerates — as the second figure's ledger shows.

Figure

The cosmic energy-density ledger on log-log axes: radiation dilutes as a⁻⁴, matter as a⁻³, and the cosmological constant Λ stays flat, so each reigns in turn. Radiation and matter are equal at a ≈ 2.9×10⁻⁴ (z ≈ 3400), matter and Λ at a ≈ 0.77 (z ≈ 0.3); today Ω ≈ 9×10⁻⁵ radiation, 0.3 matter, 0.7 Λ

Because the Friedmann equations are Einstein's equations with the tensor machinery already integrated out by symmetry, working cosmology never requires a Christoffel symbol: you track one number, $a(t)$ , and every observable — redshift, luminosity distance, angular size, cosmic age, particle horizon — is a definite integral over $H(a)$ . That economy is why these equations, written in 1922, remain the quantitative engine of the field a century later.

The core equations

The first Friedmann equation is the $00$ (time–time) component of Einstein's field equations on the FLRW metric. Read it as an energy budget: the left side is the squared expansion rate, the right balances gravitating energy density against curvature, and everything the universe contains enters only through the single total $\varepsilon$ .

H^2=\left(\frac{\dot a}{a}\right)^2=\frac{8\pi G}{3c^2}\,\varepsilon-\frac{kc^2}{a^2} \qquad (1)

where: $a(t)$ — scale factor, normalised $a(t_0)=1$ today; $H=\dot a/a$ — Hubble parameter; $\varepsilon$ — total energy density of all components; $G$ — gravitational constant; $c$ — speed of light; $k$ — curvature constant ( $+1$ closed, $0$ flat, $-1$ open). Reading: fix the contents $\varepsilon$ and geometry $k$ and the rate is set; curvature is the only place the global shape of space enters. The second, or acceleration, equation comes from the spatial components. Its content is the relativistic surprise that pressure sources gravity: the combination $\varepsilon+3P$ is the active gravitational density, so what decelerates the expansion is not density alone.

\frac{\ddot a}{a}=-\frac{4\pi G}{3c^2}\left(\varepsilon+3P\right) \qquad (2)

where: $\ddot a$ — second time derivative of $a$ ; $P$ — total pressure of the cosmic fluid; $\varepsilon+3P$ — effective gravitating density in GR. Reading: ordinary matter ( $P\!\ge\!0$ ) always decelerates, but any component with $P<-\varepsilon/3$ makes $\ddot a>0$ and accelerates. The fluid, or continuity, equation is local energy conservation — $\nabla_\mu T^{\mu\nu}=0$ written out for a homogeneous fluid — the third member of the family, though not an independent one.

\dot\varepsilon+3\frac{\dot a}{a}\left(\varepsilon+P\right)=0 \qquad (3)

where: $\dot\varepsilon$ — time rate of change of energy density. Reading: density falls both because volume grows ( $\varepsilon$ term) and because pressure does work on the expansion ( $P$ term) — the origin of the $3(1+w)$ exponent. Closing the system requires a relation between pressure and density. For each cosmic component this is the equation of state $P=w\varepsilon$ with constant $w$ ; substituting it into the fluid equation and integrating gives the dilution law that ranks the components.

P=w\,\varepsilon\;\Rightarrow\;\varepsilon(a)=\varepsilon_0\,a^{-3(1+w)} \qquad (4)

where: $w$ — equation-of-state parameter (constant for a given component); $\varepsilon_0$ — present-day density of that component. Reading: radiation ( $w=\tfrac13$ ) gives $a^{-4}$ , matter ( $w=0$ ) $a^{-3}$ , a cosmological constant ( $w=-1$ ) $a^{0}$ . Setting the curvature term of equation (1) to zero defines the critical density that makes space exactly flat; normalising every component by it gives the dimensionless density parameters that dominate cosmology's vocabulary.

