= Cosmological density parameter flow
{title2=$d\Omega/d\log a$}
For an expanding <Friedmann-Lemaitre-Robertson-Walker metric> with positive <energy density>, put $\Omega=8\pi G\rho/(3H^2)$ and $w=P/\rho$, in units $c=1$. The <Friedmann equation>, <Friedmann acceleration equation> and <cosmological perfect-fluid continuity equation> give
$$
\frac{\dot H}{H^2}=-1-\frac12(1+3w)\Omega,\qquad
\frac{d\Omega}{d\log a}=(1+3w)\Omega(\Omega-1).
$$
For constant $w$, linearizing at $\Omega=1$ gives $\Omega-1\propto a^{1+3w}$. For variable $w$, replace the power by $\exp\int(1+3w(a))\,d\log a$. This describes the <Flatness problem> for decelerating expansion, away from epochs at which $H=0$.
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