Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 59 / 4

Let $p^0=d\tau /d\lambda$. A comoving observer measures photon energy $p=a{p}^0$, so $q=ap=a^2{p}^0$. The null condition is $({\delta }_{ij}+h_{ij})p^i{p}^j=(p^0{)}^2$. Using the stated Christoffel symbols in the time geodesic equation givesDivide by $p^0$ to differentiate with respect to conformal time. The background redshift term cancels when differentiating $a^2{p}^0$:In the perturbation term use the background relation $p^i/p^0={\hat{n}}^i$, givingWithout metric tensor perturbations, both $q$ and the propagation direction are constant along an unperturbed ray. Thus $q^{\prime }$ and $d{\hat{n}}^i/d\tau$ are first order. Multiplying them by derivatives of $f_1$, which is also first order, gives second-order terms. The term $q^{\prime }d{f}_0/dq$ is first order and must be retained. At linear order the Collisionless Boltzmann equation therefore readsTaking the Fourier transform in the spatial coordinates provesA first-order correction to the propagation velocity multiplying a spatial derivative of $f_1$ is likewise second order.

Write $T=\overline{T}(1+\Theta )$, with $\Theta =\Delta T/\overline{T}$ at linear order. Since $\overline{T}\propto a^{-1}$, the background Planck function depends only on comoving momentum. Expanding its temperature-shifted argument givesUsing $T$ rather than $\overline{T}$ in the perturbation denominator changes only higher-order terms.

The photon brightness perturbation is the energy-weighted angular perturbationIntegration by parts gives $\int q^4{f}_0^{\prime }dq=-4\int q^3{f}_{0}dq$; the boundary term vanishes at both endpoints for a Planck spectrum. Therefore $\Delta =4\Theta$. Multiplying the transport equation by $q^3$ and integrating yields the synchronous photon brightness equationThe PDF's definition is $4\Delta T/T$; the TeX's denominator $\hslash$ is a transcription error.

Expand $\Delta$ in Legendre moments in $\mu$. The monopole is the density/temperature perturbation, and the dipole is the bulk velocity. Before decoupling, rapid Thomson scattering isotropizes photons in the electron rest frame. In the tight-coupling approximation, the scattering rate is much larger than $k$ and $\mathcal{H}$, so the quadrupole and higher moments are suppressed by the mean-free-path parameter, with the quadrupole typically of order $k/\dot{\kappa }$ times the dipole. To leading order only monopole and dipole are needed. This suppression uses the collision term restored before decoupling; it does not follow from the collisionless equation by itself. Finite quadrupole corrections generate polarization and diffusion damping.

The emission-density term is the intrinsic temperature fluctuation: blackbody radiation has $\delta {\rho }_{\gamma }/{\rho }_{\gamma }=4\Delta T/T$, hence the factor $1/4$. Adiabatic compression and rarefaction of the photon-baryon fluid create coherent density extrema at last scattering. Their projection produces acoustic peaks, with angular scale set by the sound horizon divided by the distance to last scattering. In a familiar nearly flat cosmology the first prominent temperature peak is around degree scales, $\ell$ of order a few hundred; precise positions depend on the background.

The velocity term is a Doppler effect from the emitting plasma's line-of-sight motion. The acoustic velocity is approximately a quarter-cycle out of phase with the density oscillation. Its projected contribution is strongest on acoustic and smaller angular scales and interleaves with density extrema, with projection broadening its features. Both sources are suppressed on sufficiently small scales by Silk damping and the finite thickness of last scattering.

The metric tensor integral describes photon-energy shifts during propagation through the changing perturbed geometry. In synchronous coordinates its split from the intrinsic endpoint term is gauge dependent. Rewriting the observable in Newtonian variables produces the usual emission Sachs-Wolfe effect and the Integrated Sachs-Wolfe effect from evolving potentials, with the appropriate endpoint terms. The ordinary large-scale Sachs-Wolfe combination gives a low-multipole plateau for nearly scale-invariant initial curvature power. Late-time evolving potentials chiefly affect large angles, often $\ell \lesssim 20$; evolution near radiation-matter equality also drives an early integrated contribution near the first acoustic peak. Static matter-era potentials do not produce that Newtonian integrated contribution.

All emission quantities are evaluated at the last-scattering endpoint and the metric tensor source along the unperturbed photon ray, not at a single fixed spatial point throughout the integral. Observer monopole/dipole conventions do not change this physical interpretation.

The sketch uses arbitrary units and representative scales. It displays source auto-power shapes, not measured data or an assertion that the total power is their simple sum. The terms are correlated; their cross-power contributions and the gauge-consistent endpoint combination are essential in an actual angular-spectrum calculation.