\varepsilon_c=\frac{3c^2H^2}{8\pi G},\qquad \Omega_i\equiv\frac{\varepsilon_i}{\varepsilon_c} \qquad (5)

where: $\varepsilon_c$ — critical energy density for exact flatness; $\Omega_i$ — density parameter of component $i$ . Reading: $\Omega_{\rm tot}=1$ is flat, $>1$ closed, $<1$ open — geometry reduces to whether the total density hits critical. Dividing equation (1) by $H_0^2$ and inserting each component's scaling turns it into the master expansion history — the formula from which distances, ages, and horizons are computed.

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

where: $H_0$ — Hubble constant (present $H$ ); $\Omega_{r,0},\Omega_{m,0}$ — present radiation and matter density parameters; $\Omega_{k,0}=-kc^2/(a_0^2H_0^2)$ — effective curvature parameter; $\Omega_{\Lambda,0}$ — present dark-energy parameter. Reading: each term carries the $a$ -power of its equation of state; setting $a=1$ forces the closure relation below. Two closure relations complete the set: evaluating equation (6) at $a=1$ gives the sum rule that ties the density parameters together (spatial flatness when $\Omega_k=0$ ), and the cosmological constant is just a vacuum energy with $w=-1$ .

\Omega_{m,0}+\Omega_{r,0}+\Omega_{\Lambda,0}+\Omega_{k,0}=1,\qquad \Lambda=\frac{8\pi G}{c^2}\varepsilon_\Lambda=3H^2\Omega_\Lambda \qquad (7)

where: $\Omega_{k,0}=0$ for a flat universe (observed $|\Omega_{k,0}|\lesssim0.005$ ); $\Lambda$ — cosmological constant (inverse time squared here, the coefficient of a $\Lambda/3$ term in equation (1); $\Lambda/c^2$ is the geometric inverse-length-squared form); $\varepsilon_\Lambda$ — constant vacuum density. Reading: near-flatness forces the parameters to sum to one, and a nonzero $\Lambda$ acts as a fluid that never dilutes.

Derivation AThe first Friedmann equation from a Newtonian energy argument

The goal is to recover the exact form of equation (1) — coefficient $8\pi G/3c^2$ and curvature term $-kc^2/a^2$ — from nothing but Newtonian energy conservation applied to an expanding sphere of fluid, and to expose what the curvature constant $k$ physically is: the conserved mechanical energy of a comoving shell, with Birkhoff's theorem guaranteeing the same coefficient as full GR.

Step 1. Consider a test galaxy of mass $m$ on a sphere of physical radius $r=a(t)\chi$ , with $\chi$ a fixed comoving coordinate. By Birkhoff's theorem all mass outside the sphere cancels, so only the enclosed $M=\tfrac43\pi r^3\rho$ acts, with $\rho=\varepsilon/c^2$ , and its total mechanical energy $U=\tfrac12 m\dot r^2-GMm/r$ is conserved. Why: reducing the cosmos to one shell is legitimate because the exterior exerts no net force in a spherically symmetric, homogeneous medium.

Step 2. Substitute $M=\tfrac43\pi(a\chi)^3\rho$ and $r=a\chi$ to get $\tfrac12 m\dot a^2\chi^2-\tfrac{4\pi G\rho}{3}m a^2\chi^2=U$ . Every term carries the same factor $m\chi^2$ , the signature that the comoving coordinate $\chi$ drops out and leaves a statement about $a$ alone. Why: homogeneity means the dynamics cannot depend on which shell we picked, so $\chi$ must cancel — a consistency check on the Cosmological Principle.

Step 3. Divide through by $\tfrac12 m a^2\chi^2$ and identify $H=\dot a/a$ , giving $\left(\dot a/a\right)^2=\tfrac{8\pi G\rho}{3}+\tfrac{2U}{m\chi^2 a^2}$ ; writing $\rho=\varepsilon/c^2$ converts the first term into the relativistic coefficient $8\pi G\varepsilon/3c^2$ . Why: the mass-to-energy substitution is where Newtonian bookkeeping meets the relativistic form; the coefficient emerges without any input from GR.

