Friedmann Equations from the FLRW Metric

Derivation

Friedmann Equations from the FLRW Metric

D-399 Home PU-403 Threads energy · fields · symmetry Depends on Field Equations from the Einstein-Hilbert Action, Stress-Energy Tensor as the Conserved Source

Statement

Substituting the maximally symmetric Friedmann–Lemaître–Robertson–Walker (FLRW) line element and a comoving perfect fluid into the Einstein field equations \(G_{\mu\nu}+\Lambda g_{\mu\nu}=\frac{8\pi G}{c^4}T_{\mu\nu}\) reduces the ten coupled field equations to two independent ordinary differential equations for the single scale factor \(a(t)\): the constraint (energy) equation \(\left(\dot a/a\right)^2=\frac{8\pi G}{3}\rho-\frac{kc^2}{a^2}+\frac{\Lambda c^2}{3}\) and the evolution (acceleration) equation \(\ddot a/a=-\frac{4\pi G}{3}\left(\rho+3p/c^2\right)+\frac{\Lambda c^2}{3}\).

Why it matters

The Friedmann equations are physical cosmology at the background level: every statement about the age of the Universe, the expansion history \(H(z)\), the critical density, big-bang nucleosynthesis timing, and the eventual fate of the cosmos is a solution of these two equations for a specified matter content. They convert the geometric content of general relativity into a Newtonian-looking energy balance for a single scalar degree of freedom.

They also demonstrate the power of symmetry reduction. Imposing spatial homogeneity and isotropy collapses the full nonlinear Einstein system — in general an intractable set of coupled PDEs for ten metric functions — into two ODEs. Cosmology is tractable precisely because the Copernican principle is so restrictive.

Assumptions

Spatial homogeneity and isotropy about every point (Cosmological Principle).If dropped, the metric acquires position-dependent and directional structure; one leaves FLRW for Bianchi or inhomogeneous (Lemaître–Tolman–Bondi) models, and no single \(a(t)\) suffices.

Matter is a perfect fluid, \(T_{\mu\nu}=(\rho+p/c^2)u_\mu u_\nu+p\,g_{\mu\nu}\), comoving with the cosmic rest frame.If anisotropic stress or bulk momentum flux is present, \(T_{0i}\neq0\) and the off-diagonal field equations no longer vanish, forbidding the FLRW ansatz.

A four-dimensional pseudo-Riemannian spacetime with a metric-compatible, torsion-free (Levi-Civita) connection.If dropped, the Ricci tensor used below is not the connection's curvature; in torsionful or metric-affine theories extra terms enter and the identifications \(G_{tt}\leftrightarrow\rho\) change.

The cosmological constant \(\Lambda\) is a fixed geometric term (equivalently, vacuum energy with \(w=-1\)).If \(\Lambda\) is promoted to a dynamical field (quintessence), it must be carried inside \(T_{\mu\nu}\) with its own equation of motion, and the constant-\(\Lambda\) term below is replaced by an evolving component.

Derivation

Convention. Throughout the derivation we set \(c=1\) to keep the tensor algebra clean, restoring factors of \(c\) by dimensional analysis in the boxed results. Coordinates are \((t,r,\theta,\phi)\); overdot is \(d/dt\).

\[ ds^2=-dt^2+a(t)^2\left[\frac{dr^2}{1-kr^2}+r^2\left(d\theta^2+\sin^2\theta\,d\phi^2\right)\right] \]

The unique (up to normalisation) metric with maximally symmetric spatial slices of constant curvature \(k\in\{-1,0,+1\}\); \(a(t)\) is the only free function permitted by homogeneity and isotropy. A

\[ \Gamma^{t}{}_{ij}=a\dot a\,\gamma_{ij},\qquad \Gamma^{i}{}_{tj}=\frac{\dot a}{a}\,\delta^{i}{}_{j},\qquad \Gamma^{i}{}_{jk}=\tilde\Gamma^{i}{}_{jk} \]

