The Friedmann Equations

The Friedmann equations are the equations of motion of the universe as a whole. The cosmological principle — that space is homogeneous and isotropic on the largest scales — forces the metric into the single-function FLRW form and the matter into a perfect fluid described only by an energy density $\rho(t)$ and an isotropic pressure $p(t)$ . Feed that symmetry into Einstein's ten coupled field equations and something remarkable happens: nine of them vanish or duplicate one another, and the whole of gravitational dynamics collapses to two ordinary differential equations for one function, the scale factor $a(t)$ — a constraint tying the expansion rate to what fills space, and an acceleration law showing that in general relativity pressure gravitates. Alexander Friedmann wrote them down in 1922, five years before Hubble's redshifts and while Einstein was still defending a static cosmos propped up by the cosmological constant $\Lambda$ . Their message was radical and unavoidable: a universe of ordinary matter cannot stand still — it must expand or contract, and any static balance is unstable to the smallest nudge. Georges Lemaître rediscovered them in 1927 and connected them directly to the observed recession of galaxies. Everything in modern cosmology is downstream of these two lines. Together with an equation of state $p=w\rho c^2$ they partition cosmic history into a radiation era ( $a\propto t^{1/2}$ ), a matter era ( $a\propto t^{2/3}$ ), and a dark-energy era ( $a\propto e^{Ht}$ ); they define the critical density separating open, flat, and closed geometries; and they supply the integrand $1/H(a)$ behind every measured cosmic distance and the age of the universe itself.

Key takeaways

The Friedmann equations are the two independent components of Einstein's field equations evaluated on the homogeneous, isotropic FLRW metric — a complete dynamical theory of the scale factor $a(t)$ .
The first (constraint) equation, $H^2=\tfrac{8\pi G}{3}\rho-\tfrac{kc^2}{a^2}+\tfrac{\Lambda c^2}{3}$ , is an energy budget: the squared expansion rate balances total energy density against spatial curvature.
The second (acceleration) equation, $\tfrac{\ddot a}{a}=-\tfrac{4\pi G}{3}\!\left(\rho+\tfrac{3p}{c^2}\right)+\tfrac{\Lambda c^2}{3}$ , shows pressure is a source of gravity — only $w<-\tfrac13$ can make the expansion accelerate.
The continuity equation $\dot\rho+3H(\rho+p/c^2)=0$ follows from the Bianchi identities, so only two of the three relations are independent.
An equation of state $p=w\rho c^2$ closes the system, giving $\rho\propto a^{-3(1+w)}$ and the succession of radiation, matter, and dark-energy eras.
Key idea: symmetry does almost all the work — imposing homogeneity and isotropy on Einstein's equations reduces gravity to two scalar equations for the single function $a(t)$ , with curvature, energy density, pressure, and $\Lambda$ as the only inputs.

The core insight

Symmetry does almost all the work: imposing homogeneity and isotropy on Einstein's equations reduces gravitational dynamics to two scalar equations for the single function $a(t)$ , with curvature, energy density, pressure, and $\Lambda$ as the only inputs.

Where this lives today

The Friedmann equations are not a historical monument; they are the fitting function of modern cosmology. Every time Planck, the Dark Energy Survey, or DESI reports a value of $\Omega_m$ , $\Omega_\Lambda$ , $H_0$ , or $w$ , it is constraining the coefficients of equation (5) against data — the CMB acoustic peaks, the baryon acoustic oscillation ruler in galaxy clustering, and thousands of Type Ia supernovae. The age of the universe $t_0=\int_0^1\mathrm{d}a/[aH(a)]\approx13.8\ \mathrm{Gyr}$ and every cosmological distance $\chi=c\int\mathrm{d}z/H(z)$ are integrals of $1/H(a)$ from the first equation. The live frontier is the equation of state of dark energy: is $w$ exactly $-1$ , or does it evolve? DESI's 2024–2025 results hint mildly at evolving dark energy, a crack in $\Lambda$ CDM visible only through precision fits to $H(z)$ . Meanwhile the persistent "Hubble tension" — early- and late-universe measurements of $H_0$ disagreeing at $\sim5\sigma$ — is a disagreement about how to run the same Friedmann equation from the CMB to today, and may signal new physics in the expansion history.

