Cosmological Solutions

General relativity is not only a theory of stars, black holes, and gravitational waves, it is the framework for understanding the universe as a whole. When we apply Einstein's field equations to the largest scales, assuming the universe is homogeneous and isotropic, we discover that the universe must be dynamic: expanding or contracting. This was the first great prediction of relativistic cosmology, confirmed by Hubble's observation of receding galaxies in 1929. In this lesson we derive the FLRW metric, the Friedmann equations, and explore the different eras in the history of our universe.

The FLRW Metric

The most general metric consistent with homogeneity and isotropy is the Friedmann-Lemaitre-Robertson-Walker (FLRW) metric:

The coordinates $(r, \theta, \phi)$ are comoving coordinates: galaxies at rest in the cosmological expansion have fixed comoving coordinates. The physical distance between two comoving galaxies separated by $\Delta r$ is $d(t) = a(t) \Delta r$: it scales with $a(t)$.

The three values of $k$ correspond to three spatial geometries:

• $k = +1$: positively curved space, like the surface of a 3-sphere. Finite volume.
• $k = 0$: flat Euclidean space. The geometry we are most familiar with. Infinite extent.
• $k = -1$: negatively curved space, like a 3D saddle. Infinite extent.

Observations (primarily of the cosmic microwave background) show that $k$ is very close to zero, the universe is spatially flat to within measurement precision.

The Friedmann Equations

Substituting the FLRW metric into Einstein's field equations with a perfect fluid source yields two independent equations for $a(t)$:

Here $H = \dot{a}/a$ is the Hubble parameter, $\rho$ is the energy density, $p$ is the pressure, and $\Lambda$ is the cosmological constant. The first equation is a constraint (an energy equation); the second describes how the expansion accelerates or decelerates.

Energy conservation ($\nabla^\mu T_{\mu\nu} = 0$) gives the fluid equation:

$$\dot{\rho} + 3H\left(\rho + \frac{p}{c^2}\right) = 0$$

This is not independent, it follows from the two Friedmann equations. To close the system, we need an equation of state relating $p$ to $\rho$.

The Hubble Parameter

The Hubble parameter $H(t) = \dot{a}/a$ measures the expansion rate. Its present value is:

$$H_0 \approx 70 \text{ km/s/Mpc}$$

A galaxy at distance $d$ recedes with velocity $v = H_0 d$: this is Hubble's law. The Hubble time $t_H = 1/H_0 \approx 14$ billion years gives a rough estimate of the age of the universe.

Hubble's law does not mean we are at the center of the expansion. Every observer in a homogeneous, expanding universe sees the same law. The expansion is of space itself, not an explosion into preexisting space.

Cosmological Eras

Different components of the universe have different equations of state $p = w\rho c^2$, and each dilutes differently as the universe expands:

Component	$w$	$\rho \propto$	$a(t) \propto$
Radiation	$1/3$	$a^{-4}$	$t^{1/2}$
Matter (dust)	$0$	$a^{-3}$	$t^{2/3}$
Dark energy ($\Lambda$)	$-1$	$\text{const}$	$e^{Ht}$

Radiation dilutes as $a^{-4}$: one factor of $a^{-3}$ from the expansion of volume, plus one factor of $a^{-1}$ from the cosmological redshift of each photon's wavelength. Matter dilutes as $a^{-3}$ (volume only). Dark energy density remains constant, it is a property of spacetime itself.

This means the universe has passed through distinct eras:

• Radiation-dominated era ($a \propto t^{1/2}$): the first ~50,000 years, when radiation (photons and neutrinos) dominated the energy budget
• Matter-dominated era ($a \propto t^{2/3}$): from ~50,000 years to ~10 billion years, when ordinary and dark matter dominated
• Dark-energy-dominated era ($a \propto e^{Ht}$): the current era (last ~4 billion years), when the cosmological constant dominates and the expansion accelerates

Critical Density and Omega

The first Friedmann equation (with $\Lambda = 0$ and $k = 0$) defines the critical density:

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

The density parameter $\Omega = \rho / \rho_c$ determines the geometry:

• $\Omega > 1$: closed universe ($k = +1$)
• $\Omega = 1$: flat universe ($k = 0$)
• $\Omega < 1$: open universe ($k = -1$)

Today's critical density is $\rho_{c,0} \approx 9.5 \times 10^{-27}$ kg/m$^3$: about 6 hydrogen atoms per cubic meter. The total density parameter is measured to be $\Omega_{\text{total}} \approx 1.00$, decomposed as:

• $\Omega_\Lambda \approx 0.68$ (dark energy)
• $\Omega_{\text{DM}} \approx 0.27$ (dark matter)
• $\Omega_b \approx 0.05$ (ordinary baryonic matter)
• $\Omega_r \approx 10^{-4}$ (radiation, negligible today)

The ordinary matter we see, stars, planets, gas, makes up only about 5% of the energy content of the universe.

The Expanding Universe

A photon traveling through expanding space has its wavelength stretched by the expansion. The cosmological redshift $z$ is defined by:

$$1 + z = \frac{a(t_0)}{a(t_e)} = \frac{\lambda_{\text{observed}}}{\lambda_{\text{emitted}}}$$

where $t_e$ is the time of emission and $t_0$ is now. A galaxy at $z = 1$ emitted its light when the universe was half its present size. The most distant known galaxies have $z > 10$, meaning we see them when the universe was less than one-eleventh its present size.

Age and Fate of the Universe

The age of the universe is obtained by integrating the Friedmann equation backward from $a = 1$ (today) to $a = 0$ (the big bang). With the measured cosmological parameters:

$$t_0 = \int_0^1 \frac{da}{a H(a)} \approx 13.8 \text{ billion years}$$

The fate depends on the nature of dark energy. If $\Lambda$ is truly constant, the expansion will accelerate forever, and the universe will become increasingly cold, dilute, and dark, the "heat death." Galaxies beyond our local group will eventually recede beyond our cosmological horizon, disappearing from view forever.

Cosmological Horizons

The finite age and finite speed of light create natural boundaries on what we can observe or influence.

The particle horizon is the maximum distance from which light could have reached us since the big bang:

$$d_{\text{particle}}(t) = a(t) \int_0^t \frac{c \, dt'}{a(t')}$$

This defines the size of the observable universe: currently about 46 billion light-years in radius (larger than 13.8 billion light-years because the universe has expanded while the light was traveling).

The event horizon is the maximum distance from which light emitted now will ever reach us:

$$d_{\text{event}}(t) = a(t) \int_t^\infty \frac{c \, dt'}{a(t')}$$

In a universe with accelerating expansion, this integral converges, there are events happening now that we will never observe, no matter how long we wait.

Key Insights

• The cosmological principle (homogeneity + isotropy) restricts the metric to the FLRW form, parameterized by the scale factor $a(t)$ and spatial curvature $k$
• The Friedmann equations, derived from Einstein's equations, govern the expansion of the universe
• The Hubble parameter $H = \dot{a}/a$ measures the expansion rate; Hubble's law $v = Hd$ is an exact result for comoving observers
• The universe has passed through radiation-dominated, matter-dominated, and dark-energy-dominated eras, each with characteristic expansion rates
• Observations show $\Omega \approx 1$ (flat universe), with ~68% dark energy, ~27% dark matter, and ~5% ordinary matter
• The universe is about 13.8 billion years old. Eternal expansion is the prediction of $\Lambda$CDM if dark energy remains a cosmological constant, not a model-independent certainty
• Cosmological horizons define the boundary between what is observable and what is forever beyond our reach

Looking Ahead

You have now completed Phase 6: General Relativity. From the mathematical foundations of differential geometry to the Einstein equations, from black holes to gravitational waves, and now to the expanding universe, you have seen how geometry and gravity are one and the same.

The cosmological solutions we derived here are the starting point for the detailed study of cosmology in Phase 9. But first, we turn to the other pillar of modern physics: quantum field theory. In Phase 7, we will see how quantum mechanics and special relativity combine to give us the framework for understanding particles, forces, and the vacuum itself.