Step 4. Match the remaining term to curvature by defining $-kc^2/a^2\equiv 2U/(m\chi^2 a^2)$ . The sign of the conserved energy $U$ now is the geometry: $U<0$ (bound) gives $k=+1$ closed, $U=0$ gives $k=0$ flat, $U>0$ (unbound) gives $k=-1$ open. Why: spatial curvature in GR is the cosmic analogue of binding energy, so "will the universe recollapse" is literally an escape-velocity question.

Go deeper · Derivation B · 1 minThe acceleration equation and why pressure gravitates

Derivation BThe acceleration equation and why pressure gravitates

The aim is to derive the second Friedmann equation, equation (2), by combining the first with the fluid equation, and to make transparent the feature with no Newtonian analogue — the appearance of $3P$ alongside $\varepsilon$ . The Newtonian argument was blind to pressure; recovering $\varepsilon+3P$ is what marks this result as relativistic.

Step 1. Differentiate the first Friedmann equation $H^2=\tfrac{8\pi G}{3c^2}\varepsilon-\tfrac{kc^2}{a^2}$ with respect to time, giving $2H\dot H=\tfrac{8\pi G}{3c^2}\dot\varepsilon+\tfrac{2kc^2}{a^2}H$ . Why: a time derivative turns the static energy constraint into a statement about how the rate changes — what an acceleration equation must be.

Step 2. Insert the fluid equation $\dot\varepsilon=-3H(\varepsilon+P)$ to eliminate $\dot\varepsilon$ , and use $\dot H=\ddot a/a-H^2$ to rewrite the left side in terms of $\ddot a/a$ . Pressure enters here, carried in by energy conservation. Why: the fluid equation is the only source of $P$ — pressure enters because the work it does on the expanding volume is part of the energy balance.

Step 3. Replace the leftover $kc^2/a^2$ using the first equation ( $kc^2/a^2=\tfrac{8\pi G}{3c^2}\varepsilon-H^2$ ); every $H^2$ and curvature term cancels, collapsing to exactly $\ddot a/a=-\tfrac{4\pi G}{3c^2}(\varepsilon+3P)$ . Why: the cancellation reflects that only two of the three equations are independent, so the third falls out with no new information.

Step 4. Read off the physics. For matter ( $P=0$ ) the source is $\varepsilon$ and the expansion decelerates; for radiation ( $P=\varepsilon/3$ ) it is $2\varepsilon$ , decelerating even harder per unit energy; for vacuum ( $P=-\varepsilon$ ) it is $-2\varepsilon$ , which flips the sign and accelerates. Why: the sign of $\varepsilon+3P$ is the sign of cosmic destiny — why dark energy speeds up the universe while everything else slows it.

Go deeper · Derivation C · 1 minDensity scaling and the master ΛCDM expansion law

Derivation CDensity scaling and the master ΛCDM expansion law

The objective is to turn the three equations into the concrete formula cosmologists integrate, equation (6): solve the fluid equation for how each component dilutes, then assemble the pieces into one expansion history normalised to present-day observables. This is the step that converts general relativity into a predictive tool, since once we know $H(a)$ every distance and age is an integral over it.

Step 1. Take the fluid equation with a constant equation of state $P=w\varepsilon$ , so $\dot\varepsilon=-3H(1+w)\varepsilon$ . Dividing by $\varepsilon$ and using $H=\dot a/a$ turns it into the separable form $\mathrm{d}\varepsilon/\varepsilon=-3(1+w)\,\mathrm{d}a/a$ . Why: a constant $w$ makes the equation integrable, and each real component has an essentially constant $w$ over the relevant epochs.

Step 2. Integrate to get $\varepsilon(a)=\varepsilon_0\,a^{-3(1+w)}$ , then read the exponent for each species: radiation $w=\tfrac13$ gives $a^{-4}$ , matter $w=0$ gives $a^{-3}$ , curvature acts as an effective $w=-\tfrac13$ fluid giving $a^{-2}$ , and vacuum $w=-1$ gives $a^{0}$ . Why: radiation's extra power of $a$ beyond the $a^{-3}$ volume dilution is the redshift of each photon — observable as the CMB scaling $T\propto a^{-1}$ .

