CMB Blackbody Spectrum and Spectral Distortions

The cosmic microwave background carries the most perfect blackbody spectrum ever measured in nature. Above redshift $z\sim2\times10^6$ the photon–baryon–electron plasma was so dense and hot that photon-creating and energy-shuffling reactions drove the radiation into true thermal equilibrium — a Planck spectrum with no adjustable shape. As the universe expanded and cooled through recombination ( $z\approx1100$ ) the photons decoupled and have free-streamed ever since. The decisive fact is that expansion cannot spoil a blackbody: it redshifts every frequency by the same factor, mapping one Planck curve exactly onto a cooler one, $T\propto(1+z)$ . When COBE's FIRAS spectrometer looked, it found a blackbody at $T_0=2.7255\pm0.0006\,\mathrm{K}$ with fractional departures below $10^{-4}$ across the peak — the cleanest thermodynamic measurement in cosmology. That very perfection is a weapon. Anything that dumps energy into the photon bath — decaying particles, dissipating sound waves, hot cluster gas — perturbs the equilibrium, and whether it heals back into a Planck curve depends entirely on when it happened. Two microphysical clocks govern the healing: Compton scattering, which reshuffles photon energies but conserves photon number, and double Compton scattering with bremsstrahlung, which create and destroy photons. When both run fast the blackbody is restored; when only the number-changing processes have frozen out the plasma relaxes to a Bose–Einstein spectrum with a chemical potential $\mu$ ; when even Compton scattering cannot finish, a $y$ -type distortion survives. These $\mu$ - and $y$ -distortions are a frozen fossil record of energy release across the window $10^3\lesssim z\lesssim2\times10^6$ that no other observable can reach — and the $y$ -distortion is literally the spectrum the thermal Sunyaev–Zel'dovich effect prints across galaxy clusters today.

Key takeaways

The CMB is the most perfect blackbody known in nature: $T_0=2.7255\,\mathrm{K}$ , with any departure from Planckian shape bounded by COBE/FIRAS at $|\Delta I_\nu/I_\nu|\lesssim10^{-4}$ .
Expansion preserves a blackbody because it rescales frequency and temperature by the same factor $1+z$ , so $x=h\nu/k_BT$ is invariant — a Planck curve maps onto a cooler Planck curve.
Only photon-number-changing reactions (double Compton, bremsstrahlung) can build a blackbody; they freeze out near $z_{\rm th}\approx2\times10^6$ , so later energy injection cannot be fully erased.
Injection at $5\times10^4\lesssim z\lesssim2\times10^6$ leaves a $\mu$ -distortion (Bose–Einstein, $\mu\neq0$ ); at $z\lesssim5\times10^4$ it leaves a $y$ -distortion.
FIRAS limits are $|\mu|<9\times10^{-5}$ , $|y|<1.5\times10^{-5}$ ; the same $y$ -shape, null at $217\,\mathrm{GHz}$ , is the thermal Sunyaev–Zel'dovich effect of hot cluster gas.
Key idea: cosmic expansion preserves a blackbody but only photon-number-changing reactions can create one from a distorted spectrum, so because those reactions freeze out around $z\sim2\times10^6$ , energy injected afterward is frozen into $\mu$ - and $y$ -type distortions that record both the amount and the timing of the injection.

The core insight

Cosmic expansion preserves a blackbody but only photon-number-changing reactions can create one from a distorted spectrum; because those reactions freeze out around $z\sim2\times10^6$ , energy injected after that epoch is frozen into $\mu$ - and $y$ -type spectral distortions that encode both the amount and the timing of the injection.

Where this lives today

Spectral distortions are now a central pillar of observational cosmology along two fronts. The first is the thermal Sunyaev–Zel'dovich effect: Planck, the South Pole Telescope, and the Atacama Cosmology Telescope have each catalogued thousands of galaxy clusters by their $y$ -distortion, exploiting its redshift-independence to find and weigh clusters for tests of dark energy, and stacking sightlines even maps the "missing baryons" in the warm cosmic web. The second front is the search for the primordial monopole distortions: FIRAS set the standing limits in the 1990s and no experiment has beaten them, but proposed spectrometers such as PIXIE aim for $\mu\sim10^{-8}$ , three to four orders of magnitude deeper. Reaching that level would detect the guaranteed $\Lambda$ CDM signals from Silk damping and reionization, turning distortions into a probe of inflationary power on scales a thousandfold smaller than the anisotropies reach, and a sharp test of decaying dark matter.

