What are the cosmological flatness and horizon problems? Explain what form of time evolution of the cosmological expansion scale factor $a(t)$ must occur during a period of inflationary expansion in a Friedmann universe. How can inflation solve the horizon and flatness problems? [You may assume an equation of state where pressure $P$ is proportional to density $\rho$.]

The universe has Hubble expansion rate $H=\dot{a} / a$ and contains only a scalar field $\phi$ with self-interaction potential $V(\phi)>0$. The density and pressure are given by

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

in units where $c=\hbar=1$. Show that the conservation equation

$$\dot{\rho}+3 H(\rho+P)=0$$

requires

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

If the Friedmann equation has the form

$$3 H^{2}=8 \pi G \rho$$

and the scalar-field potential has the form

$$V(\phi)=V_{0} e^{-\lambda \phi}$$

where $V_{0}$ and $\lambda$ are positive constants, show that there is an exact cosmological solution with

$$\begin{aligned} &a(t) \propto t^{16 \pi G / \lambda^{2}} \\ &\phi(t)=\phi_{0}+\frac{2}{\lambda} \ln (t), \end{aligned}$$

where $\phi_{0}$ is a constant. Find the algebraic relation between $\lambda, V_{0}$ and $\phi_{0}$. Show that a solution only exists when $0<\lambda^{2}<48 \pi G$. For what range of values of $\lambda^{2}$ does inflation occur? Comment on what happens when $\lambda \rightarrow 0$.