Step 3. Write the total as $\varepsilon=\sum_i\varepsilon_{i,0}\,a^{-3(1+w_i)}$ , insert it into the first equation, and divide by $H_0^2$ . With $\Omega_{i,0}=\varepsilon_{i,0}/\varepsilon_{c,0}$ and $\Omega_{k,0}=-kc^2/(a_0^2H_0^2)$ this gives $H^2(a)=H_0^2[\Omega_{r,0}a^{-4}+\Omega_{m,0}a^{-3}+\Omega_{k,0}a^{-2}+\Omega_{\Lambda,0}]$ . Why: normalising to $H_0$ and the $\Omega$ 's replaces unmeasurable absolute densities with the dimensionless fractions surveys actually constrain.

Step 4. Set $a=1$ to obtain the closure relation $\sum_i\Omega_{i,0}=1$ , which fixes one parameter in terms of the rest and, for near-zero curvature, reads as flatness. Why: this sum rule is how a measurement of $\Omega_m$ plus flatness pins down $\Omega_\Lambda$ without observing dark energy directly.

Limiting cases

Solving the flat single-component first Friedmann equation $H^2=\tfrac{8\pi G}{3c^2}\varepsilon$ with $\varepsilon\propto a^{-3(1+w)}$ gives characteristic power-law (or exponential) expansion for each era, plus the empty-universe sanity check.

Case	Result	Meaning
Radiation-dominated ( $w=1/3$ )	$a(t)\propto t^{1/2}$	$\varepsilon\propto a^{-4}$ ; the first $\sim50{,}000$ years, dominated by photons and relativistic neutrinos.
Matter-dominated ( $w=0$ , Einstein–de Sitter)	$a(t)\propto t^{2/3}$	$\varepsilon\propto a^{-3}$ ; the flat pressureless model, good from matter–radiation equality until dark energy took over.
Vacuum / dark-energy ( $w=-1$ )	$a(t)\propto e^{Ht},\ H=\text{const}$	$\varepsilon=\text{const}$ ; de Sitter exponential expansion — the fate of $\Lambda$ CDM and the template for inflation.
Empty curved (Milne, $\varepsilon\to0$ )	$a(t)\propto t,\ k=-1$	Open, coasting universe with $\ddot a=0$ ; curvature alone sets the kinematics.

Common trapThinking the three equations are three independent laws — and that H greater than 0 forces the universe to decelerate

Two traps hide here. First, the two Friedmann equations plus the fluid equation are not three independent relations — only two are. The fluid equation is the contracted Bianchi identity $\nabla_\mu T^{\mu\nu}=0$ in disguise, so it follows automatically; any two of the three imply the third, and treating them as three constraints over-determines the system. Second, a positive expansion rate says nothing about the acceleration: the signs of $\dot a$ and $\ddot a$ are set by different equations — $\dot a>0$ by the first, the sign of $\ddot a$ by $\varepsilon+3P$ in the second. A universe can be expanding ( $H>0$ ) while accelerating ( $\ddot a>0$ ), exactly what we observe today.

Worked examples

1 · The critical density and the emptiness of the cosmos (fully worked)

Take a Hubble constant $H_0=70\ \mathrm{km\,s^{-1}\,Mpc^{-1}}$ . Find the critical mass density $\rho_c=\varepsilon_c/c^2$ today, and express it as a number of hydrogen atoms per cubic metre.

Step 1. Convert $H_0$ to SI. One megaparsec is $3.086\times10^{22}\ \mathrm{m}$ , so $H_0=\dfrac{70\times10^{3}\ \mathrm{m\,s^{-1}}}{3.086\times10^{22}\ \mathrm{m}}=\mathbf{2.27\times10^{-18}\ s^{-1}}$ . The reciprocal is the Hubble time $1/H_0=4.41\times10^{17}\ \mathrm{s}=\mathbf{13.97\ Gyr}$ — an order-of-magnitude estimate of the age of the universe.

