2. Consider a massless scalar field; i.e., a scalar field $\phi(t,x)$ with a potential-energy density $V(\phi) = 0$. Suppose that this scalar field is initially rolling, so that $\dot\phi\neq 0$, and that the kinetic-energy density associated with this rolling dominates the energy density of the Universe. Show that this implies that $\rho\propto a^{-6}$, where $a$ is the scale factor, in two ways: (1) by recalling how the energy density of matter with an equation of state P = wρ scales with a; and (2) by solving the equation of motion $\phi$ in an expanding Universe.
3. Study the evolution of a scalar field (under the slow-roll assumption) if the inflationary potential is $\lambda \Phi^4$.
4. Show that $d(det(M))=det(M) M^{ik} dM_{ik}$, and that $T^{\mu\nu}=-\frac{2}{\sqrt{-g}}\frac{\partial(\sqrt{-g}{\mathcal L})}{\partial g_{\mu\nu}}$ leads to the usual energy-momentum tensor in the case of a scalar field with a generic potential $V(\Phi)$ and for a Minkowski space-time $g_{\mu\nu}=diag(1,-1,-1,-1)$. Work out $T^{00}$ and $T^{ii}$ in the case of a FLRW metric $g_{\mu\nu}=diag(1,-a^2(t),-a^2(t),-a^2(t))$