A homogeneous and isotropic universe, with scale factor $a$, curvature parameter $k$, energy density $\rho$ and pressure $P$, satisfies the Friedmann and energy conservation equations

$$\begin{aligned} &H^{2}+\frac{k c^{2}}{a^{2}}=\frac{8 \pi G}{3} \rho \\ &\dot{\rho}+3 H\left(\rho+P / c^{2}\right)=0 \end{aligned}$$

where $H=\dot{a} / a$, and the dot indicates a derivative with respect to cosmological time $t$.

(i) Derive the acceleration equation

$$\frac{\ddot{a}}{a}=-\frac{4 \pi G}{3}\left(\rho+3 P / c^{2}\right)$$

Given that the strong energy condition $\rho c^{2}+3 P \geqslant 0$ is satisfied, show that $(a H)^{2}$ is a decreasing function of $t$ in an expanding universe. Show also that the density parameter $\Omega=8 \pi G \rho /\left(3 H^{2}\right)$ satisfies

$$\Omega-1=\frac{k c^{2}}{a^{2} H^{2}}$$

Hence explain, briefly, the flatness problem of standard big bang cosmology.

(ii) A flat $(k=0)$ homogeneous and isotropic universe is filled with a radiation fluid $\left(w_{R}=1 / 3\right)$ and a dark energy fluid $\left(w_{\Lambda}=-1\right)$, each with an equation of state of the form $P_{i}=w_{i} \rho_{i} c^{2}$ and density parameters today equal to $\Omega_{R 0}$ and $\Omega_{\Lambda 0}$ respectively. Given that each fluid independently obeys the energy conservation equation, show that the total energy density $\left(\rho_{R}+\rho_{\Lambda}\right) c^{2}$ equals $\rho c^{2}$, where

$$\rho(t)=\frac{3 H_{0}^{2}}{8 \pi G} \frac{\Omega_{R 0}}{a^{4}}\left(1+\frac{1-\Omega_{R 0}}{\Omega_{R 0}} a^{4}\right)$$

with $H_{0}$ being the value of the Hubble parameter today. Hence solve the Friedmann equation to get

$$a(t)=\alpha(\sinh \beta t)^{1 / 2}$$

where $\alpha$ and $\beta$ should be expressed in terms $\Omega_{R 0}$ and $\Omega_{\Lambda 0}$. Show that this result agrees with the expected asymptotic solutions at both early $(t \rightarrow 0)$ and late $(t \rightarrow \infty)$ times.

[Hint: $\int d x / \sqrt{x^{2}+1}=\operatorname{arcsinh} x$.]