The CMB Power Spectrum: Sound Waves Frozen at Recombination

A baby picture of the universe

About 380,000 years after the Big Bang, the universe cooled enough for electrons and protons to combine into neutral hydrogen — an event called recombination. Before that moment, photons scattered constantly off free electrons and the universe was an opaque plasma fog; after it, photons streamed freely, and they are still streaming today as the Cosmic Microwave Background (CMB), a near-perfect blackbody glow at 2.725 K that fills the entire sky. Because it is a snapshot of the universe at one specific, very early moment, tiny temperature variations across the CMB — only about 1 part in 100,000 — encode the density fluctuations that would go on to seed every galaxy, cluster, and void we see today.

Sound waves in the early universe

Before recombination, ordinary (baryonic) matter and photons were tightly coupled into a single fluid, and that fluid supported real sound waves: gravity pulled matter into overdense regions, and photon pressure pushed back, setting up standing acoustic oscillations. Each initial density perturbation from inflation rang like a bell at a frequency set by its size, and the pattern of who was at maximum compression or maximum rarefaction at the moment of recombination — when the sound waves were suddenly frozen in place as photons decoupled — is imprinted directly onto the temperature map of the sky.

Reading the power spectrum

Rather than look at the raw temperature map, cosmologists decompose the tiny temperature fluctuations into spherical harmonics (the sky's version of a Fourier transform) and plot the variance at each angular scale — the CMB power spectrum, temperature variance versus multipole moment ℓ, where larger ℓ means smaller angular scale on the sky.

ΔT(θ,φ) = Σ a_lm Y_lm(θ,φ)      spherical-harmonic decomposition
C_l = ⟨|a_lm|²⟩                  power at multipole l  (≈ angular scale 180°/l)

acoustic peaks appear at l ≈ 220, 540, 800, ...  (roughly harmonic series)

The result is a series of bumps — acoustic peaks — and their positions and heights are a goldmine. The first peak's location (around ℓ ≈ 220) fixes the overall geometry of space: a flat universe puts it exactly there, while a closed or open universe would shift it, which is how the CMB provided the first strong evidence that space is geometrically flat to high precision.

What the peak heights reveal

The relative heights of successive peaks separate out the different kinds of matter. Baryons (ordinary matter) add inertia to the photon-baryon fluid, which enhances compression peaks (odd-numbered) relative to rarefaction peaks (even-numbered) — so the ratio between the first and second peak heights pins down the baryon density. Dark matter doesn't interact with photons at all, but it does sit in the gravitational potential wells driving the oscillation, and more dark matter suppresses the peaks at higher ℓ relative to the first — so the overall peak amplitudes and how quickly they damp constrain the total dark matter density. And the exact multipole spacing between peaks depends on the angular size of the sound horizon, which depends on the expansion history of the universe — giving a handle on the Hubble constant independent of local distance-ladder measurements.

This is exactly what the Planck satellite, and WMAP before it, spent years measuring to sub-percent precision, and it's why turning the dials on baryon density, dark matter density and the Hubble constant in a CMB simulation visibly moves the peaks: each parameter has a distinct, separable fingerprint on the same curve, which is what makes the CMB one of the most information-dense datasets in all of cosmology.

Frequently asked questions

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