The physical picture

Strip away the tensor machinery and the first Friedmann equation is a bookkeeping statement you already know. Multiply it by $\tfrac12 a^2$ and it reads $\tfrac12\dot a^2-\tfrac{4\pi G}{3}\rho a^2=-\tfrac12 kc^2$ : kinetic energy of expansion plus gravitational potential energy equals a constant fixed by the curvature $k$ . A shell of galaxies flung outward by the Big Bang either has enough energy to escape to infinity or it does not, exactly as a rocket does against a planet, and a flat universe ( $k=0$ ) is the knife-edge marginally-bound case where expansion energy equals escape energy. This is not a loose analogy; it is what the first equation is, and it gives the three classical fates at a glance.

Figure

Reading the first Friedmann equation as a cosmic energy budget ½ȧ² − 4πGρa²/3 = −½kc²: a closed universe (k = +1) is gravitationally bound and recollapses to a Big Crunch, a flat universe (k = 0) is marginally bound and coasts to rest at infinity, an open universe (k = −1) has surplus energy and expands forever, while a cosmological constant Λ overrides all three and drives late-time acceleration

The second equation carries the genuinely relativistic surprise. Its source is not the energy density $\rho$ alone but the combination $\rho c^2+3p$ — the "active gravitational mass" density that also appears in the Tolman mass of a static star. Pressure gravitates. Radiation, with $p=\tfrac13\rho c^2$ , therefore pulls twice as hard per unit energy as pressureless dust. Turn that around and you find the only way to make the expansion speed up: a component whose pressure is so negative that $\rho c^2+3p<0$ , that is $w<-\tfrac13$ . A cosmological constant, with $w=-1$ , does exactly this, and the discovery of cosmic acceleration in 1998 is the statement that this term wins today. The second figure shows how the three ingredients trade dominance as the universe expands, using the measured $\Lambda$ CDM budget.

Figure

The ΛCDM energy budget on a log scale: each species dilutes as its own power of the scale factor — radiation ρ ∝ a⁻⁴, matter ρ ∝ a⁻³, dark energy ρ = constant — so the universe passes through matter–radiation equality near a ≈ 3×10⁻⁴ (z ≈ 3400) and matter–Λ equality near a ≈ 0.77 (z ≈ 0.3), with today (a = 1) sitting just inside the dark-energy era. Present values Ωr ≈ 9×10⁻⁵, Ωm = 0.31, ΩΛ = 0.69

The core equations

The first Friedmann equation is the time-time component of Einstein's equations on the FLRW metric. It is a constraint, not an evolution law: it fixes the expansion rate $H\equiv\dot a/a$ once you specify what fills space and how curved it is. Read left to right, it says the squared Hubble rate is the sum of a density term, a curvature term, and a vacuum term.

\left(\frac{\dot a}{a}\right)^2 = \frac{8\pi G}{3}\rho - \frac{kc^2}{a^2} + \frac{\Lambda c^2}{3} \qquad (1)

where: $a(t)$ — scale factor of the FLRW metric; $\dot a=\mathrm{d}a/\mathrm{d}t$ , and $H\equiv\dot a/a$ is the Hubble parameter, $[\mathrm{s^{-1}}]$ ; $\rho$ — total mass-energy density summed over all species, $[\mathrm{kg\,m^{-3}}]$ ; $k$ — spatial curvature constant normalized to $\{-1,0,+1\}$ (open, flat, closed); $\Lambda$ — cosmological constant, $[\mathrm{m^{-2}}]$ ; $G,c$ — Newton's constant and the speed of light. The second Friedmann equation is the acceleration law obtained by combining the spatial Einstein components with the first equation. Here is where pressure enters, and enters tripled: the source is $\rho+3p/c^2$ , so a fluid with large positive pressure decelerates the expansion harder than its energy alone would suggest.

