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A deliberately restricted forced harmonic oscillator shows two effects of the baryon loading parameter without duplicating the full photon Boltzmann hierarchy. Use conformal time $d\eta =dt/a$ in seconds, a comoving wavenumber $k$ in inverse metres, constant $R=3{\rho }_b/(4{\rho }_{\gamma })$, and equal constant dimensionless Newtonian gauge gravitational potentials $\Phi =\Psi \ne 0$. The energy densities ${\rho }_b$ and ${\rho }_{\gamma }$ use the same units. Neglect expansion drag, Silk damping and evolution of the potentials. The photon-baryon sound speed is $c_s=c/\sqrt{3(1+R)}$. Writing ${\Theta }_0$ for the photon monopole temperature perturbation, the photon-baryon acoustic oscillator reduces toFor $y=({\Theta }_0+\Psi )/\Psi$ this gives the displayed forced equation. Constant loading and the neglected expansion drag make this a local teaching model, not a self-consistent expanding cosmology with $R$ fixed forever.

Choose $y(0)=1/3$ and $y^{\prime }(0)=0$, the static matter-era Sachs-Wolfe combination normalized by $\Psi$. With $q=k{c}_s\eta$, direct substitution givesAt $R=0$, successive extrema have equal magnitudes $1/3$. At the stipulated $R=0.6$, $y(\pi )=-23/15$ whereas $y(2\pi )=1/3$. The shifted equilibrium $y=-R$ makes alternating extrema unequal. This is an elementary illustration of how the baryon loading parameter changes acoustic amplitudes; these extrema are neither measured peak heights nor a complete Cosmic microwave background power spectrum. Comparing curves at fixed $q$ isolates the response; at fixed $k$ and $\eta$, increasing $R$ also reduces the photon-baryon sound speed and hence the phase.

For example, holding $c{\eta }_{\ast }=300\mathrm{Mpc}$ gives sound horizons $r_s=173.2\mathrm{Mpc}$ at $R=0$ and $136.9\mathrm{Mpc}$ at $R=0.6$. Their ratio is $1/\sqrt{1.6}$. Each term of the forced equation has units of inverse seconds squared because $y$ and $R$ are dimensionless and $k{c}_s$ is an inverse time. In the actual photon-baryon fluid, $R$ grows with the scale factor and the sound horizon is the integral of the evolving photon-baryon sound speed.

There is first an angular-scale typo in the PDF: smaller angles correspond to larger multipoles, approximately $\theta \sim \pi /\ell$. Small scales relative to the first acoustic peak therefore have $\ell \gtrsim 200$, not the printed inequality in the other direction.

The photon-baryon acoustic oscillator produces rapidly varying density and velocity sources around and below the sound horizon. Treating the emission density as a simple long-wavelength gravitational source misses these acoustic peaks and their phases. Silk damping suppresses short-scale perturbations through diffusion before decoupling. The cosmological visibility function has finite width, so different emission times and acoustic phases are averaged rather than sampled on one sharp surface. As tight coupling breaks down, the photon quadrupole and higher moments are generated; Thomson scattering of the quadrupole produces polarization. These require the full photon Boltzmann hierarchy, rather than exactly vanishing emission moments above the dipole. They are the angular-scale limits of instantaneous photon decoupling.

The distinction is important: acoustic evolution and diffusion can already be included in correctly calculated emission ${\delta }_{\gamma }$ and $\mathbf{v}$. They do not themselves invalidate collisionless propagation after decoupling. The two-moment, single-time initial condition and omission of scattering after that surface are the approximations used in the requested line-of-sight formula.

For the stated assumptions, multiply the synchronous photon brightness equation by $e^{ik\mu \tau }$ and integrate:Here ${\tau }_d$ is the decoupling time. Inverse Fourier transformation shifts each source to the unperturbed rayInsert the initial brightness monopole and Doppler dipole, and divide by four:This is the synchronous Sachs-Wolfe line-of-sight formula. The first term is intrinsic emission temperature, the second is the Doppler effect, and the last is the accumulated gravitational redshift in this gauge. The derivative $h_{ij}^{\prime }$ is the partial conformal-time derivative evaluated on the ray, not its total derivative along the ray. Its integral contains endpoint gravitational effects as well as evolving-potential effects when rewritten in Newtonian gauge; it should not be identified solely with the Integrated Sachs-Wolfe effect.

