Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 3 / 14D (Cosmology)

Since $P=P(\rho )$ and $\rho =\bar{\rho }(1+\delta )$,To first order, continuity and the divergence of Euler giveEliminating $\theta$ and using $\nabla \mapsto i\mathbf{k}/a$ yieldsThe second term is Hubble damping, gravity drives growth, and pressure produces oscillatory restoration. Their balance definesModes longer than the Jeans length are gravitationally unstable; shorter modes are pressure supported.

HereSince $\bar{\rho }\propto t^{-2}$ and $a\propto t^{2/3}$, $c_s^2/a^2=L_0^2/t^2$ for a constant $L_0^2$. With $\dot{a}/a=2/(3t)$ and $4\pi G\bar{\rho }/c^2=2/(3t^2)$, substitution of $\delta =t^n$ givessoFor $k\ll {\tilde{k}}_J=5/(6L_0)$, the modes approach $t^{2/3}$ and $t^{-1}$. For $k\gg {\tilde{k}}_J$, the exponents are complex with real part $-1/6$, giving logarithmic oscillations with decaying envelope $t^{-1/6}$. Taking $L_0\simeq 50\mathrm{Mpc}$ would pressure-suppress structure on enormous, cosmologically important scales and is incompatible with the observed growth of smaller-scale structure.