The early universe is described by equations (with units such that $c=8 \pi G=\hbar=1$ )

$$3 H^{2}=\rho, \quad \dot{\rho}+3 H(\rho+p)=0$$

where $H=\dot{a} / a$. The universe contains only a self-interacting scalar field $\phi$ with interaction potential $V(\phi)$ so that the density and pressure are given by

$$\begin{aligned} \rho &=\frac{1}{2} \dot{\phi}^{2}+V(\phi) \\ p &=\frac{1}{2} \dot{\phi}^{2}-V(\phi) . \end{aligned}$$

Show that

$$\ddot{\phi}+3 H \dot{\phi}+V^{\prime}(\phi)=0$$

Explain the slow-roll approximation and apply it to equations (1) and (2) to show that it leads to

$$\sqrt{3} \int \frac{\sqrt{V}}{V^{\prime}} d \phi=-t+\text { const. }$$

If $V(\phi)=\frac{1}{4} \lambda \phi^{4}$ with $\lambda$ a positive constant and $\phi(0)=\phi_{0}$, show that

$$\phi(t)=\phi_{0} \exp \left[-\sqrt{\frac{4 \lambda}{3}} t\right]$$

and that, for small $t$, the scale factor $a(t)$ expands to leading order in $t$ as

$$a(t) \propto \exp \left[\sqrt{\frac{\lambda}{12}} \phi_{0}^{2} t\right]$$

Comment on the relevance of this result for inflationary cosmology.