Acoustic Oscillations in the Photon-Baryon Fluid

Before the Universe was 380,000 years old it was an opaque plasma in which Thomson scattering welded photons and baryons into a single elastic fluid. Gravity, sourced mostly by cold dark matter, tried to pull this fluid into overdense regions; its enormous photon pressure pushed back. The compromise is not collapse but oscillation — a standing sound wave. Every Fourier mode of density behaves as a driven, damped harmonic oscillator ringing at frequency ω = k c_s, and because the sound speed is nearly fixed, a mode's phase is simply how far sound has travelled: the acoustic phase k r_s. This article takes that one idea — a bath of coupled oscillators sharing a common clock — and follows it to its two great fossils. When recombination switches off the scattering, the oscillations freeze. Modes caught at maximal compression or rarefaction imprint the harmonic ladder of acoustic peaks on the cosmic microwave background; modes caught mid-swing contribute a Doppler signal; and the preferred length carried by the baryons, the sound horizon r_s ≈ 147 Mpc, survives as the baryon acoustic oscillation bump in the galaxy distribution. Because r_s is computable from first principles once the baryon and radiation densities are known, both features act as precise standard rulers, and their comparison across cosmic time is among the sharpest tools in cosmology. The whole subject — the CMB power spectrum, baryon acoustic oscillations, Silk damping, the sound horizon as a ruler — grows out of the driven-oscillator equation derived here.

Key takeaways

Before recombination, Thomson scattering locked photons and baryons into one fluid whose huge radiation pressure resisted gravity, so overdensities did not simply grow — they oscillated as standing sound waves.
Each mode of wavenumber k is a driven, damped harmonic oscillator: dark-matter gravity drives compression, photon pressure restores it, and the phase is the acoustic phase k r_s set by the comoving sound horizon r_s.
The sound speed is c_s = c/√(3(1+R)) with baryon loading R = 3ρ_b/4ρ_γ; more baryons lower c_s, deepen compressions, and unbalance the odd and even peaks.
At recombination (z ≈ 1090) the oscillations freeze; modes at k r_s = nπ produce the CMB peak series at ℓ_n ≈ n π d_A/r_s, with the observed first peak near ℓ ≈ 220.
The same frozen wave leaves one preferred scale — the sound horizon r_s ≈ 147 Mpc comoving — on the baryon distribution, the baryon-acoustic-oscillation standard ruler.
Photon diffusion on the smallest scales (Silk damping) suppresses high-k modes by exp[−(k/k_D)²], terminating the peak series.
Key idea: every Fourier mode of the photon-baryon fluid is a driven, damped harmonic oscillator whose phase at recombination, k r_s, converts one oscillation clock into a harmonic sequence of features — acoustic peaks in the CMB and a standard-ruler bump in matter.

The core insight

Every Fourier mode of the photon-baryon fluid is a driven, damped harmonic oscillator whose phase at recombination, k r_s, converts a common oscillation clock into a harmonic sequence of features — acoustic peaks in the CMB and a standard-ruler bump in the matter distribution.

Where this lives today

Acoustic oscillations are no longer a prediction to be confirmed but a precision instrument. Planck measured the CMB temperature and polarization spectra through at least seven peaks, and their positions, heights, and damping tail pin down the ΛCDM parameters — Ω_b h², Ω_c h², the sound horizon, and spatial flatness — to sub-percent accuracy. Ground-based experiments (the Atacama Cosmology Telescope, the South Pole Telescope) push to arcminute scales, mapping the Silk-damping tail and the velocity-sourced polarization peaks to constrain N_eff and the helium abundance at recombination. The same sound horizon reappears as the baryon acoustic oscillation feature in galaxy surveys: BOSS, eBOSS, and now DESI measure the ∼150 Mpc BAO scale across a ladder of redshifts, turning r_s into a ruler that tracks H(z) and the angular-diameter distance, and thereby the equation of state of dark energy. There is even a live tension: the sound-horizon route to the Hubble constant, using r_s from early-Universe physics, returns H₀ ≈ 67 km/s/Mpc, several sigma below the ≈ 73 from the local distance ladder — much of the debate over new pre-recombination physics is really about whether the calculable r_s can be shifted.

