Friedmann Equations

The Friedmann equations are a pair of ordinary differential equations that follow from Einstein's field equations once one assumes the universe is homogeneous (the same at every point) and isotropic (the same in every direction) on large scales. Their single unknown is the scale factor a(t), which measures how stretched space is relative to today, where a = 1. The first equation, H² = (ȧ/a)² = (8πG/3)ρ − kc²/a² + Λc²/3, sets the square of the expansion rate H equal to the gravitating energy density ρ, a spatial-curvature term fixed by k ∈ {−1, 0, +1}, and a cosmological-constant term. It is an energy-conservation statement for the expansion.

The second Friedmann equation, ä/a = −(4πG/3)(ρ + 3p/c²) + Λc²/3, governs whether the expansion accelerates or decelerates. Its defining relativistic feature is that pressure gravitates: the combination ρ + 3p/c² means a gas of radiation decelerates expansion more strongly than the same energy density of pressureless matter, and a component with sufficiently negative pressure — such as vacuum energy — can make the expansion accelerate. Together the two equations form a complete equation of motion: specify the contents of the universe today and they determine its entire past and future.

In practice cosmologists write the first equation in dimensionless form as E(a)² = Ω_r a⁻⁴ + Ω_m a⁻³ + Ω_k a⁻² + Ω_Λ, where each density parameter Ω_i is a present-day energy density measured against the critical density. Because radiation, matter, curvature, and the cosmological constant each scale with a different power of a, the universe passes through a radiation-dominated era, a matter-dominated era, and finally a Λ-dominated era — and the long-term fate (recollapse, perpetual coasting, or runaway acceleration) is decided by which term wins.

Derived by Alexander Friedmann in 1922 and 1924, then independently rediscovered and tied to galaxy redshifts by Georges Lemaître in 1927. Einstein initially dismissed Friedmann's expanding solutions as erroneous before retracting the objection.