The CMB thermal spectrum and possible distortions

Spectral shape and angular structure are different information

The cosmic microwave background has a spectrum close to a thermal blackbody, established through calibrated comparisons such as FIRAS. Close agreement is a testable spectral statement, rather than proof of an exactly perfect shape at every frequency.

A spectrum describes intensity as a function of frequency, while an anisotropy map describes variation across directions. An angular average can also mix radiation from different directions or temperatures, so the distinction matters when interpreting a spectrum.

Distortions can arise when a process changes the spectral form without restoring complete thermal equilibrium. Their interpretation depends on the physical mechanism, propagation and measurement model.

The numerical example below contains no measured CMB samples, experimental residuals, current distortion limits or inferred parameters. It is an exact mixture of two ideal spectra at deliberately exaggerated supplied temperature contrasts.

Compare every component at the same frequency

Choose an arbitrary positive reference temperature T_ref and x = hν/(k_B T_ref). With C = 2(k_B T_ref)³/(h²c²), the Planck specific intensity for temperature T = tT_ref is I_ν/C = P(x,t) = x³/[exp(x/t) − 1].

The same C and the same x are used for every component. Rescaling each component with its own temperature-dependent normalization would not be the stated intensity mixture at a fixed observed frequency.

Average equal linear intensity weights at t_+ = 1.1 and t_− = 0.9: I_mix/C = [P(x,1.1) + P(x,0.9)]/2. Assume no later thermalization. The comparison reference is the single Planck spectrum P(x,1).

Display 0.25 ≤ x ≤ 10. The chosen ten percent offsets are teaching inputs, not measured CMB anisotropies. The arithmetic evaluates the full Planck functions and does not replace them with a small-distortion expansion.

A frequency-dependent brightness temperature reveals the mismatch

At a single frequency, one can define a brightness temperature by inverting the Planck function for the mixed intensity. If the mixture were one blackbody, that inferred temperature would be the same at every frequency.

For this construction, the effective temperature is about 1.000827 T_ref at x = 1, 1.010596 T_ref at x = 4 and 1.035253 T_ref at x = 10. Its dependence on frequency shows that a single temperature cannot describe the entire mixed spectrum.

The mixed/reference intensity ratio also varies with frequency, from near one at low x to about 1.40571 at x = 10. These are ratios of exact synthetic functions, rather than observed residuals or instrumental calibration errors.

A fitted temperature chosen from one part of the spectrum can therefore leave a mismatch elsewhere. The fitted band and the definition of a comparison temperature matter when discussing a spectral difference.

Number and energy select different temperatures

In the low-frequency Rayleigh–Jeans limit, intensity is linear in temperature. The equal-weight mixture therefore has limiting brightness temperature [(1.1 + 0.9)/2]T_ref = T_ref.

For a full Planck spectrum, photon number scales as T³ and energy density as T⁴. The mixture’s full-spectrum moment ratios to the reference are [(1.1)³ + (0.9)³]/2 = 1.03 and [(1.1)⁴ + (0.9)⁴]/2 = 1.0601.

A number-matched single blackbody would have T/T_ref = 1.03^(1/3) ≈ 1.009902, while an energy-matched one would have 1.0601^(1/4) ≈ 1.014698. Neither equals the Rayleigh–Jeans value 1.

These moment identities integrate over the entire frequency range, not just the displayed finite x window. They provide another check that one Planck temperature cannot reproduce all properties of the constructed mixture.

An exact teaching mixture is not a measured distortion

For sufficiently small temperature contrasts, a perturbative mixing calculation can be expressed using familiar distortion shapes under stated conditions. A truncated expansion should not be treated as exactly equal to the finite-contrast mixture at every frequency.

Here the Planck functions were averaged without that truncation. The six percent energy excess relative to the T_ref reference belongs to the chosen large contrast, not to a claim about the actual CMB spectrum.

Testing a real distortion requires calibrated spectral data, foreground and instrument modelling and a physical mechanism with its own parameter and time dependence. A mathematical mismatch alone cannot identify the source of an observed residual.

The useful skill is to separate the measured spectrum, the comparison temperature, the full frequency shape and the physical process that could alter it. Each step has an explicit definition and a separate evidential role.

Worked reasoning

Try the reasoning

1. Are the displayed curves measured FIRAS residuals?

   Show the answer and reasoning

   No.

   They are exact Planck-function evaluations at supplied temperatures and weights.

2. Is this exact finite-contrast mixture identical to a second-order distortion formula at every x?

   Show the answer and reasoning

   No.

   A small-contrast expansion truncates terms that the exact mixture retains.

3. Do the quoted 1.03 and 1.0601 moment ratios integrate only over the displayed x interval?

   Show the answer and reasoning

   No.

   They are whole-spectrum photon-number and energy identities.

What to carry forward

• A mixture of unequal-temperature blackbodies generally is not a single blackbody.
• Low-frequency, number-matched and energy-matched temperatures can differ.
• Exact synthetic spectral differences must remain separate from measured CMB distortions.