Critical density

At Hubble parameter $H$, the critical energy density is

The density parameter is $\Omega =\rho /{\rho }_{\mathrm{crit}}$. In a Friedmann universe, its departure from one measures the spatial-curvature term relative to the density or expansion term.

The cosmic baryon fraction is the ratio of mean baryonic matter density to total matter density. Assigning this fraction to a halo is a simplifying assumption; cooling, ejection and other baryonic processes can change the halo's retained gas and stellar fractions. A baryon conversion efficiency becomes a stellar-to-total-halo mass ratio only after multiplying by the applicable baryon fraction.

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 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 $H=0$.

In a decelerating expanding universe, $|\Omega -1|$ grows, so its small present value requires extremely fine-tuned early initial data unless an earlier mechanism drives $\Omega$ toward one.

The continuity equation givesDuring expansion with $\rho +3P<0$, the product $\rho a^2$ grows and $|{\Omega }^{-1}-1|\propto (\rho a^2{)}^{-1}$ decreases.