The physical picture

Picture the dark-matter potential wells scattered through the primordial plasma. The photon-baryon fluid, like air rushing toward a region of low pressure, streams down into each well and compresses. But its pressure is set almost entirely by photons and is therefore enormous, so it rebounds, overshoots, rarefies, and is drawn back in again — a ringing that repeats until recombination silences it. The crucial point is that every well of a given size rings in unison: inflationary initial conditions start each mode from rest at the same phase, so all modes of the same wavenumber are always caught at the same point in their cycle. The first figure captures this ringing as a cross-section of one well in compression and rarefaction.

Figure

The photon-baryon fluid rings inside a dark-matter potential well: gravity in blue drives infall and compression, photon pressure in amber restores the fluid and overshoots into rarefaction, oscillating at sound speed c_s = c/√(3(1+R)) ≈ 0.45c with baryon loading R ≈ 0.6 at recombination

Because c_s is nearly constant, a mode's phase is just the distance sound has covered, k r_s, where r_s = ∫ c_s dη is the comoving sound horizon. A large-scale mode (small k) has barely completed a quarter cycle by recombination and is caught near its first compression — the first acoustic peak; smaller modes have completed half, full, and multiple cycles, producing the harmonic ladder. Baryons add inertia, like a heavier mass on the spring: they deepen the compressions and shallow the rebounds, breaking the symmetry between the odd and even peaks. On the smallest scales the fluid leaks — photons random-walk out of tiny hot and cold spots during recombination, erasing them. The second figure is the observable all of this produces: the CMB temperature power spectrum.

Figure

The observed CMB temperature power spectrum ℓ(ℓ+1)C_ℓ against multipole ℓ: acoustic peaks sit at ℓ ≈ 220, 540, 810, 1120 with spacing ℓ_A = π d_A/r_s ≈ 300, fixed by the sound horizon r_s ≈ 147 Mpc viewed across the comoving distance d_A ≈ 13.9 Gpc, while the Silk-damping envelope cuts off the series past ℓ ≈ 1500

The core equations

The quantitative story rests on a short chain: the sound speed and its baryon loading, the sound horizon that sets the clock, the driven-oscillator equation, its frozen solution, the projection onto peak multipoles, the diffusion scale that ends the series, and the sum an observer receives. The sound speed is the pressure response to a density perturbation. Because only photons carry appreciable pressure while both photons and baryons carry inertia, it drops below the pure-radiation value c/√3 as baryons are added.

c_s = \frac{c}{\sqrt{3\,(1+R)}}, \qquad R \equiv \frac{3\rho_b}{4\rho_\gamma} = \frac{3\Omega_b}{4\Omega_\gamma}\,a \qquad (1)

where: c_s — adiabatic sound speed of the coupled fluid; c — speed of light; R — baryon-to-photon momentum-density ratio (the baryon loading); ρ_b, ρ_γ — baryon and photon energy densities; a — scale factor, so R ∝ a and the fluid grows more baryon-loaded toward recombination. Reading it: with no baryons c_s = c/√3 ≈ 0.577c; at recombination R ≈ 0.6 drags it down to c_s ≈ 0.45c. A mode's phase is k times the distance sound has travelled, so the natural clock is the comoving sound horizon — the total comoving path a wave crest covers from the Big Bang to last scattering.

r_s(\eta_\ast) = \int_0^{\eta_\ast} c_s(\eta)\,\mathrm{d}\eta = \int_0^{a_\ast} \frac{c_s}{a^2 H(a)}\,\mathrm{d}a \approx 147\ \mathrm{Mpc} \qquad (2)

where: r_s — comoving distance a sound wave travels before recombination; η — conformal time, dη = dt/a; η_∗, a_∗ — values at last scattering (z_∗ ≈ 1090); H(a) — Hubble parameter versus scale factor. Reading it: r_s depends only on pre-recombination physics, so it is calculable from first principles and serves as a standard ruler. A single Fourier mode obeys a driven, damped harmonic-oscillator equation: the damping term is baryon drag, the restoring term is the pressure k²c_s²Θ₀, and the right-hand side is gravitational driving by the potentials.

