A closed universe contains black-body radiation, has a positive cosmological constant $\Lambda$, and is governed by the equation

$$\frac{\dot{a}^{2}}{a^{2}}=\frac{\Gamma}{a^{4}}-\frac{1}{a^{2}}+\frac{\Lambda}{3},$$

where $a(t)$ is the scale factor and $\Gamma$ is a positive constant. Using the substitution $y=a^{2}$ and the boundary condition $y(0)=0$, deduce the boundary condition for $\dot{y}(0)$ and show that

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

and hence that

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

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

Sketch the graphs of $a(t)$ for the cases $\lambda>1$ and $0<\lambda<1$.