Christoffel symbols \(\Gamma^\lambda{}_{\mu\nu}=\tfrac12 g^{\lambda\sigma}(\partial_\mu g_{\sigma\nu}+\partial_\nu g_{\sigma\mu}-\partial_\sigma g_{\mu\nu})\); here \(\gamma_{ij}\) is the comoving 3-metric and \(\tilde\Gamma\) its own connection. Only the time-mixing terms carry \(a\). B

\[ R_{tt}=-3\frac{\ddot a}{a},\qquad R_{ij}=\left(\frac{\ddot a}{a}+2\frac{\dot a^2}{a^2}+2\frac{k}{a^2}\right)g_{ij} \]

Contract the Riemann tensor, \(R_{\mu\nu}=\partial_\lambda\Gamma^{\lambda}{}_{\mu\nu}-\partial_\nu\Gamma^{\lambda}{}_{\mu\lambda}+\Gamma^{\lambda}{}_{\lambda\sigma}\Gamma^{\sigma}{}_{\mu\nu}-\Gamma^{\lambda}{}_{\nu\sigma}\Gamma^{\sigma}{}_{\mu\lambda}\). The 3-space curvature contributes the \(2k/a^2\) term; isotropy forces \(R_{ij}\propto g_{ij}\) and \(R_{ti}=0\). C

\[ R=g^{\mu\nu}R_{\mu\nu}=6\left(\frac{\ddot a}{a}+\frac{\dot a^2}{a^2}+\frac{k}{a^2}\right) \]

Trace with \(g^{tt}=-1,\ g^{ij}R_{ij}=3(\ddot a/a+2\dot a^2/a^2+2k/a^2)\); the two contributions add. B

\[ G_{tt}=R_{tt}-\tfrac12 R\,g_{tt}=3\left(\frac{\dot a^2}{a^2}+\frac{k}{a^2}\right) \]

Definition \(G_{\mu\nu}=R_{\mu\nu}-\tfrac12 R g_{\mu\nu}\) with \(g_{tt}=-1\). The \(\ddot a\) terms cancel — the \(tt\) Einstein equation is a first-order constraint, not an evolution equation. B

\[ u^\mu=(1,0,0,0),\quad u_\mu=(-1,0,0,0)\ \Rightarrow\ T_{tt}=\rho,\qquad T_{ij}=p\,g_{ij} \]

A perfect fluid at rest in comoving coordinates: substitute the normalised timelike four-velocity into \(T_{\mu\nu}=(\rho+p)u_\mu u_\nu+p g_{\mu\nu}\). Off-diagonal components vanish, matching \(G_{\mu\nu}\). B

\[ 3\frac{\dot a^2+k}{a^2}-\Lambda=8\pi G\,\rho \;\;\Longrightarrow\;\; \frac{\dot a^2}{a^2}=\frac{8\pi G}{3}\rho-\frac{k}{a^2}+\frac{\Lambda}{3} \]

The \(tt\) field equation \(G_{tt}+\Lambda g_{tt}=8\pi G\,T_{tt}\) with \(g_{tt}=-1\); divide by 3. This is the first Friedmann (constraint) equation. A

\[ G_{ij}=\left(-2\frac{\ddot a}{a}-\frac{\dot a^2}{a^2}-\frac{k}{a^2}\right)g_{ij} \]

Insert \(R_{ij}\) and \(R\) into \(G_{ij}=R_{ij}-\tfrac12 R g_{ij}\); all three spatial equations collapse to one scalar equation by isotropy. C

\[ -2\frac{\ddot a}{a}-\frac{\dot a^2}{a^2}-\frac{k}{a^2}+\Lambda=8\pi G\,p \]

The \(ij\) field equation \(G_{ij}+\Lambda g_{ij}=8\pi G\,T_{ij}\) with \(T_{ij}=p g_{ij}\); cancel the common factor \(g_{ij}\). B

\[ -2\frac{\ddot a}{a}-\Big(\tfrac{8\pi G}{3}\rho+\tfrac{\Lambda}{3}\Big)+\Lambda=8\pi G p \;\;\Longrightarrow\;\; \frac{\ddot a}{a}=-\frac{4\pi G}{3}\left(\rho+3p\right)+\frac{\Lambda}{3} \]

