 Solution

A scalar cosmological perturbation has two scalar gauge freedoms. In Newtonian gauge in cosmology, these remove the scalar shift and spatial shear, leaving the lapse potential $\Phi$ and spatial-curvature potential $\Psi$:A perfect fluid has no scalar anisotropic stress. The traceless spatial Einstein field equations therefore set $\Phi =\Psi =\varphi$ for nonzero spatial Fourier modes, producing the line element in the question. Homogeneous perturbations include background and gauge changes and require separate treatment. The single potential describes the physical scalar perturbation in this gauge.