Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 312 / 3

For nonzero momentum, the Fourier transform of the local bias expansion isThe subtracted variance contributes only at $\mathbf{k}=0$. Because $\delta$ is a Gaussian random field, its three-point function vanishes and its four-point function factorizes by Wick theorem. At leading order one of the three galaxy fields supplies the quadratic term and the other two supply linear terms. The two cross-contractions cancel the factor $1/2$, givingThus even Gaussian matter fluctuations acquire a nonzero galaxy bispectrum through local quadratic galaxy bias.

Along the unperturbed photon path, combine the temperature and gravitational-redshift terms and use ${\mathcal{T}}^{\prime }=-\Gamma$:The integrating factor is $e^{-\mathcal{T}}$, and $g=\Gamma e^{-\mathcal{T}}$. Neglecting the exponentially hidden initial boundary term gives the Cosmic microwave background line-of-sight solutionFor instantaneous recombination, $g(\tau )=\delta (\tau -{\tau }_{\mathrm{dec}})$, and the observer potential contributes only an unobservable monopole. ThereforeThe first two terms form the ordinary Sachs-Wolfe effect, the velocity term is the Doppler CMB anisotropy at last scattering, and the integral is the Integrated Sachs-Wolfe effect produced by evolving potentials.

Insert the photon temperature multipole expansion into the Fourier-space transport equation. The Legendre polynomial recurrence relation turns multiplication by $\mu$ into nearest-neighbour multipole couplings, while Orthogonality of Legendre polynomials projects onto a fixed $\ell$. The monopole projection has no collision term because Thomson scattering conserves photon number:The dipole projection receives the electron-velocity source,For every $\ell \ge 2$, the collision term damps the anisotropic multipole and the photon Boltzmann hierarchy isThus the Free-streaming photon Boltzmann equation moves angular structure between neighbouring multipoles, whereas Thomson scattering suppresses all multipoles above the dipole in the tight-coupling approximation.