The physical picture

Picture the photon gas as a fixed pile of coins, each stamped with an energy value. Cosmic expansion lowers every value by the same factor, so the shape of the distribution never changes — it stays a Planck curve, only cooler. To distort the curve you must either heat the coins (add energy) or mint and melt them (change their number). Compton scattering off hot electrons shakes value from coin to coin without changing how many there are; double Compton scattering and bremsstrahlung are the mint and furnace that create and destroy the low-value coins needed to rebuild a true blackbody. The whole story is a race between these processes and the expansion, and which wins is set by the epoch of energy release — laid out in the first figure.

Figure

The fossil-record timeline of CMB thermalization: energy injected above z_th ≈ 2×10⁶ is erased into a pure blackbody, injection in the μ-era 5×10⁴ ≲ z ≲ 2×10⁶ leaves a Bose–Einstein chemical potential μ, and injection in the y-era z ≲ 5×10⁴ leaves a Compton y-distortion

The distortions have a definite spectral fingerprint. A $\mu$ -distortion is a low-frequency photon deficit — a Bose–Einstein spectrum. A $y$ -distortion is the familiar cluster one: hot electrons push photons up in frequency, robbing the Rayleigh–Jeans side and piling up the Wien side, with a null in between at the fixed frequency $x_0=3.83$ — for the CMB temperature, $217\,\mathrm{GHz}$ , where every tSZ survey places a channel. The second figure draws that signature from the exact expression.

Figure

The spectral signature of a Compton y-distortion, ΔI_ν/I_ν = y·g(x) with x = hν/k_BT: a decrement at low frequency with ΔT/T → −2y in the Rayleigh–Jeans limit, an exact null at x₀ = 3.83 which is ν ≈ 217 GHz, and an increment in the Wien tail — the thermal Sunyaev–Zel'dovich shape toward clusters

The core equations

Everything begins with the equilibrium spectrum. The Planck function gives the specific intensity of blackbody radiation versus frequency and temperature — the shape the early universe reached and the one every distortion is measured against.

B_\nu(T)=\frac{2h\nu^3}{c^2}\,\frac{1}{e^{h\nu/k_BT}-1} \qquad (1)

where: $B_\nu$ — specific intensity per unit frequency; $h$ — Planck constant; $\nu$ — frequency; $c$ — speed of light; $k_B$ — Boltzmann constant; $T$ — blackbody temperature, $T_0=2.7255\,\mathrm{K}$ today. All temperature dependence enters through $x=h\nu/k_BT$ . It is cleaner to track the dimensionless photon occupation number, the mean number of photons per mode. It strips away the kinematic $\nu^3$ prefactor and depends on frequency and temperature only via $x$ .

n(x)=\frac{1}{e^{x}-1},\qquad x\equiv\frac{h\nu}{k_BT} \qquad (2)

where: $n$ — photon occupation (mean number per mode); $x$ — dimensionless frequency $h\nu/k_BT$ , invariant under redshifting. Any process that leaves $x$ unchanged leaves the spectrum a blackbody — why expansion cannot distort it. When Compton scattering reaches kinetic equilibrium but photon number is frozen, the occupation relaxes not to a Planck form but to a Bose–Einstein form with a non-zero chemical potential. This is the $\mu$ -distortion, the endpoint of photon diffusion at fixed number.

n_\mu(x)=\frac{1}{e^{x+\mu}-1} \qquad (3)

where: $\mu$ — dimensionless chemical potential in units of $k_BT$ , zero for a true blackbody and positive for a photon deficit; $x$ — $h\nu/k_BT_e$ . A positive $\mu$ suppresses the occupation most severely at low frequency — exactly the deficit photon creation would fill. If instead energy is injected late, when Compton scattering can only partially redistribute it, the surviving spectrum is a $y$ -distortion. Its fractional shape is parameter-free apart from the amplitude $y$ — the curve in the second figure.

