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

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.