Step 2. Apply the critical-density formula from equation (5) in mass form, $\rho_c=\dfrac{3H_0^2}{8\pi G}$ . Insert $G=6.674\times10^{-11}\ \mathrm{m^3\,kg^{-1}\,s^{-2}}$ :

\rho_c=\frac{3\,(2.27\times10^{-18})^2}{8\pi\,(6.674\times10^{-11})}=\mathbf{9.2\times10^{-27}\ kg\,m^{-3}}.

Step 3. Divide by the hydrogen-atom mass $m_H=1.673\times10^{-27}\ \mathrm{kg}$ : $\rho_c/m_H\approx\mathbf{5.5\ atoms\,m^{-3}}$ — a spatially flat universe needs, on average, only about five and a half hydrogen atoms per cubic metre, a better vacuum than any laboratory achieves. The critical density is $\rho_c\approx9.2\times10^{-27}\ \mathrm{kg\,m^{-3}}$ , roughly 5–6 hydrogen atoms per cubic metre, and everything we call "matter" makes up only $\Omega_m\approx0.3$ of it.

2 · The three handover epochs of cosmic history

With concordance values $\Omega_{r,0}\approx9\times10^{-5}$ , $\Omega_{m,0}\approx0.31$ , $\Omega_{\Lambda,0}\approx0.69$ , locate (a) matter–radiation equality, (b) matter– $\Lambda$ equality, and (c) the onset of acceleration.

Step 1. Matter–radiation equality is where $\Omega_{m,0}a^{-3}=\Omega_{r,0}a^{-4}$ , i.e. $a_{\rm eq}=\Omega_{r,0}/\Omega_{m,0}=9\times10^{-5}/0.31=\mathbf{2.9\times10^{-4}}$ , or $1+z_{\rm eq}\approx\mathbf{3400}$ . Before this the universe was radiation-dominated ( $a\propto t^{1/2}$ ); after it, matter took over and structure could grow.

Step 2. Matter– $\Lambda$ equality is where $\Omega_{m,0}a^{-3}=\Omega_{\Lambda,0}$ , giving $a=(\Omega_{m,0}/\Omega_{\Lambda,0})^{1/3}=(0.31/0.69)^{1/3}=\mathbf{0.77}$ , or $z\approx\mathbf{0.3}$ — when dark energy became the largest single contributor to the budget.

Step 3. Acceleration begins earlier than $\Lambda$ -dominance, when $\ddot a=0$ , i.e. $\varepsilon+3P=0$ . For matter plus $\Lambda$ this is $\varepsilon_m=2\varepsilon_\Lambda$ , so $1+z_{\rm acc}=(2\Omega_{\Lambda,0}/\Omega_{m,0})^{1/3}=(2\times0.69/0.31)^{1/3}\approx1.65$ , i.e. $z_{\rm acc}\approx\mathbf{0.65}$ . Acceleration set in at $z\approx0.65$ , before dark energy even became the dominant component ( $z\approx0.3$ ) — because the pressure, through the $3P$ term, turns the corner ahead of the density crossover. Type Ia supernova Hubble diagrams detected exactly this turnover, the first direct evidence for dark energy.

3 · The age crisis that pointed to dark energy

Compare the age of a flat, pressureless (Einstein–de Sitter) universe with $\Lambda$ CDM, both at $H_0=70\ \mathrm{km\,s^{-1}\,Mpc^{-1}}$ .

Step 1. In an Einstein–de Sitter universe $a\propto t^{2/3}$ , so $H=\tfrac{2}{3t}$ inverts to the analytic age $t_0=\tfrac{2}{3H_0}=\tfrac23\times13.97=\mathbf{9.3\ Gyr}$ — younger than the oldest globular-cluster stars ( $\sim13\ \mathrm{Gyr}$ ), a real crisis for matter-only cosmology in the 1990s. Integrating $t_0=\int_0^1\mathrm{d}a/(aH)$ with the full $\Lambda$ CDM $H(a)$ instead stretches the age to $\approx13.8\ \mathrm{Gyr}$ . The dark-energy term raises the age from a too-young $9.3\ \mathrm{Gyr}$ to $13.8\ \mathrm{Gyr}$ — one of the independent pointers to $\Lambda$ .