The direction $\mathbf{n}$ is photon propagation toward the observer. The observer-to-source sky direction is its negative, so changing that convention also changes the displayed Doppler sign and the ray parametrization. Correct emitter positions and this direction convention are needed to interpret the formula.

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 $\vartheta =-k^2\theta$.

All dots in this solution denote differentiation with respect to conformal time $\eta$. This is required by comoving $k$, the streaming term $k{\Theta }_{\ell }$, and $\dot{\tau }=-a{\bar{n}}_e{\sigma }_T$. The PDF's reference to proper-time derivatives is inconsistent with these coefficients. Proper-time equations follow by dividing the conformal-time equations by $a$, replacing streaming $k$ by $k/a$ and the collision rate by $-{\bar{n}}_e{\sigma }_T$.

The monopole equation of the photon Boltzmann hierarchy is ${\dot{\Theta }}_0+k{\Theta }_1/3=\dot{\varphi }$; its collision term vanishes. Consequently the photon continuity equation isFor the dipole the hierarchy givesUsing the relations from the preceding part, the photon Euler equation becomesBecause $\dot{\tau }<0$, the collision term drives the photon velocity towards the baryon velocity, as Thomson scattering should.

For pressureless baryons, stress-energy conservation without energy exchange gives $\boxed{{\dot{\delta }}_b-k{v}_b-3\dot{\varphi }=0}$. At this order Thomson scattering changes momentum but not the local photon energy: energy transfer by the relative bulk motion is second order, and recoil/thermal energy exchange is neglected. Total momentum conservation fixes the baryon force to be the negative of the photon force. The momentum densities are proportional to their enthalpy densities, $4{\bar{\rho }}_{\gamma }/3$ and ${\bar{\rho }}_b$. ThusIncluding the pressureless expansion and gravitational terms yields the baryon Euler equation with Thomson dragThe expansion coefficient for photons cancels against the changing background pressure, while pressureless baryons retain the Hubble drag term. The opposite drag signs and the enthalpy ratio ensure that the combined photon-baryon fluid conserves momentum.

In tight coupling, the scattering time $|\dot{\tau }{|}^{-1}$ is short compared with an acoustic period and an expansion time: $k/|\dot{\tau }|\ll 1$ and $\mathcal{H}/|\dot{\tau }|\ll 1$. Photons and baryons have nearly the same velocity; higher photon multipoles and the photon-baryon velocity slip are suppressed by powers of this small ratio. Here retain the collision operator as printed, which ignores CMB polarization. Its diffusion coefficient differs from the polarized result.

Neglect gravity and expansion, so $R$ is constant on the timescale under consideration. The quadrupole equation of the photon Boltzmann hierarchy isTo first order in $k/|\dot{\tau }|$, ${\dot{\Theta }}_2$ and $k{\Theta }_3$ are subleading relative to the dipole source. HenceThis is the temperature-only tight-coupling quadrupole. The negative collision rate is essential to its sign.

Put $\Delta =v_{\gamma }-v_b$. Subtracting the baryon Euler equation with Thomson drag from the photon Euler equation givesThe zeroth-order common velocity obeys $(1+R)\dot{v}=-k{\delta }_{\gamma }/4$. Solving the slip equation to its first nonzero order therefore givesThe second expression assumes that $R$ and $\dot{\tau }$ vary only on the neglected background timescale. Terms from their variation would need retaining if cosmic expansion were restored. Multiplying the baryon Euler equation with Thomson drag by $R$ and adding it to the photon equation eliminates drag. Since $v_b=v_{\gamma }-\Delta$,Using $v_{\gamma }=3{\dot{\delta }}_{\gamma }/(4k)$ from the photon continuity equation, the quadrupole term contributes $(2/9)k{\dot{\delta }}_{\gamma }/\dot{\tau }$, and the slip term contributes $R^{2}k{\dot{\delta }}_{\gamma }/[4(1+R)\dot{\tau }]$. HenceDifferentiating the photon continuity equation gives the photon-baryon diffusion damping equationThe sound speed reflects baryon inertia; the $R^2$ term is heat conduction through velocity slip, and the $8/9$ term is photon shear viscosity for the unpolarized hierarchy.

