The CMB Anisotropy Power Spectrum

The cosmic microwave background is the most perfect blackbody ever measured — $T_0 = 2.7255\,\mathrm{K}$ to five decimal places across the whole sky — yet its faint texture, temperature ripples of only $\Delta T/T \sim 10^{-5}$ , carries the fossil record of everything. Those ripples are the imprint of density perturbations frozen at recombination ( $z_\ast \approx 1090$ ), the instant photons stopped scattering off free electrons and streamed toward us in straight lines. Because the same perturbations later grew into galaxies and clusters, the CMB hands us the linear-theory initial conditions of cosmic structure, uncontaminated by the messy nonlinear astrophysics that follows. The natural way to describe a random field painted on a sphere is its angular power spectrum. Expand $\Delta T/T$ in spherical harmonics $Y_{\ell m}$ ; the variance of the coefficients, $C_\ell = \langle |a_{\ell m}|^2\rangle$ , is — for a statistically isotropic Gaussian field — the complete statistical description. Multipole $\ell$ reads as angular scale $\theta \approx 180^\circ/\ell$ : low $\ell$ is the whole-sky Sachs–Wolfe plateau, while $\ell \gtrsim 100$ resolves the sub-degree acoustic structure. The signature of that structure is a harmonic sequence of peaks. Before recombination the photon–baryon fluid oscillated — gravity pulling matter into dark-matter wells, photon pressure pushing back — and every Fourier mode rang like an organ pipe. When the coupling switched off, each mode froze mid-phase, and those caught at maximum compression or rarefaction contribute the most variance. The first peak sits at $\ell_1 \approx 220$ ; the rest follow at a spacing set by the sound horizon at last scattering. Because those positions depend on the sound horizon projected through the distance to last scattering, the spectrum is exquisitely sensitive to geometry and contents: peak locations fix curvature, amplitudes weigh baryons and dark matter, the tail measures diffusion. Fit the whole $C_\ell$ curve — COBE, WMAP, Planck — and out drops the six-parameter $\Lambda$ CDM model at percent precision.

Key takeaways

The CMB anisotropy power spectrum $C_\ell$ is the variance of temperature fluctuations $\Delta T/T$ decomposed into spherical harmonics of multipole $\ell$ , with $\ell \approx 180^\circ/\theta$ ; for a Gaussian sky it holds all the cosmological information.
A harmonic series of acoustic peaks arises from sound waves in the primordial photon–baryon fluid, frozen at recombination; the first peak lies at $\ell_1 \approx 220$ , an angular scale near $1^\circ$ .
Peak positions fix the geometry: the sound-horizon ruler $r_s$ projected through the distance to last scattering sets the acoustic scale $\ell_A \approx 300$ , and the observed peaks pin the universe flat, $\Omega_k \approx 0$ .
Peak heights weigh matter: baryon loading enhances odd over even peaks (fixing $\Omega_b h^2 = 0.0224$ ), and the peak envelope fixes the total matter $\Omega_m h^2 = 0.143$ — most of it non-baryonic.
Silk (diffusion) damping cuts the spectrum off exponentially above $\ell \sim 1000$ , and cosmic variance $\sqrt{2/(2\ell+1)}$ sets an irreducible noise floor at low $\ell$ .
Key idea: Sound waves in the photon–baryon fluid, frozen at recombination, imprint a harmonic series of peaks in the angular power spectrum; the peak positions encode geometry (curvature) while their relative heights weigh the baryonic and dark-matter content.

Where this lives today

The acoustic-peak spectrum is the single most constraining measurement in cosmology, and the field is far from done reading it. Planck's 2018 release measured the temperature spectrum out to $\ell\approx 2500$ and the E-mode polarization alongside it, delivering the six $\Lambda$ CDM parameters — $\Omega_b h^2 = 0.0224$ , $\Omega_c h^2 = 0.120$ , $100\theta_\ast = 1.0411$ , $\tau = 0.054$ , $n_s = 0.965$ , $\ln(10^{10}A_s) = 3.04$ — to sub-percent precision, and with them $H_0 = 67.4\ \mathrm{km\,s^{-1}\,Mpc^{-1}}$ and $\Omega_k = 0.001\pm0.002$ . Ground-based instruments now push past Planck at small scales: the Atacama Cosmology Telescope and the South Pole Telescope map the damping tail beyond $\ell\sim 3000$ and, through gravitational lensing of the peaks, weigh the intervening matter and neutrino mass. The next generation — the Simons Observatory and CMB-S4 — targets the primordial B-mode polarization that inflationary gravitational waves would imprint, with limits already at $r < 0.036$ from BICEP/Keck. And the peaks sit at the centre of cosmology's sharpest open problem: the Hubble tension, where the $H_0 = 67.4$ inferred from the CMB acoustic scale disagrees at $\sim 5\sigma$ with the $\sim 73$ measured locally from Cepheids and supernovae. Whether this crack signals new physics beyond $\Lambda$ CDM — early dark energy, extra relativistic species — or unrecognized systematics is the question the spectrum is now being interrogated to answer.