Physics of the portrait

The Friedmann equations carry the name of a man who solved them, was told by Einstein he was wrong, was proved right, and died before anyone measured the expansion he had predicted on paper.

Origins · 1 minAlexander Friedmann and how this was discovered

Alexander Friedmann

1888–1925 · relativistic cosmology and the dynamical universe

Alexander Alexandrovich Friedmann was a St. Petersburg mathematician and meteorologist who spent the First World War computing bombing tables for the Russian air force and once ballooned to a near-record altitude to take atmospheric readings. In 1922, working from Einstein's 1915 field equations, he did what no one had dared: he treated the cosmological constant not as a prop for a static universe but as one parameter among many, and asked what solutions the equations permitted. The answer was that the universe could not be static — it must expand or contract, its scale factor $a(t)$ obeying the two differential equations that now bear his name. Einstein's first published reaction was that Friedmann had made an arithmetic error; he soon retracted, conceding the solutions were correct even as he disliked their non-static conclusion. A second paper in 1924 added the negatively curved (open) case, completing the classification into closed, flat, and open universes. Friedmann never saw his mathematics become physics: he died of typhoid fever in 1925 at thirty-seven, four years before Hubble's redshift–distance measurements and five before Lemaître independently rediscovered the same dynamics. That the framework of modern cosmology traces back to a wartime balloonist trusting his own equations over the greatest physicist of the age is one of the quiet triumphs of the field.

History

1915

— Einstein completes the general theory of relativity and its field equations

1917

— Einstein adds the cosmological constant Λ to force a static universe

1922

— Friedmann derives the expanding and contracting solutions for a(t)

1927

— Lemaître independently finds the same dynamics and links them to galaxy redshifts

1998

— supernova surveys reveal accelerating expansion, reviving Λ as dark energy

Going further

Texts. Ryden, Introduction to Cosmology, 2nd ed., Ch. 4–5 (cleanest first pass) · Carroll & Ostlie, An Introduction to Modern Astrophysics, 2nd ed., §29.2 · Dodelson, Modern Cosmology, Ch. 2 (relativistic derivation) · Mo, van den Bosch & White, Galaxy Formation and Evolution, §3.1 · Binney & Tremaine, Galactic Dynamics, 2nd ed., cosmological appendix.

Provenance. Equations cross-checked against Ryden Ch. 4–5, Carroll & Ostlie §29.2, and Dodelson Ch. 2. A verifier should confirm: (1) the coefficient $8\pi G/3c^2$ and curvature term $-kc^2/a^2$ in equation (1) and $\ddot a/a=-(4\pi G/3c^2)(\varepsilon+3P)$ in equation (2); (2) the dilution law $\varepsilon\propto a^{-3(1+w)}$ and the flat solutions $a\propto t^{1/2},\,t^{2/3},\,e^{Ht},\,t$ ; (3) $\rho_c\approx9.2\times10^{-27}\ \mathrm{kg\,m^{-3}}\approx5.5$ H atoms per m³ and Hubble time $13.97$ Gyr for $H_0=70$ ; (4) the epoch redshifts $z_{\rm eq}\approx3400$ , $z(\text{matter–}\Lambda)\approx0.3$ , $z_{\rm acc}\approx0.65$ , and Einstein–de Sitter age $2/(3H_0)\approx9.3$ Gyr versus $\Lambda$ CDM $13.8$ Gyr. Concordance $\Omega$ values are standard.

Prerequisites. The FLRW Metric · The Cosmological Principle · Spatial Curvature of the Universe · The Expansion of the Universe and the Scale Factor. Leads to. The Fluid and Acceleration Equations · Equation of State in Cosmology · Critical Density and Density Parameters · Hubble–Lemaître Law.

Ask the tutor: Ask me to derive the flat-universe age integral, work out any epoch's redshift for your own $\Omega$ values, or show why an inflationary de Sitter phase flattens the curvature away.

