(a) Consider a homogeneous and isotropic universe filled with relativistic matter of mass density $\rho(t)$ and scale factor $a(t)$. Consider the energy $E(t) \equiv \rho(t) c^{2} V(t)$ of a small fluid element in a comoving volume $V_{0}$ where $V(t)=a^{3}(t) V_{0}$. Show that for slow (adiabatic) changes in volume, the density will satisfy the fluid conservation equation

$$\dot{\rho}=-3 \frac{\dot{a}}{a}\left(\rho+P / c^{2}\right)$$

where $P$ is the pressure.

(b) Suppose that a flat $(k=0)$ universe is filled with two matter components:

(i) radiation with an equation of state $P_{\mathrm{r}}=\frac{1}{3} \rho_{\mathrm{r}} c^{2}$.

(ii) a gas of cosmic strings with an equation of state $P_{\mathrm{s}}=-\frac{1}{3} \rho_{\mathrm{s}} c^{2}$.

Use the fluid conservation equation to show that the total relativistic mass density behaves as

$$\rho=\frac{\rho_{\mathrm{r} 0}}{a^{4}}+\frac{\rho_{\mathrm{s} 0}}{a^{2}},$$

where $\rho_{\mathrm{r} 0}$ and $\rho_{\mathrm{s} 0}$ are respectively the radiation and string densities today (that is, at $t=t_{0}$ when $a\left(t_{0}\right)=1$ ). Assuming that both the Hubble parameter today $H_{0}$ and the ratio $\beta \equiv \rho_{\mathrm{r} 0} / \rho_{\mathrm{s} 0}$ are known, show that the Friedmann equation can be rewritten as

$$\left(\frac{\dot{a}}{a}\right)^{2}=\frac{H_{0}^{2}}{a^{4}}\left(\frac{a^{2}+\beta}{1+\beta}\right) .$$

Solve this equation to find the following solution for the scale factor

$$a(t)=\frac{\left(H_{0} t\right)^{1 / 2}}{(1+\beta)^{1 / 2}}\left[H_{0} t+2 \beta^{1 / 2}(1+\beta)^{1 / 2}\right]^{1 / 2} .$$

Show that the scale factor has the expected asymptotic behaviour at early times $t \rightarrow 0$.

Hence show that the age of this universe today is

$$t_{0}=H_{0}^{-1}(1+\beta)^{1 / 2}\left[(1+\beta)^{1 / 2}-\beta^{1 / 2}\right]$$

and that the time $t_{\mathrm{eq}}$ of equal radiation and string densities $\left(\rho_{\mathrm{r}}=\rho_{\mathrm{s}}\right)$ is

$$t_{\mathrm{eq}}=H_{0}^{-1}(\sqrt{2}-1) \beta^{1 / 2}(1+\beta)^{1 / 2}$$