The physical picture

Picture the primordial plasma as a single fluid with a stiff restoring pressure — supplied by photons — sitting inside gravitational wells supplied by dark matter. Gravity tries to compress the fluid into each well; the photon pressure pushes back out. The result is an oscillation, and because the fluid is continuous, every Fourier mode of the density field rings independently at the sound speed $c_s = c/\sqrt{3(1+R)}$ . A short-wavelength mode oscillates fast and has time to go through many compressions and rarefactions before recombination; a long-wavelength mode oscillates slowly. At the moment of recombination the photons decouple and the pressure vanishes — the ringing stops everywhere at once, freezing each mode at whatever phase it happened to occupy. Modes frozen at maximum compression or maximum rarefaction have the largest temperature contrast, and it is these that pile up into the peaks.

Figure

Gravity pulls the photon–baryon fluid into a dark-matter well while photon pressure pushes back, so each mode rings at the sound speed c_s = c/√(3(1+R)) ≈ 0.45c until recombination freezes it. Sound has travelled one comoving sound horizon r_s ≈ 147 Mpc — the standard ruler stamped on the sky

The sky is a two-dimensional slice at one enormous fixed distance, so a three-dimensional sound wave of comoving wavelength $\lambda$ subtends a definite angle $\theta \sim \lambda/d_A$ . Every mode sharing the fundamental acoustic wavelength lands at the same angular scale — so instead of a smear, $C_\ell$ shows a sharp peak. This is exactly why the CMB is a precision instrument: one physical length, the sound horizon, maps to one angle on the sky. Odd peaks (1st, 3rd, …) are compressions and even peaks (2nd, 4th, …) rarefactions; adding baryons weighs the fluid down, deepening compressions relative to rarefactions and so enhancing the odd peaks — the trick by which the CMB weighs ordinary matter without ever detecting a single baryon. Below the diffusion scale, photons random-walk out of overdensities during the finite duration of recombination, washing fluctuations out and imposing the exponential Silk cutoff at high $\ell$ . The second figure is the object all of this produces — the measured temperature power spectrum itself.

Figure

The measured CMB temperature power spectrum D_ℓ = ℓ(ℓ+1)C_ℓ/2π in μK²: a flat Sachs–Wolfe plateau near 1000 μK² at low ℓ, a first acoustic peak at ℓ ≈ 220 reaching ≈ 5750 μK², further peaks at ℓ ≈ 537 and ℓ ≈ 813 spaced by the acoustic scale ℓ_A ≈ 300, and a Silk-damping tail falling away beyond ℓ ≈ 1000

The core equations

The starting point is representation: any temperature map on the sphere is a sum of spherical harmonics, the sphere's analogue of a Fourier series. The coefficients $a_{\ell m}$ are what an experiment measures; everything cosmological is downstream of them.

\frac{\Delta T}{T}(\hat{n}) = \sum_{\ell=0}^{\infty}\sum_{m=-\ell}^{\ell} a_{\ell m}\, Y_{\ell m}(\hat{n}) \qquad (1)

where: $\Delta T/T(\hat{n})$ — fractional temperature fluctuation in sky direction $\hat{n}$ ; $a_{\ell m}$ — multipole coefficients; $Y_{\ell m}$ — spherical harmonics on the sphere; $\ell,\,m$ — multipole degree and order. Here $\ell$ counts angular wiggles ( $\theta \approx 180^\circ/\ell$ ), and each $\ell$ carries $2\ell+1$ orientations $m$ . The physics does not care about orientation, only about scale. Averaging the squared coefficients over $m$ collapses the map into a single curve — the angular power spectrum — which is the only statistic a Gaussian, isotropic sky can offer.

C_\ell \equiv \langle |a_{\ell m}|^2 \rangle = \frac{1}{2\ell+1}\sum_{m=-\ell}^{\ell} |a_{\ell m}|^2 \qquad (2)

where: $C_\ell$ — ensemble variance of $a_{\ell m}$ at multipole $\ell$ ; $2\ell+1$ — number of independent $m$ modes averaged over. Reading it: statistical isotropy forces $\langle a_{\ell m} a^\ast_{\ell'm'}\rangle = C_\ell\,\delta_{\ell\ell'}\delta_{mm'}$ , so no direction is special and $C_\ell$ alone survives. Because $C_\ell$ falls steeply with $\ell$ , cosmologists almost never plot it raw. The convention is to plot the power per logarithmic interval, which flattens the Sachs–Wolfe plateau into a horizontal line and makes the peaks jump out.

