(i) Consider a homogeneous and isotropic universe with mass density $\rho(t)$, pressure $P(t)$ and scale factor $a(t)$. As the universe expands its energy $E$ decreases according to the thermodynamic relation $d E=-P d V$ where $V$ is the volume. Deduce the fluid conservation law

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

Apply the conservation of total energy (kinetic plus gravitational potential) to a test particle on the edge of a spherical region in this universe to obtain the Friedmann equation

$$\left(\frac{\dot{a}}{a}\right)^{2}=\frac{8 \pi G}{3} \rho-\frac{k c^{2}}{a^{2}}$$

where $k$ is a constant. State clearly any assumptions you have made.

(ii) Our universe is believed to be flat $(k=0)$ and filled with two major components: pressure-free matter $\left(P_{\mathrm{M}}=0\right)$ and dark energy with equation of state $P_{\mathrm{Q}}=-\rho_{\mathrm{Q}} c^{2}$ where the mass densities today $\left(t=t_{0}\right)$ are given respectively by $\rho_{\mathrm{M} 0}$ and $\rho_{\mathrm{Q} 0}$. Assume that each component independently satisfies the fluid conservation equation to show that the total mass density can be expressed as

$$\rho(t)=\frac{\rho_{\mathrm{M} 0}}{a^{3}}+\rho_{\mathrm{Q} 0},$$

where we have set $a\left(t_{0}\right)=1$.

Now consider the substitution $b=a^{3 / 2}$ in the Friedmann equation to show that the solution for the scale factor can be written in the form

$$a(t)=\alpha(\sinh \beta t)^{2 / 3}$$

where $\alpha$ and $\beta$ are constants. Setting $a\left(t_{0}\right)=1$, specify $\alpha$ and $\beta$ in terms of $\rho_{\mathrm{M} 0}, \rho_{\mathrm{Q} 0}$ and $t_{0}$. Show that the scale factor $a(t)$ has the expected behaviour for an Einstein-de Sitter universe at early times $(t \rightarrow 0)$ and that the universe accelerates at late times $(t \rightarrow \infty)$.

[Hint: Recall that $\int d x / \sqrt{x^{2}+1}=\sinh ^{-1} x$.]