Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 57 / 3

The collision rate $\kappa =a{n}_e{\sigma }_T$ is the scattering probability per unit conformal time. Thomson scattering approximately conserves photon number and energy in the electron rest frame, but redistributes propagation directions. Consequently the collision term drives the photon angular distribution toward an isotropic monopole plus the dipole associated with the baryon/electron bulk velocity. It damps the photon-baryon relative velocity and the quadrupole and higher directional anisotropies. Photon and baryon momentum-transfer terms have equal and opposite forces when weighted by their background enthalpies.

The metric terms describe the change in photon energy along a geodesic in a time-dependent spatial metric. The trace perturbation supplies an isotropic redshift, while the scalar trace-free perturbation supplies a direction-dependent redshift proportional to $3{\mu }^2-1$. Brightness is proportional to the fourth power of temperature, accounting for the factor four relative to the temperature shift. These are gravitational sources, not scattering terms.

When $\kappa \gg k,\mathcal{H}$, angular anisotropy is scattered away faster than free streaming or background expansion can regenerate it. In the photon Boltzmann hierarchy, a moment of order $\ell \ge 2$ has damping of order $\kappa {\Delta }_{\ell }$ and is sourced by neighboring moments with coefficients of order $k$. Successive moments are therefore suppressed by powers of $k/\kappa =k{\tau }_c$. This is the tight-coupling approximation. At leading perfect-fluid order the quadrupole can be neglected; its first nonzero correction must be retained to calculate the diffusion damping in part (iii).

The printed equations do not all share a consistent velocity convention. To fix the intended one, take the Fourier velocity to be $v^i=i{k}^i\theta$. Ignoring the metric source, the synchronous perfect-fluid density equation then givesThe PDF's baryon continuity sign is opposite to this convention. Keeping it literally would make the adiabatic ratio and common velocity in part (ii) incompatible for any nonstationary perturbation. Likewise, the momentum-transfer terms for these potentials are $-\kappa ({\theta }_{\gamma }-{\theta }_b)$ and $-R\kappa ({\theta }_b-{\theta }_{\gamma })$, without the printed $k^{-2}$.

For isotropic scattering in this convention the brightness source is $\kappa [{\delta }_{\gamma }+4ik\mu {\theta }_b-\Delta ]$. Its dipole relaxes toward the baryon velocity. The printed source involving ${\theta }_{\gamma }-{\theta }_b$ is not this source, and cannot yield the stated Euler drag by taking its dipole. The following solutions use the consistent continuity and momentum equations. These repairs do not change the requested tight-coupling reduction because the opposite drag forces cancel from its total-momentum equation. The velocity-divergence convention in the standard photon and baryon evolution equations is converted to this potential convention by $\vartheta =-k^2\theta$.

An initially adiabatic cosmological perturbation leaves the local photon-to-baryon number ratio unchanged. Since radiation density scales as the fourth power of temperature and nonrelativistic particle number as the third, its density contrasts obey ${\delta }_b=3{\delta }_{\gamma }/4$. With the corrected common-velocity continuity equations,so the adiabatic relation persists in tight coupling.

Use the paper's ratio $R=4{\bar{\rho }}_{\gamma }/(3{\bar{\rho }}_b)$, which is the reciprocal of the commonly used baryon loading parameter. Multiply the photon Euler equation by $R$ and add the baryon Euler equation. The collision forces cancel exactly:This cancellation works even with the common erroneous $k^{-2}$ in the printed forces, but a physical collision rate uses the repaired forces in part (i). At leading tight-coupling order put ${\theta }_{\gamma }={\theta }_b=\theta$ and substitute ${\delta }_b=3{\delta }_{\gamma }/4$. ThenDefine the pressure-to-inertia sound speed, neglecting the very small baryon thermal pressure, byThe combined velocity equation consequently becomesThis is the requested photon-baryon acoustic oscillator system. Photon pressure supplies the restoring force, while both photon enthalpy and baryon mass supply inertia. The retained $c_{sb}^2$ term adds the small thermal-baryon correction rather than being part of the quoted leading $c_s^2$.

For photon-dominated inertia, $R\gg 1$ and $c_s^2\simeq 1/3$. The baryon expansion and pressure terms are suppressed by $1/R$. Ignoring metric driving as stipulated, the common-velocity Euler equation is ${\theta }^{\prime }=-{\delta }_{\gamma }/4+{\sigma }_{\gamma }$. Differentiating the photon continuity equation givesThis limit is compatible with a matter-dominated background: cold dark matter can dominate the expansion while photons dominate the inertia of the photon-baryon subsystem.

In the first nonzero quadrupole approximation, neglect ${\sigma }_{\gamma }^{\prime }$ relative to its collision damping. The shear equation givesSubstitution yields the shear-only photon diffusion damping equationThe sign of the first-derivative term is positive, so finite photon mean free time damps acoustic energy.