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

where: $\ddot a$ — second time derivative of the scale factor, with $\ddot a>0$ meaning accelerating expansion; $p$ — isotropic pressure of the cosmic fluid, $[\mathrm{Pa}]$ ; $\rho+3p/c^2$ — the active gravitational (Tolman) mass density whose sign decides deceleration versus acceleration; absorbing $\Lambda$ into a fluid with $p_\Lambda=-\rho_\Lambda c^2$ reproduces its term. Energy conservation appears as the continuity equation. It is the $\nu=0$ component of $\nabla_\mu T^{\mu\nu}=0$ and is equivalently the first law of thermodynamics for a comoving volume $V\propto a^3$ expanding adiabatically — no heat can flow because isotropy forbids a preferred direction.

\dot\rho + 3\frac{\dot a}{a}\left(\rho + \frac{p}{c^2}\right) = 0 \qquad (3)

where: $\dot\rho$ — rate of change of energy density; $3\dot a/a=\dot V/V$ — the fractional rate of volume expansion. The two terms balance density dilution against the work $-p\,\mathrm{d}V$ the fluid does on the expanding volume. To make the system determinate one supplies an equation of state. The simplest is the linear barotropic form with a constant parameter $w$ ; inserting it into the continuity equation integrates immediately to a power-law dilution.

p = w\,\rho c^2 \qquad\Longrightarrow\qquad \rho \propto a^{-3(1+w)} \qquad (4)

where: $w$ — dimensionless equation-of-state parameter: $w=0$ for pressureless dust, $w=\tfrac13$ for radiation, $w=-1$ for vacuum energy; the exponent $-3(1+w)$ is how that species' density dilutes as space expands. Normalizing to the present epoch ( $a_0=1$ ) and defining density parameters $\Omega_{i,0}=\rho_{i,0}/\rho_{c,0}$ turns the first equation into the working form of observational cosmology — one line that traces the entire expansion history.

H^2 = H_0^2\!\left[\Omega_{r,0}a^{-4}+\Omega_{m,0}a^{-3}+\Omega_{k,0}a^{-2}+\Omega_{\Lambda,0}\right] \qquad (5)

where: $H_0$ — present Hubble constant; $\Omega_{r,0},\Omega_{m,0},\Omega_{\Lambda,0}$ — present density parameters of radiation, matter, and $\Lambda$ ; $\Omega_{k,0}=-kc^2/(a_0^2H_0^2)$ — the curvature parameter; the identity $\sum_i\Omega_{i,0}+\Omega_{k,0}=1$ holds by construction, and each term's power of $a$ makes a different species dominate at each epoch. Two derived quantities anchor the observations. The critical density is the value that makes space exactly flat at a given expansion rate, and the deceleration parameter packages the acceleration equation into a single dimensionless number that supernova surveys measure.

\rho_c = \frac{3H^2}{8\pi G},\qquad q_0 \equiv -\frac{\ddot a\, a}{\dot a^2}\bigg|_0 = \frac{1}{2}\sum_i \Omega_{i,0}\,(1+3w_i) \qquad (6)

where: $\rho_c$ — critical density, the density giving $k=0$ at expansion rate $H$ ; $q_0$ — present deceleration parameter, with $q_0<0$ signalling accelerating expansion, as the negative- $w$ term of $\Lambda$ CDM makes it.

Derivation ADeriving the two Friedmann equations from Einstein's field equations

The goal is to show explicitly how ten coupled nonlinear field equations become two scalar ODEs. We take the FLRW metric as the only input, grind out its Christoffel symbols, Ricci tensor, and Ricci scalar, insert a perfect-fluid source and the cosmological constant, and read off the two components that survive the symmetry. Every intermediate quantity below is fixed uniquely by homogeneity and isotropy — there is no freedom to choose, which is why cosmology is so predictive. We work in comoving coordinates and restore $c$ at the end.

Step 1. Start from the FLRW line element $\mathrm{d}s^2=-c^2\mathrm{d}t^2+a^2(t)\!\left[\tfrac{\mathrm{d}r^2}{1-kr^2}+r^2\mathrm{d}\Omega^2\right]$ , whose maximally symmetric spatial slices leave $a(t)$ as the sole metric function. Why: the cosmological principle forbids any dependence on position or direction, so time-dependence can enter only through one overall scale.

Step 2. Compute the non-vanishing Christoffel symbols; the ones that matter are $\Gamma^0_{ij}=\tfrac{a\dot a}{c^2}\tilde g_{ij}$ and $\Gamma^i_{0j}=\tfrac{\dot a}{a}\delta^i_j$ , where $\tilde g_{ij}$ is the comoving spatial metric, plus the intrinsic connection of the curved 3-space. Why: these mixed time–space components encode how comoving separations stretch with $a(t)$ and are what feed expansion into the curvature.

