The Friedmann Equations and Cosmic Dynamics

The FLRW metric leaves the scale factor $a(t)$ undetermined; symmetry fixes the form of the metric but not its evolution. The Einstein equation supplies the missing dynamics. With a homogeneous isotropic source — a perfect fluid whose density and pressure depend only on cosmic time — the ten Einstein equations collapse to two ordinary differential equations for $a(t)$, the Friedmann equations. This lesson derives them, extracts the critical density that separates the spatial geometries, and follows the expansion through the radiation, matter, and vacuum-dominated eras.

The perfect-fluid source

Homogeneity and isotropy constrain the stress–energy tensor as tightly as they constrain the metric. The most general $T^{\mu \nu }$ compatible with isotropy in the comoving frame is that of a perfect fluid,

where $\rho$ is the mass-energy density (so $\rho c^2$ is the energy density), $p$ is the pressure, and $U^{\mu }$ is the four-velocity of the comoving fluid. In the comoving frame $U^{\mu }=(c,0,0,0)$ and the tensor is diagonal,

The single time–time component is the energy density; the three equal spatial components are the isotropic pressure. No other structure survives isotropy: heat flux would pick out a direction, and anisotropic stress would distinguish axes.

Conservation ${\nabla }_{\mu }{T}^{\mu \nu }=0$ applied to this source, using the Christoffel symbols of the FLRW metric, gives the $\nu =0$ component

This is the fluid equation. Its two terms have a thermodynamic reading: $\dot{\rho }$ is the change in density, the $3(\dot{a}/a)\rho$ term is dilution by the growth of proper volume $V\propto a^3$, and the pressure term is the work $p\mathbf{d}V$ done by the fluid as the volume expands. The fluid equation is the first law of thermodynamics for a comoving volume of the cosmic fluid.¹

The Friedmann equations

Evaluating the Einstein equation $G_{\mu \nu }+\Lambda g_{\mu \nu }=\frac{8\pi G}{c^4}{T}_{\mu \nu }$ on the FLRW metric requires the Einstein tensor of that metric. The time–time component gives the first Friedmann equation,

and any spatial diagonal component gives the second, the acceleration equation,

The three relations — the two Friedmann equations and the fluid equation — are not independent: differentiating the first Friedmann equation and using the fluid equation reproduces the acceleration equation. Any two of the three determine the third. The standard practice is to take the first Friedmann equation together with the fluid equation and an equation of state relating $p$ to $\rho$.

Two features of the acceleration equation deserve emphasis.

• Pressure gravitates. The source of deceleration is $\rho +3p/c^2$, not $\rho$ alone. In general relativity pressure carries weight; a fluid with large positive pressure decelerates the expansion more than its energy density alone would suggest.
• Negative pressure accelerates. If $\rho +3p/c^2<0$, that is $p<-\rho c^2/3$, the expansion accelerates even without the explicit $\Lambda$ term. A cosmological constant is one way to supply such a source.

The cosmological constant can be absorbed into the fluid as a component with energy density and pressure

This vacuum component has equation of state $p=-\rho c^2$ and satisfies $\rho +3p/c^2=-2{\rho }_{\Lambda }{c}^2<0$, so it drives acceleration. From here on $\Lambda$ is treated as one contribution to the total $\rho$ and $p$, and the explicit $\Lambda$ terms are dropped from the Friedmann equations.²

Critical density and the density parameters

Set $\Lambda$ into the fluid and evaluate the first Friedmann equation today, at $H_0=(\dot{a}/a{)}_0$. Solving for the density that would make the spatial curvature vanish defines the critical density,

Numerically ${\rho }_{c,0}\approx 9\times 10^{-27}\text{kg}{\text{m}}^{-3}$, a few hydrogen atoms per cubic metre. Express each density as a fraction of critical with the density parameter

one for each component: ${\Omega }_m$ for matter, ${\Omega }_r$ for radiation, ${\Omega }_{\Lambda }$ for the cosmological constant. Divide the first Friedmann equation by $H^2$ to write it as a sum rule,

Defining a curvature density parameter ${\Omega }_k=-k{c}^2/(a^2{H}^2)$ makes the sum exact,

The total matter-energy density parameter $\Omega ={\Omega }_m+{\Omega }_r+{\Omega }_{\Lambda }$ fixes the spatial geometry directly:

• $\Omega >1$: density above critical, $k=+1$, a closed universe.
• $\Omega =1$: density exactly critical, $k=0$, a flat universe.
• $\Omega <1$: density below critical, $k=-1$, an open universe.