Are the Friedmann equations exact, or an approximation?

Within their assumptions they are exact — Einstein's field equations, with no approximation beyond the symmetry input. The one assumption is the Cosmological Principle: that on large scales the universe is homogeneous and isotropic, which forces the FLRW metric. Given that metric the field equations reduce exactly to equations (1) and (2); nothing is dropped or linearised, so the approximation lives entirely in the premise. On small scales the universe is manifestly lumpy — galaxies, clusters, voids — and there the smooth description fails and you need perturbation theory on top. But averaged over hundreds of megaparsecs the universe really is uniform to about one part in $10^5$ , which is why the exact equations describe the real cosmos so well.

If only two of the three equations are independent, why write the fluid equation at all?

Convenience and transparency. The fluid equation $\dot\varepsilon+3H(\varepsilon+P)=0$ is the most direct expression of local energy conservation, and it is far easier to integrate for the density scaling $\varepsilon\propto a^{-3(1+w)}$ than to extract that result from the two Friedmann equations. In practice cosmologists pick the first Friedmann equation (for $H$ ) plus the fluid equation (for how each component dilutes) and treat the acceleration equation as a derived consequence — but any two of the three would do. The redundancy is the Bianchi identity $\nabla_\mu T^{\mu\nu}=0$ showing up as a built-in consistency guarantee.

How can pressure cause gravitational repulsion — isn't pressure a push outward?

This conflates two roles of pressure. The mechanical push requires a gradient, and a perfectly homogeneous universe has none, so that push is identically zero everywhere. What enters the acceleration equation is different: pressure's contribution to the gravitational source, the term $3P$ in $\varepsilon+3P$ . In general relativity every energy and momentum flux gravitates, and pressure is momentum flux. Positive pressure adds to the pull (extra deceleration); sufficiently negative pressure subtracts and can reverse it. So dark energy does not push space apart mechanically — its negative pressure makes gravity itself repulsive on cosmic scales.

Why does radiation dilute faster than matter as the universe expands?

Two effects instead of one. Matter dilutes purely by volume: a fixed particle number in a volume growing as $a^3$ gives density $\propto a^{-3}$ . Radiation suffers the same volume dilution plus an extra loss — every photon's wavelength stretches, so its energy $E=hc/\lambda$ drops as $a^{-1}$ . Multiplying the $a^{-3}$ from volume by the $a^{-1}$ from redshift gives $a^{-4}$ , exactly the $-3(1+w)$ exponent with $w=1/3$ . The observable consequence is the CMB temperature scaling $T\propto a^{-1}$ : once a $3000\ \mathrm{K}$ glow, the CMB is a chilly $2.73\ \mathrm{K}$ today.

Does the expansion of space mean galaxies fly apart through space?

No — and the distinction matters for reading the equations correctly. In the FLRW picture galaxies sit at essentially fixed comoving coordinates; it is the scale factor $a(t)$ multiplying those coordinates that grows, so the proper distance between galaxies increases even though nothing moves through space locally. This is why recession velocities can exceed $c$ for distant galaxies without violating relativity: no object overtakes a light signal in its own neighbourhood; the metric itself stretches. The Friedmann equations govern $a(t)$ , not the peculiar motion of any galaxy. Bound systems like the Milky Way do not expand, because local gravity overwhelms the cosmic stretching.

Do the Friedmann equations describe the Big Bang singularity itself?

Run them backward and a flat radiation universe has $a\propto t^{1/2}$ , so $a\to0$ and $\varepsilon\to\infty$ at $t=0$ : a genuine singularity where the classical equations break down. They are perfectly valid for the hot, dense early universe from a tiny fraction of a second onward, but cannot be trusted at the singularity itself, where densities reach the Planck scale and quantum gravity must take over. The standard resolution is cosmic inflation: a very early near-de Sitter phase ( $w\approx-1$ ) that precedes the radiation era, flattens the universe, and replaces the singular "beginning" with exponential growth. Inflation is itself a Friedmann solution, just with a dominant negative-pressure component.