Past exam of the general relativity course of the University of Oxford / 2021 / 4 / a / Solution

On large scales the appropriate background is the spatially flat FLRW metric,Here $t$ is cosmic time, the proper time of a comoving observer; $R(t)$ is the scale factor; and $(r,\theta ,\varphi )$ are dimensionless comoving coordinates. With this convention $R$ has units of length. A fixed comoving radial interval has instantaneous proper length $R(t)dr$. The cosmological principle assumes spatial homogeneity and cosmological isotropy on the scales where this background description applies.

The general FLRW metric instead has $d{r}^2/(1-k{r}^2)$ in its radial part, with normalized $k=+1,0,-1$ describing positive, zero or negative spatial curvature of an FLRW universe. The best simple description takes $k=0$; observations constrain spatial curvature to be small rather than proving it identically zero. The late accelerated expansion is encoded by the time dependence of $R$, determined by the stress-energy tensor through the Friedmann equations, rather than by an additional term in the FLRW metric. Small-scale galaxies and other inhomogeneities require perturbations of this background.