Relativity: Relativistic Cosmology

Relativity » Relativistic Cosmology

Relativistic Cosmology

Cosmology applies the Einstein field equations to the universe as a whole. On scales larger than a few hundred megaparsecs the matter distribution is observed to be homogeneous (the same everywhere) and isotropic (the same in every direction). That single empirical input — the cosmological principle — collapses the ten coupled field equations into two ordinary differential equations for a single function of time, the scale factor $a(t)$. This page derives the FLRW metric, the Friedmann equations, and the cosmological constant; works out the expansion history of the standard $\Lambda$CDM model; defines the cosmological horizons; and closes with a brief account of inflation. It assumes Tensor Formalism & the Field Equations and the physics of General Relativity.

Conventions. We work in geometric units with $G = c = 1$ except where a constant is restored for clarity. The metric signature is (−,+,+,+). The scale factor is normalized so that today $a(t_0) = a_0 = 1$, and a subscript $0$ denotes a present-day value. The Hubble parameter is $H \equiv \dot{a}/a$ with present value $H_0 \approx 67$–$73~\text{km s}^{-1}\text{Mpc}^{-1}$; the dimensionless density parameters are $\Omega_i \equiv \rho_i/\rho_{\rm crit}$ with $\rho_{\rm crit} = 3H^2/8\pi G$. An overdot denotes $d/dt$ in cosmic time.

The Cosmological Principle

The observable universe contains rich structure — stars, galaxies, clusters, voids — but averaged over scales of order $100~\text{Mpc}$ these inhomogeneities wash out. Two observations support this:

Isotropy. The cosmic microwave background (CMB) is the same temperature in every direction to about one part in $10^5$ after subtracting the dipole due to our own motion. Galaxy counts and radio-source counts are likewise direction-independent on large scales.
Homogeneity. Combined with the Copernican assumption that we occupy no special location, isotropy about every point implies homogeneity. Large galaxy redshift surveys (SDSS, 2dF) confirm directly that the matter distribution becomes statistically uniform on large scales.

Mathematically, the cosmological principle states that space is maximally symmetric: at each instant of cosmic time the spatial slices are homogeneous and isotropic three-spaces. A maximally symmetric 3-space has constant curvature, and there are exactly three of them up to scale: the 3-sphere (positive curvature), flat Euclidean 3-space (zero curvature), and the hyperbolic 3-space (negative curvature). This is what forces the metric into the FLRW form below.

The FLRW Metric

The Friedmann–Lemaître–Robertson–Walker (FLRW) metric is the most general line element describing a spatially homogeneous and isotropic spacetime. Writing the spatial part in comoving spherical coordinates $(r, \theta, \phi)$ and pulling all the time dependence into the single function $a(t)$:

\[ds^2 = -dt^2 + a(t)^2\left[\frac{dr^2}{1-kr^2} + r^2\left(d\theta^2 + \sin^2\theta\, d\phi^2\right)\right]\]

Here:

$t$ is cosmic time, the proper time of an observer comoving with the cosmic fluid (a galaxy at fixed $r,\theta,\phi$).
$a(t)$ is the dimensionless scale factor; physical distances between comoving points scale as $a(t)$, so the universe expands or contracts uniformly.
$k \in {-1, 0, +1}$ is the (normalized) spatial curvature, distinguishing the three geometries:

$k$	Geometry	Spatial volume	Fate (matter-only)
$+1$	Closed (3-sphere)	finite	recollapse
$0$	Flat (Euclidean)	infinite	expands forever, $\dot a \to 0$
$-1$	Open (hyperbolic)	infinite	expands forever

A useful alternative uses a rescaled radial coordinate $\chi$ defined by $d\chi = dr/\sqrt{1-kr^2}$, which puts the metric in the form

\[ds^2 = -dt^2 + a(t)^2\left[d\chi^2 + S_k(\chi)^2\, d\Omega^2\right], \qquad S_k(\chi) = \begin{cases} \sin\chi & k=+1 \\ \chi & k=0 \\ \sinh\chi & k=-1 \end{cases}\]

with $d\Omega^2 = d\theta^2 + \sin^2\theta\, d\phi^2$. In this form the radial coordinate $\chi$ is the comoving distance and $a S_k(\chi)$ is the transverse comoving (angular-diameter-related) distance.

