--- title: "Why primordial sound leaves a series of peaks" description: "Derive a restricted photon-baryon oscillator, calculate the sound horizon, and distinguish peak spacing from the first temperature peak." canonical_url: "https://sandboxphysics.com/stories/cosmology/acoustic-oscillations/" language: "en" page_type: "story" alternate_languages: zh-CN: "https://sandboxphysics.com/zh/stories/cosmology/acoustic-oscillations/" last_modified: "2026-10-28" --- [Cosmology from expansion to inference](https://sandboxphysics.com/stories/cosmology/) L2 · Chapter 3 # Why primordial sound leaves a series of peaks Sandbox Physics · 2026-10-28 Before recombination, photon pressure pushed baryons out of compressed regions and gravity pulled them back into potential wells. Different wavelengths reached different phases by last scattering. Modes near compression or rarefaction extrema left peaks in the angular power spectrum. ## Before you begin Read chapter two. You need sine, cosine and second derivatives. We first hold the baryon loading and gravitational potential constant to isolate the oscillator, then explain what a realistic calculation adds. [Review the preceding chapter](https://sandboxphysics.com/stories/cosmology/linear-perturbations/) - Separate photon pressure from baryon inertia. - Solve an oscillator whose equilibrium shifts under gravity. - Distinguish the acoustic scale from the first temperature peak. ## Photons push and baryons add inertia Frequent scattering couples photons to electrons, and electromagnetic interactions couple electrons to nuclei. In tight coupling the photon-baryon fluid moves almost together. Photon pressure supplies the restoring force; baryon mass resists acceleration. Define the loading $R=3\rho_b/(4\rho_\gamma)$. Both densities are energy per physical volume. The sound speed is $c_s=c/\sqrt{3(1+R)}$. More baryons lower it. Baryons and photons dilute at different rates, so $R$ grows as the universe expands. Sound therefore travels a different comoving distance in each interval. $$ r_s(z_*)=\int_0^{\eta_*}c_s(\eta)\,d\eta $$ The sound horizon is the comoving distance sound travels before last scattering. Here conformal time retains time units and sound speed retains length per time. Reading: Hu & Sugiyama ## Solve one oscillator inside a fixed well Let $\Theta_0=\delta T/T$ be the photon temperature monopole and use the Newtonian-gauge time potential $\Psi$. Set light speed to one. With constant potential and baryon loading, and omitting fluid-drag terms, the compression mode obeys the forced-oscillator equation below. Primes denote conformal-time derivatives. Gravitational redshift contributes to the observed temperature, so define $X=\Theta_0+\Psi$. Moving the constant force into a shifted equilibrium gives an ordinary cosine when the initial velocity vanishes. $$ \Theta_0''+k^2c_s^2\Theta_0=-\frac{k^2}{3}\Psi,\qquad X''+k^2c_s^2(X+R\Psi)=0 $$ In this equation sound speed is measured in units of light speed. The fixed-potential, fixed-loading approximation isolates the equilibrium shift. $$ X(\eta)+R\Psi=[X(0)+R\Psi]\cos(k r_s) $$ This is the zero-initial-velocity solution. The effective temperature oscillates about minus the baryon loading times the time potential. Reading: Hu & Sugiyama ## Calculate the two turning points Choose a fixed well with $\Psi=-10^{-5}$, $R=1/2$, initial effective temperature $X(0)=\Psi/3$ and zero initial velocity. The potential in the well is negative, so the baryon-induced equilibrium effective temperature is positive. Substitution into the cosine solution gives the dimensionless expression below. After half an oscillation, $kr_s=\pi$ and $X=(4/3)\times10^{-5}$. After a full oscillation, $X=-(1/3)\times10^{-5}$. Their absolute amplitudes differ by a factor of four. This shows the effect of the shifted equilibrium for the specified initial conditions. A temperature spectrum also combines velocity, driving and projection, as the next section explains. $$ \frac{X}{10^{-5}}=\frac12-\frac56\cos(kr_s) $$ This restricted example calculates the equilibrium shift and both turning-point amplitudes for the chosen initial conditions. ## Baryons make compression and rarefaction unequal Without baryon loading, $X$ oscillates around zero. Adding baryons shifts the equilibrium and deepens compression inside potential wells. For the usual adiabatic initial conditions, compression peaks rise relative to rarefaction peaks, producing the familiar odd-even temperature-peak pattern. The complete temperature spectrum also contains Doppler velocities, changing potentials and photon diffusion. Increasing baryon density changes recombination, sound speed and the sound horizon together. A realistic comparison therefore needs the coupled calculation. The frozen CAMB grid retains these simultaneous effects. Reading: CAMB source and documentation · Hu & Sugiyama ## Estimate the separation between peaks Modes with $k r_s\simeq n\pi$ reach oscillator extrema at last scattering. Projection places them roughly at $\ell\simeq kD_M$. Combining these relations gives the main acoustic spacing. For an illustrative $r_s=144\;\mathrm{Mpc}$ and $D_M=14000\;\mathrm{Mpc}$, the angle is about $0.0103\;\mathrm{rad}$ and the acoustic multipole scale is about $\ell_A=305$. This estimates a spacing scale. Driving, phase shifts and projection move the first temperature peak away from that simple value. $$ \theta_* = \frac{r_s}{D_M},\qquad \ell_A=\frac{\pi}{\theta_*},\qquad \Delta\ell\approx\ell_A $$ Both lengths are comoving. The next chapter replaces the rough projection with the transfer-function integral. Observer Equal comoving sound horizons Increasing comoving distance to the ruler → $$ \theta_* = r_s(z_*) / D_M(z_*) $$ Schematic geometry, with both lengths comoving. Temperature peaks also carry phase shifts and projection effects. ## Try the prediction ### [Connect compression to a Boltzmann spectrum](https://sandboxphysics.com/labs/cmb-acoustic-peaks/) 1. Locate the first three peaks in the reference preset and predict which respond most to more baryons. 2. Change baryon density and compare with the fixed reference. 3. Change late-time expansion separately to distinguish altered peak heights from a shift of the peak sequence. ## Exercises and explanations 1. At fixed sound horizon, increase transverse comoving distance to last scattering by ten percent. How do the acoustic angle and multipole scale change? Read the explanation The angle becomes ten-elevenths as large and the multipole scale rises by ten percent. The same physical pattern looks smaller and projects to higher multipoles. 2. Why do peak positions alone fail to determine baryon density and late expansion uniquely? Read the explanation They mainly constrain a ratio of sound horizon to transverse distance. Different parameter combinations can preserve that ratio. Relative heights, damping, polarization, lensing and external distances provide additional information. [Previous: How small fluctuations become structure](https://sandboxphysics.com/stories/cosmology/linear-perturbations/) [Next: From three-dimensional waves to an angular spectrum](https://sandboxphysics.com/stories/cosmology/transfer-and-degeneracy/) ## Sources and calculation scope 1. [Hu & Sugiyama, Small-Scale Cosmological Perturbations: An Analytic Approach, arXiv:astro-ph/9510117v2 (1996)](https://arxiv.org/abs/astro-ph/9510117v2) Tight-coupling acoustic physics and baryon loading. The constant-potential, constant-baryon-loading oscillator in chapter three is a restricted analytic example; the interactive spectra come from the frozen CAMB calculation. 2. [CAMB source and documentation](https://github.com/cmbant/CAMB/tree/1.6.6) The acoustic-grid generator pins CAMB 1.6.6, saves its full cosmological settings and checks interpolation at off-grid points. The browser interpolates a fixed grid; it does not run a Boltzmann hierarchy.