CMB Anisotropies and Cosmological Parameters

The dipole removed, the cosmic microwave background is uniform to about one part in $10^5$. The residual fluctuations are not noise; they are the density perturbations present at last scattering, imaged directly across the sky. Their statistical pattern — a series of acoustic peaks in the angular power spectrum — is set by the physics of sound waves in the photon-baryon plasma, and the peak positions and heights measure the geometry, the baryon density, and the dark-matter density with percent precision. This lesson decomposes the anisotropy field into its power spectrum, derives the acoustic oscillations, reads off what each peak constrains, and states the Planck concordance parameters.

The anisotropy field and the power spectrum

The temperature measured in a direction $\hat{n}$ on the sky differs from the mean by a small fractional amount,

Because the field lives on a sphere, its natural decomposition is in spherical harmonics,

with the monopole ($\ell =0$, the mean) and dipole ($\ell =1$, the kinematic term) removed. The multipole $\ell$ corresponds to an angular scale $\theta \approx 180^{\circ }/\ell$: low $\ell$ is large angles, high $\ell$ small angles.

For a statistically isotropic, Gaussian field, all the information is in the variance of the coefficients, the angular power spectrum

independent of $m$. It is conventionally plotted as ${\mathcal{D}}_{\ell }=\ell (\ell +1)C_{\ell }/2\pi$, which is flat for a scale-invariant spectrum. Because there are only $2\ell +1$ independent modes at each $\ell$, the power spectrum at low $\ell$ is measured with irreducible uncertainty, cosmic variance,

which dominates the error budget on the largest scales, where we have only one sky to observe.

Acoustic oscillations of the photon-baryon fluid

Before recombination, photons and baryons are locked into a single fluid by Thomson scattering. Dark matter, which does not scatter, has already begun to form potential wells. Baryons fall into the dark-matter wells; the photon pressure resists compression and pushes back. The competition drives acoustic oscillations: the photon-baryon fluid rings like sound waves in the primordial potential wells.

For a Fourier mode of comoving wavenumber $k$, the density perturbation ${\delta }_{\gamma }$ obeys, in the tight-coupling limit, a driven oscillator equation

where the sound speed of the photon-baryon fluid is

and $R$ measures the baryon loading. A mode oscillates as $\cos (k{c}_{s}t)$ until recombination freezes it. The key scale is the sound horizon at last scattering, the comoving distance a sound wave travels before decoupling,

Modes caught at an extremum of their oscillation at recombination — maximum compression or maximum rarefaction — have the largest temperature contrast. These occur at wavenumbers

producing a harmonic series of peaks in the power spectrum. The first peak is the mode that had just reached maximum compression at last scattering; higher peaks are successive compressions and rarefactions.

Reading the peaks

Each feature of the power spectrum constrains a cosmological quantity. The logic is that the sound horizon $r_s$ is a known physical length (a standard ruler), and its angular size on the sky, ${\theta }_s=r_s/d_A$, depends on the angular-diameter distance $d_A$ to last scattering, which depends on the geometry and expansion history.

• First-peak position → spatial curvature. The angular scale of the first peak, ${\ell }_1\approx 220$, corresponds to the sound horizon seen at last scattering. In a flat universe the geodesics are straight and ${\theta }_s\approx 0.6^{\circ }$, placing the peak at $\ell \approx 220$. Positive curvature (closed) would magnify the scale and shift the peak to lower $\ell$; negative curvature (open) would shift it higher. The observed ${\ell }_1\approx 220$ fixes the universe as spatially flat to about half a percent, ${\Omega }_{\text{tot}}=1.00\pm 0.005$.
• Odd/even peak ratio → baryon density. Baryon loading $R$ breaks the symmetry between compression and rarefaction: gravity plus baryon inertia deepens the compressions (odd peaks: first, third) relative to the rarefactions (even peaks: second). A larger ${\Omega }_b$ enhances the odd peaks over the even. The observed first-to-second peak ratio fixes ${\Omega }_b{h}^2\approx 0.0224$, in agreement with the value from nucleosynthesis.
• Third peak → dark-matter density. The heights of the higher peaks depend on how much the potential wells decayed while a mode oscillated, which is governed by the dark-matter density (through the redshift of matter-radiation equality). A larger ${\Omega }_c$ (cold dark matter) raises the third peak relative to the first. The measured third peak requires substantial non-baryonic matter, ${\Omega }_c{h}^2\approx 0.120$, several times the baryon density.
• Damping tail → the diffusion scale. Above $\ell \sim 1000$ the peaks are progressively suppressed by Silk damping: recombination is not instantaneous, and photons diffuse out of small-scale perturbations during it, erasing power. The damping scale further constrains the baryon density and $N_{\text{eff}}$.

On the largest scales, below the first peak, the plateau is the Sachs-Wolfe regime: photons climbing out of potential wells at last scattering are gravitationally redshifted, and $\Delta T/T=\frac{1}{3}\Delta \Phi /c^2$ maps the primordial potential directly. This plateau, nearly flat in ${\mathcal{D}}_{\ell }$, reflects the near-scale-invariant primordial spectrum from inflation. Along the line of sight the potentials also evolve as dark energy begins to dominate, and a decaying potential adds an integrated Sachs-Wolfe contribution at the lowest multipoles, a small late-time boost that correlates the largest-scale CMB anisotropies with the nearby matter distribution.

Polarization

Thomson scattering of an anisotropic radiation field produces linear polarization, so the CMB is polarized at the few-percent level. The polarization pattern decomposes into two geometrically distinct parts:

• E-modes: a curl-free pattern, aligned with or perpendicular to the density gradients. E-modes are generated by the velocity field of the photon-baryon fluid at last scattering (scalar density perturbations) and correlate with the temperature peaks, providing an independent confirmation of the acoustic physics. They have been measured precisely.
• B-modes: a curl pattern that scalar density perturbations cannot produce at linear order. Primordial B-modes would be the signature of gravitational waves from inflation; a secondary B-mode signal is generated by gravitational lensing of E-modes by intervening structure. The lensing B-modes are detected; a primordial component has not been, setting limits on the tensor-to-scalar ratio.

The temperature-polarization cross-spectrum and the E-mode spectrum add independent constraints that tighten the parameter determination and break degeneracies present in temperature alone.

The concordance parameters

Fitting the temperature and polarization power spectra with a six-parameter $\Lambda$CDM model yields the concordance cosmology. The Planck 2018 values are:

Four points stand out. First, the total density ${\Omega }_{\text{tot}}=1.000\pm 0.005$: the universe is spatially flat. Second, the matter budget is dominated by non-baryonic dark matter, ${\Omega }_c\approx 5{\Omega }_b$, confirming the nucleosynthesis argument from a completely different measurement. Third, the spectral index $n_s=0.965$ is close to but significantly below unity, the mild tilt from scale invariance predicted by inflation. Fourth, the CMB value of the Hubble constant, $H_0=67.4$, disagrees at the several-sigma level with the local distance-ladder value near $73{\text{km s}}^{-1}{\text{Mpc}}^{-1}$ — the Hubble tension — one of the open problems of the concordance model.¹

The acoustic peaks measure the same baryon density as Big Bang nucleosynthesis and the same flat geometry that inflation predicts; the near-scale-invariant plateau is the fingerprint of the primordial spectrum whose origin the inflation lesson derives.