Past exam of the mathematics course of the University of Cambridge / 2017 / iii / Paper 310 / 2 / a / Solution

Here primes mean conformal time derivatives, $d\tau =dt/a$, and spatial derivatives use comoving coordinates in Newtonian gauge in cosmology. Set ${\theta }_m=\nabla \cdot {\mathbf{v}}_m$. The linear cosmological Euler equation gives ${\theta }_m^{\prime }=-\mathcal{H}{\theta }_m-{\nabla }^2\Phi$, while the continuity equation gives ${\theta }_m=-{\delta }_m^{\prime }+3{\Phi }^{\prime }$.

Differentiate the continuity equation and substitute:Thus the linear matter density equation in Newtonian gauge isThe common potential in these supplied equations assumes that the relevant scalar anisotropic stress is neglected. No Fourier-sign convention is needed for the derivation in position space.