Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 53 / 2

The one-particle distribution function is a scalar function on the future mass shell. Choose its normalization so that, in a local orthonormal tetrad, $f{d}^{3}x{d}^{3}p$ is the expected photon number in a phase-space cell on a constant-time slice. Spin degeneracy and any phase-space normalization factors are included in $f$; with an occupation-number convention they would instead appear explicitly in the measure.

More covariantly, the photon number crossing a spacelike element is $f{p}^{\mu }d{\Sigma }_{\mu }{d}^{3}p/E$. The factor $p^{\mu }d{\Sigma }_{\mu }$ reduces to $E{d}^{3}x$ on a local constant-time slice, recovering the cell interpretation. The measure $d^{3}p/E$ is Lorentz invariant on the mass shell.

A photon carries energy $E$ and momentum $p^i$, while its velocity is $p^i/E$. Energy density, momentum density and momentum flux are thereforeThey assemble into the stress-energy tensorThe tetrad momenta are related to coordinate components by the tetrad basis. For an isotropic distribution, angular averaging gives $T^{\hat{i}\hat{j}}={\rho }_{\gamma }{\delta }^{ij}/3$ and $T^{\hat{0}\hat{i}}=0$. Thus the photon gas has pressure ${\rho }_{\gamma }/3$, consistent with its massless equation of state.

A small direction-dependent temperature change $T=\bar{T}(1+\Theta )$ shifts a thermal spectrum according to $\bar{f}(ϵ/(1+\Theta ))=\bar{f}(ϵ)-\Theta ϵ{\bar{f}}^{\prime }(ϵ)$ at first order. Hence $\boxed{\Theta =\delta T/\bar{T}}$ is the angular photon temperature perturbation. For a general spectrum the same ansatz describes an energy-independent brightness dilation; a literal thermodynamic temperature interpretation additionally assumes a thermal shape without spectral distortions.

Use $E=ϵ/a$ in the tetrad stress-energy tensor. The background photon energy density isIntegration by parts gives $-\int dϵ{ϵ}^4{\bar{f}}^{\prime }=4\int dϵ{ϵ}^3\bar{f}$, assuming the boundary term vanishes. Thus each directional energy perturbation is four times its temperature perturbation. Writing $⟨\cdot {⟩}_{\Omega }=(4\pi {)}^{-1}\int d\Omega$, we haveIn the paper's convention for photon angular temperature moments, there is no $(2\ell +1)$ factor multiplying ${\Theta }_{\ell }$. Legendre polynomial orthogonality gives $⟨\Theta {⟩}_{\Omega }={\Theta }_0$ and $⟨e^i\Theta {⟩}_{\Omega }=-i{\hat{k}}^i{\Theta }_1/3$. Comparing with the stated density and velocity conventions yieldsFor completeness, $⟨e^i{e}^j{P}_2(\hat{\mathbf{k}}\cdot \mathbf{e}){⟩}_{\Omega }=({\hat{k}}^i{\hat{k}}^j-{\delta }^{ij}/3)/5$. The $(-i{)}^2$ quadrupole phase then gives $\boxed{{\Pi }_{\gamma }=-3{\Theta }_2/5}$. These moment relations are photon angular temperature moments and keep the anisotropic stress convention consistent with the subsequent hierarchy.

All dots in this solution denote differentiation with respect to conformal time $\eta$. This is required by comoving $k$, the streaming term $k{\Theta }_{\ell }$, and $\dot{\tau }=-a{\bar{n}}_e{\sigma }_T$. The PDF's reference to proper-time derivatives is inconsistent with these coefficients. Proper-time equations follow by dividing the conformal-time equations by $a$, replacing streaming $k$ by $k/a$ and the collision rate by $-{\bar{n}}_e{\sigma }_T$.