The connection between the density of the universe and its spatial curvature is the content of the first Friedmann equation: geometry is not a free choice but is set by how much the universe contains. Observations put the total $\Omega$ within a few percent of unity, so the spatial geometry is close to flat.³

Equation of state and how densities dilute

Close the system with a linear equation of state $p=w\rho c^2$, with $w$ a constant for each component. Substituting into the fluid equation,

which integrates to a power law in the scale factor,

The three components of the standard model have different $w$ and therefore dilute at different rates.

• Matter (pressureless dust: galaxies, dark matter), $w=0$: ${\rho }_m\propto a^{-3}$. Density falls as the inverse proper volume; particle number is conserved and spread through a growing volume.
• Radiation (photons, relativistic particles), $w=\frac{1}{3}$: ${\rho }_r\propto a^{-4}$. Three powers from volume dilution plus one more from the redshift of each photon's energy, $E\propto 1/a$.
• Vacuum (cosmological constant), $w=-1$: ${\rho }_{\Lambda }\propto a^0=\text{const}$. The vacuum energy density does not dilute as the universe expands; new volume comes with its own fixed energy density.

Because the exponents differ, the composition of the universe changes with scale. Running the clock backward, radiation grows fastest and dominated the earliest era; matter, diluting more slowly, took over next; the constant vacuum term, negligible early, comes to dominate late. The crossovers are the radiation–matter equality and the matter–$\Lambda$ equality.

Expansion histories of the single-component eras

Within an era dominated by one component, the flat first Friedmann equation $H^2=\frac{8\pi G}{3}\rho$ with $\rho \propto a^{-3(1+w)}$ integrates to a definite growth law. For $w\ne -1$,

The three eras follow.

• Radiation era ($w=\frac{1}{3}$): $a(t)\propto t^{1/2}$. The expansion decelerates, $\ddot{a}<0$, and the Hubble parameter falls as $H=1/(2t)$.
• Matter era ($w=0$): $a(t)\propto t^{2/3}$. Still decelerating, with $H=2/(3t)$.
• Vacuum era ($w=-1$): the power law fails and the solution is exponential. With ${\rho }_{\Lambda }$ constant, $H=\sqrt{8\pi G{\rho }_{\Lambda }/3}$ is constant and

The vacuum-dominated phase is de Sitter expansion: constant $H$, accelerating $a(t)$, never recollapsing. A universe that becomes vacuum-dominated expands forever at an exponential rate. The present universe is entering this phase, with ${\Omega }_{\Lambda }\approx 0.7$ already exceeding ${\Omega }_m$.⁴

The fate of the expansion

Whether the universe expands forever or recollapses depends on the balance between the density content and the curvature term. Turning points of $a(t)$ occur where $\dot{a}=0$, that is where the first Friedmann equation gives

With no cosmological constant, only $k=+1$ (a closed, over-critical universe) admits such a turning point: the density term falls off more slowly than required and the right side catches up, halting the expansion and driving a recollapse to a big crunch. Flat and open universes ($k=0,-1$) with ordinary matter expand forever, decelerating but never stopping.

A positive cosmological constant changes the ending. Because ${\rho }_{\Lambda }$ stays constant while matter and radiation dilute, the vacuum term eventually dominates the first Friedmann equation regardless of $k$, and the expansion tends to the de Sitter exponential. A closed universe with sufficient $\Lambda$ expands forever rather than recollapsing; the fate is no longer read off from the spatial curvature alone. The observed universe, with ${\Omega }_m\approx 0.3$ and ${\Omega }_{\Lambda }\approx 0.7$, is spatially near-flat and headed for eternal accelerating expansion.

The hand-off to cosmology

The Friedmann equations turn the geometric scale factor into a dynamical variable whose evolution is fixed by the matter content, and the density parameters compress the whole model into a handful of numbers. This is the point where relativity as a theory of spacetime geometry becomes cosmology as a quantitative science. A dedicated cosmology subject develops what follows: the thermal history of the hot early universe, primordial nucleosynthesis and the origin of the light elements, the microwave background as a snapshot of the last scattering surface, the growth of structure from small perturbations, and the evidence for dark matter and dark energy that fixes the density parameters.

The FLRW metric and the Friedmann equations are the shared foundation of all of it. The expanding-universe kinematics and cosmological dynamics carry directly into a full treatment of the FRW metric and cosmological redshift, the Friedmann equations and cosmic dynamics, and dark energy and the accelerating universe, where the same equations derived here from general relativity are matched against the observations.