Comoving vs. proper distance

A galaxy at fixed comoving coordinate $\chi$ sits at proper distance $d_{\rm prop}(t) = a(t)\,\chi$ from us. Differentiating,

\[\dot d_{\rm prop} = \dot a\, \chi = \frac{\dot a}{a}\, d_{\rm prop} = H(t)\, d_{\rm prop}\]

which is exactly Hubble’s law, $v = H d$: the recession velocity is proportional to distance, with proportionality constant the Hubble parameter. Crucially this is not motion through space — comoving galaxies are at rest in the cosmic rest frame — but the stretching of space itself.

Cosmological redshift

Light emitted at time $t_e$ with wavelength $\lambda_e$ and observed today at $t_0$ is stretched in proportion to the expansion. Following a radial null geodesic ($ds^2 = 0$) one finds that the ratio of observed to emitted wavelength equals the ratio of scale factors:

\[1 + z \equiv \frac{\lambda_0}{\lambda_e} = \frac{a(t_0)}{a(t_e)} = \frac{1}{a(t_e)}\]

using $a_0 = 1$. Redshift is therefore a direct, model-independent measure of the scale factor at emission: a quasar at $z = 6$ is seen as it was when the universe was $1/(1+z) = 1/7$ of its present size. This is the workhorse relation of observational cosmology.

The Friedmann Equations

To find $a(t)$ we feed the FLRW metric into the Einstein field equations $G_{\mu\nu} + \Lambda g_{\mu\nu} = 8\pi G\, T_{\mu\nu}$. Homogeneity and isotropy force the stress–energy tensor to be that of a perfect fluid,

\[T^{\mu}{}_{\nu} = \mathrm{diag}(-\rho,\, p,\, p,\, p)\]

with energy density $\rho(t)$ and pressure $p(t)$ that depend only on time. Computing the Einstein tensor for the FLRW metric (the Christoffel symbols and Ricci tensor follow the recipe on the tensor-formalism page) and reading off the $tt$ and spatial components gives the two Friedmann equations:

\[\left(\frac{\dot{a}}{a}\right)^2 = \frac{8\pi G\,\rho}{3} - \frac{k}{a^2} + \frac{\Lambda}{3}\] \[\frac{\ddot{a}}{a} = -\frac{4\pi G}{3}\left(\rho + 3p\right) + \frac{\Lambda}{3}\]

The first Friedmann equation is a constraint relating the expansion rate to the energy content and curvature; it is essentially energy conservation for the expansion. The second Friedmann equation (the acceleration equation) governs $\ddot a$: because ordinary matter and radiation have $\rho + 3p > 0$, they decelerate the expansion — gravity is attractive. Only a sufficiently negative-pressure component or a positive $\Lambda$ can drive acceleration.

The continuity equation

The two Friedmann equations are not independent of how $\rho$ evolves. The conservation law $\nabla_\mu T^{\mu\nu} = 0$ (equivalently, combining the two equations above) yields the fluid/continuity equation:

\[\dot{\rho} + 3\frac{\dot{a}}{a}\left(\rho + p\right) = 0\]

This is the first law of thermodynamics for a comoving volume: $d(\rho V) = -p\, dV$ with $V \propto a^3$. Given an equation of state $p = w\rho$ with constant $w$, it integrates immediately to

\[\rho \propto a^{-3(1+w)}\]

The three components that dominate cosmic history correspond to three values of $w$:

Component	$w$	Scaling $\rho(a)$	Why
Radiation (photons, relativistic neutrinos)	$1/3$	$\rho \propto a^{-4}$	number density $\propto a^{-3}$ and each quantum redshifts $\propto a^{-1}$
Matter (cold dark matter + baryons)	$0$	$\rho \propto a^{-3}$	pressureless dust simply dilutes with volume
Dark energy ($\Lambda$)	$-1$	$\rho = \text{const}$	vacuum energy density is fixed