\mathcal{D}_\ell \equiv \frac{\ell(\ell+1)}{2\pi}\,C_\ell \qquad (3)

where: $\mathcal{D}_\ell$ — power per logarithmic $\ell$ interval, quoted in $\mu\mathrm{K}^2$ . Reading it: a scale-invariant spectrum gives $\mathcal{D}_\ell \approx$ const at low $\ell$ ; the first peak reaches $\mathcal{D}_{220} \approx 5750\,\mu\mathrm{K}^2$ . To turn a multipole into a scale on the sky, use the rule of thumb that $\ell$ resolves features of size $\theta$ . It is not the inverse of the angle in radians but $\pi$ over it.

\theta \approx \frac{180^\circ}{\ell}, \qquad \ell \approx \frac{\pi}{\theta_{\rm rad}} \qquad (4)

where: $\theta$ — angular scale probed by multipole $\ell$ ; e.g. $\ell = 220$ corresponds to $\theta \approx 0.8^\circ$ , roughly two full-moon widths. The peaks are not arbitrary. Their spacing is fixed by the sound horizon $r_s$ (a calculable standard ruler) projected through the comoving distance $d_A$ to last scattering; the ratio defines the acoustic scale.

\ell_A \approx \pi\,\frac{d_A}{r_s}, \qquad \ell_n \approx \ell_A\,(n-\varphi) \qquad (5)

where: $\ell_A$ — acoustic scale ( $\approx 300$ ), equal to the peak spacing; $d_A$ — comoving angular-diameter distance to last scattering ( $\approx 13.9\,\mathrm{Gpc}$ ); $r_s$ — comoving sound horizon at decoupling ( $\approx 147\,\mathrm{Mpc}$ ); $\varphi \approx 0.27$ — a phase shift from acoustic driving that pulls the first peak down to $\ell_1 \approx 220$ . On the largest scales, above the acoustic horizon, the temperature is set not by sound but by gravity alone: photons climb out of potential wells and are redshifted, the Sachs–Wolfe effect.

\left[\frac{\Delta T}{T}\right]_{\rm SW} = \frac{1}{3}\,\Phi \qquad (6)

where: $\Phi$ — Newtonian gravitational potential at last scattering; the factor $\tfrac13$ is the net of gravitational redshift and cosmological time dilation. Reading it: a scale-invariant $\Phi$ gives the flat $\ell(\ell+1)C_\ell$ plateau seen at $\ell \lesssim 30$ . Formally, the whole spectrum is a projection of the primordial power spectrum through a radiation transfer function that carries all the acoustic physics.

C_\ell = 4\pi \int_0^\infty \frac{\mathrm{d}k}{k}\, \mathcal{P}_\mathcal{R}(k)\, |\Delta_\ell(k)|^2 \qquad (7)

where: $\mathcal{P}_\mathcal{R}(k)$ — primordial curvature power spectrum; $\Delta_\ell(k)$ — transfer function projecting mode $k$ onto multipole $\ell$ . Reading it: since $\Delta_\ell(k)\propto j_\ell(k d_A)$ peaks at $k d_A \approx \ell$ , wavenumber maps to multipole and the oscillations in $\Delta_\ell$ become the peaks. Two effects bound the usable range. At high $\ell$ , Silk damping suppresses power exponentially; at low $\ell$ , cosmic variance sets an irreducible statistical floor.

C_\ell \to C_\ell\, e^{-2(\ell/\ell_D)^2}\ \ (\ell_D \approx 1400), \qquad \frac{\Delta C_\ell}{C_\ell} = \sqrt{\frac{2}{(2\ell+1)f_{\rm sky}}} \qquad (8)

where: $\ell_D$ — damping multipole set by the photon diffusion length; $f_{\rm sky}$ — fraction of sky observed. Reading it: the damping envelope kills the tail beyond $\ell \sim 1000$ , while cosmic variance dominates the error budget below $\ell \sim 1000$ .

Derivation ACosmic variance from finite mode counting

The goal is to show that even a flawless, noise-free, full-sky experiment cannot measure $C_\ell$ to arbitrary precision, and to compute how large that irreducible uncertainty is. The obstruction is statistical, not instrumental: the theory predicts an ensemble variance $C_\ell$ , but we possess only one universe and so only $2\ell+1$ independent samples of that distribution at each multipole. We build the natural estimator, confirm it is unbiased, and compute its variance under the Gaussian assumption, arriving at the floor $\Delta C_\ell/C_\ell = \sqrt{2/(2\ell+1)}$ that dominates the low- $\ell$ error budget of every CMB experiment ever flown.

Step 1. Form the estimator by averaging the squared coefficients over the available orientations, $\hat{C}_\ell = \frac{1}{2\ell+1}\sum_{m=-\ell}^{\ell}|a_{\ell m}|^2$ . This is the only rotationally invariant quadratic we can build from the $a_{\ell m}$ at fixed $\ell$ . Why: with no preferred direction on the sky, the estimator must weight every $m$ equally, and squaring is what turns a zero-mean coefficient into a positive measure of power.

