Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 346 / 3

Before recombination, tight coupling makes photons and baryons share an adiabatic perturbation. Since $\delta {\rho }_b/{\rho }_b=(3/4)\delta {\rho }_r/{\rho }_r$ and only radiation supplies appreciable pressure,Thus the photon-baryon sound speed isBecause ${\rho }_r\propto a^{-4}$ and ${\rho }_b\propto a^{-3}$,After cosmological recombination, baryons decouple from radiation; for adiabatically cooling nonrelativistic gas, $T_b\propto a^{-2}$ and $c_s\propto a^{-1}$.

The Comoving Jeans length isDuring radiation domination $\bar{\rho }\propto a^{-4}$, so ${\lambda }_{\mathrm{com}}\propto a$. During matter domination before recombination, $\bar{\rho }\propto a^{-3}$ and $c_s\propto a^{-1/2}$, so ${\lambda }_{\mathrm{com}}$ is approximately constant. Recombination causes a sharp downward jump in sound speed, after which ${\lambda }_{\mathrm{com}}\propto a^{-1/2}$. The requested log-log sketch therefore rises with slope one, reaches a plateau after equality, drops at recombination, and then declines with slope $-1/2$.

For collisionless particles, replace $c_s$ by their one-dimensional velocity dispersion $v_{\mathrm{rms}}$:For a thermally produced cold relic, identify four epochs:

Its plot rises through the two relativistic epochs, turns onto a plateau at the nonrelativistic transition, and falls after equality. Recombination does not dynamically affect collisionless dark matter.

Treat the Milky Way and M31 as a radial Kepler orbit of total mass $M$. At maximum separation $r=2a$ the radial speed vanishes, so conservation of energy givesPut $r=a(1-\cos 2\eta )=2a{\sin }^2\eta$. Substitution into the energy equation and integration from the Big Bang gives the cycloidal orbitSince $r^3=8a^3{\sin }^6\eta$,Differentiating the parametric solution givesEliminating $\eta$ numerically gives the stated accurate fit

A remote dwarf sees the Local Group primarily through its total monopole mass, so the same radial timing relation applies with its own $r_i,v_i$. Dimensional consistency requiresrather than the cube-root exponent printed in the converted statement. Let $C=2^{-3/2}\pi$. At the zero-velocity radius $r_0$, ${\omega }_0=0$ and ${\Omega }_{0}t=C$. For the dwarf, ${\Omega }_{i}t=C-0.85{\omega }_{i}t$. Since $\Omega \propto r^{-3/2}$,Therefore