Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 1 / 15A (Cosmology)

The homogeneous part of the continuity equation gives $\dot{\bar{n}}+3H\bar{n}=0$. At first order,For the density contrast $\delta =\delta n/\bar{n}$, the background equation cancels the expansion terms, leaving

The homogeneous Euler equation removes the background force. Products of perturbations are second order, while differentiating the peculiar velocity inside the expanding flow produces the Hubble drag term. The first-order equation is therefore

Take the divergence of the linear Euler equation, use $\delta P=c_s^{2}m\bar{n}\delta$, and differentiate the continuity equation. Eliminating $\nabla \cdot \delta \mathbf{v}$ givesThe perturbed Poisson equation isThusHere $k_J$ is the comoving Jeans wavenumber.

Neglecting the pressure and self-gravity bracket leavesDuring radiation domination, $a(t)\propto t^{1/2}$ and $H=1/(2t)$, soConsequentlyMatter perturbations therefore grow only logarithmically while radiation dominates.