The expansion of the universe during inflation is governed by the Friedmann equation

$$\left(\frac{\dot{a}}{a}\right)^{2}=\frac{8 \pi G}{3}\left[\frac{1}{2} \dot{\phi}^{2}+V(\phi)\right]$$

and the equation of motion for the inflaton field $\phi$,

$$\ddot{\phi}+3 \frac{\dot{a}}{a} \dot{\phi}+\frac{\mathrm{d} V}{\mathrm{~d} \phi}=0 .$$

Consider the potential

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

with $V_{0}>0$ and $\lambda>0$.

(a) Show that the inflationary equations have the exact solution

$$a(t)=\left(\frac{t}{t_{0}}\right)^{\gamma} \quad \text { and } \quad \phi=\phi_{0}+\alpha \log t$$

for arbitrary $t_{0}$ and appropriate choices of $\alpha, \gamma$ and $\phi_{0}$. Determine the range of $\lambda$ for which the solution exists. For what values of $\lambda$ does inflation occur?

(b) Using the inflaton equation of motion and

$$\rho=\frac{1}{2} \dot{\phi}^{2}+V$$

together with the continuity equation

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

determine $P$.

(c) What is the range of the pressure energy density ratio $\omega \equiv P / \rho$ for which inflation occurs?