Step 2. Check that it is unbiased. Since statistical isotropy gives $\langle|a_{\ell m}|^2\rangle = C_\ell$ for every $m$ , the average of $2\ell+1$ such terms returns $\langle\hat{C}_\ell\rangle = C_\ell$ exactly. Why: an unbiased estimator centres on the truth, so any remaining spread is genuine sampling noise rather than a systematic offset we could calibrate away.

Step 3. Compute the variance of a single term. Each $|a_{\ell m}|^2$ is the square of a Gaussian, so it follows a $\chi^2$ distribution and carries $\mathrm{Var}(|a_{\ell m}|^2) = 2C_\ell^2$ . Why: Gaussianity makes every higher moment collapse to products of $C_\ell$ (Wick's theorem), so the entire error propagation is fixed by the two-point function alone.

Step 4. Add the independent modes and normalise. The $2\ell+1$ orientations are statistically independent, so their variances add and the pre-factor $1/(2\ell+1)^2$ cancels one power of the count,

\mathrm{Var}(\hat{C}_\ell) = \frac{1}{(2\ell+1)^2}\sum_{m} 2C_\ell^2 = \frac{2C_\ell^2}{2\ell+1}\quad\Rightarrow\quad \frac{\Delta C_\ell}{C_\ell} = \sqrt{\frac{2}{2\ell+1}}.

At $\ell=2$ this is $\sqrt{2/5}\approx 63\%$ ; by $\ell=1000$ it has fallen to $\approx 3\%$ . Why: this is why the Sachs–Wolfe plateau scatters wildly in every map — there are simply too few large-scale modes in the observable universe to pin it down, and no telescope can fix that.

Go deeper · Derivation B · 2 minThe acoustic scale as a standard ruler and curvature probe

Derivation BThe acoustic scale as a standard ruler and curvature probe

The goal is to turn the qualitative statement "peaks come from the sound horizon" into a quantitative, geometry-sensitive prediction, and to see why the first peak's position measures spatial curvature. The sound horizon $r_s$ is a physical length calculable from the pre-recombination expansion history; the distance $d_A$ to last scattering depends on the geometry light has crossed since. Because curvature bends light paths, it changes $d_A$ and hence the angle a fixed ruler subtends — so the measured peak angle reads off $\Omega_k$ directly. We assemble the ruler, project it to an angle, expose the curvature dependence, and read off the flat-universe value.

Step 1. Build the ruler. The comoving sound horizon is the distance a sound wave travels before decoupling,

r_s = \int_0^{t_\ast}\frac{c_s}{a}\,\mathrm{d}t = \int_{z_\ast}^{\infty}\frac{c_s\,\mathrm{d}z}{H(z)} \approx 147\ \mathrm{Mpc},

with $c_s = c/\sqrt{3(1+R)}$ set by the baryon loading $R$ . Why: this integral depends only on the well-measured densities $\Omega_b h^2$ and $\Omega_m h^2$ , making $r_s$ a calibrated standard length rather than a free parameter.

Step 2. Project the ruler to an angle. A comoving length $r_s$ at comoving distance $d_A$ subtends $\theta_\ast = r_s/d_A$ , and the corresponding acoustic multipole is

\theta_\ast = \frac{r_s}{d_A}, \qquad \ell_A = \frac{\pi}{\theta_\ast} = \pi\,\frac{d_A}{r_s}.

Why: this is the ruler-to-angle step that converts a length we can compute into an angle we can observe — the heart of the CMB as a geometric probe.

Step 3. Expose the curvature dependence. The comoving distance to last scattering runs through the curved geometry,

d_A = \frac{c}{H_0\sqrt{|\Omega_k|}}\,S_k\!\left(\sqrt{|\Omega_k|}\int_0^{z_\ast}\frac{\mathrm{d}z}{E(z)}\right), \qquad E(z)=\frac{H(z)}{H_0},

with $S_k = \sin,\ \mathrm{id},\ \sinh$ for closed, flat, open. In an open universe ( $\Omega_0<1$ ) geodesics diverge, $d_A$ grows, $\theta_\ast$ shrinks and the peaks slide to higher $\ell$ ; a closed universe does the reverse. Why: this single dependence is what makes the peak position a curvature meter — the same ruler looks smaller in an open universe and larger in a closed one.

Step 4. Read off the flat value, remembering the driving phase shift. With $r_s\approx 147\,\mathrm{Mpc}$ and $d_A\approx 13.9\,\mathrm{Gpc}$ one gets $\theta_\ast\approx 0.6^\circ$ and $\ell_A\approx 300$ — the observed peak spacing. The first peak itself sits at

\ell_1 \approx \ell_A\,(1-\varphi) \approx 0.74\,\ell_A \approx 220 \quad\Longleftrightarrow\quad \Omega_k \approx 0.

