[You may work in units of the speed of light, so that $c=1$.]

Consider a spatially-flat FLRW universe with a single, canonical, homogeneous scalar field $\phi(t)$ with a potential $V(\phi)$. Recall the Friedmann equation and the Raychaudhuri equation (also known as the acceleration equation)

$$\begin{aligned} \left(\frac{\dot{a}}{a}\right)^{2} &=H^{2}=\frac{8 \pi G}{3}\left[\frac{1}{2} \dot{\phi}^{2}+V\right] \\ \frac{\ddot{a}}{a} &=-\frac{8 \pi G}{3}\left(\dot{\phi}^{2}-V\right) \end{aligned}$$

(a) Assuming $\dot{\phi} \neq 0$, derive the equations of motion for $\phi$, i.e.

$$\ddot{\phi}+3 H \dot{\phi}+\partial_{\phi} V=0 .$$

(b) Assuming the special case $V(\phi)=\lambda \phi^{4}$, find $\phi(t)$, for some initial value $\phi\left(t_{0}\right)=\phi_{0}$ in the slow-roll approximation, i.e. assuming that $\dot{\phi}^{2} \ll 2 V$ and $\ddot{\phi} \ll 3 H \dot{\phi}$.

(c) The number $N$ of efoldings is defined by $d N=d \ln a$. Using the chain rule, express $d N$ first in terms of $d t$ and then in terms of $d \phi$. Write the resulting relation between $d N$ and $d \phi$ in terms of $V$ and $\partial_{\phi} V$ only, using the slow-roll approximation.

(d) Compute the number $N$ of efoldings of expansion between some initial value $\phi_{i}<0$ and a final value $\phi_{f}<0$ (so that $\dot{\phi}>0$ throughout).

(e) Discuss qualitatively the horizon and flatness problems in the old hot big bang model (i.e. without inflation) and how inflation addresses them.