= Solution
The <cosmological principle> idealizes spatial slices as homogeneous and isotropic. The corresponding <Friedmann-Lemaître-Robertson-Walker metric> can be written
$$
ds^2=c^2dt^2-a(t)^2\left[\frac{d\chi^2}{1-K\chi^2}
+\chi^2(d\theta^2+\sin^2\theta\,d\phi^2)\right],\qquad K=+1,0,-1.
$$
Here $t$ 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 $1+z=a(t_{\rm obs})/a(t_{\rm em})$.
For a <perfect fluid> of mass-equivalent <energy density> $\rho$ and <pressure> $p$, Einstein's equations give
$$
\boxed{H^2=\frac{8\pi G\rho}{3}-\frac{Kc^2}{a^2}+\frac{\Lambda c^2}{3},}
$$
$$
\boxed{\frac{\ddot a}{a}=-\frac{4\pi G}{3}\left(\rho+\frac{3p}{c^2}\right)
+\frac{\Lambda c^2}{3},\qquad
\dot\rho+3H\left(\rho+\frac p{c^2}\right)=0.}
$$
The 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 $H=c\sqrt{\Lambda/3}$.
\Image[/past-exam-of-the-relativity-course-of-the-university-of-cambridge/2015/paper-1-flrw-expansion.png]
{title=Original comparison of normalized scale factors in flat radiation, matter and positive-vacuum models; the vacuum model has no finite-time big-bang origin in this chart}
\b[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 $\Omega_K=-Kc^2/(a_0^2H_0^2)$. These models describe the large-scale background, while local structure and inhomogeneities require perturbations or more general metrics.
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