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First integral of the Friedmann acceleration equation

Combining the perfect-fluid continuity equation with

\frac{a ¨}{a} = - \frac{4 π G}{3 c ^{2}} (ρ + 3 P)

shows that

\frac{8 π G}{3 c ^{2}} ρ a^{2} - \overset{a}{˙}^{2}

is constant in time. This constant is the spatial-curvature integration constant in the first Friedmann equation.

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