= Solution
Introduce the positive <dimensionless quantity> $x=(1-\Omega)a/\Omega$ and the time scale $A=\Omega/[H_0(1-\Omega)^{3/2}]$. The <Friedmann equation> gives the separable relation
$$
dt=\frac{da}{aH(a)}=A\sqrt{\frac{x}{1+x}}\,dx.
$$
For $x=(\cosh\theta-1)/2=\sinh^2(\theta/2)$, $\theta\geq0$, the <hyperbolic function> identities give
$$
\sqrt{\frac{x}{1+x}}\,dx
=\tanh\frac\theta2\,\frac{\sinh\theta}{2}\,d\theta
=\frac{\cosh\theta-1}{2}\,d\theta.
$$
Choose the <constant of integration> so that $t=0$ at $a=0$. The resulting <parametric equation> for the <open matter-dominated Friedmann solution> is
$$
\boxed{t=\frac{\Omega}{2H_0(1-\Omega)^{3/2}}(\sinh\theta-\theta),\qquad a=\frac{\Omega}{2(1-\Omega)}(\cosh\theta-1).}
$$
For $\theta\ll1$, the <Taylor series> give
$$
t=\frac{\Omega\theta^3}{12H_0(1-\Omega)^{3/2}}[1+O(\theta^2)],\qquad a=\frac{\Omega\theta^2}{4(1-\Omega)}[1+O(\theta^2)].
$$
Eliminating $\theta$ yields $a^{3/2}\simeq(3/2)H_0\sqrt\Omega\,t$, exactly the early <matter-dominated universe> result. For $\theta\gg1$, $\sinh\theta\sim\cosh\theta\sim e^\theta/2$, whereas $\theta/e^\theta\to0$. The <ratio> of the two parametric expressions tends to $a/t=H_0\sqrt{1-\Omega}$, recovering the <curvature-dominated universe> result including its coefficient.
Back to article page
