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A CMB power spectrum is a map of how strongly the microwave background varies on different angular scales. Read the horizontal multipole axis from low to high values (large sky patterns to fine detail), identify whether the vertical axis is Cℓ or Dℓ = ℓ(ℓ+1)Cℓ/(2π), then distinguish measured bandpowers from the fitted theoretical curve. The first acoustic peak near ℓ ≈ 200 (about 1–2 degrees) marks sound waves in the early photon–baryon plasma; the sequence of later peaks, polarization spectra and the damping tail carry the parameter information used in modern cosmology.
What a CMB power-spectrum plot represents
The cosmic microwave background (CMB) is almost uniform, with a mean temperature of about 2.73 K. Its temperature anisotropies are only about one part in 100,000, yet their angular pattern records conditions in the young universe. A power spectrum compresses a full-sky map into the amount of variance associated with each spherical-harmonic multipole, conventionally written ℓ (ell).
Instead of showing where a hot or cold patch sits on the sky, the spectrum answers “how much structure exists at this angular scale?” Two maps can have different-looking spots but nearly the same spectrum if they contain the same statistical power by scale. That statistical description is why spectra, rather than a single image, are used to estimate cosmological parameters.
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Read the axes before interpreting any peak
The horizontal axis: multipole ℓ
On a sphere, temperature or polarization is expanded in spherical harmonics. The multipole index ℓ is the angular analogue of a spatial frequency:
- Low
ℓmeans broad patterns covering a large fraction of the sky. - High
ℓmeans smaller, finer features. - A useful scale estimate is
θ ≈ 180°/ℓ. It is an approximation, not a sharp boundary for a feature.
Thus ℓ = 2 describes the largest non-monopole pattern that can be measured, while ℓ = 2000 corresponds to structure roughly a tenth of a degree across. The U.S. Planck Data Center describes the plot as the power obtained when the sky image is decomposed into spherical harmonics of order “L”; the symbol differs, but the quantity is the same multipole concept.
The vertical axis: Cℓ or Dℓ
The mathematically direct spectrum is Cℓ, the variance assigned to each multipole. Popular plots instead show
Dℓ = ℓ(ℓ+1)Cℓ/(2π).
Dℓ roughly displays contribution per logarithmic interval in angular scale and makes acoustic peaks easier to see. Both conventions contain the same information, but their vertical shapes differ, especially at low ℓ. A line that appears flat in one convention can slope strongly in the other. Check the axis label and units before comparing figures. Temperature spectra are commonly reported in microkelvin squared (µK²); polarization spectra use the corresponding squared units and cross-spectra can be positive or negative.
Separate measurements from the model
Points, bins and error bars
Individual points may be estimates at single multipoles, but most published plots show bandpowers: averages over a range of ℓ. Binning reduces visual scatter and correlates neighboring points through the window functions used in the estimate. The horizontal position is therefore an effective multipole for a bin, not necessarily every mode in that interval.
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Error bars combine several effects. Instrumental noise, finite resolution, calibration uncertainty and map-making choices contribute at all scales. At low multipoles, the dominant irreducible term is often cosmic variance: we observe one sky containing only a finite number of independent modes. ESA’s Planck explanation states that the red measurement dots carry errors accounting for measurement uncertainty and an estimate of the uncertainty caused by the limited number of sky positions available for the measurement. A low-ℓ point far from a line is therefore not automatically an instrumental failure or evidence for new physics.
The smooth curve and residuals
The smooth curve is generally a theoretical spectrum generated for a parameter set, often the maximum-likelihood or best-fit Lambda-CDM model. It may be fitted to the same data, taken from another release, or supplied as an external comparison. The caption should tell you which. “Close to the line” is a visual description, not a statistical result.
To assess a discrepancy, inspect residuals (data minus model) together with the covariance matrix. Neighboring bandpowers can be correlated, and the covariance includes noise, beam uncertainty, foreground parameters and sample variance. A run of points on one side of the curve can be less significant than it looks if those points are highly correlated; a single apparently large deviation can be unimportant if its uncertainty is large.
Read the spectrum from left to right
Largest angles: ISW rise and Sachs–Wolfe plateau
The far-left region covers tens of degrees and larger. Standard plots often label an integrated Sachs–Wolfe (ISW) rise followed by the Sachs–Wolfe plateau. These modes probe gravitational potentials on very large scales and are few in number, so their error bars are broad and cosmic variance is substantial. The exact low-ℓ appearance also depends on sky masks and how the monopole and dipole are removed.
