---
title: CMB Anisotropies and Cosmological Parameters
draft: false
module: The Hot Big Bang
moduleNumber: 12
lessonNumber: 4
order: 1204
summary: >
  The cosmic microwave background carries temperature fluctuations at the
  ten-parts-per-million level, imprinted by sound waves in the photon-baryon plasma
  before recombination. Decomposed into spherical harmonics, the fluctuations form
  an angular power spectrum whose acoustic peaks encode the geometry and contents of
  the universe: the first peak fixes spatial flatness, the odd-even peak ratio the
  baryon density, and the third peak the dark-matter density. Polarization adds an
  independent channel, and the Planck measurements pin the concordance parameters.
topics: [The Hot Big Bang]
sources:
  - book: Ryden
    ref: "Ch. 8 — The Cosmic Microwave Background; §8.4–8.5"
  - book: Planck 2018
    ref: "Planck 2018 results VI. Cosmological parameters, A&A 641, A6"
  - book: WMAP
    ref: "NASA WMAP mission — temperature anisotropies and the acoustic power spectrum"
---

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,

$$
\frac{\Delta T}{T}(\hat n) = \frac{T(\hat n) - T_0}{T_0}
\sim 10^{-5}.
$$

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

$$
\frac{\Delta T}{T}(\hat n) = \sum_{\ell=2}^{\infty} \sum_{m=-\ell}^{\ell}
a_{\ell m}\,Y_{\ell m}(\hat n),
$$

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**

$$
C_\ell = \langle |a_{\ell m}|^2 \rangle,
$$

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**,

$$
\frac{\Delta C_\ell}{C_\ell} = \sqrt{\frac{2}{2\ell + 1}},
$$

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

$$
\ddot\delta_\gamma + c_s^2 k^2 \delta_\gamma = F(k),
$$

where the sound speed of the photon-baryon fluid is

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

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,

$$
r_s = \int_0^{t_{\text{dec}}} c_s\,\frac{\d t}{a}
\approx 147\ \text{Mpc (comoving)}.
$$

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

$$
k_n = \frac{n\pi}{r_s}, \qquad n = 1, 2, 3, \dots,
$$

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.

$$
% caption: In a dark-matter potential well the photon-baryon fluid compresses and
% rarefies as a standing sound wave; photon pressure pushes out against gravity,
% and the mode caught at maximum compression at recombination becomes the first
% acoustic peak.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
% potential well
\draw[black, very thick] (0,3.4) .. controls (1.6,3.4) and (2.2,0.5) .. (4.0,0.5)
  .. controls (5.8,0.5) and (6.4,3.4) .. (8.0,3.4);
\node[black, anchor=south] at (4.0,3.4) {dark-matter potential well};
% infalling baryons (gravity arrows)
\draw[->, very thick] (2.0,2.6) -- (3.2,1.2);
\draw[->, very thick] (6.0,2.6) -- (4.8,1.2);
\node[anchor=east] at (2.0,2.4) {gravity in};
% photon pressure out
\draw[->, black, very thick] (4.0,0.9) -- (4.0,2.1);
\node[black!70, anchor=south] at (4.0,2.1) {pressure out};
% compressed fluid blob at bottom
\fill[acc!18] (4.0,0.9) ellipse (0.9 and 0.35);
\draw[acc] (4.0,0.9) ellipse (0.9 and 0.35);
\node[acc, anchor=north] at (4.0,0.5) {compression};
\end{tikzpicture}
$$

## 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 $N_{\text{eff}}$.

$$
% caption: The sound horizon is a standard ruler of fixed length at last
% scattering. In a flat universe it subtends about half a degree; positive
% curvature magnifies the angle and shifts the first peak to lower multipole,
% negative curvature shrinks it and shifts the peak higher.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
% observer at bottom
\fill[black!70] (4.5,0) circle (2pt);
\node[black!70, anchor=north] at (4.5,-0.1) {observer};
% the ruler at top (fixed length) drawn three times conceptually via one bar
\draw[acc, very thick] (3.3,4.0) -- (5.7,4.0);
\node[acc, anchor=south] at (4.5,4.05) {sound horizon (same length)};
% flat: straight sightlines
\draw[black] (4.5,0) -- (3.3,4.0);
\draw[black] (4.5,0) -- (5.7,4.0);
\node[black!70, anchor=west] at (5.75,2.0) {Euclidean: straight rays};
% closed: converging (bowed inward) sightlines to a wider apparent angle
\draw[dashed] (4.5,0) .. controls (2.6,1.6) and (2.2,2.8) .. (3.3,4.0);
\draw[dashed] (4.5,0) .. controls (6.4,1.6) and (6.8,2.8) .. (5.7,4.0);
\node[anchor=east] at (2.2,1.4) {closed: wider};
% open: diverging (bowed outward) to a narrower apparent angle
\draw[black, densely dotted] (4.5,0) .. controls (4.3,1.8) and (3.9,3.0) .. (3.3,4.0);
\draw[black, densely dotted] (4.5,0) .. controls (4.7,1.8) and (5.1,3.0) .. (5.7,4.0);
\node[black, anchor=west] at (4.9,1.1) {open: narrower};
\end{tikzpicture}
$$

