Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 346 / 4

For a statistically homogeneous and isotropic density contrast,Using the Fourier transform conventionand translational invariance giveswithAt one point,

For a spherical top-hat volume $V=4\pi R^3/3$,The overlap of two radius-$R$ balls separated by $r\le 2R$ isTherefore

For $\xi (r)=(r/r_0{)}^{-1.8}$,ThusThis calculation assumes that the observed galaxies trace the matter field with unit, scale-independent galaxy bias, and that the fitted power law remains valid throughout the top-hat separations.

The photon mean free path isIn time $t$ a photon takes about $ct/{\ell }_{\gamma }$ independent steps. Isotropy assigns one third of the mean-square displacement to any coordinate, giving the Silk damping diffusion scale

In the matter era, $a\propto t^{2/3}$ and $n_e=n_{\mathrm{rec}}(a_{\mathrm{rec}}/a{)}^3$. Accumulating the diffusion variance while converting each displacement to comoving length givesHenceThe order-one coefficient depends on the precise diffusion-length convention; the expression uses the convention stated in the question.

In a universe containing only baryons and radiation, photon diffusion erases coupled baryon-photon perturbations below this scale before recombination. Baryons begin subsequent matter-era growth from a strongly smoothed field, suppressing small-scale structure. With cold dark matter, collisionless dark-matter perturbations are not Silk damped and provide surviving gravitational wells. After recombination baryons fall into those wells, so baryonic acoustic structure is damped but small-scale total-matter structure can continue to grow.