The first acoustic peak near ℓ ≈ 200
The most recognizable landmark is a broad maximum around ℓ ≈ 200, corresponding to roughly 1–2 degrees. It is the angular projection of the sound horizon at the time photons last scattered. Before recombination, photons and baryons behaved as a tightly coupled fluid. Dark-matter potential wells compressed and rarefied that fluid, launching acoustic oscillations. The first peak records the mode that had reached its first compression by last scattering.
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Peak position is primarily a geometric measurement. It depends on the physical sound horizon and on how that ruler is projected through the universe’s expansion history and spatial curvature. Moving the entire peak pattern horizontally is therefore not equivalent to changing one local property of the plasma; a full model fit is required to separate geometry from other parameters.
Alternating acoustic peaks and troughs
Successive maxima and minima are acoustic harmonics: modes that completed different fractions of an oscillation before photons decoupled. Their spacing and relative heights encode several effects at once:
- Baryon loading: baryons add inertia to the photon fluid and enhance compressional (odd-numbered) peaks relative to rarefaction (even-numbered) peaks.
- Total matter density: it changes the expansion rate and the gravitational driving of the oscillations.
- Primordial amplitude and tilt: the initial fluctuation spectrum sets the overall normalization and scale dependence.
- Geometry and projection: the angular scale of the sound horizon shifts the peak locations.
Do not infer a parameter from one peak height in isolation. The same height can result from compensating changes in several parameters, and the information is distributed across all spectra and their covariance.
The high-ℓ damping tail
At sufficiently fine scales, power declines in the damping tail. Photon diffusion (often called Silk damping) erases fluctuations below a characteristic length before last scattering. The observed tail is also shaped by the instrument beam, calibration, unresolved foreground emission and secondary anisotropies such as the Sunyaev–Zel’dovich effect. Consequently, a high-ℓ falloff is physical only after beam and foreground models have been accounted for.
Why there are TT, TE and EE curves
Temperature and linear polarization are different fields on the sky. Their auto- and cross-spectra are complementary, not duplicate versions of the same plot.
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| Spectrum | What is correlated | How to read its sign and features |
|---|---|---|
| TT | Temperature anisotropy with temperature anisotropy | The familiar acoustic-peak pattern; it has the highest signal-to-noise over much of the measured range. |
| EE | E-mode (parity-even) polarization with E-mode polarization | Peaks are phase-shifted relative to TT because polarization is sourced mainly by the fluid’s velocity, which is offset from its density extrema. |
| TE | Temperature with E-mode polarization | A cross-correlation that can be positive or negative, so it crosses the zero line. Its alternating sign is expected from the relative phase of temperature and velocity. |
| BB | B-mode (parity-odd) polarization with B-mode polarization | Often dominated by gravitational lensing at small scales; a large-scale primordial component would be relevant to tensor perturbations, but interpretation is especially sensitive to foregrounds and systematics. |
| TB and EB | Temperature–B and E–B cross-correlations | Usually used as null or systematics tests; unexpected nonzero signals can indicate calibration or leakage problems. |
Planck’s 2018 cosmology-product documentation reports TT coverage from ℓ = 2 to 2508, while TE, EE, EB and BB products extend to about ℓ = 1996. The exact range in a figure can be shorter because of masking, signal-to-noise cuts or a particular likelihood.
Because temperature and E-mode skies are correlated, the Planck and Particle Data Group treatments emphasize that TT alone does not contain all available information. Combining TT, TE and EE helps break parameter degeneracies and provides internal consistency checks.
What peak positions and heights can—and cannot—tell you
Position: an angular ruler
Locate the first peak and compare the spacing of later peaks. A coherent horizontal shift is mainly a change in the projected acoustic scale, which depends on spatial geometry and the distance to last scattering. The approximate ℓ ≈ 200 landmark is useful for orientation, not a precision estimate of curvature.
Heights: a coupled set of physical effects
Compare odd and even temperature peaks, the overall normalization, and the rate at which power declines toward high ℓ. These patterns jointly constrain baryon density, matter density, the primordial amplitude and spectral tilt, recombination physics and, with polarization and lensing information, additional parameters. A peak ratio is not a one-parameter meter: changing the primordial tilt can alter several peaks at once, while foreground subtraction can change the apparent tail.
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EE has lower raw signal than TT but responds differently to the same plasma physics. Its phase relative to TT tests the acoustic interpretation. TE supplies a signed cross-check: positive and negative lobes should occur where the model predicts them. Agreement among TT, TE and EE is stronger evidence than a visually good TT fit alone.
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A practical workflow for reading any plot
- Read the caption and axes. Record the experiment, release date, sky fraction, multipole range, vertical convention (
CℓorDℓ) and units. - Identify the data product. Determine whether points are unbinned estimates or bandpowers, and whether they are an auto-spectrum or a noise-bias-reduced cross-spectrum.