Why: the observed $\ell_1 = 220$ (not the $\sim 400$ an open low-density universe would give) is the clean, model-independent statement that space is flat — one of the pillars of $\Lambda$ CDM.

Go deeper · Derivation C · 2 minBaryon loading and the odd-even peak asymmetry

Derivation CBaryon loading and the odd-even peak asymmetry

The goal is to explain why the CMB weighs ordinary baryonic matter through the relative heights of the acoustic peaks, and to derive the sign of the effect. The key fact is that baryons contribute mass but no pressure, so loading the photon fluid with baryons is like hanging a weight on a spring: it lowers the sound speed and displaces the equilibrium about which the fluid oscillates. That displacement breaks the symmetry between the compression and rarefaction halves of each oscillation, enhancing the odd (compression) peaks over the even (rarefaction) peaks by an amount that grows with $\Omega_b h^2$ . We model the oscillation as a driven harmonic oscillator with a baryon-shifted zero-point and extract the observable ratio.

Step 1. Write the oscillator equation for the temperature perturbation $\Theta = \Delta T/T$ of a mode $k$ , driven by gravity,

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

Why: this is the acoustic-oscillation article's result recalled here — a forced oscillator whose restoring term is photon pressure and whose forcing is the gravitational potential.

Step 2. Introduce the baryon loading $R\equiv 3\rho_b/4\rho_\gamma$ , the ratio of baryon to photon momentum density,

R = \frac{3\rho_b}{4\rho_\gamma} \approx 0.6\left(\frac{\Omega_b h^2}{0.022}\right)\left(\frac{1090}{1+z}\right).

More baryons mean larger $R$ , a slower sound speed, and — the important part — more inertia. Why: $R$ is the single number through which the baryon density enters the acoustic physics, so any baryon signature in the spectrum must be a function of $R$ .

Step 3. Complete the square: the combination $\Theta + (1+R)\Phi$ oscillates about zero, so $\Theta$ itself oscillates about the shifted equilibrium $-(1+R)\Phi$ . The oscillation amplitude is symmetric about that shifted centre, but the observed temperature is measured from zero, so compressions (into the well, deepened by the shift) reach $\propto (1+3R)$ times the rarefaction amplitude. Why: this asymmetric zero-point is the entire origin of baryon loading — gravity plus baryon inertia make the fluid fall harder than it rebounds.

Step 4. Translate to the spectrum. Odd peaks are compressions and even peaks rarefactions, so the enhanced compression amplitude raises the 1st and 3rd peaks relative to the 2nd,

\frac{\mathcal{D}_{\ell_1}}{\mathcal{D}_{\ell_2}}\ \text{increases with}\ \Omega_b h^2.

The measured first-to-second peak ratio fixes $\Omega_b h^2 = 0.0224$ . Why: this is how a map of light temperature weighs invisible ordinary matter — the peak-height ratio is a baryometer, and it agrees with the independent value from Big Bang nucleosynthesis.

Limiting cases

The spectrum is really three physical regimes stitched together, each governed by different physics and each yielding a different measurement.

Case	Result	Meaning
Large scales ( $\ell \lesssim 30$ )	$\ell(\ell+1)C_\ell \approx \text{const}$	Sachs–Wolfe plateau: photons climb out of static potential wells, $\Delta T/T = \tfrac13\Phi$ ; a scale-invariant primordial spectrum makes $\ell(\ell+1)C_\ell$ flat. Cosmic-variance limited.
Acoustic regime ( $100 \lesssim \ell \lesssim 1000$ )	$\ell_n \approx \ell_A(n-\varphi),\ \ell_A\approx 300$	Harmonic peaks from frozen sound waves; spacing set by the sound horizon $r_s$ projected through $d_A$ ; positions fix curvature, heights weigh baryons and dark matter.
Damping tail ( $\ell \gtrsim 1000$ )	$C_\ell \propto e^{-2(\ell/\ell_D)^2}$	Silk (diffusion) damping exponentially suppresses power below the photon mean-free-path scale, $\ell_D \sim 1400$ ; measures the thickness of the last-scattering surface.
Very low $\ell$ ( $\ell = 2,3$ )	$\Delta C_\ell/C_\ell \approx 50\text{–}60\%$	Only $2\ell+1$ modes exist; the quadrupole is intrinsically uncertain, so apparent "anomalies" here are largely cosmic variance.

