Cosmological density parameter flow ($d\Omega /d\log a$)

For an expanding Friedmann-Lemaître-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 giveFor 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 $H=0$.

For separately conserved pressureless matter, ${\rho }_m\propto a^{-3}$. The deceleration parameter identity $\dot{H}/H^2=-(1+q)$ and ${\Omega }_m\propto {\rho }_m/H^2$ give ${\dot{\Omega }}_m=H{\Omega }_m(2q-1)$. A dust-plus-vacuum universe has $q={\Omega }_m/2-{\Omega }_{\Lambda }$; conversion to conformal time yields the displayed flow. The familiar matter-only equation requires negligible cosmological constant. In a spatially flat dust-plus-vacuum universe, ${\Omega }_m+{\Omega }_{\Lambda }=1$ and ${\Omega }_m^{\prime }=-3\mathcal{H}{\Omega }_m{\Omega }_{\Lambda }$, so flatness does not require ${\Omega }_m=1$.