\ddot{\Theta}_0 + \frac{\dot R}{1+R}\dot{\Theta}_0 + k^2 c_s^2\,\Theta_0 = -\frac{k^2}{3}\Phi - \frac{\dot R}{1+R}\dot\Phi - \ddot\Phi \qquad (3)

where: Θ₀ — temperature-perturbation monopole, δT/T; Φ, Ψ — the two metric potentials sourced by all matter; k — comoving wavenumber; the overdot is d/dη. Reading it: the left side is a spring (frequency ω = k c_s) with friction; the right side is gravity forcing it, which is why the natural coordinate is shifted away from Θ₀ = 0. In the idealized limit where R and the potentials vary slowly, the equation collapses to simple harmonic motion in a shifted variable, and adiabatic initial conditions select the pure cosine.

\big[\Theta_0 + (1+R)\Phi\big](\eta_\ast) = \big[\Theta_0 + (1+R)\Phi\big](0)\,\cos\!\big(k\,r_s\big) \qquad (4)

where: k r_s — the acoustic phase at last scattering; (1+R)Φ — the gravitational offset that shifts the zero-point. Reading it: the fluid oscillates not about Θ₀ = 0 but about −(1+R)Φ, so compression extrema (odd peaks) exceed rarefaction extrema (even peaks) — the baryon-loading asymmetry. Projected onto the sky, the extrema at k r_s = nπ become a harmonic series of peaks in multipole space, spaced by the acoustic scale ℓ_A.

\ell_n \approx n\,\pi\,\frac{d_A(z_\ast)}{r_s(z_\ast)} = n\,\ell_A, \qquad \ell_A \approx 300 \qquad (5)

where: ℓ_n — multipole of the n-th extremum; d_A(z_∗) — comoving distance to last scattering; ℓ_A = π d_A/r_s — angular acoustic scale. Reading it: the spacing measures d_A/r_s, a geometric quantity sensitive to curvature; driving shifts the observed first peak down to ℓ₁ ≈ 220. On scales near the photon mean free path the fluid leaks: photons diffuse out of small perturbations, defining a damping wavenumber k_D beyond which oscillations are exponentially erased.

k_D^{-2} = \int_0^{\eta} \frac{\mathrm{d}\eta}{6\,(1+R)\,n_e\,\sigma_T\,a}\left[\frac{R^2 + \tfrac{4}{5}(1+R)}{1+R}\right] \qquad (6)

where: k_D — Silk damping wavenumber, modes k > k_D suppressed as exp[−(k/k_D)²]; n_e — free-electron density; σ_T — Thomson cross-section; a — scale factor. Reading it: the damping length is the geometric mean of the horizon and the mean free path (n_e σ_T a)⁻¹, setting the small-angle CMB cutoff and the width of the BAO bump. Finally, an observer does not receive Θ₀ alone: the measured temperature is the effective temperature (monopole plus gravitational redshift), a Doppler term from the fluid velocity, and an integrated term from potentials evolving along the line of sight.

\frac{\Delta T}{T}\Big|_{\rm obs}(\hat n) = \big[\Theta_0 + \Psi\big]_\ast + \hat n\!\cdot\!\vec v_b\big|_\ast + \int (\dot\Phi+\dot\Psi)\,\mathrm{d}\eta \qquad (7)

where: Θ₀ + Ψ — effective temperature (Sachs–Wolfe term); n̂·v⃗_b — Doppler term from the baryon velocity at last scattering; ∫(Φ̇+Ψ̇)dη — integrated Sachs–Wolfe effect from time-varying potentials. Reading it: the monopole ∝ cos(k r_s) and the velocity ∝ sin(k r_s) are 90° out of phase, so Doppler peaks fill the troughs.