Common trapThe peaks are peaks in temperature, and ℓ is one over the angle

Both halves of this are wrong, and both trip up newcomers. First, $\ell$ is not the inverse of the angle in radians — it is roughly $\ell \approx \pi/\theta_{\rm rad} \approx 180^\circ/\theta_{\rm deg}$ , so the $\ell=220$ peak sits near $0.8^\circ$ , not near $1/220$ of a radian. Second, and more subtly, the peaks are peaks in the variance $C_\ell \propto \langle|a_{\ell m}|^2\rangle$ , not in temperature. A mode frozen at maximum rarefaction has a negative temperature contrast, yet it contributes just as much power as one at maximum compression, because power is amplitude squared. That is why an unbroken harmonic series appears — odd peaks (compressions) and even peaks (rarefactions) both show up, and their relative heights carry the baryon information. A third trap: the naive $\ell_1 = \pi d_A/r_s$ gives $\sim 300$ , but that is the peak spacing $\ell_A$ ; the first peak is dragged down to 220 by the driving phase shift, so never quote $\pi d_A/r_s$ as the first-peak multipole.

Worked examples

1 · Cosmic variance across the spectrum (fully worked)

A perfect, noise-free experiment maps the full sky. What is the irreducible fractional error on $C_\ell$ at the quadrupole $\ell=2$ , at the first peak $\ell=220$ , and at $\ell=1000$ ? How does masking the galaxy to $f_{\rm sky}=0.7$ change the $\ell=220$ value?

Step 1. Use the full-sky floor $\Delta C_\ell/C_\ell = \sqrt{2/(2\ell+1)}$ . At the quadrupole, $\sqrt{2/5} = \mathbf{0.63}$ — a 63% uncertainty on the single most cosmological large-scale number, no matter how good the detector.

Step 2. At the first acoustic peak, $\sqrt{2/441} = \sqrt{0.00454} = \mathbf{0.067}$ , a 6.7% error per multipole. In practice one bins many neighbouring $\ell$ together, beating this down by $\sqrt{N_{\rm bins}}$ , which is why the peak amplitude is known to better than a percent.

Step 3. At $\ell=1000$ , $\sqrt{2/2001} = \mathbf{0.032}$ — only 3.2% per multipole, and thousands of modes are available, so cosmic variance is negligible here and instrumental noise and foregrounds dominate instead.

Step 4. Masking the galaxy throws away sky and independent modes: $\Delta C_\ell/C_\ell = \sqrt{2/[(2\ell+1)f_{\rm sky}]}$ . At $\ell=220$ with $f_{\rm sky}=0.7$ , the error rises to $\sqrt{2/(441\times0.7)} = \sqrt{0.00648} = \mathbf{0.081}$ , an 8.1% floor — a 20% inflation over the full-sky value, the direct price of the galactic mask. Cosmic variance dominates low ℓ (63% at ℓ = 2), is modest at the first peak (6.7%), and vanishes into the noise by ℓ = 1000 (3.2%).

2 · The acoustic scale and the flatness of space (fully worked)

Given the comoving sound horizon at decoupling $r_s \approx 147\,\mathrm{Mpc}$ and the comoving distance to last scattering $d_A \approx 13.9\,\mathrm{Gpc}$ , predict the acoustic scale, the spacing of the peaks, and the first-peak multipole. What would an open, matter-only universe with $d_A$ larger by 40% give instead?

Step 1. Sound-horizon angle: $\theta_\ast = r_s/d_A = 147/13900 = 0.0106\,\mathrm{rad} = \mathbf{0.61^\circ}$ , close to the precisely measured Planck value $100\theta_\ast = 1.0411$ (i.e. $\theta_\ast = 0.597^\circ$ ). The tiny residual is the difference between the ruler at photon decoupling ( $r_\ast\approx 144\,\mathrm{Mpc}$ ) and at baryon drag ( $r_{\rm drag}\approx 147\,\mathrm{Mpc}$ ).

Step 2. Acoustic scale: $\ell_A = \pi/\theta_\ast = 3.1416/0.0106 = \mathbf{297 \approx 300}$ . This is the spacing between successive peaks.

Step 3. First peak with the driving phase shift $\varphi\approx0.27$ : $\ell_1 \approx \ell_A(1-\varphi) = 300\times0.74 = \mathbf{220}$ , corresponding to an angular feature $\theta \approx 180^\circ/220 = 0.8^\circ$ . The predicted higher peaks $\ell_2\approx 537$ and $\ell_3\approx 813$ match the observed spacing $\Delta\ell\approx 300$ .

Step 4. Open universe: if geodesic divergence inflated $d_A$ by 40%, then $\theta_\ast$ shrinks by 40% and $\ell_A$ rises to $\approx 420$ , pushing the first peak to $\ell_1\approx 310$ . The observed $\ell_1=220$ therefore rules this out and demands near-flat geometry, $\Omega_k\approx 0$ . A 147 Mpc ruler at 13.9 Gpc subtends 0.6°, giving ℓ_A ≈ 300 and a first peak at ℓ₁ ≈ 220 — the signature of a flat universe.

3 · Weighing the baryons from the peak ratio

Sketch how the first-to-second peak height ratio pins the baryon density, using the baryon loading $R \approx 0.6$ at recombination.

