= Solution
The <cosmological principle> permits a <Friedmann-Lemaître-Robertson-Walker metric>. The <Friedmann equation>, with vanishing <cosmological constant>, is
$$
H^2=\frac{8\pi G}{3}\bar\rho_m-\frac{k}{a^2}.
$$
For separately conserved <pressureless matter>, the <cosmological continuity equation> gives $\bar\rho_m=\bar\rho_{m0}a^{-3}$: its rest <energy> is conserved while its physical <volume> grows as $a^3$. Define the present <critical density> and <matter density parameter> by
$$
\rho_{\mathrm{crit},0}=\frac{3H_0^2}{8\pi G},\qquad \Omega=\frac{\bar\rho_{m0}}{\rho_{\mathrm{crit},0}}.
$$
Evaluating the <Friedmann equation> today gives $-k=H_0^2(1-\Omega)>0$. Thus the model has negative <spatial curvature of an FLRW universe>; the second contribution to $H^2$ represents that <spatial curvature of an FLRW universe>, rather than another material fluid. Taking the expanding branch gives
$$
\boxed{H(a)=H_0\sqrt{\Omega a^{-3}+(1-\Omega)a^{-2}}.}
$$
Here $0<\Omega<1$. At early times, $a\ll\Omega/(1-\Omega)$, <matter domination> gives $\dot a\simeq H_0\sqrt\Omega\,a^{-1/2}$, hence, measuring <cosmic time> from the <Big Bang>,
$$
\boxed{a(t)\simeq\left(\frac32H_0\sqrt\Omega\,t\right)^{2/3}.}
$$
At late times the $a^{-2}$ term dominates the <Friedmann equation>. This <curvature-dominated universe> has $\dot a\to H_0\sqrt{1-\Omega}$ and therefore
$$
\boxed{a(t)\sim H_0\sqrt{1-\Omega}\,t.}
$$
The late expansion approaches the coasting <Milne model>. The early limit belongs to this idealized matter-only model; including <radiation in cosmology> would alter the sufficiently early evolution.
Back to article page
