A fluid with pressure $P$ sits in a volume $V$. The change in energy due to a change in volume is given by $d E=-P d V$. Use this in a cosmological context to derive the continuity equation,

$$\dot{\rho}=-3 H(\rho+P),$$

with $\rho$ the energy density, $H=\dot{a} / a$ the Hubble parameter, and $a$ the scale factor.

In a flat universe, the Friedmann equation is given by

$$H^{2}=\frac{8 \pi G}{3 c^{2}} \rho .$$

Given a universe dominated by a fluid with equation of state $P=w \rho$, where $w$ is a constant, determine how the scale factor $a(t)$ evolves.

Define conformal time $\tau$. Assume that the early universe consists of two fluids: radiation with $w=1 / 3$ and a network of cosmic strings with $w=-1 / 3$. Show that the Friedmann equation can be written as

$$\left(\frac{d a}{d \tau}\right)^{2}=B \rho_{\mathrm{eq}}\left(a^{2}+a_{\mathrm{eq}}^{2}\right)$$

where $\rho_{\mathrm{eq}}$ is the energy density in radiation, and $a_{\mathrm{eq}}$ is the scale factor, both evaluated at radiation-string equality. Here, $B$ is a constant that you should determine. Find the solution $a(\tau)$.