Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 310 / 3

During matter domination, $4\pi G{\bar{\rho }}_m=3H^2/2$ and $a\propto t^{2/3}$. Substitution of ${\delta }_m\propto t^p$ gives $p=2/3,-1$, henceDuring radiation domination, neglecting the rapidly oscillating radiation perturbation and the subdominant matter source leavesThereforeThis logarithmic behavior is the Mészáros effect for subhorizon cold dark matter.

For an approximately scale-invariant primordial spectrum, the late cosmological density power spectrum behaves schematically asModes with $k\ll k_{\mathrm{eq}}$ enter the horizon after matter-radiation equality and retain $T(k)\simeq 1$, so $P(k)\sim k^{n_s}$. Modes with $k\gg k_{\mathrm{eq}}$ enter during radiation domination and grow only logarithmically until equality. Their transfer function behaves as $T(k)\sim \log (k/k_{\mathrm{eq}})/(k/k_{\mathrm{eq}}{)}^2$, giving a strongly falling small-scale spectrum. The result is a turnover near the matter-radiation equality scale $k_{\mathrm{eq}}$.

The Fourier-space cosmological Poisson equation givesInsert this into the line-of-sight expression for the CMB lensing potential, Fourier transform ${\delta }_m$, and use the Rayleigh plane-wave expansion. Projection onto $Y_{lm}$ and spherical-harmonic orthogonality give

Define the radial transfer integralwhere ${\delta }_m(\mathbf{k},\tau )=D_i(\tau ){\delta }_m(\mathbf{k},{\tau }_i)$ and $D_i({\tau }_i)=1$. Withthe angular and radial integrations giveThis expression makes the lensing signal a weighted projection of the evolving matter power spectrum.