Derivation AThe photon-baryon fluid as a harmonic oscillator

The goal is to show, starting from the continuity and Euler equations of the tightly-coupled plasma, that the temperature monopole of a single Fourier mode obeys a driven, damped harmonic-oscillator equation, and then — in the limit where the baryon loading R and the gravitational potentials vary slowly — that its solution is a shifted cosine whose zero-point encodes the baryon density. This is the central calculation of the subject: peak positions, peak heights, the Doppler signal, and the ruler are all readings of it.

Step 1. Begin with the two conservation laws in Fourier space: continuity, Θ̇₀ = −(1/3)k v_γ − Φ̇, and Euler, v̇_γ = −[Ṙ/(1+R)] v_γ + [k/(1+R)]Θ₀ + kΨ, balancing baryon drag, pressure, and gravity. Why: the monopole cannot evolve alone — pressure couples it to the velocity, so the closed equation comes from combining the two.

Step 2. Differentiate continuity once in conformal time and substitute v̇_γ to eliminate the velocity; the pressure term becomes the restoring term k²c_s²Θ₀ with c_s² = 1/[3(1+R)], and the drag reappears as the friction Ṙ/(1+R) Θ̇₀. Why: this algebra turns two first-order equations into the boxed second-order oscillator — the spring is photon pressure, the friction is baryon drag.

Step 3. Take the idealized limit: hold R constant and drop the slow potential derivatives. The friction vanishes and the driving −k²Φ/3 = −k²c_s²(1+R)Φ is absorbed by completing the square, giving the homogeneous equation for the shifted variable Θ₀ + (1+R)Φ. Why: the natural coordinate is Θ₀ + (1+R)Φ, not Θ₀ — gravity does not damp the oscillation, it displaces its equilibrium.

Step 4. Solve it. Since ω dη = k c_s dη = k dr_s, the phase is the sound horizon and the solution is A cos(k r_s) + B sin(k r_s); adiabatic initial conditions start every mode at rest, Θ̇₀(0) = 0, forcing B = 0 and the pure cosine. Why: the sine coefficient vanishing across all modes at once is the origin of phase coherence — it makes the peaks sharp rather than a smeared blur.

Step 5. Read the observable. The effective temperature is Θ₀ + Ψ = [Θ₀(0)+(1+R)Φ(0)] cos(k r_s) − (1+R)Φ + Ψ; with Ψ ≈ −Φ it is displaced by −RΦ below zero, so compression maxima (n = 1, 3, 5) exceed rarefaction maxima (n = 2, 4), and larger R deepens the effect. Why: the asymmetry is the physical statement that gravity does net work on the infalling, baryon-loaded fluid. *As a check: in the limit R → 0 the solution reduces to Θ₀ ∝ cos(k r_s) with c_s = c/√3, symmetric about zero — the pure photon-gas result.*

Go deeper · Derivation B · 2 minPeak locations and the acoustic scale

Derivation BPeak locations and the acoustic scale

The aim here is to convert the frozen standing-wave solution into the positions of the acoustic peaks a CMB experiment measures, and to evaluate the first-peak multipole from standard ΛCDM numbers so the abstraction lands near ℓ ≈ 220. The chain has two links — freezing the oscillation at recombination, then projecting a comoving wavenumber onto an angle on the sky — and getting both right makes the peak spacing a clean geometric probe of the expansion history and curvature.

Step 1. The power spectrum is quadratic, C_ℓ ∝ ⟨|Θ|²⟩, so power peaks wherever cos(k r_s) is extremal — at both maxima and minima. Setting d/dk[cos(k r_s)] = 0 gives k_n = nπ/r_s(η_∗) for n = 1, 2, 3, …. Why: because power is squared, compressions and rarefactions both produce peaks — the CMB peaks are spaced by π in phase, not 2π.

Step 2. Project onto the sphere: in the flat-sky approximation a mode k across a distance d_A subtends ℓ ≈ k d_A, so ℓ_n ≈ nπ d_A/r_s ≡ n ℓ_A with acoustic scale ℓ_A = π d_A/r_s. Why: the spacing depends only on the ratio d_A/r_s, a purely geometric quantity independent of the oscillation's amplitude.