For constant coefficients the characteristic roots are $-D/2\pm \sqrt{D^2/4-c_s^2{k}^2}$. On the acoustic branch where $D<2c_{s}k$,Thus the solutions are damped acoustic oscillations, with positive diffusion damping growing as $k^2$. This is Silk damping. Slowly varying coefficients give an approximate envelope $\exp [-\frac{1}{2}\int Dd\eta ]$ and acoustic phase $k\int c_{s}d\eta$. The mathematical critically damped and overdamped cases follow from the roots, but extrapolation beyond the tight-coupling range is not reliable. For very large $R$, the acoustic and damping scales must be compared explicitly rather than inferring underdamping from $k/|\dot{\tau }|\ll 1$ alone.

A scalar cosmological perturbation at fixed wavevector carries no helicity or distinguished transverse direction. Its metric and scalar sources are unchanged by rotations around $\mathbf{k}$, and the scalar baryon peculiar velocity is longitudinal. With scalar initial conditions, rotational invariance of the linear evolution therefore makes $\Theta$ a function only of $\mu =\hat{\mathbf{k}}\cdot \mathbf{e}$. Expand it using the stated normalization, without an extra $(2\ell +1)$ factor:In the Thomson kernel, $1+v^2=4/3+(2/3)P_2(v)$. The spherical harmonic addition theorem and Orthogonality of Legendre polynomials giveOnly the monopole and quadrupole survive; the quadrupole's $(-i{)}^2=-1$ gives the angular scattering integralThis also explains why an isotropic distribution has zero net collision term.

The streaming term $ik\mu \Theta$ and the Legendre polynomial recurrence relation yield, after division by $(-i{)}^{\ell }$, $k[(\ell +1){\Theta }_{\ell +1}/(2\ell +3)-\ell {\Theta }_{\ell -1}/(2\ell -1)]$. With ${\mathbf{v}}_b=i\hat{\mathbf{k}}{v}_b$, the velocity collision source is $+{\tau }^{\prime }{v}_b$ in the dipole equation; the gravitational gradient source is $+k\psi$. Therefore the photon Boltzmann hierarchy isSet the absent ${\Theta }_{-1}$ contribution to zero. This is algebraically the printed hierarchy. In particular,Since ${\tau }^{\prime }<0$, these collision terms damp the nonzero multipoles and drive the dipole to ${\Theta }_1=-v_b$. The latter sign follows from the paper's velocity and $(-i{)}^{\ell }$ conventions, not a reversed physical photon velocity.

Throughout this question dots denote conformal time derivatives, and $\mathcal{H}=\dot{a}/a$ is the conformal Hubble parameter. The monopole of the photon Boltzmann hierarchy gives ${\dot{\Theta }}_0+k{\Theta }_1/3=\dot{\varphi }$. With the specified velocity convention, the photon continuity equation is thereforeThe dipole equation is ${\dot{\Theta }}_1+k(2{\Theta }_2/5-{\Theta }_0)=\dot{\tau }({\Theta }_1+v_b)+k\psi$. Thus the photon Euler equation isIn particular, $\dot{\tau }<0$ makes the scattering term damp the photon-baryon velocity slip; it does not amplify it.

The linear momentum density of the photon-baryon fluid is weighted by enthalpy, not merely energy density:Elastic Thomson scattering exchanges momentum internally. Its photon force density is $(4{\bar{\rho }}_{\gamma }/3)\dot{\tau }(v_{\gamma }-v_b)$, so the baryon force density is its negative. Dividing by ${\bar{\rho }}_b$ gives the baryon Euler equation with Thomson dragThis $R$ is the baryon loading parameter, the baryon-to-photon inertia ratio. The two collision terms cancel exactly when the equations are weighted by their enthalpies. Background conservation gives ${\bar{\rho }}_b\propto a^{-3}$ and ${\bar{\rho }}_{\gamma }\propto a^{-4}$, so $\dot{R}=\mathcal{H}R$.

