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Friedmann Equations

These equations describe the expansion of the universe based on general relativity, relating expansion rate to energy density and curvature: ȧ/a = - (4πG/3)ρ + (Λ/3).
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The statement of the theorem

Let a(t)a(t) be the scale factor of the universe, ρ\rho the total energy density, PP the pressure, kk the curvature parameter (k∈{−1,0,1}k\in\{-1, 0, 1\}), and Λ\Lambda the cosmological constant. The Friedmann equations are:\n\n(a˙a)2=8πG3ρ−kc2a2+Λc23(First Friedmann Equation)\left(\frac{\dot{a}}{a}\right)^2 = \frac{8\pi G}{3}\rho - \frac{kc^2}{a^2} + \frac{\Lambda c^2}{3} \quad \text{(First Friedmann Equation)}\n\na¨a=−4πG3(ρ+3P)+Λc23(Second Friedmann Equation)\frac{\ddot{a}}{a} = -\frac{4\pi G}{3}(\rho + 3P) + \frac{\Lambda c^2}{3} \quad \text{(Second Friedmann Equation)}\n\nThese equations govern the evolution of the scale factor a(t)a(t) based on the energy-momentum tensor TμνT_{\mu\nu} via Einstein's field equations.
Source: Wikipedia