= Solution
The two scalings give $\rho=\rho_{r0}a^{-4}+\rho_{s0}a^{-2}$. Flatness and the <Friedmann equation> give $H^2=8\pi G\rho/3$, with $H_0^2=(8\pi G/3)(\rho_{r0}+\rho_{s0})$. Thus
$$
\boxed{\left(\frac{\dot a}{a}\right)^2=\frac{H_0^2}{a^4}\frac{a^2+\beta}{1+\beta}.}
$$
On the expanding branch, putting the <Big Bang> at zero, separation gives
$$
t=\frac{\sqrt{1+\beta}}{H_0}\bigl(\sqrt{a^2+\beta}-\sqrt\beta\bigr),\qquad a^2=\frac{H_0^2t^2}{1+\beta}+\frac{2H_0t\sqrt\beta}{\sqrt{1+\beta}}.
$$
Its positive <square root> is the required solution. If $\beta>0$, its early behaviour is $a\propto t^{1/2}$, as for radiation. The values $a=1$ and $a^2=\beta$ give
$$
\boxed{t_0=\frac{\sqrt{1+\beta}}{H_0}(\sqrt{1+\beta}-\sqrt\beta),\qquad t_{\rm eq}=\frac{\sqrt\beta\sqrt{1+\beta}}{H_0}(\sqrt2-1).}
$$
For $\beta=0$ there is only the string component, with linear expansion and no radiation era.
Back to article page
