Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 312 / 3

A scalar Fourier mode selects one spatial direction, $\hat{\mathbf{k}}$, and has no transverse vector or tensor polarization. Rotations about $\mathbf{k}$ therefore leave its temperature and polarization patterns unchanged. Only azimuthal number $m=0$ occurs in the corresponding spherical harmonic expansion.

Let $\mu =\hat{\mathbf{k}}\cdot \mathbf{e}$ and use the paper's conventionwhich omits the $(2\ell +1)$ factor used in some other definitions of the photon temperature multipole. In Fourier space the transport term is $ik\mu \Theta$. The Legendre polynomial recurrence relation makes its coefficient at multipole $\ell$ equal toThe gravitational time derivative is purely monopolar, and the potential gradient and baryon velocity are dipolar. Thomson scattering removes the monopole from its relaxation term and adds the quadrupole polarization correction. Thus the photon Boltzmann hierarchy isThe $\ell =0$ term containing ${\Theta }_{-1}$ is understood as zero.

Dropping gravity and expansion from the photon Boltzmann hierarchy givesThe first equation is photon energy continuity: the photon dipole transports the photon monopole. The second is the photon Euler equation: a pressure gradient accelerates the radiation, its quadrupole contributes shear stress, and Thomson scattering drags the velocity toward the baryon velocity.

With $v_{\gamma }=-{\Theta }_1$, the scattering contribution to ${\dot{v}}_{\gamma }$ is $\dot{\tau }(v_{\gamma }-v_b)$. Radiation momentum density is $(4{\bar{\rho }}_{\gamma }/3){\mathbf{v}}_{\gamma }$, whereas baryon momentum density is ${\bar{\rho }}_b{\mathbf{v}}_b$. Conservation of total momentum therefore requiresorSince the cosmological optical depth is measured to the observation time, $\dot{\tau }<0$; the two drag terms reduce their relative velocity.

Write $\Gamma =-\dot{\tau }>0$. The tight-coupling approximation requires $\Gamma \gg k,\mathcal{H}$: frequent Thomson scattering makes the photon-baryon velocity slip small and suppresses higher multipoles. Work with constant $R$ on the short scales of the question.

Set $s=v_{\gamma }-v_b$ and $A=k{\Theta }_0-2k{\Theta }_2/5$. The dipole and baryon equations implyTo leading order, $s=RA/[(1+R)\dot{\tau }]$. Differentiating this and keeping the first correction in $k/\Gamma$ yieldsTerms involving derivatives of ${\Theta }_2$ contribute at the next order here.

The leading $\ell =2$ polarization balance isThe $\ell =2$ temperature equation then becomesso the photon quadrupole in tight coupling with polarization isFinally substitute ${\Theta }_1=-3{\dot{\Theta }}_0/k$ into the corrected dipole equation. This gives the photon-baryon diffusion damping equationThe oscillation frequency agrees with the photon-baryon sound speed. The positive damping coefficient comes from velocity slip and shear viscosity. Its growth as $k^2$ suppresses small-scale oscillations, producing the Silk damping tail and reducing the high-multipole Cosmic microwave background acoustic peaks.