Step 3. Assemble the Ricci tensor: $R_{00}=-\tfrac{3\ddot a}{c^2 a}$ and $R_{ij}=\tfrac{1}{c^2}\!\left(a\ddot a+2\dot a^2+2kc^2\right)\tilde g_{ij}$ . Why: the $R_{00}$ component measures how the volume of a comoving ball accelerates, while the $2kc^2$ piece injects the intrinsic curvature of the spatial slices.

Step 4. Contract to the Ricci scalar $R=\tfrac{6}{c^2}\!\left(\tfrac{\ddot a}{a}+\tfrac{\dot a^2}{a^2}+\tfrac{kc^2}{a^2}\right)$ using $R=g^{\mu\nu}R_{\mu\nu}$ . Why: the scalar is needed to build the Einstein tensor $G_{\mu\nu}=R_{\mu\nu}-\tfrac12 R g_{\mu\nu}$ , the geometric side of the field equations.

Step 5. Supply the source as a perfect fluid at rest in comoving coordinates, $T_{\mu\nu}=(\rho c^2+p)\tfrac{u_\mu u_\nu}{c^2}+p\,g_{\mu\nu}$ with $u^\mu=(c,0,0,0)$ , so that $T^0{}_0=-\rho c^2$ and $T^i{}_j=p\,\delta^i_j$ . Why: isotropy permits no heat flux or anisotropic stress, so the most general allowed matter is a perfect fluid with just $\rho$ and $p$ .

Step 6. Write the field equations with the cosmological term, $R_{\mu\nu}-\tfrac12 R g_{\mu\nu}+\Lambda g_{\mu\nu}=\tfrac{8\pi G}{c^4}T_{\mu\nu}$ . By symmetry only a time-time and a single spatial component are independent; every off-diagonal equation is $0=0$ . Why: the ten equations of a general spacetime are reduced to two by the same symmetry that reduced the metric to one function.

Step 7. The $00$ equation gives $3\tfrac{\dot a^2+kc^2}{a^2}-\Lambda c^2=8\pi G\rho$ ; dividing by three and isolating $H^2$ yields the first Friedmann equation exactly as in (1). Why: this is the constraint — it contains only first derivatives of $a$ and fixes the expansion rate from the content of space.

Step 8. The $ii$ equation gives $\tfrac{2\ddot a}{a}+\tfrac{\dot a^2+kc^2}{a^2}-\Lambda c^2=-\tfrac{8\pi G}{c^2}p$ ; subtract the $00$ result to cancel $\dot a^2+kc^2$ and out drops $\tfrac{\ddot a}{a}=-\tfrac{4\pi G}{3}\!\left(\rho+\tfrac{3p}{c^2}\right)+\tfrac{\Lambda c^2}{3}$ , the acceleration equation (2). Why: eliminating the first-derivative terms isolates the genuine second-order dynamics and exposes the $\rho+3p/c^2$ source that makes pressure gravitate. *Verified: the Ricci components, both field-equation components, and the final pair match Carroll §8.2 eqs (8.66)–(8.67), MTW §27.2, and Wald eqs (5.2.14)–(5.2.15).*

Go deeper · Derivation B · 1 minThe continuity equation from the Bianchi identity and the first law

Derivation BThe continuity equation from the Bianchi identity and the first law

The goal is to prove that energy conservation is not an extra assumption but a theorem, and that it is identical to elementary thermodynamics. The contracted Bianchi identity $\nabla_\mu G^{\mu\nu}=0$ is a geometric identity satisfied by every metric; through the field equations it forces $\nabla_\mu T^{\mu\nu}=0$ on the matter. We evaluate the time component on FLRW, recognize the result as the first law $\mathrm{d}(\rho c^2 a^3)=-p\,\mathrm{d}(a^3)$ , and then show it is not independent of equations (1) and (2).

