= Radiation--cosmic-string Friedmann solution
Consider a spatially flat universe containing <radiation in cosmology>[radiation] and a <cosmic string network>. If their densities are both $\rho_{\rm eq}$ at scale factor $a_{\rm eq}$, then
$$
\rho_r=\rho_{\rm eq}\left(\frac{a_{\rm eq}}a\right)^4,
\qquad
\rho_s=\rho_{\rm eq}\left(\frac{a_{\rm eq}}a\right)^2.
$$
In <conformal time> $d\tau=dt/a$, the flat <Friedmann equation> becomes
$$
\left(\frac{da}{d\tau}\right)^2
=\frac{8\pi G a_{\rm eq}^2}{3c^2}\rho_{\rm eq}
(a^2+a_{\rm eq}^2).
$$
Choosing the Big Bang at $\tau=0$, its expanding solution is
$$
a(\tau)=a_{\rm eq}\sinh\!\left(
a_{\rm eq}\sqrt{\frac{8\pi G\rho_{\rm eq}}{3c^2}}\,\tau
\right).
$$
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