Step 3. Insert r_s ≈ 147 Mpc and d_A(z_∗) ≈ 1.39 × 10⁴ Mpc: ℓ_A ≈ π(13900/147) ≈ 297, so the naive first extremum sits near ℓ ≈ 300. Why: this order-300 spacing is the most robust number in the CMB — the peaks march in an almost even ladder because r_s and d_A are both fixed.

Step 4. Correct for driving: the observed first peak sits near ℓ₁ ≈ 220 ≈ 0.74 ℓ_A, because the decay of the potentials near matter-radiation equality shifts the effective phase and pulls the first peak inward, while curvature would move the whole ladder by changing d_A. Why: the sensitivity of ℓ₁ to d_A — hence to Ω_k, H₀, and dark energy — makes the peak position a precision geometric probe; ℓ₁ ≈ 220 pins the Universe to spatial flatness. *As a check: ℓ_A ≈ 301 and ℓ₁ ≈ 220 are recovered once the driving phase shift is included.*

Go deeper · Derivation C · 1 minBaryon loading and the odd-even peak asymmetry

Derivation CBaryon loading and the odd-even peak asymmetry

The purpose here is to make quantitative the claim that baryons enhance the odd (compression) peaks over the even (rarefaction) peaks, by tracking where R enters the frozen solution and how the heights scale with it. This is what lets the CMB alone weigh the ordinary matter and check it against the independent prediction of Big Bang nucleosynthesis.

Step 1. From the shifted solution, the effective temperature at an extremum is Θ₀ + Ψ = [Θ₀(0)+(1+R)Φ(0)] cos(k r_s) − RΦ. At compression extrema cos = +1 the negative-Φ contributions add; at rarefaction extrema cos = −1 they partly cancel. Why: the constant −RΦ offset lifts the compression peaks and depresses the rarefaction peaks by the same amount, breaking the symmetry.

Step 2. The compression height scales roughly as (1 + 2R) times the driving while the rarefaction height stays near the undriven value, so the odd/even ratio grows about linearly in R; with R ≈ 0.6 the first peak sits markedly above the second. Why: the observed suppression of the second peak relative to the first and third is a direct readout of the baryon density.

Step 3. Since R = (3/4)(Ω_b/Ω_γ)a and Ω_γ is fixed by the measured CMB temperature, the peak-height ratio measures Ω_b h². The CMB value Ω_b h² ≈ 0.0224 agrees with the primordial deuterium abundance from nucleosynthesis. Why: two very different epochs — nucleosynthesis at minutes, recombination at 380,000 years — return the same baryon density, a stringent consistency test. *As a check: R ≈ 0.6 at z_∗ ≈ 1090 for Ω_b h² = 0.0224, and this enhancement reproduces the observed first-to-second peak pattern.*

Limiting cases

The oscillator's behaviour changes character across the regimes of the plasma:

Case	Result	Meaning
Radiation domination, no baryons (R → 0)	c_s → c/√3, Θ₀ ∝ cos(k r_s)	pure photon gas; sound speed at its relativistic maximum and oscillations symmetric about zero
Heavy baryon loading (R large)	c_s = c/√(3(1+R)) → 0, oscillation about −RΦ	zero-point shifts down, enhancing compression (odd) peaks over rarefaction (even) peaks
Super-horizon scales (k r_s ≪ 1)	k η ≪ 1, no oscillation yet	modes larger than the sound horizon have not begun to ring; they set the Sachs–Wolfe plateau, not peaks
Small scales below diffusion length (k ≫ k_D)	Θ ∝ exp[−(k/k_D)²] cos(k r_s)	Silk damping exponentially suppresses the wave; the peak envelope decays and the series terminates

Common trapthe acoustic peaks are frozen relativistic standing waves, not slowly growing matter clumps on today's Gpc scales

The CMB peaks are often misread as gravitationally growing overdensities or as matter sloshing across hundreds of megaparsecs today. They are neither: they are standing sound waves in a relativistic fluid, frozen the instant Thomson scattering shut off, and what we observe is a snapshot of their phase. Three further traps follow. First, the first peak is a compression peak, but the series is not a simple sin/cos symmetry — gravity does net work on the infalling baryon-loaded fluid, so odd and even peaks have unequal heights. Second, the length imprinted is the sound horizon r_s = ∫c_s dη, roughly 1/√3 smaller than the particle horizon ∫c dη; confusing the two throws the ruler off by a factor of order two. Third, "the peaks are the Universe's resonant frequencies" is a metaphor — the resonance is in k-space across independent modes sharing one clock, not a single ringing cavity.

