 Solution

The cosmological principle idealizes spatial slices as homogeneous and isotropic. The corresponding Friedmann-Lemaître-Robertson-Walker metric can be writtenHere $t$ is cosmic time, $a(t)$ has dimensions of length in this convention, and $\chi$ is a dimensionless comoving coordinate. The sign of $K$ gives positive, zero or negative spatial curvature. It does not alone determine global topology. A comoving pair at fixed radial coordinates has proper separation scaling with $a(t)$, so the Hubble parameter is $H=\dot{a}/a$, and cosmological redshift satisfies $1+z=a(t_{\mathrm{obs}})/a(t_{\mathrm{em}})$.

For a perfect fluid of mass-equivalent energy density $\rho$ and pressure $p$, Einstein's equations giveThe last equation is local energy conservation for constant $\Lambda$, with vacuum energy kept separate from $\rho ,p$. A component with $p=w\rho c^2$ has $\rho \propto a^{-3(1+w)}$: pressureless matter dilutes as $a^{-3}$ and radiation as $a^{-4}$, the additional radiation factor being photon redshift. With $K=\Lambda =0$, matter domination gives $a\propto t^{2/3}$ and radiation domination gives $a\propto t^{1/2}$. Positive vacuum energy alone gives exponential expansion, $a\propto e^{Ht}$ with $H=c\sqrt{\Lambda /3}$.

Geometry, contents and initial conditions together determine the expansion history. A closed matter-only model recollapses; an open matter-only model expands indefinitely. With a cosmological constant, positive spatial curvature no longer by itself implies recollapse. Present critical density is ${\rho }_c=3H_0^2/(8\pi G)$ and the density parameters obey ${\Omega }_m+{\Omega }_r+{\Omega }_{\Lambda }+{\Omega }_K=1$, where ${\Omega }_K=-K{c}^2/(a_0^2{H}_0^2)$. These models describe the large-scale background, while local structure and inhomogeneities require perturbations or more general metrics.