Cosmology: FLRW & the Expanding Universe
Friedmann equations, scale factor, and the late-time acceleration.
On large scales the universe is homogeneous and isotropic — the cosmological principle. The metric of such a spacetime is FLRW:
$$ds^2 = -c^2 dt^2 + a(t)^2 \left[\frac{dr^2}{1 - k r^2} + r^2\, d\Omega^2\right],$$with $k \in \{-1, 0, +1\}$ for hyperbolic, flat, and spherical spatial geometry. The scale factor $a(t)$ describes the expansion.
Substituting into Einstein's equations yields the Friedmann equations:
$$H^2 = \left(\frac{\dot a}{a}\right)^2 = \frac{8\pi G}{3}\rho - \frac{k c^2}{a^2} + \frac{\Lambda c^2}{3},$$ $$\frac{\ddot a}{a} = -\frac{4\pi G}{3}(\rho + 3p/c^2) + \frac{\Lambda c^2}{3}.$$Energy densities scale as: matter $\rho_m \propto a^{-3}$, radiation $\rho_r \propto a^{-4}$, cosmological constant $\rho_\Lambda$ = const. The present universe is dominated by dark energy ($\Omega_\Lambda \approx 0.68$) and matter (mostly dark, $\Omega_m \approx 0.32$), with $\Omega_k \approx 0$.