Step 1. Sound speed with loading: $c_s = c/\sqrt{3(1+R)} = c/\sqrt{4.8} = \mathbf{0.46\,c}$ , slowed from the radiation-only $c/\sqrt{3}=0.58c$ . The compression enhancement scales as $(1+3R) = \mathbf{2.8}$ : gravity plus baryon inertia make the fluid fall into wells harder than it rebounds, deepening the odd peaks.

Step 2. So the first (odd, compression) peak is boosted relative to the second (even, rarefaction): a larger $\mathcal{D}_{\ell_1}/\mathcal{D}_{\ell_2}$ means a larger $\Omega_b h^2$ . Matching Planck's ratio yields $\mathbf{\Omega_b h^2 = 0.0224}$ — ordinary matter is about $5\%$ of the cosmic energy budget, agreeing with the independent value from Big Bang nucleosynthesis. The odd/even peak asymmetry is a baryometer: raising Ω_b h² lifts the odd peaks, and the measured ratio gives Ω_b h² = 0.0224.

Physics of the portrait

The acoustic-peak spectrum is a rare case where theory ran three decades ahead of data: the peaks were predicted from pen-and-paper Boltzmann physics in 1970, long before any experiment could see the anisotropies at all. The figure most associated with turning the young cosmic microwave background into quantitative cosmology is James Peebles.

Origins · 1 minJames Peebles and how this was discovered

James Peebles

1935– · physical cosmology and the acoustic peaks of the CMB

Phillip James Edwin Peebles, born in Winnipeg in 1935 and long at Princeton, did more than anyone to make cosmology a precision science rather than a branch of speculation. When Penzias and Wilson stumbled on excess microwave noise in 1965, it was Peebles — in Robert Dicke's group — who had already computed that a hot Big Bang must leave a relic radiation bath at a few kelvin, and who interpreted the signal correctly. Through the late 1960s he laid out how photons and baryons behave as a coupled fluid, and in a landmark 1970 paper with his student Jer Yu he solved the linearized Boltzmann and Einstein equations to predict a series of oscillations — acoustic peaks — in the angular spectrum, their spacing set by the sound horizon and their heights by the baryon and matter densities. The same conclusion was reached independently that year by Sunyaev and Zel'dovich in Moscow. For a generation these predictions sat almost untested: COBE saw only the large-scale plateau in 1992, and it took BOOMERANG, MAXIMA and finally WMAP and Planck to resolve the peaks Peebles had drawn. He also pioneered the statistical treatment of galaxy clustering and the cold-dark-matter paradigm. In 2019 he received the Nobel Prize in Physics "for theoretical discoveries in physical cosmology" — recognition that the $\Lambda$ CDM model read off the peaks is, in large part, the working out of a framework he built.

History

1965

— Penzias and Wilson detect the microwave background; Dicke and Peebles interpret it as relic radiation

1968

— Silk predicts photon-diffusion damping of small-scale fluctuations

1970

— Peebles and Yu, and independently Sunyaev and Zeldovich, predict acoustic peaks in the angular spectrum

1992

— COBE DMR detects the anisotropies and the Sachs-Wolfe plateau at the 10⁻⁵ level

2000

— BOOMERANG and MAXIMA locate the first acoustic peak near ℓ = 200, showing a flat universe

2013

— Planck resolves the full peak sequence and delivers six-parameter ΛCDM

Going further

Texts. Ryden, Introduction to Cosmology, 2nd ed., Ch. 8 (The Cosmic Microwave Background) · Dodelson & Schmidt, Modern Cosmology, 2nd ed., Ch. 8–9 (Anisotropies and the power spectrum) · Mo, van den Bosch & White, Galaxy Formation and Evolution, Ch. 6 · Carroll & Ostlie, An Introduction to Modern Astrophysics, 2nd ed., Ch. 30 · Hu & Dodelson, "CMB Anisotropies," Ann. Rev. Astron. Astrophys. 40, 171 (2002) · Planck 2018 Results VI: Cosmological Parameters, A&A 641, A6 (2020). The anchor numbers used throughout — $\ell_1\approx 220$ , $\ell_A\approx 300$ , $r_s\approx 147\,\mathrm{Mpc}$ ( $r_\ast\approx 144\,\mathrm{Mpc}$ at decoupling), $d_A\approx 13.9\,\mathrm{Gpc}$ , $T_0=2.7255\,\mathrm{K}$ , $z_\ast\approx 1090$ , $\Omega_b h^2=0.0224$ , $\Omega_m h^2=0.143$ , $n_s=0.965$ , $100\theta_\ast=1.0411$ , $\Omega_k=0.001\pm0.002$ , $H_0=67.4$ — are Planck 2018 (A&A 641, A6) values. The driving phase shift $\varphi\approx 0.27$ sets $\ell_1\approx 0.74\,\ell_A\approx 220$ , and the peak spacing 220→537→813 corresponds to $\Delta\ell\approx 300$ .

