Acoustic Oscillations and Peak Structure

The acoustic peaks are not a row of independent rulers. Their approximate spacing is fixed by the sound horizon projected across the comoving distance to last scattering, while their phases and relative heights retain the effects of initial conditions, baryon inertia, potential decay, photon diffusion, and the finite visibility window. The useful question is therefore which physical scale controls each feature, not which single parameter “sets the peaks.”

Required background. Probability and Inference for Gravity supplies Probability and Statistical Inference for power spectra and correlated observables.

Helpful background. The Einstein Equation supplies Einstein Equations and potential evolution; Relativistic Matter and Compact Objects supplies Relativistic Matter and Compact Objects fluid dynamics; Relativistic Cosmology supplies the thermal and perturbation history; and The Wave Equation and Characteristics supplies Waves, Green Functions, and Geometric Optics for oscillation and projection.

The photon–baryon acoustic degree of freedom

Before recombination, Thomson drag makes the photon and baryon velocities equal at leading tight-coupling order. Work in conformal time $\eta$, with primes denoting $d/d\eta$, and use the Newtonian-gauge metric potentials $\Phi$ (lapse) and $\Psi$ (spatial curvature). Define

$$R(\eta)=\frac{3\rho_b}{4\rho_\gamma}, \qquad c_s^2=\frac{1}{3(1+R)}, \qquad r_s(\eta)=\int_0^\eta c_s(\tilde\eta)\,d\tilde\eta.$$

Here $R$ measures baryon inertia relative to photon momentum density, $c_s$ is the coupled-fluid sound speed, and $r_s$ is a comoving distance. This $R$ is neither the curvature perturbation $\mathcal R$ nor a radius. For an adiabatic compression, $\delta\rho_b=3\rho_b\delta\rho_\gamma/(4\rho_\gamma)$, so $\delta p/\delta\rho=(\delta\rho_\gamma/3)/ [(1+R)\delta\rho_\gamma]=1/[3(1+R)]$: photons supply pressure while baryons add inertia. Neglecting photon shear and baryon pressure at this order, combine photon continuity with total photon–baryon momentum conservation to obtain

$$\frac{1}{1+R} \left[(1+R)(\Theta_0'-\Psi')\right]' +k^2c_s^2\Theta_0=-\frac{k^2}{3}\Phi.$$

For a direct derivation, write the common velocity divergence as $\theta$. Photon continuity gives $\Theta_0'-\Psi'=-\theta/3$. Adding the two momentum equations with their momentum-density weights gives

$$[(1+R)\theta]'=k^2\Theta_0+(1+R)k^2\Phi,$$

where $R'=\mathcal HR$ follows from $\rho_b\propto a^{-3}$ and $\rho_\gamma\propto a^{-4}$. Substituting for $\theta$ produces the displayed oscillator. Photon pressure supplies the restoring force, baryons lower the sound speed and add time-dependent inertia, and the metric potentials drive the oscillator. Ma and Bertschinger derive the tight-coupling slip and quadrupole expansion in Ma and Bertschinger 1995, §5.7, printed pp. 23–25, eqs. (70)–(75) (Open PDF).

For constant $R$, $\Phi=\Psi$, and constant potentials, the solution is

$$\Theta_0+(1+R)\Phi =A\cos(kr_s)+B\sin(kr_s).$$

Adiabatic growing-mode initial data select the cosine phase in the simplest limit; a velocity-displacement or isocurvature idealization can supply a sine component. The observed effective temperature is $\Theta_0+\Phi=A\cos(kr_s)+B\sin(kr_s)-R\Phi$. Baryons therefore shift the oscillation equilibrium as well as changing $c_s$: compressions in potential wells and rarefactions no longer have equal amplitudes.

Four scales behind the peak pattern

Let $\eta_*$ denote the visibility peak and $D_M(\eta_*)$ the transverse comoving distance to it. The acoustic angular scale is

$$\ell_A=\pi\frac{D_M(\eta_*)}{r_s(\eta_*)}, \qquad k_mr_s(\eta_*)\simeq m\pi, \qquad \ell_m\simeq k_mD_M.$$

This gives nearly uniform spacing, not exact peak locations. Projection, driving, Doppler terms, finite last-scattering width, and lensing produce phase shifts. Spatial curvature changes $D_M$; it does not change the local acoustic wavelength by the same operation.