Critical density and density parameters

Set $k = \Lambda = 0$ in the first Friedmann equation; the energy density required to make space exactly flat is the critical density:

\[\rho_{\rm crit} = \frac{3H^2}{8\pi G}, \qquad \rho_{\rm crit,0} \approx 8.5\times 10^{-27}~\text{kg m}^{-3} \approx 5~\frac{\text{proton masses}}{\text{m}^3}\]

Dividing the first Friedmann equation by $H^2$ and defining $\Omega_i = \rho_i/\rho_{\rm crit}$, plus a curvature term $\Omega_k = -k/(a^2 H^2)$ and $\Omega_\Lambda = \Lambda/(3H^2)$, gives the clean cosmic sum rule:

\[\Omega_m + \Omega_r + \Omega_\Lambda + \Omega_k = 1\]

Measurements (Planck) give a spatially flat universe, $\Omega_{k,0} \approx 0$, partitioned roughly as $\Omega_{\Lambda,0} \approx 0.69$, $\Omega_{m,0} \approx 0.31$ (of which only $\sim 0.05$ is baryonic), and $\Omega_{r,0} \approx 9\times 10^{-5}$. This is the $\Lambda$CDM (“Lambda cold dark matter”) concordance model.

The Cosmological Constant and de Sitter Space

Einstein introduced the cosmological constant $\Lambda$ in 1917 to permit a static universe; after Hubble’s discovery of expansion he reportedly called it his “biggest blunder.” It returned in 1998 when type Ia supernovae revealed that the expansion is accelerating, which the second Friedmann equation requires a positive $\Lambda$ (or equivalent $w \approx -1$ dark energy) to explain.

$\Lambda$ as vacuum energy

A cosmological constant is mathematically identical to a perfect fluid with

\[\rho_\Lambda = \frac{\Lambda}{8\pi G}, \qquad p_\Lambda = -\rho_\Lambda \quad (w = -1)\]

Its energy density does not dilute as space expands, so it eventually dominates over matter ($\propto a^{-3}$) and radiation ($\propto a^{-4}$). Once it does, the first Friedmann equation (neglecting the now-tiny matter and curvature terms) reads $H^2 \to \Lambda/3 = \text{const}$, giving exponential expansion:

\[a(t) \propto e^{Ht}, \qquad H = \sqrt{\frac{\Lambda}{3}}\]

This is the late-time fate of our universe in $\Lambda$CDM: an ever-emptier, exponentially expanding de Sitter space.

The de Sitter and anti-de Sitter metrics

The maximally symmetric solutions of the vacuum Einstein equations with $\Lambda \neq 0$ are de Sitter ($\Lambda > 0$) and anti-de Sitter ($\Lambda < 0$). In static coordinates they take a Schwarzschild-like form with a cosmological horizon:

\[ds^2 = -\left(1-\frac{r^2}{\alpha^2}\right)dt^2 + \left(1-\frac{r^2}{\alpha^2}\right)^{-1}dr^2 + r^2\, d\Omega^2 \quad \text{(de Sitter, } \Lambda > 0\text{)}\] \[ds^2 = -\left(1+\frac{r^2}{\alpha^2}\right)dt^2 + \left(1+\frac{r^2}{\alpha^2}\right)^{-1}dr^2 + r^2\, d\Omega^2 \quad \text{(anti-de Sitter, } \Lambda < 0\text{)}\]

with curvature radius $\alpha = \sqrt{3/

\Lambda

}$. In de Sitter space $r = \alpha$ is a cosmological event horizon: an observer is causally cut off from events beyond it, and the horizon carries a Gibbons–Hawking temperature $T = H/2\pi$, the cosmological analogue of Hawking radiation. Anti-de Sitter space, by contrast, has no horizon and a timelike boundary at infinity; it is the arena of the AdS/CFT correspondence in string theory.

The cosmological constant problem. Quantum field theory predicts a vacuum energy density from zero-point fluctuations that, cut off at the Planck scale, exceeds the observed $\rho_\Lambda$ by roughly 120 orders of magnitude — the largest discrepancy between theory and observation in physics. Why $\Lambda$ is so small but nonzero is unsolved; proposals range from quintessence (a slowly rolling scalar field with $w$ near but not exactly $-1$) to anthropic selection in a multiverse to modifications of gravity.