Prerequisites. Density Perturbations and the Density Contrast · Acoustic Oscillations in the Photon–Baryon Fluid · The Primordial Power Spectrum · The Sound Horizon and the Standard Ruler. Leads to. The ΛCDM Concordance Model · Baryon Acoustic Oscillations · Silk Damping · Cold Dark Matter.

Ask the tutor: Ask me to walk through how a single wiggle in the peak heights would move $\Omega_b h^2$ , or how the phase shift drags the first peak from $\ell_A\approx 300$ down to $\ell_1\approx 220$ .

Why is the CMB power spectrum a curve of ℓ instead of a map of the sky?

Because for a statistically isotropic Gaussian random field, the map holds far more numbers than there is physics to explain them — most of the detail is one random realization, not a law of nature. Statistical isotropy means no direction is special, so all the cosmological information lives in how much power sits at each angular scale, which is exactly what $C_\ell$ records. Compressing millions of pixels into a few thousand $C_\ell$ values loses nothing physical for a Gaussian sky, because Gaussianity guarantees (via Wick's theorem) that every higher-order statistic is already fixed by $C_\ell$ . The map is one draw; the power spectrum is the underlying distribution.

What is the difference between the acoustic scale ℓ_A and the first peak ℓ₁?

They are close but not identical, and conflating them is a common error. The acoustic scale $\ell_A = \pi d_A/r_s \approx 300$ is the pure geometric projection of the sound horizon — it equals the spacing between successive peaks. The first peak, however, sits lower, at $\ell_1\approx 220$ . The offset comes from the acoustic driving: the gravitational potentials that force the oscillations are themselves decaying as the modes enter the horizon, adding a phase shift $\varphi\approx 0.27$ so that $\ell_n\approx\ell_A(n-\varphi)$ . That is why the peaks fall at $\ell\approx 220,\,537,\,813$ — spaced by $\ell_A\approx 300$ but each shifted down from a clean harmonic. Quoting $\pi d_A/r_s\approx 300$ as the first-peak position is wrong by that phase shift.

How can the CMB weigh dark matter if dark matter emits no light?

Through gravity and inertia, both of which leave fingerprints in the peaks. Dark matter provides the potential wells the photon–baryon fluid falls into, so its density controls how deeply modes compress and hence the envelope of peak heights — the third peak relative to the second is boosted when dark matter is abundant. Baryons add inertia that deepens compressions and enhances the odd peaks. Fitting both gives the total matter $\Omega_m h^2=0.143$ and the baryonic part $\Omega_b h^2=0.0224$ separately, and the large gap between them is direct gravitational evidence that most matter is non-baryonic cold dark matter — never seen, only weighed.

What physically sets the damping tail at high ℓ?

Silk damping, the diffusion of photons out of small-scale overdensities. Recombination is not instantaneous — it takes a finite stretch of cosmic time during which photons still scatter off the dwindling free electrons. In that window a photon executes a random walk, and over the diffusion length it carries heat from hot compressions into cold rarefactions, averaging the temperature contrast toward zero. Any acoustic mode shorter than this diffusion scale is smeared out, so the spectrum is multiplied by an envelope $\propto e^{-2(\ell/\ell_D)^2}$ with $\ell_D\sim 1400$ , cutting off the peaks beyond $\ell\sim 1000$ . The tail therefore measures the photon mean free path and the physical thickness of the last-scattering surface.

Why is the low-ℓ end of the spectrum so scattered even in the best data?

Cosmic variance. At each multipole $\ell$ the sky offers only $2\ell+1$ statistically independent modes, so the power there is estimated from a tiny sample. At the quadrupole $\ell=2$ that is just five numbers, giving a fractional uncertainty $\sqrt{2/5}\approx 63\%$ — larger than the signal's own features. No improvement in detector sensitivity or integration time can beat this floor, because it is set by the finite number of large-scale patterns that fit inside the observable universe. This is why apparent "anomalies" at low $\ell$ (the low quadrupole, the hemispherical asymmetry) are viewed cautiously: with so few modes, moderately unusual draws are statistically ordinary.

Does the same sound horizon appear anywhere other than the CMB?

Yes — it reappears at low redshift as the baryon acoustic oscillation (BAO) feature in the clustering of galaxies. The same pressure waves that froze into the CMB left a slight excess of galaxy pairs separated by the sound horizon, a bump in the correlation function near $r_{\rm drag}\approx 147\,\mathrm{Mpc}$ (the ruler evaluated at the slightly later baryon-drag epoch, rather than the $r_\ast\approx 144\,\mathrm{Mpc}$ at photon decoupling relevant to the CMB). Measuring the same ruler at $z\approx 1090$ in the CMB and at $z\lesssim 1$ in galaxy surveys, and finding them consistent, is a powerful validation of the standard-ruler paradigm and one of the tightest handles on dark energy and the expansion history.