Three further scales or ratios organize the heights:

• $R_*$ fixes baryon loading and the alternating compression–rarefaction modulation, conventionally enhancing odd compression peaks in adiabatic TT.
• $k_{\rm eq}$ marks horizon entry near matter–radiation equality. Decaying potentials during radiation domination drive modes that enter earlier, so equality changes the envelope rather than merely translating every peak.
• $k_D$ is the comoving diffusion wavenumber. Including polarization in the leading tight-coupling estimate,

$$k_D^{-2}(\eta)=\int_0^\eta \frac{d\tilde\eta}{6\dot\kappa(1+R)} \left[\frac{R^2}{1+R}+\frac{16}{15}\right].$$

Here $\dot\kappa\equiv a n_e\sigma_T\gt0$ denotes the positive differential Thomson optical depth. Sources that instead define a decreasing optical-depth variable with $\kappa'=-a n_e\sigma_T$ require $\lvert\kappa'\rvert$ in this diffusion integral.

Temperature amplitudes are suppressed approximately as $\exp[-(k/k_D)^2]$ in the regime where the diffusion approximation applies; the power receives the square of that factor. Hu and Dodelson develop the oscillator, baryon loading, radiation driving, and damping in Hu and Dodelson 2002, §§3.1–3.6, preprint printed pp. 9–21, eqs. (6)–(20) (Open PDF).

Phases, heights, and projection must be read together

Feature	Leading physical control	What prevents a one-parameter reading
Spacing	$r_s/D_M$	free-streaming phase shifts, projection, finite visibility
Odd/even contrast	baryon loading $R_*$	radiation driving, tilt, damping, foreground model
Early-peak envelope	equality and potential decay	neutrino stress and initial-mode content
Damping tail	diffusion and visibility width	helium/recombination history, lensing, beams
Trough filling	Doppler and projected sources	the $j_\ell'$ kernel is not the monopole kernel

The current review-level physical picture is summarized by the Particle Data Group in Scott and Smoot 2026, §29.5.2, printed pp. 7–8 (official PDF). That qualitative stability does not make a peak-height parameter table evergreen: precision predictions require a recombination history, neutrino sector, primordial spectrum, full Einstein–Boltzmann evolution, lensing, and a versioned measurement model. The CMB line-of-sight page supplies the projection that turns these local acoustic sources into angular multipoles.

Limiting checks and common pitfalls

Set $R=0$ and hold $\Phi=\Psi$ constant. Then $X=\Theta_0+\Phi$ obeys

$$X''+\frac{k^2}{3}X=0,$$

so an initially displaced, zero-velocity mode has $X=X_i\cos(k\eta/\sqrt3)$. Successive extrema are separated by $\Delta k=\pi/r_s$, and projection gives $\Delta\ell\simeq\ell_A$. This simultaneously checks the sound speed, phase, and factor of $\pi$. Hu and Dodelson give the corresponding scale dictionary in Hu and Dodelson 2002, §3.9, preprint printed pp. 26–28, eq. (24) (Open PDF).

The sound horizon is not the Hubble radius, and a peak is not created by a single Fourier mode. “Baryons raise odd peaks” is a controlled adiabatic TT statement, not a universal rule for arbitrary initial modes or polarization. Silk damping is physical diffusion before last scattering; beam smoothing is an instrumental convolution and belongs to a different layer of the model.

Exercises

Solve the constant-$R$, constant-potential oscillator and show explicitly how baryon loading changes its equilibrium point but not its cosine phase for zero initial velocity.

Solution

With $R'=\Phi'=\Psi'=0$, the equation becomes

$$\Theta_0''+k^2c_s^2\Theta_0=-\frac{k^2}{3}\Phi.$$

A constant particular solution is $\Theta_{0,{\rm eq}}=-(1+R)\Phi$. Writing $Y=\Theta_0-\Theta_{0,{\rm eq}}$ gives $Y''+k^2c_s^2Y=0$. Hence

$$\Theta_0(\eta)=-(1+R)\Phi +[\Theta_0(0)+(1+R)\Phi]\cos(kc_s\eta)$$

when $\Theta_0'(0)=0$. The equilibrium shift changes $\Theta_0+\Phi$ by $-R\Phi$, while the extrema remain at $kc_s\eta=m\pi$. A varying $R$ or potential invalidates this constant-coefficient solution and supplies driving or adiabatic-amplitude corrections.

References

• Hu, W., and S. Dodelson, “Cosmic Microwave Background Anisotropies,” Annual Review of Astronomy and Astrophysics 40, 171–216 (2002), DOI, Open PDF.
• Ma, C.-P., and E. Bertschinger, “Cosmological Perturbation Theory in the Synchronous and Conformal Newtonian Gauges,” The Astrophysical Journal 455, 7–25 (1995), DOI, Open PDF.
• Scott, D., and G. F. Smoot, “Cosmic Microwave Background,” in F. Takahashi et al. (Particle Data Group), Review of Particle Physics, International Journal of Modern Physics A 41, 2630011 (2026), Official PDF.