Expansion History

Putting the components together, the first Friedmann equation for a flat ($k=0$) universe can be written entirely in terms of present-day density parameters and redshift. Using $\rho_i \propto a^{-3(1+w_i)}$ and $a = 1/(1+z)$:

\[H(z)^2 = H_0^2\left[\Omega_{r,0}(1+z)^4 + \Omega_{m,0}(1+z)^3 + \Omega_{k,0}(1+z)^2 + \Omega_{\Lambda,0}\right]\]

Because the four terms redshift at different rates, the universe passes through distinct epochs, each dominated by one component:

Radiation-dominated era ($z \gtrsim 3400$, $t \lesssim 50{,}000$ yr). With $\rho \propto a^{-4}$ the Friedmann equation gives $a \propto t^{1/2}$. Nucleosynthesis (forging helium and deuterium) occurs here at $t \sim 1$–$3$ min.

Matter-dominated era ($3400 \gtrsim z \gtrsim 0.3$). With $\rho \propto a^{-3}$ one finds $a \propto t^{2/3}$. Recombination and the release of the CMB occur at $z \approx 1100$; gravitational structure formation proceeds throughout.

Dark-energy-dominated era ($z \lesssim 0.3$, beginning a few Gyr ago). $\Lambda$ takes over, $\ddot a > 0$, and the expansion accelerates toward the de Sitter regime $a \propto e^{Ht}$.

For a single component with equation of state $w \neq -1$ in a flat universe, the generic power-law solution is worth memorizing:

\[a(t) \propto t^{\frac{2}{3(1+w)}} \quad\Longrightarrow\quad \begin{cases} a \propto t^{1/2} & \text{radiation } (w=1/3) \\ a \propto t^{2/3} & \text{matter } (w=0) \end{cases}\]

The age of the universe

The age follows from integrating $dt = da/\dot a = da/(aH)$:

\[t_0 = \int_0^1 \frac{da}{a\,H(a)} = \frac{1}{H_0}\int_0^\infty \frac{dz}{(1+z)\,E(z)}, \qquad E(z) \equiv H(z)/H_0\]

For the Planck $\Lambda$CDM parameters this gives $t_0 \approx 13.8$ billion years. The combination $1/H_0 \approx 14.4$ Gyr is the Hubble time; that $t_0$ comes out close to it is a coincidence of the present epoch, when deceleration (matter) and acceleration ($\Lambda$) roughly balance in the integral.

Horizons

Because light travels at finite speed in an expanding universe, there are fundamental limits on what we can see and influence. Two distinct horizons matter.

The particle (comoving) horizon

The particle horizon is the comoving distance light could have traveled since the Big Bang ($t = 0$) — the boundary of the observable universe:

\[\chi_{\rm ph}(t) = \int_0^{t} \frac{dt'}{a(t')} = \int_0^{a} \frac{da'}{a'^2 H(a')}\]

Its present proper radius, $a_0\,\chi_{\rm ph}$, is about $46$ billion light-years — far larger than the naive $c\,t_0 \approx 14$ billion light-years, because space expanded while the light was in transit. Regions beyond the particle horizon have never been in causal contact with us.

The event horizon

The cosmological event horizon is the comoving distance light emitted now can ever reach in the infinite future:

\[\chi_{\rm eh}(t) = \int_{t}^{\infty} \frac{dt'}{a(t')}\]

In a matter-only or radiation-only universe this integral diverges — there is no event horizon, and given enough time we could signal any comoving point. But with dark energy driving $a \propto e^{Ht}$, the integral converges: distant galaxies are accelerating away faster than light can cross the growing gap, and an event horizon forms at proper radius $\approx 1/H_0 \approx 16$ billion light-years. Galaxies beyond it are receding superluminally (a consequence of stretching space, not local motion) and their light will never reach us. Over the coming $\sim 100$ Gyr nearly all galaxies outside the Local Group will redshift away and disappear.

