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Adiabatic initial conditions select coherent cosine phases of the photon-baryon acoustic oscillator. Its extrema at last scattering satisfy $k{r}_s\simeq n\pi$ and project to $\ell \simeq k{\chi }_{\ast }$. Thus ${\ell }_n\simeq n\pi {\chi }_{\ast }/r_s({\eta }_{\ast })$, with comoving sound horizon $r_s=\int d\eta /\sqrt{3(1+R)}$. Projection, driving and finite last-scattering width modify the exact peak locations.

Approximating decoupling by a single time and retaining only a photon monopole and Doppler dipole omits the finite cosmological visibility function width, diffusion or Silk damping, quadrupole anisotropy and polarization generated as tight coupling fails. The photon-baryon acoustic oscillator also make the emission density and velocity strongly scale dependent. They can be included in the initial monopole and dipole, so their existence alone does not invalidate the subsequent collisionless line-of-sight solution for free-streaming photons. Smaller angles correspond to larger angular multipole $\ell$.

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.

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 $R\gg 1$, $3{\tilde{c}}_s^2\simeq 1$, 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.

Expand the fractional CMB temperature fluctuation as $\Theta (\mathbf{n})={\sum }_{\ell m}{a}_{\ell m}{Y}_{\ell m}(\mathbf{n})$. Under statistical isotropy, its CMB angular power spectrum is defined byThe normalization is for dimensionless temperature, rather than temperature in kelvin. The spherical harmonics separate angular scales, with large $\ell$ corresponding to small angles.

For approximately constant $R,\varphi ,\psi$, a useful solution of the photon-baryon acoustic oscillator isHere $r_s$ is the comoving sound horizon. With slowly varying $R$, the oscillatory phase is still $k{r}_s$ and its homogeneous amplitude varies approximately as $(1+R{)}^{-1/4}$ under the short-period approximation. Evolving potentials add gravitational driving rather than erase this phase coherence.

The regular growing adiabatic initial conditions produce an initial density displacement and negligible initial acoustic velocity outside the horizon. They therefore select the cosine phase, with $B$ approximately zero, for all modes. At last scattering, extrema of compression and rarefaction occur at $k{r}_s({\eta }_{\ast })\simeq n\pi$. The observed monopole-plus-potential Sachs-Wolfe combination is ${\Theta }_0+\psi ={\delta }_{\gamma }/4+\psi$, so baryon loading shifts its equilibrium and makes alternating peaks unequal. The velocity produces a phase-shifted Doppler CMB anisotropy.

Projection from the Cosmic microwave background last-scattering surface places a mode near $\ell \simeq k{\chi }_{\ast }$, where ${\chi }_{\ast }={\eta }_0-{\eta }_{\ast }$ is its comoving radial distance in a flat universe. The resulting adiabatic acoustic-peak spacing isCoherent acoustic phases turn successive compressions and rarefactions into the CMB peak series. Finite-width last scattering, Doppler projection, evolving potentials and diffusion shift or broaden actual peaks, so this is the requested approximate spacing, not an exact prediction for every peak.

The tight-coupling approximation requires the positive scattering rate $\kappa =-\dot{\tau }$ to exceed both $k$ and the conformal Hubble parameter. The photon-baryon velocity slip and photon quadrupole are then small, and the leading fluid motion has $v_b=v_{\gamma }=v$. One must combine the equations before setting the slip to zero: a small slip times a large collision rate can exert a finite force.

Add the photon Euler equation to $R$ times the baryon Euler equation with Thomson drag. The collisions cancel and, neglecting ${\Theta }_2$ at this order, the result isSince $\dot{R}=\mathcal{H}R$, the common velocity obeysThe inertia of the baryons reduces the photon-baryon sound speed to $c_s^2=1/[3(1+R)]$.

Differentiate the photon continuity equation and substitute this velocity equation:Thus the photon-baryon acoustic oscillator isThe terms on the right drive the oscillation gravitationally; the first-derivative term comes from changing baryon inertia. Diffusion damping enters only beyond this leading tight-coupling approximation.

P. J. E. Peebles made the connection between microscopic processes and cosmological observables quantitative. His 1968 paper treated cosmological recombination as a kinetic problem. Capturing an electron directly into the hydrogen ground state releases a photon that can ionize another atom. Net neutralization depends on escape of resonance radiation through cosmological redshift and on two-photon spontaneous emission from the $2s$ state. An equilibrium Saha equation alone does not give that history.^[ref]

This matters for an acoustic prediction because the hydrogen ionization fraction fixes the supply of free electrons for Thomson scattering. In physical units the scattering rate per photon is ${\Gamma }_T=n_e{\sigma }_{T}c$. The cosmological visibility function describes the distribution of last-scattering times. Thus atomic physics helps determine when the photon-baryon acoustic oscillator is read out, as well as how rapidly tight coupling fails.

In 1970 P. J. E. Peebles and J. T. Yu followed adiabatic cosmological perturbations by numerically integrating the photon collision equation through cosmological recombination. Their calculation connected acoustic evolution and dissipation to residual matter structure and microwave brightness fluctuations. It did not assume that a perfect photon-baryon fluid remained valid throughout last scattering. Their illustrative cosmological models and normalization were historical choices, rather than the present Lambda-CDM model; their predicted brightness variance versus angular resolution should not be relabelled as a modern measured $C_{\ell }$ spectrum.^[ref]

The continuing idea is a forward prediction: specify initial conditions, evolve the coupled matter and radiation, project onto the sky, and compare the result with an instrument's readout. Coherent initial phases are needed for acoustic variance peaks explains why a collection of oscillating modes does not automatically produce a peaked power spectrum. The BOOMERANG acoustic-peak measurement provides a later observational test of this physical programme.