What is the flatness problem? Show by reference to the Friedmann equation how a period of accelerated expansion of the scale factor $a(t)$ in the early stages of the universe can solve the flatness problem if $\rho+3 P<0$, where $\rho$ is the mass density and $P$ is the pressure.

In the very early universe, where we can neglect the spatial curvature and the cosmological constant, there is a homogeneous scalar field $\phi$ with a vacuum potential energy

$$V(\phi)=m^{2} \phi^{2},$$

and the Friedmann energy equation (in units where $8 \pi G=1$ ) is

$$3 H^{2}=\frac{1}{2} \dot{\phi}^{2}+V(\phi),$$

where $H$ is the Hubble parameter. The field $\phi$ obeys the evolution equation

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

During inflation, $\phi$ evolves slowly after starting from a large initial value $\phi_{i}$ at $t=0$. State what is meant by the slow-roll approximation. Show that in this approximation,

$$\begin{aligned} \phi(t) &=\phi_{i}-\frac{2}{\sqrt{3}} m t \\ a(t) &=a_{i} \exp \left[\frac{m \phi_{i}}{\sqrt{3}} t-\frac{1}{3} m^{2} t^{2}\right]=a_{i} \exp \left[\frac{\phi_{i}^{2}-\phi^{2}(t)}{4}\right] \end{aligned}$$

where $a_{i}$ is the initial value of $a$.

As $\phi(t)$ decreases from its initial value $\phi_{i}$, what is its approximate value when the slow-roll approximation fails?