(a) Consider a closed universe endowed with cosmological constant $\Lambda>0$ and filled with radiation with pressure $P$ and energy density $\rho$. Using the equation of state $P=\frac{1}{3} \rho$ and the continuity equation

$$\dot{\rho}+\frac{3 \dot{a}}{a}(\rho+P)=0,$$

determine how $\rho$ depends on $a$. Give the physical interpretation of the scaling of $\rho$ with

(b) For such a universe the Friedmann equation reads

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

What is the physical meaning of $R ?$

(c) Making the substitution $a(t)=\alpha \tilde{a}(t)$, determine $\alpha$ and $\Gamma>0$ such that the Friedmann equation takes the form

$$\left(\frac{\dot{\tilde{a}}}{\tilde{a}}\right)^{2}=\frac{\Gamma}{\tilde{a}^{4}}-\frac{1}{\tilde{a}^{2}}+\frac{\Lambda}{3} .$$

Using the substitution $y(t)=\tilde{a}(t)^{2}$ and the boundary condition $y(0)=0$, deduce the boundary condition for $\dot{y}(0)$.

Show that

$$\ddot{y}=\frac{4 \Lambda}{3} y-2$$

and hence that

$$\tilde{a}^{2}(t)=\frac{3}{2 \Lambda}\left[1-\cosh \left(\sqrt{\frac{4 \Lambda}{3}} t\right)+\lambda \sinh \left(\sqrt{\frac{4 \Lambda}{3}} t\right)\right]$$

Express the constant $\lambda$ in terms of $\Lambda$ and $\Gamma$.

Sketch the graphs of $\tilde{a}(t)$ for the cases $\lambda>1, \lambda<1$ and $\lambda=1$.