Eliminate \(\dot a^2/a^2+k/a^2\) using Step 7, then solve for \(\ddot a/a\). This is the second Friedmann (acceleration) equation. A

\[ \nabla_\mu T^{\mu}{}_{t}=0\;\;\Longrightarrow\;\; \dot\rho+3\frac{\dot a}{a}\left(\rho+p\right)=0 \]

The contracted Bianchi identity \(\nabla_\mu G^{\mu\nu}=0\) forces \(\nabla_\mu T^{\mu\nu}=0\); its time component is the fluid continuity equation. It is not independent — differentiating Step 7 and using Step 10 reproduces it — so only two of the three equations are dynamically independent. B

Result

\[ \left(\frac{\dot a}{a}\right)^2=\frac{8\pi G}{3}\rho-\frac{kc^2}{a^2}+\frac{\Lambda c^2}{3},\qquad \frac{\ddot a}{a}=-\frac{4\pi G}{3}\left(\rho+\frac{3p}{c^2}\right)+\frac{\Lambda c^2}{3} \]

Reading. The first equation says the squared expansion rate \(H^2\equiv(\dot a/a)^2\) is sourced additively by mass–energy density, spatial curvature (acting like a negative energy for \(k>0\)), and the cosmological constant. The second says gravity decelerates expansion whenever \(\rho+3p/c^2>0\): pressure gravitates, so radiation (\(p=\rho c^2/3\)) decelerates more strongly than dust at equal density, while any component with \(p<-\rho c^2/3\) (e.g. \(\Lambda\), \(w=-1\)) drives acceleration.

Units check. \([G\rho]=(\mathrm{m^3\,kg^{-1}\,s^{-2}})(\mathrm{kg\,m^{-3}})=\mathrm{s^{-2}}=[\dot a/a]^2\). With \(a\) dimensionless and \(k\) carrying \(\mathrm{m^{-2}}\), \([kc^2/a^2]=\mathrm{m^{-2}\cdot m^2\,s^{-2}}=\mathrm{s^{-2}}\); and \([\Lambda c^2]=\mathrm{m^{-2}\cdot m^2\,s^{-2}}=\mathrm{s^{-2}}\). All three terms share units \(\mathrm{s^{-2}}\). In the acceleration equation \(p/c^2\) has units \(\mathrm{kg\,m^{-3}}\), matching \(\rho\).

Limiting cases

Flat, matter-dominated (Einstein–de Sitter): \(k=\Lambda=p=0\), \(\rho\propto a^{-3}\Rightarrow a\propto t^{2/3}\), \(H=2/(3t)\).
Radiation-dominated: \(p=\rho c^2/3\), \(\rho\propto a^{-4}\Rightarrow a\propto t^{1/2}\), \(H=1/(2t)\).
Vacuum / de Sitter: \(\rho=p=k=0\), \(\Lambda>0\Rightarrow H=c\sqrt{\Lambda/3}\) constant, \(a\propto e^{Ht}\).
Empty curved (Milne): \(\rho=p=\Lambda=0\), \(k=-1\Rightarrow \dot a^2=c^2\), \(a\propto t\); flat spacetime in disguise.
Static (Einstein) universe: \(\dot a=\ddot a=0\) forces \(\Lambda c^2=4\pi G(\rho+3p/c^2)\) and \(kc^2/a^2=\Lambda c^2/3+\ldots\); unstable to perturbation.

Breaks when

Inhomogeneity becomes significant. On scales below \(\sim100\,\mathrm{Mpc}\) (galaxies, clusters, voids) the density field is not homogeneous; FLRW describes only the smoothed background, and structure requires perturbation theory or full numerical relativity on top of it.
Planck-scale curvature (\(t\to0\)). As \(a\to0\) the classical equations predict a curvature singularity \(H,\rho\to\infty\); at \(\rho\sim c^5/(\hbar G^2)\) quantum-gravitational effects invalidate the classical Einstein equations entirely.
Anisotropic stress or vorticity present. Free-streaming neutrinos, magnetic fields, or rotation give \(T_{\mu\nu}\) off-diagonal/shear terms that violate the perfect-fluid ansatz, so the reduction to a single \(a(t)\) fails (Bianchi cosmologies needed).

