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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.

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.

For $R\gg 1$, $3{\tilde{c}}_s^2\simeq 1$, and the baryon-inertia corrections to the common Euler equation vanish. Differentiating the photon continuity equation givesSet aside the gravitational driving terms and balance the quadrupole streaming source against Thomson scattering. With ${\tau }_c={\Gamma }_T^{-1}$ and ${\sigma }_{\gamma }^{\prime }\simeq 0$,The positive friction is Silk damping; this particular truncation gives shear-only photon diffusion damping. Its coefficient follows the supplied simplified quadrupole equation; adding polarization and the full velocity-slip correction changes that coefficient, as in the general photon-baryon diffusion damping equation.

For completeness, put ${\omega }_0=k/\sqrt{3}$, $b=4k^2{\tau }_c/15$ and $D=\frac{1}{2}{\int }_{{\tau }_0}^{\tau }b(s)ds$. Substitution ${\delta }_{\gamma }=e^{-D}y$ gives the exact transformed equationFor constant ${\tau }_c$ the two independent solutions are $e^{-b(\tau -{\tau }_0)/2}\cos [\sqrt{{\omega }_0^2-b^2/4}(\tau -{\tau }_0)]$ and the corresponding sine. Their frequency correction begins at second order in ${\tau }_c$. For variable ${\tau }_c$, retaining first order alone does not automatically remove $b^{\prime }/2$: a general first-order representation isThe intended cosmological approximation additionally uses slow variation on an acoustic period, $\mathcal{H}/k\ll 1$, with tight coupling $k{\tau }_c\ll 1$. Then the $b^{\prime }$ correction to the frequency is subleading and the two independent photon-baryon acoustic oscillator solutions take the resummed damping formA shift of the time origin is absorbed in $A,B$. If the ionization fraction is constant, $n_e\propto a^{-3}$ gives ${\tau }_c\propto a^2\propto {\tau }^4$ during matter domination, and an initial origin at zero gives $k_D^{-2}=2\tau {\tau }_c/75$. Across recombination the ionization fraction varies, so the integral is the general answer. The PDF prints $O({\tau }_c^{-2})$, while the TeX aid prints $O({\tau }_c^2)$; the small-mean-free-time expansion requires the latter interpretation.

The intrinsic photon temperature fluctuation is $\delta T/T={\delta }_{\gamma }/4$. The cosine and sine produce the Cosmic microwave background acoustic peaks; the envelope suppresses their small-scale amplitudes. The corresponding contribution to the Cosmic microwave background power spectrum contains $e^{-2k^2/k_D^2}$, before angular projection and the other temperature sources are included. Thus Silk damping erases the high-multipole acoustic structure progressively rather than shifting every peak to a new frequency.