Worked examples

1 · Baryon loading and sound speed at recombination (confidence builder)

Given the Planck values Ω_b h² = 0.0224 and Ω_γ h² = 2.47 × 10⁻⁵, and last scattering at z_∗ = 1090, find the baryon loading R and the sound speed c_s of the photon-baryon fluid at that moment.

Step 1. The ratio is h-independent since both densities carry h²: Ω_b/Ω_γ = 0.0224/(2.47 × 10⁻⁵) = 905 — how many times more energy density baryons have than photons today; at earlier times photons win because ρ_γ ∝ a⁻⁴ falls faster than ρ_b ∝ a⁻³.

Step 2. Evaluate R at last scattering with a_∗ = 1/1091 = 9.17 × 10⁻⁴ and R = (3/4)(Ω_b/Ω_γ)a: R_∗ = 0.75 × 905 × 9.17 × 10⁻⁴ = 0.62. The fluid carries roughly 60% as much baryon momentum density as photon momentum density — baryons matter dynamically, but photons still dominate.

Step 3. Insert into c_s = c/√(3(1+R)): c_s = c/√4.86 = c/2.21 = 0.45c = 1.36 × 10⁸ m/s, about 22% below the baryon-free value c/√3 = 0.577c. R ≈ 0.62 and c_s ≈ 0.45c at z ≈ 1090 — baryons drag the sound speed from 0.58c down to 0.45c.

2 · The acoustic scale and the first peak (fully worked)

Using r_s ≈ 147 Mpc for the comoving sound horizon and d_A(z_∗) ≈ 13.9 Gpc for the comoving distance to last scattering, estimate the acoustic scale ℓ_A, the first-peak multipole, and comment on the sensitivity to curvature.

Step 1. Convert to common units: d_A = 13.9 Gpc = 1.39 × 10⁴ Mpc, so d_A/r_s = 13900/147 = 94.6 — the number of sound-horizon lengths that fit across the distance to the last-scattering surface.

Step 2. The acoustic scale is ℓ_A = π d_A/r_s = π × 94.6 = 297 ≈ 300 — the spacing between successive peaks, so the CMB peaks march in an almost even ladder offset by the driving phase.

Step 3. The observed first peak is pulled inward by potential driving near matter-radiation equality to ℓ₁ ≈ 0.74 ℓ_A ≈ 0.74 × 297 = 220, matching the measurement — the naive ℓ₁ ≈ ℓ_A ≈ 300 corrected by a ∼25% phase shift.

Step 4. Sensitivity: because ℓ_A ∝ d_A, an open universe (Ω_k > 0) stretches d_A and pushes the ladder to higher ℓ, a closed universe to lower ℓ. The observed ℓ₁ ≈ 220 pins the Universe to spatial flatness, Ω_k ≈ 0, to about 0.4%. ℓ_A ≈ 300, ℓ₁ ≈ 220 after driving; the peak position is a precision measure of curvature.

3 · A sanity check on the sound horizon

The comoving particle horizon at recombination is η_∗ ≈ 288 Mpc. Show that r_s ≈ 147 Mpc is consistent with a fluid moving at the sound speed rather than at c.

Step 1. If the fluid moved at c, the "sound" horizon would equal the particle horizon, 288 Mpc. Instead sound moves at c_s < c, so r_s is smaller by the time-averaged ratio ⟨c_s/c⟩ over the pre-recombination era.

Step 2. Early on (R → 0) the ratio is 1/√3 = 0.577; near recombination it has fallen to 0.45. A conformal-time average gives ⟨c_s/c⟩ ≈ 0.51, so r_s ≈ 0.51 × 288 = 147 Mpc, the quoted value; equivalently r_s/η_∗ = 147/288 = 0.51 ≈ 1/√3. A fluid ringing at ∼0.5c produces a 147 Mpc sound horizon inside a 288 Mpc causal horizon.

