CMB Spectral Distortions

A tape recorder you cannot rewind

The cosmic microwave background is the most perfect blackbody anyone has ever measured. Its spectrum follows the Planck curve for a temperature of T₀ = 2.725 K to better than 50 parts per million across more than a decade in frequency. That perfection is not an accident — it is a fossil of a time when the universe was a hot, dense plasma in which photons scattered constantly off free electrons. In that bath, any lump of excess energy was smeared out, any photon-number imbalance refilled, and the spectrum relaxed to the one distribution that maximises entropy at fixed energy: the blackbody.

But that relaxation machinery does not run forever. As the universe expands and cools, the scattering rates that maintain a perfect blackbody fall off and eventually freeze out. After they freeze, the spectrum can still be nudged — but it can no longer be fully repaired. Any energy injected from then on leaves a permanent, frequency-dependent scar. Those scars are the spectral distortions. They come in two canonical flavours — the chemical-potential μ-distortion and the Compton y-distortion — and because each flavour switches on at a definite cosmic epoch, the CMB acts as a tape recorder whose erase head was switched off when the universe was roughly a month old. Read the tape carefully enough and you recover a history of energy releases reaching back to redshifts of a few million.

Why the blackbody freezes in: the thermalisation clock

Maintaining a true Planck spectrum requires two distinct kinds of process. Compton scattering off electrons redistributes photon energies but conserves photon number. To create or destroy photons — necessary to fix the normalisation of the spectrum — you need double-Compton scattering (e + γ → e + 2γ) and thermal bremsstrahlung (free-free emission). All three rates scale with density and temperature and therefore decline as the universe expands.

The number-changing processes are the first to lose the race. Double-Compton scattering, dominant in the early radiation era because of the enormous photon-to-baryon ratio (η⁻¹ ≈ 1.6×10⁹), becomes too slow to keep up around a redshift of

z_μ ≈ 2 × 10⁶      (t ≈ a few weeks after the Big Bang)

This is the thermalisation redshift. Before it, any injected energy is fully digested and the blackbody is restored. After it, the plasma can still redistribute photon energies by Compton scattering, but it can no longer manufacture the extra soft photons needed to rebuild the Rayleigh-Jeans tail. Energy injected in this window relaxes instead to the most general equilibrium that conserves photon number — a Bose-Einstein distribution with a non-zero chemical potential:

n(x) = 1 / (exp(x + μ) − 1),    x = hν / kT

A small μ > 0 represents a photon deficit relative to a blackbody of the same energy density — exactly what you get when energy is added without new photons. This is the μ-distortion era, running from z ≈ 2×10⁶ down to about z ≈ 5×10⁴, where Compton scattering itself can no longer keep pace.

The two distortion shapes

The shape of a distortion encodes the epoch that produced it. The two pure types have closed-form spectral signatures.

The y-distortion (Zel'dovich & Sunyaev 1969) arises when scattering is too rare to thermalise: hot electrons up-scatter CMB photons by the inverse-Compton effect, conserving photon number but pushing energy from low to high frequencies. Its fractional intensity change is

ΔI/I = y · g(x),   g(x) = (x e^x)/(e^x − 1) · [ x·coth(x/2) − 4 ]
y = ∫ (k T_e / m_e c²) σ_T n_e  dl       (the Compton-y parameter)

The function g(x) is negative at low frequency (a deficit on the Rayleigh-Jeans side) and positive at high frequency (an excess on the Wien tail), crossing zero at x ≈ 3.83, i.e. ν ≈ 217 GHz. The μ-distortion has a different shape, set by the Bose-Einstein form; it also produces a low-frequency deficit and high-frequency excess but with its crossover near 124 GHz, distinct enough that a sensitive spectrometer can separate μ from y.

The boundary between the two eras is not sharp. Between z ≈ 5×10⁴ and z ≈ 2×10⁴ the Comptonisation is partial, producing an intermediate or "i-type" / "r-type" (residual) distortion that is neither pure μ nor pure y. This residual carries extra information — it can in principle pin down when the energy was injected, not just how much — and is a key science target for next-generation spectrometers.