Step 1. Write the covariant divergence $\nabla_\mu T^{\mu 0}=\partial_\mu T^{\mu 0}+\Gamma^\mu_{\mu\alpha}T^{\alpha 0}+\Gamma^0_{\mu\alpha}T^{\mu\alpha}=0$ , the $\nu=0$ conservation law. Why: the covariant divergence, not the ordinary one, is what the Bianchi identity guarantees, and its connection terms carry the effect of the expanding background.

Step 2. Insert the FLRW Christoffels ( $\Gamma^\mu_{\mu 0}=3\dot a/a$ and $\Gamma^0_{ij}T^{ij}=3(\dot a/a)p$ ) and the fluid $T^{\mu\nu}$ to obtain $\dot\rho c^2+3\tfrac{\dot a}{a}\rho c^2+3\tfrac{\dot a}{a}p=0$ . Why: the factor $3\dot a/a$ is exactly the fractional rate at which a comoving volume grows, so this term is the dilution of density by expansion.

Step 3. Divide by $c^2$ to reach $\dot\rho+3\tfrac{\dot a}{a}\!\left(\rho+\tfrac{p}{c^2}\right)=0$ , which rearranges to $\mathrm{d}(\rho c^2 a^3)=-p\,\mathrm{d}(a^3)$ : the change in internal energy of a comoving box equals minus the pressure times the change in its volume. Why: this is the first law of thermodynamics with no heat term, and it is what turns the abstract conservation law into a picture of a gas doing work as it expands.

Step 4. Confirm redundancy: differentiate the first Friedmann equation with respect to time, substitute the acceleration equation for $\ddot a/a$ , and the continuity equation drops out. Why: this closes the logical loop — any two of the three relations imply the third, so the true dynamical content is a single second-order ODE for $a(t)$ . *Verified: matches Carroll eq (8.68), Hartle eq (18.7), and Weinberg §1.5; the redundancy check reproduces (3) identically.*

Limiting cases

The single first Friedmann equation specializes to every regime of cosmic history through the equation-of-state parameter $w$ and the resulting density scaling.

Case	Result	Meaning
Radiation, $w=\tfrac13$	$\rho\propto a^{-4},\ \ a(t)\propto t^{1/2}$	Volume dilution ( $a^{-3}$ ) times per-quantum redshift ( $a^{-1}$ ); dominant in the hot early universe.
Matter (dust), $w=0$	$\rho\propto a^{-3},\ \ a(t)\propto t^{2/3}$	Pressureless dilution by volume alone — the Einstein–de Sitter solution that governs structure formation.
Cosmological constant, $w=-1$	$\rho=\text{const},\ \ a(t)\propto e^{Ht}$	Constant vacuum density drives exponential de Sitter expansion with $H=c\sqrt{\Lambda/3}$ ; the far-future attractor.
Empty, curvature-dominated	$a(t)\propto t\ \ (k=-1)$	With $\rho\to0$ the $a^{-2}$ term rules: the Milne universe coasts linearly, the boundary between eternal expansion and recollapse.
Closed, matter-only	$a(\eta)=\tfrac{a_{\max}}{2}(1-\cos\eta)$	For $k=+1,\ \Lambda=0$ the solution is a cycloid in conformal time $\eta$ : expansion to $a_{\max}$ , then recollapse to a Big Crunch.

Common trapForgetting that pressure gravitates

The acceleration equation contains $\rho+3p/c^2$ , not $\rho$ alone. In Newtonian gravity only mass density sources the field; in general relativity the full stress–energy does, so pressure gravitates. This has two consequences students routinely miss. First, positive pressure decelerates expansion more strongly, not less — radiation, despite being "just energy", pulls harder per unit $\rho$ than dust because of its pressure. Second, acceleration ( $\ddot a>0$ ) is impossible for any ordinary matter and requires $w<-\tfrac13$ , a genuinely negative pressure. Dropping the $3p$ term, or imagining that pressure pushes the expansion outward like a gas in a balloon, is the classic and fatal error — the cosmic fluid has no walls to push against.

Worked examples

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

Using the Planck value $H_0=67.4\ \mathrm{km\,s^{-1}\,Mpc^{-1}}$ , compute the critical density 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 $H_0=\dfrac{67.4\times10^{3}\ \mathrm{m\,s^{-1}}}{3.086\times10^{22}\ \mathrm{m}}=2.18\times10^{-18}\ \mathrm{s^{-1}}$ . The Hubble time $1/H_0\approx14.5\ \mathrm{Gyr}$ sets the rough scale of the age.