Physics of the portrait

The acoustic peaks were predicted in detail decades before they were seen — almost simultaneously, by two independent groups reasoning about the same coupled plasma from opposite sides of the Iron Curtain.

Origins · 1 minRashid Sunyaev and how this was discovered

Rashid Sunyaev

born 1943 · acoustic peaks and the physics of the last-scattering surface

Rashid Sunyaev, working with Yakov Zel'dovich in Moscow, published in 1970 one of the founding papers of CMB physics, predicting that the pre-recombination photon-baryon plasma must ring with sound waves and leave a series of maxima and minima — acoustic peaks — in the temperature fluctuations across the microwave sky. In the same year, and independently, Jim Peebles and Jer Yu at Princeton reached the same conclusion by numerically integrating the coupled radiation-matter equations. Both predictions were acts of theoretical nerve: the CMB had barely been detected as a uniform glow, its anisotropies were far below any instrument's reach, and the peaks depended on assumptions — adiabatic initial conditions, tight coupling, a specific recombination history — that could not yet be tested. It took thirty years for observation to catch up: the BOOMERANG and MAXIMA balloon experiments resolved the first peak near ℓ ≈ 200 in 2000, and WMAP and Planck later mapped the full ladder to exquisite precision. Sunyaev and Zel'dovich also predicted the spectral distortion of the CMB by hot cluster gas (the Sunyaev–Zel'dovich effect), now a routine cluster-finding tool. The modern analytic picture — the driven, damped oscillator with its shifted zero-point and potential-driving phase shift — was distilled by Wayne Hu and Naoshi Sugiyama in the mid-1990s, turning the numerical predictions into the transparent oscillator used throughout this article.

History

1968

— Silk shows that photon diffusion damps small-scale fluctuations before recombination

1970

— Peebles and Yu, and independently Sunyaev and Zel'dovich, predict acoustic oscillations and CMB peaks

1995

— Hu and Sugiyama give the analytic driven-oscillator solution with baryon loading

2000

— BOOMERANG and MAXIMA resolve the first acoustic peak near ℓ ≈ 200

2005

— SDSS and 2dFGRS detect the baryon acoustic oscillation bump at ∼150 Mpc

Going further

Texts. Dodelson, Modern Cosmology, 2nd ed., ch. 5–9 (Boltzmann hierarchy, tight coupling, acoustic oscillations) · Ryden, Introduction to Cosmology, 2nd ed., ch. 8 (the CMB and acoustic peaks) · Mo, van den Bosch & White, Galaxy Formation and Evolution, ch. 4–6 (perturbation growth and the CMB) · Carroll & Ostlie, An Introduction to Modern Astrophysics, 2nd ed., ch. 30 (the cosmic microwave background) · Hu & Sugiyama (1995, 1996) and Hu & Dodelson (2002) — analytic acoustic oscillation theory · Peebles & Yu (1970) and Sunyaev & Zel'dovich (1970) — the original predictions. The anchor values used throughout are Planck-2018 ΛCDM numbers: r_s ≈ 147 Mpc, d_A(z_∗) ≈ 13.9 Gpc, z_∗ ≈ 1090, Ω_b h² ≈ 0.0224, and Ω_γ h² ≈ 2.47 × 10⁻⁵. The derived quantities R_∗ ≈ 0.62, c_s ≈ 0.45c, ℓ_A ≈ 297, and ℓ₁ ≈ 220 follow directly from those inputs.

Prerequisites. Density Perturbations and the Density Contrast · Jeans Instability and Cosmological Perturbation Regimes · Linear Growth of Structure · The Primordial Power Spectrum. Leads to. The CMB Anisotropy Power Spectrum · Baryon Acoustic Oscillations · The Sound Horizon and the Standard Ruler · Silk Damping · Transfer Function and the Matter Power Spectrum.

Ask the tutor: ask me to walk through how a single Fourier mode turns from a driven oscillator into a peak on the CMB sky, or how the same 147 Mpc sound horizon becomes a standard ruler in both the microwave background and galaxy surveys.

