Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 59 / 2 / b / Solution

The massless scalar stress tensor in mixed-index form isIn conformal Newtonian gauge, $g^{00}=-a^{-2}(1-2\Phi )$ and $g^{ij}=a^{-2}(1+2\Psi ){\delta }^{ij}$ to first order. Spatial gradients of the background vanish, so products of two spatial perturbation gradients are second order. ConsequentlySubstitution yieldsThe potential $\Psi$ multiplies only second-order gradient products in these components. The same reasoning makes the off-diagonal spatial stress vanish at first order, consistent with $\Phi =\Psi$.