Put $\omega =k/\sqrt{3}$ and $\gamma (\tau )=4k^2{\tau }_c(\tau )/15$. For constant ${\tau }_c$, the characteristic roots are $-\gamma /2\pm i\sqrt{{\omega }^2-{\gamma }^2/4}$. The frequency shift begins at second order in the mean free time. To the requested order the general solution iswhere the error describes the frequency expansion at fixed $k$ and time interval. Keeping the exponential retains the accumulated damping rather than expanding away a possibly important effect.

For a general time-dependent coefficient, a first-order solution with prescribed initial conditions can also be written aswhere ${\delta }_0=A\cos [\omega (\tau -{\tau }_i)]+B\sin [\omega (\tau -{\tau }_i)]$. Substitution verifies that the integral solves the oscillator forced by $-\gamma {\delta }_0^{\prime }$, and its value and first derivative vanish initially. This gives the general first-order result without assuming arbitrary time dependence may be absorbed exactly into an exponential.

To obtain the slowly varying subhorizon form, set ${\delta }_{\gamma }=e^{-G}y$ with $G^{\prime }=\gamma /2$. The exact transformed equation isDropping the quadratic term alone leaves ${\omega }^2-{\gamma }^{\prime }/2$, so slow variation is also needed. When $k\gg \mathcal{H}$, $k{\tau }_c\ll 1$, and the mean free time changes on an expansion timescale, $|{\gamma }^{\prime }|/{\omega }^2\ll 1$. The WKB method solution has local frequency $\Omega =({\omega }^2-{\gamma }^{\prime }/2{)}^{1/2}$ and prefactor ${\Omega }^{-1/2}$. At leading acoustic order both reduce to constants; the small derivative corrections to phase and prefactor can be neglected. This slowly varying shear-damped photon oscillator therefore givesA change in the phase origin is absorbed into $A,B$. At zero mean free time the damping vanishes; as time passes the diffusion length grows and $k_D$ decreases. The coefficient is that of the question's isotropic-scattering, shear-only truncation, not the polarization-inclusive full Silk damping formula.

If ionization remains approximately constant during matter domination, $n_e\propto a^{-3}$, so ${\tau }_c\propto a^2\propto {\tau }^4$. Away from the lower limit this givesNear recombination the rapidly changing ionization fraction and eventual failure of tight coupling require the full hierarchy; the integral formula specifies the damping accumulated during the regime used here.

A blackbody photon density perturbation gives an intrinsic temperature fluctuation ${\Theta }_0={\delta }_{\gamma }/4$. The baryon/electron bulk velocity produces a Doppler effect at last scattering. In the common-potential convention of part (ii),Ignoring the slowly varying damping derivative at leading acoustic order, differentiating the density cosine produces a velocity sine. Thus density and velocity acoustic sources are approximately a quarter cycle out of phase. For regular adiabatic initial conditions the density oscillation has approximately a common cosine phase across modes; this phase coherence is what makes a sequence of peaks instead of a smooth incoherent superposition.

Let $D_{\ast }={\tau }_0-{\tau }_{\ast }$ be the comoving distance to last scattering and $r_s({\tau }_{\ast })={\int }^{{\tau }_{\ast }}{c}_{s}d\tau$ its sound horizon. In the present limit $r_s\simeq {\tau }_{\ast }/\sqrt{3}$. Density extrema have $k{r}_s\simeq m\pi$, whereas velocity extrema have $k{r}_s\simeq (m+1/2)\pi$. The projection from spatial wavenumber to high angular multipoles is roughly $\ell \simeq k{D}_{\ast }$, giving an acoustic spacingThe Doppler effect fills some density troughs and modifies the detailed peak shapes; its different projection prevents treating it as an identical second series of narrow peaks.

For clarity, choose the sightline to point outward from the observer and let its angle to $\mathbf{k}$ have cosine $\mu$. The intrinsic-plus-emitter-Doppler source is then $[{\delta }_{\gamma }/4-ik\mu {\theta }_b]e^{ik\mu D_{\ast }}$. Expanding its angular dependence gives the angular projection of density and Doppler acoustic sourcesThe second sign follows from the chosen outward sightline and $v_b^i=i{k}^i{\theta }_b$; it changes with the sightline convention together with the velocity projection. The $j_{\ell }$ are Spherical Bessel functions. Both sources are proportional to the primordial curvature. If ${\mathcal{T}}_{\ell }={\Theta }_{\ell }/{\zeta }_{\ast }$, their angular spectrum in the usual continuum Fourier normalization isThis formula includes the density-velocity cross term, not just a sum of independent source powers.

Photon diffusion multiplies both leading source amplitudes by $e^{-k^2/k_D^2}$, so their power carries $e^{-2k^2/k_D^2}$. The approximate angular damping scale isThis produces the damping tail in the Cosmic microwave background power spectrum. Finite last-scattering thickness adds further small-scale averaging. Gravity, baryon loading and later propagation alter detailed peak locations and heights, but the oscillatory peak sequence, phase-shifted velocity contribution and diffusion damping follow directly from the simplified solution obtained here.