Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 64 / 4

For an adiabatic mode the perturbation of baryon number per photon is zero. Since $n_{\gamma }\propto T^3$, ${\rho }_{\gamma }\propto T^4$ and nonrelativistic ${\rho }_b\propto n_b$, the common temperature/entropy perturbation gives ${\delta }_b=3\delta T/T$ and ${\delta }_{\gamma }=4\delta T/T$. Hence ${\delta }_{\gamma }=4{\delta }_b/3$.

Let $q=a{n}_e{\sigma }_T$, $\mathcal{H}=a^{\prime }/a$ and $R=4{\bar{\rho }}_{\gamma }/(3{\bar{\rho }}_b)$. In tight coupling, the relative velocity is small but its drag force need not vanish, so it must be eliminated before setting the two velocities equal. Multiply the photon Euler equation by $R$ and add the baryon Euler equation. Their opposite drag terms cancel, even with the common drag-prefactor typo in the printed equations. The result at common velocity potential $\theta$ isTogether with continuity this givesThe same photon-baryon sound speed follows from $\delta P=\delta {\rho }_{\gamma }/3$ and $\delta \rho =\delta {\rho }_{\gamma }+\delta {\rho }_b$: using the adiabatic density relation gives $\delta P/\delta \rho =c_s^2$.

The Euler bracket in the original PDF prints $a/a^{\prime }$; the derivation requires $a^{\prime }/a=\mathcal{H}$. Its baryon continuity equation also needs a prime on ${\delta }_b$. Finally, for the defined velocity potential $\mathbf{v}=i\mathbf{k}\theta$, physical Thomson drag is $-q({\theta }_{\gamma }-{\theta }_b)$, without an extra $k^{-2}$: taking a velocity divergence introduces $-k^2$ on both sides, which cancels on returning to the potential equation. This is photon opacity normalization for a velocity potential. These errors should not be concealed by reproducing a dimensionally inconsistent equation.

The inequalities are compatible: CDM can dominate the total density while photons dominate baryons. With $R\gg 1$, $c_s^2\simeq 1/3$ and the baryon-inertia expansion term is negligible. Differentiate photon continuity and substitute the combined Euler equation:Thus ${\delta }_{\gamma }^{\prime \prime }+k^2{\delta }_{\gamma }/3-4k^2{\sigma }_{\gamma }/3=0$. The quasistatic quadrupole equation givesSubstitution yields the shear-only photon diffusion damping equationTo solve it, put $\Gamma =4k^2{\tau }_c/15$ and ${\delta }_{\gamma }=u\exp [-\frac{1}{2}{\int }^{\tau }\Gamma (s)ds]$. Direct differentiation, including both derivatives of the prefactor, givesNeglect the requested ${\tau }_c^{\prime }$ and ${\tau }_c^2$ terms. The remaining equation is the free acoustic oscillator, so its general solution isA different lower limit is absorbed into the amplitudes. The damping integral is positive and increases with time: shorter wavelengths are smoothed more strongly by photon diffusion. A subhorizon but still tightly coupled mode satisfies $k\tau \gg 1$ and $k{\tau }_c\ll 1$, with slowly varying scattering time; the formula is not valid for arbitrarily short waves after tight coupling fails. The truncated shear coefficient is the one implied by the supplied equations, not the more complete damping coefficient including photon polarization and photon-baryon slip. For constant ${\tau }_c$, the exact oscillator frequency differs from $k/\sqrt{3}$ only at order ${\tau }_c^2$, consistent with this expansion.

Let $q_{\ast }=k{\tau }_{\mathrm{dec}}/\sqrt{3}$ and $E_D=\exp (-k^2/k_D^2)$ at last scattering. For random oscillator amplitudes, define spectra $P_A,P_B$ and cross-spectrum $P_{AB}$ using the same Fourier normalization as the density spectrum. Averaging the solution's squared modulus givesConsequently the intrinsic-temperature source has variance spectrum $P_{{\delta }_{\gamma }}/16$. A common acoustic phase is necessary for pronounced peaks in the variance. Equal independent sine/cosine amplitudes would erase them. A regular growing adiabatic mode begins with negligible velocity outside the sound horizon and selects approximately the cosine phase, $B\simeq 0$, preserving a ${\cos }^2{q}_{\ast }$ sequence. This illustrates coherent initial phases are needed for acoustic variance peaks.

From continuity, ${\theta }_{\gamma }=3{\delta }_{\gamma }^{\prime }/(4k^2)$. Ignoring the small damping correction in this velocity relation, the coherent cosine solution givesThe Doppler CMB anisotropy source therefore has directional variance $3{\mu }^2{P}_A{E}_D^2{\sin }^2{q}_{\ast }/16$, where $\mu =\hat{\mathbf{n}}\cdot \hat{\mathbf{k}}$. Its velocity maxima lie between density maxima: compression and bulk motion are a quarter acoustic period apart.

Angular projection must also be included. Let ${\chi }_{\ast }={\tau }_0-{\tau }_{\mathrm{dec}}$, assume a thin last-scattering surface, and choose the emission phase $e^{ik{\chi }_{\ast }\mu }$. The coherent intrinsic-plus-Doppler radial response is proportional toThe derivative follows from $i\mu e^{ix\mu }={\partial }_x{e}^{ix\mu }$ and the velocity sign above. A reversed line-of-sight convention reverses the corresponding Doppler sign consistently. The CMB angular power spectrum is $C_{\ell }=(2/\pi )\int k^{2}dk{P}_A(k)|{\Theta }_{\ell }(k){|}^2$ in this simplified model. Squaring the combined response keeps its correlated density–velocity cross term; adding two independent angular powers would discard that correlation. The derivative Bessel projection also spreads the Doppler contribution differently from the density contribution.

The sound horizon here is $r_s\simeq {\tau }_{\mathrm{dec}}/\sqrt{3}$. Intrinsic acoustic extrema occur near $k{r}_s=n\pi$, mapping roughly to ${\ell }_n\simeq n\pi {\chi }_{\ast }/r_s$, with interleaved velocity extrema and projection broadening. Cosmic microwave background diffusion damping suppresses both contributions exponentially at $k\gtrsim k_D$, or multipoles $\ell \gtrsim k_D{\chi }_{\ast }$. Modes outside the sound horizon have no completed acoustic oscillation.

The metric integral represents gravitational redshift, including contributions that in another gauge are separated into endpoint Sachs-Wolfe effects and Integrated Sachs-Wolfe effects. It is not correct to call the entire synchronous metric integral only a late-time integrated effect. Large-scale metric/curvature fluctuations remain important where photon pressure cannot redistribute matter causally and produce the broad large-angle signal rather than just the pressure-oscillation sequence. Potential evolution during radiation-to-matter transition also drives and shifts acoustic amplitudes near horizon entry; late accelerating-era evolution adds a large-angle integrated signal. These effects are excluded by the oscillator's instruction to ignore metric perturbations. Even if the Newtonian-gauge potential is constant in matter domination, synchronous $h_{ij}^{\prime }$ need not vanish, and the complete observable requires the consistent density, velocity and metric combination.