Failure modes

Dropping the \(3p\) in the acceleration equation. Students import Newtonian intuition (\(\ddot a\propto-\rho\)) and forget that pressure gravitates; this loses radiation's extra deceleration and the entire mechanism of accelerated expansion.
Treating the two Friedmann equations plus continuity as three independent equations. Only two are independent (Bianchi identity); over-counting leads to an "over-determined" system and spurious constraints.
Confusing \(k=\pm1\) (comoving curvature label) with the density parameter \(\Omega\). The sign of \(k\) is fixed by \(\Omega_{\rm tot}-1\), but \(k\) itself is a discrete topological label, not a continuously tunable number.
Using energy density \(\rho c^2\) where mass density \(\rho\) is required (or vice versa). A stray \(c^2\) between \(T_{tt}=\rho c^2\) and the \(\rho\) in \(H^2=\frac{8\pi G}{3}\rho\) is the most common dimensional slip.
Assuming \(k/a^2\) redshifts like matter. The curvature term scales as \(a^{-2}\), radiation as \(a^{-4}\), matter as \(a^{-3}\), \(\Lambda\) as \(a^{0}\); mislabelling the scaling wrecks the expansion history.

Discussion

The deepest structural fact is that the \(tt\) equation is a constraint, not an evolution equation — it contains \(\dot a\) but not \(\ddot a\). This is the cosmological face of the general relativistic split between constraint and evolution: the Hamiltonian constraint of the ADM formulation, restricted to FLRW, is exactly the first Friedmann equation. Initial data \((a,\dot a,\rho)\) cannot be chosen freely; they must satisfy this constraint, which the acceleration equation and continuity then propagate consistently in time.

Pressure appearing as a source of gravity, on equal footing with density, is the single most counter-Newtonian feature. In Newtonian gravity only mass sources the field; in GR the active gravitational mass density is \(\rho+3p/c^2\). This is why a positive-pressure early Universe decelerates, and why a negative-pressure vacuum accelerates it — the observation that grounded the discovery of dark energy in 1998.

One can recover a Newtonian caricature: for a uniform dust ball of density \(\rho\) and radius \(R=a\chi\), energy conservation of a shell gives \(\tfrac12\dot R^2-\tfrac{4\pi G}{3}\rho R^2=\text{const}\), which rearranges to \((\dot a/a)^2=\frac{8\pi G}{3}\rho-\frac{k c^2}{a^2}\) with the integration constant playing the role of curvature. This "Newtonian cosmology" reproduces the matter term and even the curvature term, but it cannot generate the \(3p\) pressure source or the \(\Lambda\) term — those are irreducibly relativistic. The agreement is a coincidence of the pressureless, \(\Lambda=0\) case, and it misleads students into thinking cosmology is Newtonian.

Common misconceptions. The expansion is not galaxies moving through space with a well-defined center; it is the growth of \(a(t)\) itself, and there is no preferred spatial origin. "The Universe expands into..." is meaningless within FLRW — the manifold is all there is. And \(k=+1\) (closed) does not by itself guarantee recollapse once \(\Lambda>0\): fate is set by the full energy budget, not curvature alone.

Worked examples

Example 1 — Critical density and the Einstein–de Sitter expansion law.

\[ k=0,\ \Lambda=0:\qquad H^2=\frac{8\pi G}{3}\rho \;\Rightarrow\; \rho_c=\frac{3H_0^2}{8\pi G} \]

Set curvature and \(\Lambda\) to zero in the first Friedmann equation and solve for the density that makes the geometry flat. A

\[ H_0=70\,\tfrac{\mathrm{km/s}}{\mathrm{Mpc}}=\frac{7.0\times10^4\,\mathrm{m\,s^{-1}}}{3.086\times10^{22}\,\mathrm{m}}=2.27\times10^{-18}\,\mathrm{s^{-1}} \]

