1. From a temperature map to a peak plot

The cosmic microwave background (CMB) is early thermal radiation shifted to longer wavelengths by expansion. Its mean temperature and small directional fluctuations convey different information. Hu and Dodelson's review explains expanding ΔT/T in spherical harmonics and representing it with an angular power spectrum.

The horizontal multipole ℓ is neither time nor observing frequency. Larger ℓ describes smaller angular structure. Check whether the vertical axis is temperature, Cℓ or ℓ(ℓ+1)Cℓ/2π; the meaning of height depends on the representation even for the same data.

2. Why there are multiple acoustic peaks

Gravity and pressure produced compressions and rarefactions in the early photon–baryon fluid. Its oscillation state while photons scattered frequently is reflected in later temperature fluctuations. The review explains how baryon loading affects alternating oscillation amplitudes, while radiation driving and diffusion damping also affect peak heights.

Do not directly convert one peak height or a two-peak ratio into the entire matter composition. Compare multiple peaks and polarization with model predictions together.

3. Reading Planck's “100θ*”

Section 3.1 of the Planck parameter paper defines θ* = r*/DM using the comoving sound horizon at recombination and the comoving distance mapping it to an angle. The TT,TE,EE+lowE combination reports 100θ*=1.04109±0.00030 (68%).

Dividing by 100 gives θ*=0.0104109 radians, approximately 0.5965 degrees. This is a unit conversion of the reported scale, not simply the location of a particular first peak. Temperature and polarization peaks can occur at different positions while constraining the same acoustic scale. Retain the data combination and uncertainty with the number.

Separate the plotted measurement from its interpretation

When reading a figure, use the legend and caption to identify whether points and error bars are an observational analysis and whether a curve is a prediction from a chosen cosmological model. For example, Planck’s TT,TE,EE+lowE label denotes a selected combination of temperature and polarization spectra plus low-multipole data. The reported 100θ*=1.04109±0.00030 is the acoustic angular scale constrained by that combination. Dividing by 100 and converting radians to degrees gives about 0.5965°, but this arithmetic does not produce H₀.

A useful reading sequence is: measurement: estimate angular temperature/polarization fluctuation spectra from sky maps; scale constraint: use their oscillatory pattern to constrain the acoustic angle; model inference: fit a model such as ΛCDM and multiple parameters to infer values such as H₀. Planck describes the acoustic angle as a robust geometrical measurement that is almost independent of the cosmological model. That statement does not make every later cosmological parameter a direct, assumption-free measurement. Keeping these stages separate helps distinguish “read from the plotted observation” from “inferred by fitting data with a model.”

4. CMB versus galaxy BAO; observations versus inference

CMB measures angular temperature/polarization fluctuations; late-time galaxy BAO is a distinct observation in galaxy distributions. They connect through early acoustic physics but are not the same plot. Planck also includes analyses combined with separate BAO data, so check whether a result is CMB-only or combined.

The paper distinguishes robust geometrical constraints on the acoustic scale from late-Universe parameters inferred assuming base ΛCDM. Describing H0 as a model-independent number directly read from a peak omits that inference.

5. Sources

Hu and Dodelson, Cosmic Microwave Background Anisotropies; Planck Collaboration, Planck 2018 results VI, §3.1. This article concerns reading CMB acoustic structure.