- Identify the curve. Check whether it is a best fit to these points, a model from another analysis, or a theory curve with foreground and secondary contributions included.
- Translate scales. Use
θ ≈ 180°/ℓto label broad, degree-scale and sub-degree structure. - Mark the landmarks. Note the low-
ℓplateau, first peak, harmonic sequence and damping tail before judging deviations. - Compare TT, TE and EE. Look for the expected phase relation and TE sign changes rather than demanding that all three curves have identical shapes.
- Check uncertainties. At low
ℓ, include cosmic variance; at highℓ, inspect beam, calibration and foreground uncertainties. - Use residuals and covariance. Decide whether an apparent feature is statistically meaningful only with the published covariance or likelihood, not by vertical distance from the line.
How to distinguish a real feature from noise or cosmic variance
Large angular scales
There are only a few independent modes at low multipoles, so estimates fluctuate from one realization of the universe to another. A broad departure at ℓ below a few tens may be a normal sample-variance fluctuation. Compare independent component-separation methods and the quoted covariance before invoking an instrumental problem.
Small angular scales
At high multipoles, ask four questions:
- Was the spectrum foreground-cleaned, or were foreground amplitudes marginalized in the likelihood?
- What beam window function and beam uncertainty were applied?
- Were calibration and polarization-efficiency nuisance parameters fitted?
- Are unresolved point sources, Galactic emission and thermal or kinetic Sunyaev–Zel’dovich terms included?
Two spectra can differ because one displays only the primary CMB while another plots the total modeled signal. Their curves are not directly comparable until the component definitions match.
Null tests and cross-spectra
Cross-spectra made from independent detector sets or map splits avoid the noise bias that affects an auto-spectrum, but they do not remove all systematics. TB and EB are especially useful null tests for polarization-angle errors and temperature-to-polarization leakage. A null result increases confidence; it does not prove that every residual is cosmological.
Comparing spectra or datasets correctly
When choosing between plots, archives or likelihood products, compare like with like:
| Comparison axis | Why it matters | What to record |
|---|---|---|
| Multipole coverage | Determines which angular scales and physical effects are tested. | Minimum and maximum ℓ, plus any excluded intervals. |
| Spectrum types | TT-only cannot provide the same degeneracy breaking as TT+TE+EE. | Included auto- and cross-spectra, including BB, TB and EB. |
| Frequency and channels | Frequency diversity separates CMB emission from Galactic and extragalactic foregrounds. | Channels used and whether maps were combined. |
| Sky fraction and mask | Masking changes mode coupling and effective variance. | Unmasked sky percentage and mask treatment. |
| Binning and covariance | Bins can correlate neighboring points and alter apparent scatter. | Bin edges, window functions and supplied covariance. |
| Foreground treatment | Cleaning and marginalization lead to different displayed curves and errors. | Components removed, fitted or left in the plot. |
| Beam and calibration | Errors grow toward fine scales and can mimic a change in damping. | Beam window, calibration priors and nuisance parameters. |
| Model status | A best fit to the same data is not an independent prediction. | Parameter source, likelihood and whether the curve is external. |
Planck’s broad frequency coverage was specifically used to separate CMB emission from Galactic and other foreground sources. That separation is part of the measurement, not a cosmetic post-processing step.
A worked visual interpretation
Suppose a plot shows binned TT points with a smooth Lambda-CDM line. First verify that the vertical axis is Dℓ in µK². The leftmost points represent very large angular patterns and should have broad uncertainties. Near ℓ = 200, identify the first maximum and translate it to about 1–2 degrees. Follow the next maxima and minima: their alternating pattern is the acoustic-harmonic signature, not random oscillation. Finally, note the decline at high ℓ, but defer a physical interpretation until the caption specifies beam and foreground treatment. If a second panel shows TE crossing zero while EE peaks are offset from TT, those are expected phase relationships. Only after these checks should you examine residuals and covariance for a statistically unusual feature.
A historical caveat about low-ℓ anomalies
An ESA/Planck explanatory graphic from the 2013 results described large-angle fluctuations between about 90° and 6° as roughly 10% weaker than the best-fit standard model. That statement belongs to that historical analysis and scale range; it is not a universal claim that current CMB data contain a confirmed anomaly. Low-ℓ cosmic variance and analysis choices remain essential context whenever an apparent large-scale deficit is discussed.
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Multipole ell converts angular size into a one-dimensional scale coordinate; the first peak near 200 is the projected sound horizon; the following peaks and damping tail encode the plasma, expansion history and diffusion physics; and TT, TE and EE provide complementary views of temperature and polarization. Reliable cosmological conclusions come from fitting the complete set of spectra, nuisance parameters and covariance—not from reading a single peak or judging how close a point looks to a line.
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