Step 2. Apply equation (6): $\rho_c=\dfrac{3H_0^2}{8\pi G}=\dfrac{3(2.18\times10^{-18})^2}{8\pi(6.674\times10^{-11})}=\dfrac{1.43\times10^{-35}}{1.68\times10^{-9}}=\mathbf{8.5\times10^{-27}\ kg\,m^{-3}}$ .

Step 3. Divide by the proton mass $1.67\times10^{-27}\ \mathrm{kg}$ : $\rho_c/m_p=\mathbf{5.1\ protons\ per\ m^3}$ . The critical density — the density that makes the whole universe spatially flat — is about five hydrogen atoms in a cubic metre, a better vacuum than any laboratory can make. Of that, ordinary baryonic matter is only $\sim5\%$ , so the atoms actually present number about one in four cubic metres. ρ_c ≈ 8.5×10⁻²⁷ kg m⁻³ ≈ 5 protons per cubic metre.

2 · When did the universe start accelerating? (fully worked)

In a flat matter-plus- $\Lambda$ universe with $\Omega_{m,0}=0.31$ and $\Omega_{\Lambda,0}=0.69$ , find the scale factor and redshift at which the expansion switched from decelerating to accelerating.

Step 1. The turnover is $\ddot a=0$ . The acceleration equation sums $\rho_i+3p_i/c^2$ over species: matter contributes $\rho_m$ (since $p_m=0$ ), and $\Lambda$ contributes $\rho_\Lambda+3(-\rho_\Lambda c^2)/c^2=-2\rho_\Lambda$ . So $\ddot a=0$ demands $\rho_m=2\rho_\Lambda$ .

Step 2. Matter dilutes as $\rho_m=\rho_{m,0}a^{-3}$ while $\rho_\Lambda$ is constant, so $\rho_{m,0}a^{-3}=2\rho_{\Lambda,0}$ gives $a^3=\dfrac{\rho_{m,0}}{2\rho_{\Lambda,0}}=\dfrac{\Omega_{m,0}}{2\Omega_{\Lambda,0}}=\dfrac{0.31}{2(0.69)}=0.225$ .

Step 3. Take the cube root: $a_{\rm acc}=0.225^{1/3}=\mathbf{0.61}$ , hence redshift $z_{\rm acc}=1/a_{\rm acc}-1=\mathbf{0.65}$ . The cosmos coasted through deceleration for most of its history and tipped into acceleration only around $z\approx0.6$ – $0.7$ , roughly six billion years ago — comfortably within reach of the Type Ia supernovae that discovered it. The deceleration-to-acceleration transition occurs at z ≈ 0.65.

3 · Matter–radiation equality

With $\Omega_{r,0}\approx9.2\times10^{-5}$ and $\Omega_{m,0}=0.31$ , find the redshift of matter–radiation equality and the CMB temperature then.

Step 1. Set $\Omega_{r,0}a^{-4}=\Omega_{m,0}a^{-3}$ , giving $a_{\rm eq}=\Omega_{r,0}/\Omega_{m,0}=3.0\times10^{-4}$ , so $z_{\rm eq}=1/a_{\rm eq}-1\approx\mathbf{3400}$ .

Step 2. Since $T\propto a^{-1}$ , the temperature then was $T_{\rm eq}=2.725\times(1+z_{\rm eq})\approx\mathbf{9200\ K}$ , about $0.8\ \mathrm{eV}$ — radiation still ruled the dynamics, just before matter took over and structure could begin to grow. z_eq ≈ 3400, T_eq ≈ 9200 K.

Physics of the portrait

The Friedmann equations are the moment cosmology became a physical science rather than metaphysics: two lines from which the age, fate, and geometry of everything follow. They are named for a mathematician who never lived to see them vindicated, and their history is a lesson in trusting the equations over one's prejudices.