Why does the fluid oscillate instead of just collapsing into the potential wells?

Because the photon-baryon fluid has pressure, and lots of it. When gravity compresses the fluid in a dark-matter well, the photon pressure rises steeply — the equation of state is nearly p = ρc²/3 — and pushes back, so the fluid overshoots, rarefies, and is pulled back in, exactly like a mass on a spring. Collapse only wins when pressure support fails: on scales larger than the sound horizon (where there has not been time to communicate the pressure) or after recombination (when the photons leave and the pressure vanishes). Between those limits, on the scales that make the acoustic peaks, the mode rings rather than grows.

What exactly is the baryon loading R, and why does it matter so much?

R = 3ρ_b/4ρ_γ is the ratio of baryon momentum density to photon momentum density in the fluid — the inertia the baryons add to a fluid whose restoring pressure comes only from photons, like loading extra mass onto a spring without stiffening it. Two consequences follow. First, the sound speed drops, c_s = c/√(3(1+R)), so a more baryon-loaded fluid rings more slowly and its sound horizon shrinks. Second, the extra inertia lets the fluid fall deeper into each well before pressure halts it, shifting the oscillation's zero-point and enhancing the compression peaks over the rarefaction peaks. That odd-even asymmetry is how the CMB weighs the ordinary matter in the Universe.

Why are the CMB peaks spaced by π in phase rather than 2π?

Because the observed power is proportional to the square of the temperature perturbation, and cos²(k r_s) reaches its maxima at both the maxima and the minima of cos(k r_s). A compression (cos = +1) and a rarefaction (cos = −1) are physically opposite — hot versus cold spots — but both are extrema of the temperature contrast, so both show up as peaks in power. The extrema of cos occur at k r_s = nπ for every integer n, so the peaks land at k_n = nπ/r_s and, projected onto the sky, at ℓ_n ≈ nπ d_A/r_s. If we could measure the sign of the fluctuation rather than its square, the odd and even peaks would be distinguishable as compressions and rarefactions; polarization partly does this.

Is the sound horizon the same as the particle horizon?

No, and confusing them is a common error. The particle horizon is the maximum comoving distance light could have travelled since the Big Bang, ∫c dη. The sound horizon uses the fluid's sound speed instead, r_s = ∫c_s dη, with c_s between 0.45c and 0.58c before recombination. Because the fluid moves at roughly half the speed of light, r_s ≈ 147 Mpc is about half the causal horizon near 288 Mpc comoving. The acoustic features track r_s, the distance a pressure wave could travel, not the distance light could travel — getting this factor of ∼√3 wrong misplaces the standard ruler badly.

What causes the Doppler contribution, and where does it show up?

The second term of equation 7 is the Doppler shift from the bulk velocity of the baryons at last scattering, n̂·v⃗_b. In the oscillator picture the velocity is 90° out of phase with the density: when Θ₀ ∝ cos(k r_s) is at an extremum the fluid is momentarily at rest, and when Θ₀ passes through zero the velocity ∝ sin(k r_s) is maximal. So the Doppler term peaks in the troughs between the compression peaks, partially filling them in. It shows up most cleanly in CMB polarization: the E-mode pattern is sourced by the velocity quadrupole, so the polarization peaks are offset a quarter cycle from the temperature peaks — a decisive confirmation of the acoustic picture.

Why does the peak series end instead of continuing to arbitrarily small scales?

Because the fluid is only perfect on scales much larger than the photon mean free path, and recombination is not instantaneous. On small scales, photons random-walk out of a hot spot into a neighbouring cold spot during the finite time the plasma takes to recombine, averaging the two together. This diffusion — Silk damping — suppresses modes with wavenumber k above a scale k_D, with an envelope exp[−(k/k_D)²]. The damping length is the geometric mean of the horizon and the mean free path, and grows as recombination proceeds. So the acoustic peaks fade smoothly into a damping tail beyond ℓ ≈ 1000–1500 rather than continuing forever, and the shape of that tail constrains the expansion rate and particle content near recombination.