Cosmology studies the universe as a whole by combining geometry, gravitation, and astrophysical observables. At advanced level, the central goal is to infer the spacetime metric and matter content from data such as distance-redshift relations, anisotropies, and large-scale structure.

On large scales the universe is well approximated by a spatially homogeneous and isotropic metric, the Friedmann-Lemaître-Robertson-Walker metric: $$ds^2 = -c^2 dt^2 + a^2(t)\left[\frac{dr^2}{1-kr^2} + r^2 d\Omega^2\right].$$ The scale factor $a(t)$ encodes the expansion history, while $k$ labels the spatial curvature.

For a perfect fluid with energy density $\rho$, pressure $p$, and cosmological constant $\Lambda$, Einstein's equations yield $$H^2 = \left(\frac{\dot a}{a}\right)^2 = \frac{8\pi G}{3}\rho - \frac{kc^2}{a^2} + \frac{\Lambda c^2}{3}$$ and $$\frac{\ddot a}{a} = -\frac{4\pi G}{3}\left(\rho + \frac{3p}{c^2}\right) + \frac{\Lambda c^2}{3}.$$ The continuity equation follows from stress-energy conservation: $$\dot\rho + 3H\left(\rho + \frac{p}{c^2}\right)=0.$$

Redshift satisfies $$1+z = \frac{a(t_0)}{a(t_e)}.$$ Cosmological distances differ: comoving distance, proper distance, luminosity distance, and angular-diameter distance are related but distinct, and the Etherington relation gives $d_L=(1+z)^2 d_A$ in metric theories with photon number conservation.