The monopole equation of the photon Boltzmann hierarchy is ${\dot{\Theta }}_0+k{\Theta }_1/3=\dot{\varphi }$; its collision term vanishes. Consequently the photon continuity equation isFor the dipole the hierarchy givesUsing the relations from the preceding part, the photon Euler equation becomesBecause $\dot{\tau }<0$, the collision term drives the photon velocity towards the baryon velocity, as Thomson scattering should.

For pressureless baryons, stress-energy conservation without energy exchange gives $\boxed{{\dot{\delta }}_b-k{v}_b-3\dot{\varphi }=0}$. At this order Thomson scattering changes momentum but not the local photon energy: energy transfer by the relative bulk motion is second order, and recoil/thermal energy exchange is neglected. Total momentum conservation fixes the baryon force to be the negative of the photon force. The momentum densities are proportional to their enthalpy densities, $4{\bar{\rho }}_{\gamma }/3$ and ${\bar{\rho }}_b$. ThusIncluding the pressureless expansion and gravitational terms yields the baryon Euler equation with Thomson dragThe expansion coefficient for photons cancels against the changing background pressure, while pressureless baryons retain the Hubble drag term. The opposite drag signs and the enthalpy ratio ensure that the combined photon-baryon fluid conserves momentum.

In tight coupling, the scattering time $|\dot{\tau }{|}^{-1}$ is short compared with an acoustic period and an expansion time: $k/|\dot{\tau }|\ll 1$ and $\mathcal{H}/|\dot{\tau }|\ll 1$. Photons and baryons have nearly the same velocity; higher photon multipoles and the photon-baryon velocity slip are suppressed by powers of this small ratio. Here retain the collision operator as printed, which ignores CMB polarization. Its diffusion coefficient differs from the polarized result.

Neglect gravity and expansion, so $R$ is constant on the timescale under consideration. The quadrupole equation of the photon Boltzmann hierarchy isTo first order in $k/|\dot{\tau }|$, ${\dot{\Theta }}_2$ and $k{\Theta }_3$ are subleading relative to the dipole source. HenceThis is the temperature-only tight-coupling quadrupole. The negative collision rate is essential to its sign.

Put $\Delta =v_{\gamma }-v_b$. Subtracting the baryon Euler equation with Thomson drag from the photon Euler equation givesThe zeroth-order common velocity obeys $(1+R)\dot{v}=-k{\delta }_{\gamma }/4$. Solving the slip equation to its first nonzero order therefore givesThe second expression assumes that $R$ and $\dot{\tau }$ vary only on the neglected background timescale. Terms from their variation would need retaining if cosmic expansion were restored. Multiplying the baryon Euler equation with Thomson drag by $R$ and adding it to the photon equation eliminates drag. Since $v_b=v_{\gamma }-\Delta$,Using $v_{\gamma }=3{\dot{\delta }}_{\gamma }/(4k)$ from the photon continuity equation, the quadrupole term contributes $(2/9)k{\dot{\delta }}_{\gamma }/\dot{\tau }$, and the slip term contributes $R^{2}k{\dot{\delta }}_{\gamma }/[4(1+R)\dot{\tau }]$. HenceDifferentiating the photon continuity equation gives the photon-baryon diffusion damping equationThe sound speed reflects baryon inertia; the $R^2$ term is heat conduction through velocity slip, and the $8/9$ term is photon shear viscosity for the unpolarized hierarchy.

For constant coefficients the characteristic roots are $-D/2\pm \sqrt{D^2/4-c_s^2{k}^2}$. On the acoustic branch where $D<2c_{s}k$,Thus the solutions are damped acoustic oscillations, with positive diffusion damping growing as $k^2$. This is Silk damping. Slowly varying coefficients give an approximate envelope $\exp [-\frac{1}{2}\int Dd\eta ]$ and acoustic phase $k\int c_{s}d\eta$. The mathematical critically damped and overdamped cases follow from the roots, but extrapolation beyond the tight-coupling range is not reliable. For very large $R$, the acoustic and damping scales must be compared explicitly rather than inferring underdamping from $k/|\dot{\tau }|\ll 1$ alone.