How perfect: the COBE/FIRAS measurement

The benchmark came from the Far-Infrared Absolute Spectrophotometer (FIRAS) aboard NASA's COBE satellite, with John Mather as principal investigator. FIRAS compared the sky to an onboard reference blackbody and found the CMB spectrum is Planckian to extraordinary precision:

T₀ = 2.7255 ± 0.0006 K
residuals  < 50 parts per million (rms), 60–600 GHz
|μ| < 9 × 10⁻⁵      (95% CL)
|y| < 1.5 × 10⁻⁵    (95% CL)

This is the most perfect blackbody ever measured in nature — a laboratory blackbody is harder to make this clean. Mather shared the 2006 Nobel Prize in Physics with George Smoot, who led the COBE anisotropy detection. The FIRAS limits remain the state of the art for the spectral monopole nearly three decades later; no instrument since has improved on the absolute spectrum, only on the anisotropies.

Numbers: limits, predictions, and what each tells you

The interesting physics lives in the gap between what FIRAS could rule out (~10⁻⁵) and what standard cosmology predicts (~10⁻⁸ for μ). Closing that three-orders-of-magnitude gap is the science case for a new mission.

Signal	Type	Amplitude	Epoch (z)	What it measures
FIRAS upper limit	μ	< 9×10⁻⁵	—	Current bound
FIRAS upper limit	y	< 1.5×10⁻⁵	—	Current bound
Silk damping (acoustic dissipation)	μ	≈ 2×10⁻⁸	5×10⁴ – 2×10⁶	Small-scale power spectrum
Silk damping (late part)	y	≈ 4×10⁻⁹	< 5×10⁴	Power spectrum, k ~ 1–50 Mpc⁻¹
Reionisation + warm IGM	y	≈ 2×10⁻⁶	< 10	Thermal history of baryons
Recombination lines (H, He)	lines	≈ 10⁻⁹	≈ 1300, 6000	Recombination dynamics, Y_p
Adiabatic cooling of baryons	μ (negative)	≈ −3×10⁻⁹	broad	Baryon density Ω_b

Two features stand out. First, the dominant guaranteed distortion is the y-signal from reionisation and the warm-hot intergalactic medium, about 2×10⁻⁶ — well below FIRAS but a factor of a few above the μ-signals. Second, the μ-distortion from Silk damping is the cleanest cosmological target: it directly probes the primordial power spectrum on comoving scales 50 ≲ k ≲ 10⁴ Mpc⁻¹, scales an order of magnitude smaller than the CMB anisotropies (k ≲ 0.2 Mpc⁻¹) or galaxy surveys can reach. A μ measurement is, in effect, a window onto inflationary physics at otherwise inaccessible wavenumbers.

Worked example: the μ-distortion from Silk damping

Small-scale sound waves in the photon-baryon plasma do not survive to recombination. Photons diffuse out of the compressions and rarefactions — Silk damping — and the wave energy is thermalised into the photon bath. If this happens during the μ-era it produces a μ-distortion. The injected fractional energy is roughly

ΔE/E ≈ ∫ d ln k  k³ P(k) · [dissipation weight](k, z)

and the resulting μ is approximately

μ ≈ 1.4 · (ΔE/E)|_μ-era

For a near-scale-invariant primordial spectrum with amplitude A_s ≈ 2.1×10⁻⁹ and tilt n_s ≈ 0.965 (the Planck best fit), integrating the dissipated power over the μ-window gives

μ ≈ 2 × 10⁻⁸    (standard ΛCDM, single-field slow-roll)

The point is the leverage on new physics. If the small-scale power spectrum were enhanced — by a feature in the inflaton potential, a phase of fast roll, or extra power that also seeds primordial black holes — the dissipated energy and hence μ would rise sharply. A measured μ a few times above 2×10⁻⁸ would be a smoking gun for enhanced small-scale power; a μ consistent with 2×10⁻⁸ would confirm that the simple power-law spectrum continues down to scales we can reach no other way. Either way, the number is a direct readout of inflationary physics at k ~ 10³ Mpc⁻¹.

