Past exam of the mathematics course of the University of Cambridge / 2003 / iii / Paper 58 / 5 / c / Solution

Vary the action with compactly supported variations and integrate by parts. Its overall normalization does not affect the equation of motion:Here $\sqrt{-g}=e^{3t}$, so the massless scalar field obeysSeparate a spatial Fourier mode as $\varphi (t,\mathbf{x})=F_k(t)e^{i\mathbf{k}\cdot \mathbf{x}}$. Its mode equation is $F_{k,tt}+3F_{k,t}+k^2{e}^{-2t}{F}_k=0$. With conformal time $\tau =-e^{-t}$, $a=-1/\tau$ and $\mathcal{H}=-1/\tau$, it becomesFor $k>0$, put $z=-k\tau =k{e}^{-t}$ and $F_k=z^{2}G(z)$. Direct differentiation reduces the equation tothe Spherical Bessel function equation with order one. Hence the complete classical mode basis isSince $z^2{j}_1(z)=\sin z-z\cos z$ and $z^2{y}_1(z)=-\cos z-z\sin z$, an equivalent complex basis isThe classical action does not select one frequency branch or a quantum vacuum. A real field has conjugate-related coefficients at opposite wavevectors. The zero mode, which is not obtained by treating these two $k>0$ basis functions as independent at $k=0$, isAt late times one independent solution freezes and the other decays as $e^{-3t}$. These are minimally coupled massless modes; the supplied action has no curvature-coupling term, so it should not be replaced by the conformally coupled wave equation.