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The Friedmann equations are the Einstein field equations applied to a homogeneous, isotropic universe described by the Robertson-Walker metric. The first Friedmann equation (ȧ/a)² = (8πG/3)ρ - kc²/a² + Λ/3 relates the expansion rate H = ȧ/a (Hubble parameter) to the total energy density ρ, the spatial curvature parameter k, and the cosmological constant Λ. The second (acceleration) equation ä/a = -(4πG/3)(ρ + 3p/c²) + Λ/3 determines whether the expansion is accelerating or decelerating. Together with an equation of state relating pressure to density, they form a complete dynamical system for the scale factor a(t). These equations are the foundation of the standard ΛCDM cosmological model and describe the expansion history of the universe from the Big Bang to the present accelerating expansion.
The Friedmann equations are what you get when you apply Einstein's field equations to a universe that is homogeneous (the same everywhere) and isotropic (the same in every direction). The cosmological principle — the assumption that we do not occupy a special place — motivated these symmetry assumptions, and observations of the cosmic microwave background confirm homogeneity and isotropy to better than one part in 10⁵ on large scales. The geometry of such a universe is described by the Robertson-Walker metric, and the Einstein equations reduce to two ordinary differential equations for the scale factor a(t), which measures how the "size" of the universe changes with time.
The first Friedmann equation, H² = (8πG/3)ρ - kc²/a² + Λ/3, is an energy-balance equation. The left side, H² = (ȧ/a)², is the square of the Hubble parameter — the expansion rate. The right side has three terms: the energy density ρ (which drives expansion), the spatial curvature k/a² (which can accelerate or decelerate depending on sign), and the cosmological constant Λ (which drives accelerated expansion). The critical density ρ_crit = 3H²/(8πG) is the density at which the universe is spatially flat (k = 0). The density parameter Ω = ρ/ρ_crit determines the spatial geometry: Ω = 1 means flat, Ω > 1 means positively curved (closed), Ω < 1 means negatively curved (open). Observations from the CMB, baryon acoustic oscillations, and supernovae consistently give Ω ≈ 1, indicating a nearly flat universe.
The second Friedmann equation (acceleration equation) ä/a = -(4πG/3)(ρ + 3p/c²) + Λ/3 determines whether the expansion is accelerating or decelerating. The crucial quantity is ρ + 3p/c²: if it is positive (as for ordinary matter and radiation), gravity decelerates the expansion (ä < 0). If negative (as for a cosmological constant, where p = -ρc² gives ρ + 3p/c² = -2ρ), the expansion accelerates (ä > 0). The 1998 discovery that the expansion is accelerating (Type Ia supernovae observations by the Supernova Cosmology Project and the High-z Supernova Search Team) implies that the dominant energy component of the universe today has negative pressure — dark energy, modeled most simply as a cosmological constant Λ.
The complete cosmological model requires an equation of state p = wρc² for each component. Matter has w = 0 (pressureless), radiation has w = 1/3, and a cosmological constant has w = -1. Each component's density scales differently with the scale factor: ρ_m ∝ a⁻³, ρ_r ∝ a⁻⁴, ρ_Λ = const. This means the universe's expansion history passes through distinct eras: radiation-dominated (early, a ∝ t1/2), matter-dominated (intermediate, a ∝ t2/3), and dark-energy-dominated (late, a ∝ exp(Ht)). The ΛCDM model — cold dark matter plus a cosmological constant — fits all current observations and describes a universe that is 13.8 billion years old, spatially flat, and composed of about 68% dark energy, 27% dark matter, and 5% ordinary (baryonic) matter.