= Friedmann effective potential for matter, radiation and vacuum
{c}
{title2=$V(a)=-M/a-Q/a^2-La^2$}
With nonnegative conserved <pressureless matter> and <radiation in cosmology>, set $M=(8\pi G/3)\rho_m a^3$, $Q=(8\pi G/3)\rho_r a^4$, and $L=(8\pi G/3)\rho_\Lambda$. The <Friedmann equation> becomes $(da/dt)^2+V(a)=-k$. For negative vacuum density the potential rises strictly from negative infinity to positive infinity, so an expanding branch has one maximum scale factor. For positive vacuum density it has one maximum: energy above that maximum gives monotonic expansion or contraction, energy below it gives disjoint recollapsing and bouncing branches, and equality admits an <Einstein static universe> together with asymptotic monotonic branches.
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