$$
% caption: The angular power spectrum of CMB temperature fluctuations. The first
% acoustic peak near multipole 220 fixes spatial flatness, the odd-even peak
% heights fix the baryon and dark-matter densities, and the small-scale tail is
% suppressed by photon diffusion.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->, black] (0,0) -- (9.8,0) node[right, black!70] {multipole (small angles right)};
\draw[->, black] (0,0) -- (0,4.6) node[above, black!70] {power};
% Sachs-Wolfe plateau then peaks with decreasing amplitude (damping)
\draw[acc, very thick]
  (0.3,1.4) .. controls (0.9,1.35) and (1.2,1.3) .. (1.5,1.5)
  .. controls (2.0,2.0) and (2.3,4.1) .. (2.7,4.1)
  .. controls (3.1,4.1) and (3.4,1.2) .. (3.9,1.1)
  .. controls (4.4,1.0) and (4.7,2.3) .. (5.1,2.35)
  .. controls (5.5,2.4) and (5.8,1.0) .. (6.3,0.95)
  .. controls (6.8,0.9) and (7.1,1.7) .. (7.5,1.7)
  .. controls (7.9,1.7) and (8.2,0.7) .. (8.7,0.5)
  .. controls (9.0,0.4) and (9.2,0.35) .. (9.4,0.3);
% peak markers
\draw[black, densely dotted] (2.7,0) node[below, black!70] {peak one near 220} -- (2.7,3.9);
\node[black!70, anchor=south] at (5.1,2.4) {second};
\node[black!70, anchor=south] at (7.5,1.75) {third};
\node[black, anchor=north east] at (0.3,1.35) {plateau};
\node[black, anchor=south east] at (9.4,0.55) {damping tail};
\end{tikzpicture}
$$

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 $\Delta T/T = \tfrac{1}{3}\,\Delta\Phi/c^2$ 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.

> **Worked example.** The first acoustic peak sits at the multipole matching the
> sound horizon: $\ell_1 \approx \pi\,d_A/r_s$, where $d_A$ is the comoving
> angular-diameter distance to last scattering. With $r_s \approx 147\ \text{Mpc}$
> and $d_A \approx 14{,}000\ \text{Mpc}$ in a flat concordance model,
> $$
> \ell_1 \approx \frac{\pi \times 14{,}000}{147} \approx 300,
> $$
> reduced to the observed $\ell_1 \approx 220$ once the driving of the oscillator
> and the projection are treated exactly. What matters is the ratio: a closed
> universe shortens $d_A$, lowering $\ell_1$, and an open universe raises it. The
> measured $\ell_1 \approx 220$ is the value for spatial flatness.

## 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.

$$
% caption: The two CMB polarization patterns. E-modes are curl-free, radial or
% tangential about a hot or cold spot, and trace the density field; B-modes are
% the curl pattern, sourced by gravitational waves or by lensing of E-modes.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
% E-mode: radial pattern
\node[acc, anchor=south] at (2.0,3.2) {E-mode};
\foreach \a in {0,45,90,135,180,225,270,315}{
  \draw[acc, very thick] ({2.0+1.0*cos(\a)},{1.6+1.0*sin(\a)}) -- ({2.0+1.5*cos(\a)},{1.6+1.5*sin(\a)});
}
% B-mode: swirl pattern (tangential-ish, rotated 45)
\node[black, anchor=south] at (7.0,3.2) {B-mode};
\foreach \a in {0,45,90,135,180,225,270,315}{
  \draw[black, very thick] ({7.0+1.25*cos(\a)},{1.6+1.25*sin(\a)})
    -- ({7.0+1.25*cos(\a)+0.55*cos(\a+45)},{1.6+1.25*sin(\a)+0.55*sin(\a+45)});
}
\end{tikzpicture}
$$

## 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:

| Parameter | Symbol | Planck 2018 value |
| --- | --- | --- |
| Baryon density | $\Omega_b h^2$ | $0.0224 \pm 0.0001$ |
| Cold dark matter density | $\Omega_c h^2$ | $0.120 \pm 0.001$ |
| Hubble constant | $H_0$ | $67.4 \pm 0.5\ \text{km s}^{-1}\text{Mpc}^{-1}$ |
| Dark-energy density | $\Omega_\Lambda$ | $0.685 \pm 0.007$ |
| Scalar spectral index | $n_s$ | $0.965 \pm 0.004$ |
| Optical depth to reionization | $\tau$ | $0.054 \pm 0.007$ |

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.[^planck-vi]

The acoustic peaks measure the same baryon density as [Big Bang
nucleosynthesis](/astrophysics-cosmology/the-hot-big-bang/big-bang-nucleosynthesis)
and the same flat geometry that [inflation](/astrophysics-cosmology/the-hot-big-bang/cosmic-inflation)
predicts; the near-scale-invariant plateau is the fingerprint of the primordial
spectrum whose origin the inflation lesson derives.

[^planck-vi]: Planck Collaboration, Planck 2018 results VI. Cosmological parameters — the acoustic peak positions and heights, the spatial flatness constraint, the baryon and cold-dark-matter densities, the concordance parameters, and the Hubble tension. https://arxiv.org/abs/1807.06209