Particle horizon vs. event horizon. The particle horizon looks backward: it bounds the part of the universe we can already see. The event horizon looks forward: it bounds the part we can ever reach or influence. A static or decelerating universe has a particle horizon but no event horizon; an accelerating (de Sitter–like) universe has both.

A Brief Overview of Inflation

The hot Big Bang model, despite its success, leaves several initial-condition puzzles unexplained:

The horizon problem. Regions of the CMB separated by more than $\sim 1°$ were never in causal contact under standard radiation/matter expansion, yet they share the same temperature to one part in $10^5$. How did they thermalize?
The flatness problem. $\Omega_k = 0$ is an unstable fixed point: any small initial curvature grows during radiation and matter domination. For $\Omega$ to be near $1$ today it must have been tuned to $1$ to better than $1$ part in $10^{60}$ at early times.
The monopole problem. Grand unified theories predict copious magnetic monopoles and other relics that are not observed.

Cosmic inflation (Guth, Linde, Albrecht & Steinhardt, c. 1980) solves all three with a brief epoch of near-exponential expansion in the very early universe, $t \sim 10^{-36}$–$10^{-32}$ s. A scalar field $\phi$ (the inflaton) with potential $V(\phi)$ dominates the energy density while “slowly rolling” down a nearly flat region of its potential. With $\dot\phi^2 \ll V(\phi)$ its equation of state approaches $w \approx -1$, so it acts like a temporary, large cosmological constant and drives quasi-de-Sitter expansion:

\[a(t) \propto e^{Ht}, \qquad H^2 \approx \frac{8\pi G}{3}V(\phi)\]

A factor of $e^{60}$ or more of expansion (at least 60 e-folds) accomplishes the following:

Horizon problem solved: the entire observable universe inflated from a single, tiny, causally connected patch, explaining its uniformity.
Flatness problem solved: the enormous stretch flattens any initial curvature, driving $\Omega_k \to 0$ regardless of where it started — a robust prediction of near-perfect spatial flatness, confirmed by Planck.
Monopoles diluted: any pre-inflationary relics are spread so thin (one per Hubble volume or fewer) as to be unobservable.

Inflation’s most striking success is structure formation: quantum fluctuations of the inflaton, stretched to cosmological scales and frozen in, seed the tiny density perturbations that grow into galaxies. The model predicts a nearly scale-invariant spectrum with a slight red tilt ($n_s \approx 0.965$), Gaussian, adiabatic primordial perturbations, and a stochastic background of primordial gravitational waves. The first two are strikingly confirmed by the CMB power spectrum; detecting the inflationary gravitational-wave signature (B-mode polarization, parameterized by the tensor-to-scalar ratio $r$) remains a major experimental goal.

How It Fits Together

graph TD
    CP["Cosmological principle<br/>(homogeneous + isotropic)"] --> FLRW["FLRW metric a(t), k"]
    FLRW --> EFE["Einstein equations<br/>+ perfect fluid"]
    EFE --> FRIED["Friedmann equations"]
    EFE --> CONT["Continuity equation"]
    CONT --> EOS["Equation of state w<br/>ρ ∝ a^-3(1+w)"]
    EOS --> HIST["Expansion history<br/>rad → matter → Λ"]
    FRIED --> HIST
    HIST --> HOR["Horizons<br/>(particle, event)"]
    HIST --> FATE["Accelerating de Sitter fate"]
    INFL["Inflation<br/>(early de Sitter phase)"] --> FLRW
    INFL --> SEEDS["Seeds for structure<br/>+ flatness, horizon, monopoles"]
    classDef geom fill:#e0f2f1,stroke:#11998e,stroke-width:2px;
    classDef phys fill:#fff3e0,stroke:#e65100,stroke-width:2px;
    class CP,FLRW,EFE,HOR geom;
    class FRIED,CONT,EOS,HIST,FATE,INFL,SEEDS phys;

The chain is tight: the cosmological principle fixes the metric up to one function and one discrete parameter; the field equations turn that into the Friedmann equations; the equation of state of each component dictates how its density dilutes; and the sum of those components, integrated, gives the entire history — from an inflationary beginning, through radiation and matter domination, to the accelerating, horizon-bounded de Sitter future we are entering now.