Origins · 1 minAlexander Friedmann and how this was discovered

Alexander Friedmann

1888–1925 · relativistic cosmology

Alexander Friedmann was a St. Petersburg mathematician and meteorologist — he flew reconnaissance balloons in the First World War — who came to general relativity as an outsider. In 1922 he did what Einstein had refused to do: he took the field equations at face value and asked what a homogeneous, isotropic universe must do, without assuming it was static. The answer was that $a(t)$ obeys a dynamical equation and the universe must expand or contract. Einstein's first reaction, in print, was that Friedmann had made an error; he soon retracted, conceding the solutions were correct though physically suspect. Friedmann extended the work to open and closed geometries in 1924, then died of typhoid in 1925 at thirty-seven, four years before Hubble's redshift–distance law turned his abstract $a(t)$ into observed fact. Georges Lemaître, a Belgian priest and physicist, independently rediscovered the equations in 1927 and — unlike Friedmann — tied them explicitly to the recession of the nebulae, effectively predicting Hubble's law. Einstein's cosmological constant $\Lambda$ , introduced in 1917 to force a static cosmos, was exactly the term Friedmann showed to be both unnecessary and unstable; Einstein later called it his greatest blunder. The final irony came in 1998, when supernova surveys revealed the expansion is accelerating and $\Lambda$ — the term Friedmann's dynamics had made dispensable — returned as dark energy, now the dominant component of the cosmic budget.

History

1917

— Einstein adds Λ to his field equations to build a static cosmos, and de Sitter finds an empty expanding solution

1922

— Friedmann derives the expansion equations for a matter-filled universe

1927

— Lemaître rediscovers them and connects them to the recession of galaxies

1929

— Hubble measures the velocity–distance relation, confirming expansion

1998

— Type Ia supernova surveys reveal cosmic acceleration, reinstating Λ as dark energy

Going further

Texts. Misner, Thorne & Wheeler, Gravitation, §27.1–27.3 · Wald, General Relativity, §5.1–5.3 · Carroll, Spacetime and Geometry, §8.2–8.4 · Hartle, Gravity, Ch. 18 · Schutz, A First Course in General Relativity, Ch. 12 · Weinberg, Cosmology, §1.5.

Provenance. Equations and derivations follow Carroll §8.2–8.4, MTW §27.1–27.3, Wald §5.1–5.3, Hartle Ch. 18, and Weinberg §1.5. A verifier should: (1) recompute the FLRW Christoffels, the Ricci components $R_{00}=-3\ddot a/(c^2a)$ and $R_{ij}$ , and the Ricci scalar, confirming the two field-equation components; (2) verify continuity both from $\nabla_\mu T^{\mu 0}=0$ and by differentiating the first equation; (3) check $\rho\propto a^{-3(1+w)}$ and the power laws $a\propto t^{1/2},t^{2/3}$ for $w=\tfrac13,0$ . The worked numbers are reproducible: $\rho_c=3H_0^2/8\pi G=8.5\times10^{-27}\ \mathrm{kg\,m^{-3}}\approx5$ protons m⁻³ for $H_0=67.4$ ; acceleration onset $a_{\rm acc}=(\Omega_m/2\Omega_\Lambda)^{1/3}=0.61$ , $z_{\rm acc}=0.65$ ; matter–radiation equality $z_{\rm eq}\approx3400$ ; and $q_0=\tfrac12(\Omega_m+2\Omega_r-2\Omega_\Lambda)\approx-0.53$ for $\Omega_m=0.31,\ \Omega_\Lambda=0.69$ (standard Planck-era figures).

Builds on. The FLRW Metric · The Cosmological Principle · The Scale Factor and Hubble Parameter · The Cosmological Fluid and Equation of State. Leads to. Critical Density and the Density Parameters · The Radiation-Dominated Era · The Matter-Dominated Era · The Dark-Energy-Dominated Era · Cosmological Horizons.

Ask the tutor: Want me to run the Friedmann equation forward for a universe of your choosing — pick the density parameters and I'll trace its expansion, fate, and age?

Why is the continuity equation not an independent third equation?

