Cosmology: The Shape of the Whole Universe
Apply Einstein’s field equations to the universe as a whole — assuming it’s homogeneous (the same everywhere) and isotropic (the same in every direction) on large scales — and you get a theory of cosmic expansion, the Big Bang, and the ultimate fate of everything.
The FLRW metric
The most general metric consistent with homogeneity and isotropy is the Friedmann-Lemaître-Robertson-Walker metric:
$$ds^2 = -c^2 dt^2 + a(t)^2 \left[\frac{dr^2}{1-kr^2} + r^2 d\Omega^2\right]$$
where: - $a(t)$ is the scale factor — it tracks how distances between galaxies change over time - $k$ determines the spatial geometry: $k = +1$ (closed, spherical), $k = 0$ (flat), $k = -1$ (open, hyperbolic) - $t$ is cosmic time — the proper time of observers at rest in the expanding flow
The scale factor is the single unknown function. Finding $a(t)$ tells you the entire expansion history of the universe.
The Friedmann equations
Plugging the FLRW metric into Einstein’s field equations with a perfect fluid stress-energy tensor gives:
$$\left(\frac{\dot{a}}{a}\right)^2 = \frac{8\pi G}{3}\rho - \frac{kc^2}{a^2} + \frac{\Lambda c^2}{3}$$
$$\frac{\ddot{a}}{a} = -\frac{4\pi G}{3}\left(\rho + \frac{3p}{c^2}\right) + \frac{\Lambda c^2}{3}$$
The first equation relates the expansion rate to the energy content. The second determines whether the expansion accelerates or decelerates.
The Hubble parameter $H(t) = \dot{a}/a$ measures the expansion rate. Today: $H_0 \approx 70$ km/s/Mpc.
The scale factor $a(t)$ for different universe compositions. Adjust the density parameters to see how matter, radiation, and dark energy compete.
Three eras
The universe’s contents dilute differently as space expands:
| Component | Density scaling | Equation of state | Dominates when |
|---|---|---|---|
| Radiation | $\rho \propto a^{-4}$ | $p = \rho c^2/3$ | Early universe ($a \ll 1$) |
| Matter | $\rho \propto a^{-3}$ | $p = 0$ | Middle era |
| Dark energy ($\Lambda$) | $\rho = \text{const}$ | $p = -\rho c^2$ | Late universe ($a \gg 1$) |
Radiation dilutes fastest (redshift stretches wavelengths), so it dominated earliest. Matter dilutes next. Dark energy doesn’t dilute at all — its density is constant — so it eventually dominates. We’re currently in the transition from matter domination to dark energy domination.
Observational pillars
1. Hubble’s law (1929): Galaxies recede with velocity proportional to distance: $v = H_0 d$. This is exactly what the FLRW metric predicts — space itself is expanding.
2. The cosmic microwave background (1965): A nearly uniform thermal radiation at 2.725 K filling all of space. The afterglow of the hot, dense early universe, emitted when the universe was 380,000 years old and cooled by the expansion from ~3000 K to 2.7 K. Its tiny anisotropies ($\sim 10^{-5}$) encode the seeds of all structure — galaxies, clusters, voids.
3. Accelerating expansion (1998): Type Ia supernovae at high redshift are dimmer than expected in a decelerating universe, implying that the expansion is speeding up. Consistent with $\Lambda > 0$ — a cosmological constant, or more generally, dark energy with $w \approx -1$.
The concordance model
Current best measurements (Planck + BAO + supernovae): $\Omega_m \approx 0.31$, $\Omega_\Lambda \approx 0.69$, $k \approx 0$ (flat to high precision). The universe is 13.8 billion years old and will expand forever, with the expansion accelerating exponentially.
About 68% dark energy, 27% dark matter, 5% ordinary matter. We understand the 5%.