Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 62 / 3 / b / Solution

For the FLRW metric in conformal time, $\sqrt{-g}=a^4$, $g^{00}=-a^{-2}$ and $g^{ij}=a^{-2}{\delta }^{ij}$. The equation for a minimally coupled massless scalar field is consequentlyFor a Fourier mode of comoving wavevector $\mathbf{k}$, the Laplacian contributes $-k^2$, where $k=|\mathbf{k}|$. ThusSince $a^{\prime }/a=-1/\tau$, this yieldsThe absence of a curvature-coupling term is material: a conformally coupled scalar field obeys a different mode equation.