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Insert the component-density laws into the Friedmann equation, using ${\rho }_{i,0}=3H_0^2{\Omega }_{i,0}/(8\pi G)$ and $a_0=1$. Multiplying by $a^2$ givesThe radiation in cosmology and pressureless matter contributions add; the printed first formula has lost the plus sign between them. In the Friedmann acceleration equation, $\rho +3P=2{\rho }_r+{\rho }_m-2{\rho }_{\Lambda }$, soEvaluating the first equation today gives $K=H_0^2({\Omega }_{r,0}+{\Omega }_{m,0}+{\Omega }_{\Lambda ,0}-1)$.

Now remove dark energy and assume a positive pressureless matter or radiation in cosmology density, starting on an expanding branch. Let $f(a)=H_0^2({\Omega }_{r,0}/a^2+{\Omega }_{m,0}/a)$. It decreases from infinity to zero, and ${\dot{a}}^2=f(a)-K$, while $\ddot{a}<0$. This immediately distinguishes all three cases of spatial curvature of an FLRW universe.

The early small-$a$ solution has $a\propto t^{1/2}$ if radiation in cosmology is present, and $a\propto t^{2/3}$ for pressureless matter alone. The figure isolates the effect of curvature in three matter-only examples, rather than comparing models with the same present density parameters.

Put $\Omega =1+\delta$ with $|\delta |\ll 1$. Linearizing the cosmological density parameter flow gives $d\delta /d\log a=(1+3\omega )\delta +O({\delta }^2)$. For constant equation-of-state parameter,For variable $\omega$, the correct replacement is $\delta (a)=\delta (a_i)\exp [{\int }_{a_i}^a(1+3\omega (\tilde{a}))d\log \tilde{a}]$. The printed power assumes constant $\omega$.

For pressureless matter, the departure from flatness grows as $a$; for radiation in cosmology, it grows as $a^2$. Hence in a conventional decelerating expanding history, maintaining $|\Omega -1|\ll 1$ at a late epoch requires extremely small departures at early times. This is the Flatness problem, not an assertion that exact flatness is inconsistent: $\Omega =1$ is an exact solution, but it is unstable to small departures in spatial curvature of an FLRW universe during these eras. With $\omega <-1/3$ the perturbation instead decays, which supplies the inflationary solution of the flatness problem.

The Hubble parameter is the fractional expansion rateHere $\rho$ is the homogeneous energy density, $k\in \{-1,0,1\}$ records the sign of the spatial curvature of an FLRW universe, and $R$ is the fixed curvature length scale used with the dimensionless scale factor $a(t)$.

Energy-momentum conservation in a homogeneous expanding universe gives the continuity equationDifferentiate the Friedmann equationUsing $\dot{a}/a=H$ and the continuity equation givesEliminating the curvature term with the Friedmann equation yieldsSince $\dot{H}+H^2=\ddot{a}/a$, this is the Raychaudhuri equation

Under the strong energy condition $\rho +3P\ge 0$, one has $\ddot{a}\le 0$. Thus $a(t)$ is concave, and for $t<t_0$ it lies below its tangent at the present time:The right side vanishes at $t=t_0-H_0^{-1}$. A positive scale factor therefore cannot extend regularly to that time; it must reach the Big Bang singularity at some $t_{\mathrm{BB}}\ge t_0-H_0^{-1}$. Equivalently,

A round three-sphere of radius $R$ embedded by $|\mathbf{x}{|}^2+u^2=R^2$ has metric $R^2\{d{r}^2/(1-r^2)+r^{2}d{\Omega }^2\}$ for $r=|\mathbf{x}|/R$. Multiplying by $b(t{)}^2$ gives a scale factor $a=Rb$ and normalized positive spatial curvature of an FLRW universe $K=1$. This radial chart covers a hemisphere; $r=\sin \chi$ gives the global polar description.