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Friedmann Equations

Governing equations of an expanding universe, incorporating dark energy and matter density.
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The statement of the theorem

Assuming the Friedmann-Lemaître-Robertson-Walker (FLRW) metric, the evolution of the scale factor a(t)a(t) is governed by the Friedmann equations:\n\n1. **First Friedmann Equation (Energy):**\n(a˙a)2=8πG3c2ρ−kc2a2+Λc23\left(\frac{\dot{a}}{a}\right)^2 = \frac{8\pi G}{3c^2} \rho - \frac{k c^2}{a^2} + \frac{\Lambda c^2}{3} \n\n2. **Second Friedmann Equation (Acceleration):**\na¨a=−4πG3c2(ρ+3Pc2)+Λ3\frac{\ddot{a}}{a} = -\frac{4\pi G}{3c^2} \left(\rho + \frac{3P}{c^2}\right) + \frac{\Lambda}{3}
Source: Wikipedia