Complete Cosmology

Observational Starting Point

Modern cosmology begins from two empirical facts. First, on sufficiently large scales the universe is statistically homogeneous and isotropic. Second, distant galaxies are redshifted in a way that is naturally described by expansion.

The homogeneous and isotropic spacetime ansatz is the FLRW metric. In units where the spatial curvature parameter is $k=0,\pm 1$,

$$ds^2 = -c^2dt^2 + a(t)^2\left[ \frac{dr^2}{1-kr^2}+r^2d\Omega^2 \right].$$

The single function $a(t)$ is the scale factor. Much of introductory cosmology is the study of how $a(t)$ evolves and how light propagates through this geometry.

Redshift and the Scale Factor

For light emitted when the scale factor was $a_{\rm em}$ and observed when it is $a_0$, the cosmological redshift is

$$1+z = \frac{a_0}{a_{\rm em}}.$$

This relation is kinematic: it follows from the stretching of wavelengths by the expanding geometry. It is the bridge between observation and cosmic time.

Hubble Expansion

The expansion rate is defined by the Hubble parameter

$$H(t) = \frac{\dot a(t)}{a(t)}.$$

For nearby galaxies, the leading relation between recession velocity and proper distance is Hubble’s law,

$$v \simeq H_0 d.$$

The law is local and approximate. At large redshift, distances must be defined more carefully because light travels through an evolving spacetime.

Friedmann Dynamics

Assuming general relativity and a perfect-fluid stress-energy tensor, the first Friedmann equation is

$$H^2 = \frac{8\pi G}{3}\rho -\frac{kc^2}{a^2} +\frac{\Lambda c^2}{3}.$$

The second Friedmann equation can be written as

$$\frac{\ddot a}{a} =-\frac{4\pi G}{3}\left(\rho+\frac{3p}{c^2}\right) +\frac{\Lambda c^2}{3}.$$

The first equation is an energy constraint; the second describes acceleration. Together with energy conservation, they determine the scale-factor history for specified matter, radiation, curvature, and cosmological constant.

Critical Density

For a spatially flat universe with $\Lambda=0$, the critical density is

$$\rho_c = \frac{3H^2}{8\pi G}.$$

Density parameters are defined by $\Omega_i=\rho_i/\rho_c$. They allow the Friedmann equation to be written as a bookkeeping equation for the cosmic contents.

Thermal History

In the early universe, radiation dominated. As the universe expanded, it cooled. The cosmic microwave background is the relic radiation from the epoch when photons decoupled from matter. Its near-blackbody spectrum and anisotropies are central evidence for hot Big Bang cosmology.

Standard Model of Cosmology

The minimal successful model is often called Lambda CDM. It contains radiation, baryonic matter, cold dark matter, and a cosmological constant. Its success is not that it explains everything from first principles, but that it gives a compact quantitative framework for the expansion history and large-scale structure.