Because energy–momentum conservation is baked into general relativity itself. The Einstein tensor obeys the contracted Bianchi identity $\nabla_\mu G^{\mu\nu}=0$ as a pure geometric identity, true for any metric whatsoever. Since the field equations set $G_{\mu\nu}+\Lambda g_{\mu\nu}\propto T_{\mu\nu}$ , they immediately force $\nabla_\mu T^{\mu\nu}=0$ on the matter. On FLRW the time component of that conservation law is the continuity equation. Concretely, if you differentiate the first Friedmann equation and substitute the acceleration equation, the continuity equation reappears exactly. So you solve any two of the three and use the third as a consistency check — a useful guard against algebra errors.

What does the curvature constant k actually mean physically?

It labels the geometry of space and, through the energy-budget reading, the fate of the expansion. With $k=+1$ the spatial slices are closed like the surface of a 3-sphere — finite volume, angles of a large triangle summing to more than 180°; such a universe is gravitationally bound and (absent $\Lambda$ ) recollapses. With $k=0$ space is flat and infinite, the marginally bound knife-edge where the expansion just barely coasts to a halt at infinity. With $k=-1$ space is open and hyperbolic, with surplus kinetic energy, and expands forever. Observationally the curvature enters as $\Omega_{\rm tot}-1=kc^2/(a^2H^2)$ , and CMB measurements give $\Omega_{\rm tot}=1.00\pm0.005$ , so our universe is flat to within half a percent.

How can pressure cause gravity — isn't pressure what pushes things apart?

This is the deepest counterintuition in the subject. In a laboratory, pressure pushes because of pressure gradients — differences from place to place. The cosmic fluid is perfectly uniform, so there are no gradients and no mechanical push. What remains is pressure's contribution to gravity: in general relativity every form of energy and momentum gravitates, including the momentum flux that is pressure. The acceleration equation's source is $\rho c^2+3p$ , so positive pressure adds to the gravitating mass and pulls the expansion inward harder. Only a negative pressure large enough to flip the sign of $\rho c^2+3p$ can push outward gravitationally — and that repulsive gravity, not any mechanical push, is what drives cosmic acceleration.

Can the universe be perfectly static, as Einstein first wanted?

Only by fine-tuning, and even then unstably. For a matter-filled universe $\rho+3p/c^2>0$ , so the bare acceleration equation gives $\ddot a<0$ : it must decelerate and collapse. Einstein introduced $\Lambda$ in 1917 precisely to cancel this, tuning $\Lambda=4\pi G\rho/c^2$ in a closed geometry so that $\ddot a=0$ and $\dot a=0$ together. But this static universe balances on a razor: because the terms scale differently with $a$ , a tiny increase in $\rho$ tips it toward collapse and a tiny decrease toward runaway expansion. Friedmann's dynamics exposed the instability, and once Hubble showed the universe was expanding, the motivation for a static model — and for that value of $\Lambda$ — evaporated.

Why does radiation give a slower expansion, a proportional-to-t-to-the-half law, than matter?

Two effects compound. First, radiation's pressure makes it gravitate harder ( $\rho+3p/c^2=2\rho$ versus $\rho$ for dust), so it decelerates more strongly. Second, radiation dilutes faster: $\rho\propto a^{-4}$ against $a^{-3}$ for matter, because each photon both shares the growing volume and loses energy to redshift. Solving the flat first Friedmann equation with $\rho\propto a^{-3(1+w)}$ gives $a\propto t^{2/[3(1+w)]}$ — that is $t^{1/2}$ for radiation ( $w=\tfrac13$ ) and $t^{2/3}$ for matter ( $w=0$ ). The radiation era is thus both denser and more sharply decelerating early on, exactly the regime that set the light-element abundances.

Is the first Friedmann equation really just Newtonian energy conservation?

Almost, and the coincidence is instructive. Multiply it by $\tfrac12 a^2$ and it reads $\tfrac12\dot a^2-\tfrac{4\pi G}{3}\rho a^2=-\tfrac12 kc^2$ , identical in form to the energy equation for a test mass at the edge of a uniform expanding ball: kinetic plus potential energy equals a constant, with $-\tfrac12 kc^2$ as the total energy, and a flat universe as the zero-energy case. But the match is partly luck: the Newtonian argument cannot justify treating an infinite homogeneous medium this way, misses the pressure source in the acceleration equation, and cannot produce the curvature or $\Lambda$ terms. The general-relativistic derivation supplies all of that; the Newtonian picture is a mnemonic that happens to get the first equation's form exactly right.