The Friedmann-Lemaître-Robertson-Walker Universe
Derivation of the Friedmann equations from the Einstein field equations applied to a homogeneous, isotropic spacetime filled with a perfect fluid
Overview
The Friedmann-Lemaître-Robertson-Walker (FLRW) metric is the most general metric consistent with spatial homogeneity and isotropy — the Cosmological Principle. From it, Einstein's field equations yield the Friedmann equations that govern the expansion history of the universe. These equations form the backbone of modern physical cosmology.
We derive the FLRW metric, compute its Christoffel symbols and Ricci tensor, insert a perfect-fluid stress-energy tensor, and obtain all three fundamental equations: the two Friedmann equations and the continuity equation. We then solve them for radiation, matter, and cosmological-constant domination.
1. The FLRW Metric
By the Cosmological Principle, the spatial part of the universe is a maximally symmetric 3-space of constant curvature \(k\). The most general line element is:
FLRW Line Element
$$ds^2 = -dt^2 + a^2(t)\left[\frac{dr^2}{1 - kr^2} + r^2\,d\Omega^2\right]$$
Here \(a(t)\) is the scale factor, \(k = -1, 0, +1\) labels open, flat, or closed spatial geometry, and \(d\Omega^2 = d\theta^2 + \sin^2\theta\,d\phi^2\).
The coordinate \(r\) is dimensionless (or carries units of length depending on convention); all physical distances scale with \(a(t)\). The Hubble parameter is defined as:
$$H(t) \equiv \frac{\dot{a}}{a}$$
2. Christoffel Symbols of the FLRW Metric
The non-vanishing Christoffel symbols \(\Gamma^\alpha{}_{\beta\gamma}\) are computed from \(\Gamma^\alpha{}_{\beta\gamma} = \frac{1}{2}g^{\alpha\delta}(\partial_\beta g_{\gamma\delta} + \partial_\gamma g_{\beta\delta} - \partial_\delta g_{\beta\gamma})\). For the FLRW metric the key results are:
$$\Gamma^0{}_{ij} = a\dot{a}\,\tilde{g}_{ij}, \qquad \Gamma^i{}_{0j} = H\,\delta^i{}_j$$
where \(\tilde{g}_{ij}\) is the spatial metric divided by \(a^2\). Additionally, the purely spatial Christoffel symbols are those of the 3-space of constant curvature \(k\).
3. Ricci Tensor and Scalar
The Ricci tensor \(R_{\mu\nu}\) follows from the Christoffel symbols. The time-time and space-space components are:
$$R_{00} = -3\frac{\ddot{a}}{a}$$
$$R_{ij} = \left(\frac{\ddot{a}}{a} + 2H^2 + \frac{2k}{a^2}\right)a^2\,\tilde{g}_{ij}$$
The Ricci scalar is the trace:
$$R = 6\left(\frac{\ddot{a}}{a} + H^2 + \frac{k}{a^2}\right)$$
4. The Perfect Fluid and Einstein's Equations
The energy-momentum tensor of a perfect fluid comoving with the Hubble flow is:
$$T^{\mu}{}_{\nu} = \mathrm{diag}(-\rho,\, p,\, p,\, p)$$
Einstein's field equations with cosmological constant read:
$$G_{\mu\nu} + \Lambda g_{\mu\nu} = 8\pi G\, T_{\mu\nu}$$
Inserting the FLRW Ricci tensor and scalar into the Einstein tensor \(G_{\mu\nu} = R_{\mu\nu} - \frac{1}{2}Rg_{\mu\nu}\)and matching components yields the Friedmann equations.
5. The Friedmann Equations
The \((0,0)\) component gives the first Friedmann equation:
First Friedmann Equation
$$\boxed{H^2 = \frac{8\pi G}{3}\rho - \frac{k}{a^2} + \frac{\Lambda}{3}}$$
This is the energy constraint: it relates the expansion rate to the total energy density, spatial curvature, and cosmological constant.
The \((i,j)\) components, combined with the first equation, give the second Friedmann equation (the acceleration equation):
Second Friedmann Equation (Acceleration)
$$\boxed{\frac{\ddot{a}}{a} = -\frac{4\pi G}{3}(\rho + 3p) + \frac{\Lambda}{3}}$$
Note the factor \((\rho + 3p)\): pressure contributes to gravity in general relativity. This is why a positive cosmological constant (with \(p_\Lambda = -\rho_\Lambda\)) drives accelerated expansion.
6. The Continuity Equation
Energy-momentum conservation \(\nabla_\mu T^{\mu 0} = 0\) gives:
Fluid Continuity Equation
$$\boxed{\dot{\rho} + 3H(\rho + p) = 0}$$
This is the cosmological analogue of the first law of thermodynamics: \(dE + p\,dV = 0\)applied to a comoving volume \(V \propto a^3\).
Only two of the three equations (first Friedmann, second Friedmann, continuity) are independent; any one can be derived from the other two.
7. Solutions for Single-Component Universes
For a barotropic fluid with equation of state \(p = w\rho\), the continuity equation integrates to:
$$\rho \propto a^{-3(1+w)}$$
Inserting into the first Friedmann equation for a flat (\(k = 0\)),\(\Lambda = 0\) universe gives \(a \propto t^{2/[3(1+w)]}\) for\(w \neq -1\).
7.1 Radiation Domination (\(w = 1/3\))
$$\rho_r \propto a^{-4}, \qquad a(t) \propto t^{1/2}, \qquad H = \frac{1}{2t}$$
The extra factor of \(a^{-1}\) beyond the \(a^{-3}\) dilution arises from the cosmological redshift of each photon's energy.
7.2 Matter Domination (\(w = 0\))
$$\rho_m \propto a^{-3}, \qquad a(t) \propto t^{2/3}, \qquad H = \frac{2}{3t}$$
This is the Einstein-de Sitter universe. It decelerates with \(\ddot{a} < 0\)since matter has \(\rho + 3p = \rho > 0\).
7.3 Cosmological Constant Domination (\(w = -1\))
$$\rho_\Lambda = \text{const}, \qquad a(t) \propto e^{Ht}, \qquad H = \sqrt{\Lambda/3}$$
The de Sitter solution: exponential expansion. This is the late-time attractor of our universe and also the model for inflation in the early universe.