= Friedmann effective potential for a constant-equation-of-state fluid
{c}
{title2=$V(a)=-\rho_0a^{-(1+3w)}/6+k/2-\Lambda a^2/6$}
= Friedmann mechanical analogy
{c}
{synonym}
For a <perfect fluid in general relativity> of constant <equation of state> $P=w\rho$, conservation gives $\rho=\rho_0a^{-3(1+w)}$. In units $8\pi G=1$, the <Friedmann equation> becomes $\dot a^2/2+V(a)=0$. Allowed scales have $V\leq0$, and the <Friedmann acceleration equation> is $\ddot a=-V'(a)$. The curvature term shifts the potential vertically; positive <cosmological constant> bends it downward and negative <cosmological constant> bends it upward.
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