Dropping gravity and expansion from the photon Boltzmann hierarchy givesThe first equation is photon energy continuity: the photon dipole transports the photon monopole. The second is the photon Euler equation: a pressure gradient accelerates the radiation, its quadrupole contributes shear stress, and Thomson scattering drags the velocity toward the baryon velocity.

With $v_{\gamma }=-{\Theta }_1$, the scattering contribution to ${\dot{v}}_{\gamma }$ is $\dot{\tau }(v_{\gamma }-v_b)$. Radiation momentum density is $(4{\bar{\rho }}_{\gamma }/3){\mathbf{v}}_{\gamma }$, whereas baryon momentum density is ${\bar{\rho }}_b{\mathbf{v}}_b$. Conservation of total momentum therefore requiresorSince the cosmological optical depth is measured to the observation time, $\dot{\tau }<0$; the two drag terms reduce their relative velocity.

A scalar Fourier mode selects one spatial direction, $\hat{\mathbf{k}}$, and has no transverse vector or tensor polarization. Rotations about $\mathbf{k}$ therefore leave its temperature and polarization patterns unchanged. Only azimuthal number $m=0$ occurs in the corresponding spherical harmonic expansion.

Let $\mu =\hat{\mathbf{k}}\cdot \mathbf{e}$ and use the paper's conventionwhich omits the $(2\ell +1)$ factor used in some other definitions of the photon temperature multipole. In Fourier space the transport term is $ik\mu \Theta$. The Legendre polynomial recurrence relation makes its coefficient at multipole $\ell$ equal toThe gravitational time derivative is purely monopolar, and the potential gradient and baryon velocity are dipolar. Thomson scattering removes the monopole from its relaxation term and adds the quadrupole polarization correction. Thus the photon Boltzmann hierarchy isThe $\ell =0$ term containing ${\Theta }_{-1}$ is understood as zero.

Insert the photon temperature multipole expansion into the Fourier-space transport equation. The Legendre polynomial recurrence relation turns multiplication by $\mu$ into nearest-neighbour multipole couplings, while Orthogonality of Legendre polynomials projects onto a fixed $\ell$. The monopole projection has no collision term because Thomson scattering conserves photon number:The dipole projection receives the electron-velocity source,For every $\ell \ge 2$, the collision term damps the anisotropic multipole and the photon Boltzmann hierarchy isThus the Free-streaming photon Boltzmann equation moves angular structure between neighbouring multipoles, whereas Thomson scattering suppresses all multipoles above the dipole in the tight-coupling approximation.

Write $\mu =\hat{\mathbf{k}}\cdot \hat{\mathbf{p}}$ and use the Legendre polynomial recurrence relation. The definition in the question is inverted byFor Newtonian-gauge potentials the scanned geodesic equation is the standard identityIts monopole and dipole are $S_0={\Phi }^{\prime }$ and $S_1=k\Psi /3$. Projecting the photon Boltzmann hierarchy onto $P_0$ and $P_1$ therefore givesThe first is the photon continuity equation; the second is the photon Euler equation, with the photon quadrupole providing the anisotropic-stress term. Indeed, with photon density contrast ${\delta }_{\gamma }=4{\Theta }_0$ and velocity divergence ${\theta }_{\gamma }=3k{\Theta }_1$,

For the expansion $\Theta (\mu )={\sum }_{\ell }(-i{)}^{\ell }{\Theta }_{\ell }{P}_{\ell }(\mu )$ without a $(2\ell +1)$ factor, the leading polarization relation is $E_2=-\sqrt{6}{\Theta }_2/4$. The photon Boltzmann hierarchy then gives ${\Theta }_2=-8k{\Theta }_1/(9\dot{\tau })$. This small photon quadrupole supplies the shear-viscosity part of Silk damping.

Here $\mu =\cos \theta$ and $a_{\ell ,m}=\sqrt{({\ell }^2-m^2)/[(2\ell +1)(2\ell -1)]}$. This follows by combining the associated Legendre function recurrence $(2\ell +1)\mu P_{\ell }^m=(\ell -m+1)P_{\ell +1}^m+(\ell +m)P_{\ell -1}^m$ with the normalized spherical harmonic constants. It gives the adjacent-multipole couplings in a photon Boltzmann hierarchy.