Where distortions show up — and where they would betray new physics

• Reionisation and the warm-hot IGM. Hot electrons (10⁴–10⁷ K) in the ionised intergalactic medium up-scatter CMB photons, giving the dominant guaranteed y-distortion, y ≈ 2×10⁻⁶. This is a direct thermometer of when and how the universe was reheated by the first stars and quasars.
• Galaxy clusters — the thermal Sunyaev-Zel'dovich effect. Toward a single rich cluster, electrons at 10⁷–10⁸ K give a line-of-sight y ~ 10⁻⁴, routinely mapped by Planck, the South Pole Telescope and the Atacama Cosmology Telescope. The sky-average of all such regions contributes to the global y-monopole.
• Silk damping of acoustic waves. The cleanest cosmological μ-signal (~2×10⁻⁸), encoding the primordial power on scales 50–10⁴ Mpc⁻¹ that no other probe reaches.
• Cosmological recombination radiation. The cascade of hydrogen and helium recombination at z ≈ 1300 (H) and z ≈ 2000–6000 (He) emits faint spectral lines, summing to ~10⁻⁹ in the CMB spectrum — a direct test of recombination physics and the helium abundance.
• Decaying or annihilating dark matter. A long-lived particle decaying in the μ- or y-era injects electromagnetic energy and shifts the distortions. Spectral-distortion limits already constrain decay lifetimes 10⁸–10¹³ s and particle masses across wide ranges, complementary to direct detection.
• Evaporating primordial black holes. Black holes with masses 10¹³–10¹⁷ g evaporate via Hawking radiation during the distortion eras, injecting energy whose absence in FIRAS already excludes part of the primordial-black-hole mass spectrum.
• Cosmic strings and other relics. A decaying string network or other exotic energy-loss channel would add to μ and y, making distortions a broad, model-independent calorimeter of the high-redshift universe.

Detecting the 10⁻⁸ signal: PIXIE and beyond

FIRAS reached ~10⁻⁵. The guaranteed standard signals sit near 10⁻⁸ (μ) to 10⁻⁶ (y). Bridging that gap needs an absolutely-calibrated spectrometer with vastly more integration time and exquisite control of foregrounds. The leading concept is NASA's proposed PIXIE (Primordial Inflation Explorer), a Fourier-transform spectrometer that, like FIRAS, compares the sky against an internal blackbody calibrator but with ~10³–10⁴ times the sensitivity, targeting μ ~ 10⁻⁸ and y ~ 10⁻⁹. Earlier European study concepts (PRISM) and the ESA Voyage 2050 spectrometer studies pursue similar goals.

The hard part is not raw sensitivity but foregrounds: galactic synchrotron, free-free, spinning and thermal dust, and the cosmic infrared background all swamp a 10⁻⁸ distortion in raw intensity. Because each foreground has a different spectral shape across many frequency channels, and the distortion shapes (μ, y, the residual, and the recombination lines) are calculable, the signal can in principle be separated — but it demands percent-level foreground modelling and absolute calibration to parts per billion of the CMB intensity. This is why no successor to FIRAS has yet flown: the measurement is technologically brutal, but the scientific payoff — a calorimetric history of the universe from a redshift of a few million, and a unique probe of inflation at small scales — keeps it near the top of the long-term cosmology roadmap.

Common misconceptions and edge cases

• Distortions are not the same as anisotropies. The famous CMB "maps" (COBE, WMAP, Planck) measure temperature fluctuations across the sky at fixed blackbody shape — ΔT/T ~ 10⁻⁵ hot and cold spots. Spectral distortions are about the shape of the spectrum itself, the same in every direction at lowest order (the monopole). You can have a perfectly Planckian spectrum that varies in temperature across the sky, and a distorted spectrum that is identical in all directions.
• A μ-distortion is not "a hotter CMB". Adding energy without adding photons does not simply raise T₀; it deforms the spectrum into a Bose-Einstein shape with μ ≠ 0. Re-fitting a single temperature to that spectrum leaves frequency-dependent residuals — that residual is the distortion.
• The y-distortion does not violate photon-number conservation. Inverse-Compton scattering moves photons up in frequency but keeps their total count fixed; the low-frequency deficit exactly balances the high-frequency excess in number, not in energy.
• Energy injected before z ≈ 2×10⁶ leaves no distortion. Earlier than the thermalisation redshift, the plasma fully rebuilds the blackbody, just at a slightly higher temperature. This is why distortions probe a specific redshift window — they are blind to the very earliest epochs (before z ≈ 2×10⁶), while energy released in the local universe shows up cleanly as a late-time y-distortion.
• The 217 GHz null is non-relativistic. For very hot cluster electrons (kT_e approaching tens of keV) relativistic corrections shift the crossover frequency by a few GHz and change the spectral shape — the relativistic SZ effect — which is itself a thermometer for the hottest clusters but must not be confused with the canonical 217 GHz null of the pure non-relativistic y-distortion.