Convert to SI so \(G\) can be used directly. A

\[ \rho_c=\frac{3\,(2.27\times10^{-18})^2}{8\pi\,(6.674\times10^{-11})}=\frac{1.55\times10^{-35}}{1.677\times10^{-9}} \]

Insert numbers with units; numerator \(\mathrm{s^{-2}}\), denominator \(\mathrm{m^3\,kg^{-1}\,s^{-2}}\), quotient \(\mathrm{kg\,m^{-3}}\). B

\[ \rho_c\approx9.2\times10^{-27}\,\mathrm{kg\,m^{-3}}\approx5.5\ \frac{m_p}{\mathrm{m^3}} \]

Reading. Flatness demands only about five hydrogen atoms per cubic metre — the Universe is extraordinarily dilute. For dust \(\rho\propto a^{-3}\), the same equation integrates to \(a\propto t^{2/3}\), giving \(H=2/(3t)\) and an Einstein–de Sitter age \(t_0=2/(3H_0)\approx9.3\,\mathrm{Gyr}\).

Units check. \(\mathrm{s^{-2}}/(\mathrm{m^3\,kg^{-1}\,s^{-2}})=\mathrm{kg\,m^{-3}}\), a mass density. ✓

Example 2 — Present deceleration parameter for \(\Lambda\)CDM.

\[ q\equiv-\frac{\ddot a\,a}{\dot a^2}=-\frac{\ddot a/a}{H^2}=\frac{1}{2}\sum_i\Omega_i\left(1+3w_i\right) \]

Divide the acceleration equation by \(H^2\) and use \(\Omega_i=\rho_i/\rho_c\), \(p_i=w_i\rho_i c^2\); \(\Lambda\) enters as a component with \(w_\Lambda=-1\). B

\[ \Omega_m=0.30\ (w=0),\quad \Omega_\Lambda=0.70\ (w=-1),\quad \Omega_r\approx0 \]

Adopt the concordance flat-model budget \(\Omega_m+\Omega_\Lambda=1\); radiation is negligible today. A

\[ q_0=\tfrac12\big[\,0.30(1+0)+0.70(1-3)\,\big]=\tfrac12\big[0.30-1.40\big] \]

Substitute \(1+3w=1\) for dust and \(1+3w=-2\) for \(\Lambda\). A

\[ q_0=-0.55\;<\;0 \]

Reading. A negative deceleration parameter means \(\ddot a>0\): the expansion is accelerating today. The dark-energy term (\(-1.40\)) overwhelms the matter term (\(+0.30\)), consistent with the 1998 supernova result. The transition \(q=0\) occurred at redshift \(z\approx0.6\), when \(\Omega_m(z)(1)+\Omega_\Lambda(z)(-2)=0\).

Units check. \(q\) is a ratio of \((\mathrm{s^{-2}})/(\mathrm{s^{-2}})\), dimensionless. ✓

Problems

Starting from the two Friedmann equations, derive the fluid continuity equation \(\dot\rho+3\tfrac{\dot a}{a}(\rho+p/c^2)=0\), confirming that it is not independent.
Solution
Differentiate the first equation (with \(c=1\)) \(\dot a^2=\tfrac{8\pi G}{3}\rho a^2-k+\tfrac{\Lambda}{3}a^2\) in time: \(2\dot a\ddot a=\tfrac{8\pi G}{3}(\dot\rho a^2+2\rho a\dot a)+\tfrac{2\Lambda}{3}a\dot a\). Divide by \(2a\dot a\): \(\tfrac{\ddot a}{a}=\tfrac{8\pi G}{3}\big(\tfrac{\dot\rho a}{2\dot a}+\rho\big)+\tfrac{\Lambda}{3}\). Set equal to the acceleration equation \(\tfrac{\ddot a}{a}=-\tfrac{4\pi G}{3}(\rho+3p)+\tfrac{\Lambda}{3}\). The \(\Lambda\) terms cancel; solving gives \(\tfrac{8\pi G}{3}\cdot\tfrac{\dot\rho a}{2\dot a}=-\tfrac{4\pi G}{3}(\rho+3p)-\tfrac{8\pi G}{3}\rho=-\tfrac{4\pi G}{3}(3\rho+3p)\). Hence \(\tfrac{\dot\rho a}{2\dot a}=-\tfrac12\cdot3(\rho+p)\), i.e. \(\dot\rho=-3\tfrac{\dot a}{a}(\rho+p)\). Restoring \(c\): \(\dot\rho+3\tfrac{\dot a}{a}(\rho+p/c^2)=0\). ∎
Compute the critical density and Einstein–de Sitter age for \(H_0=67\,\mathrm{km\,s^{-1}\,Mpc^{-1}}\).
Solution
\(H_0=6.7\times10^4/3.086\times10^{22}=2.17\times10^{-18}\,\mathrm{s^{-1}}\). \(\rho_c=3H_0^2/(8\pi G)=3(2.17\times10^{-18})^2/(1.677\times10^{-9})=1.413\times10^{-35}/1.677\times10^{-9}=8.4\times10^{-27}\,\mathrm{kg\,m^{-3}}\). Age (matter only) \(t_0=2/(3H_0)=2/(3\cdot2.17\times10^{-18})=3.07\times10^{17}\,\mathrm{s}=9.7\,\mathrm{Gyr}\). (The true \(\Lambda\)CDM age, \(\sim13.8\,\mathrm{Gyr}\), is longer because \(\Lambda\) slows early expansion.)
Use continuity plus the equation of state \(p=w\rho c^2\) (constant \(w\)) to show \(\rho\propto a^{-3(1+w)}\), and read off the scaling for dust, radiation, and \(\Lambda\).
Solution
Continuity: \(\dot\rho/\rho=-3(\dot a/a)(1+w)\). Integrate: \(\ln\rho=-3(1+w)\ln a+\text{const}\), so \(\rho\propto a^{-3(1+w)}\). Dust \(w=0\Rightarrow\rho\propto a^{-3}\); radiation \(w=1/3\Rightarrow\rho\propto a^{-4}\) (the extra power is cosmological redshift of each quantum's energy); cosmological constant \(w=-1\Rightarrow\rho\propto a^{0}=\text{const}\), a genuinely constant vacuum energy density.
For a flat radiation-dominated universe (\(k=\Lambda=0\), \(\rho\propto a^{-4}\)), solve the first Friedmann equation for \(a(t)\) and give \(H(t)\).
Solution
\(\dot a^2/a^2=\tfrac{8\pi G}{3}\rho=C a^{-4}\) with \(C\) constant, so \(\dot a=\sqrt{C}\,a^{-1}\), i.e. \(a\,\dot a=\sqrt{C}\). Integrate: \(\tfrac12 a^2=\sqrt{C}\,t\Rightarrow a\propto t^{1/2}\). Then \(H=\dot a/a=\tfrac{1}{2t}\). The energy density evolves as \(\rho=3H^2/(8\pi G)=3/(32\pi G t^2)\), diverging as \(t\to0\).
A spatially flat de Sitter universe has \(\rho=p=k=0\) and \(\Lambda=1.1\times10^{-52}\,\mathrm{m^{-2}}\). Find the constant Hubble rate \(H\) and the e-folding time \(1/H\).
Solution
First Friedmann reduces to \(H^2=\Lambda c^2/3\). \(\sqrt{\Lambda/3}=\sqrt{1.1\times10^{-52}/3}=\sqrt{3.67\times10^{-53}}=6.05\times10^{-27}\,\mathrm{m^{-1}}\). Multiply by \(c=3.0\times10^8\,\mathrm{m\,s^{-1}}\): \(H=1.82\times10^{-18}\,\mathrm{s^{-1}}\). E-folding time \(1/H=5.5\times10^{17}\,\mathrm{s}\approx17\,\mathrm{Gyr}\). The scale factor grows as \(a\propto e^{Ht}\); every \(17\,\mathrm{Gyr}\) it multiplies by \(e\).