\frac{\Delta I_\nu}{I_\nu}\bigg|_y = y\,\frac{x\,e^{x}}{e^{x}-1}\left[x\,\frac{e^{x}+1}{e^{x}-1}-4\right] \qquad (4)

where: $y$ — Compton $y$ -parameter (the integrated electron pressure the photons cross); $x$ — $h\nu/k_BT$ . The bracket vanishes at $x_0\simeq3.83$ ( $\nu\approx217\,\mathrm{GHz}$ ); negative below (a decrement, $\Delta T/T\to-2y$ as $x\to0$ ), positive above (the Wien increment). The amplitude $y$ is a line-of-sight integral of the fractional energy each scattering hands from hot electrons to photons. It is a pure number — the same quantity whether the electrons sit in the early plasma or a cluster today.

y=\int \frac{k_B(T_e-T_\gamma)}{m_e c^2}\,\sigma_T\,n_e\,c\,\mathrm{d}t \qquad (5)

where: $T_e$ — electron temperature; $T_\gamma$ — photon temperature; $m_ec^2$ — electron rest energy, $511\,\mathrm{keV}$ ; $\sigma_T$ — Thomson cross section, $6.652\times10^{-25}\,\mathrm{cm}^2$ ; $n_e$ — free-electron density. The integrand is the energy gain per unit optical depth, so $y$ is the scattering count times the per-scatter transfer. The equation governing how the occupation evolves under Compton scattering — from which both the Bose–Einstein endpoint and the $y$ -shape descend — is the Kompaneets equation.

\frac{\partial n}{\partial y_c}=\frac{1}{x^2}\frac{\partial}{\partial x}\left[x^4\left(\frac{\partial n}{\partial x}+n+n^2\right)\right] \qquad (6)

where: $n$ — photon occupation; $y_c$ — Compton time variable, $\mathrm{d}y_c=(k_BT_e/m_ec^2)\sigma_Tn_ec\,\mathrm{d}t$ ; $x$ — $h\nu/k_BT_e$ . The right side is an exact divergence in $x$ , so the photon number $\int x^2 n\,\mathrm{d}x$ is conserved; the terms $n+n^2$ carry the recoil and induced-scattering physics that make Bose–Einstein the stationary state. Finally, the practical translation: a fractional energy injection $\Delta E_\gamma/E_\gamma$ produces distortions whose amplitudes follow from moments of the distributions, while above $z_{\rm th}$ none survives.

\mu\simeq1.4\,\frac{\Delta E_\gamma}{E_\gamma},\qquad y\simeq\frac{1}{4}\frac{\Delta E_\gamma}{E_\gamma},\qquad z_{\rm th}\approx1.98\times10^{6}\left(\frac{\Omega_bh^2}{0.0224}\right)^{-2/5} \qquad (7)

where: $\Delta E_\gamma/E_\gamma$ — fractional energy added; the factor $1.4$ is a Bose–Einstein moment for pure $\mu$ -era heating; $z_{\rm th}$ — thermalization redshift; $\Omega_bh^2$ — physical baryon density controlling the double-Compton and bremsstrahlung rates. The FIRAS bounds therefore cap $\Delta E_\gamma/E_\gamma$ at a few $\times10^{-5}$ per era.

Derivation AWhy the CMB stays a blackbody under cosmic expansion

The goal is to prove rigorously that a photon gas which starts as a Planck spectrum stays a Planck spectrum as the universe expands, with only its temperature falling as $T\propto(1+z)$ , so that any observed departure from blackbody form must be blamed on non-gravitational energy injection, not on expansion. We track the occupation of a single mode using collisionless conservation of phase-space density, and show that the invariant $x=h\nu/k_BT$ carries the argument.

Step 1. Between scatterings the photon distribution function $f(\vec x,\vec p,t)$ is conserved along geodesics, $\mathrm{d}f/\mathrm{d}t=0$ — Liouville's theorem for a collisionless gas. Why: with no interactions to add, remove, or reshuffle photons, the occupation of each phase-space cell is frozen as the cell is carried along by the expansion.

Step 2. Every photon momentum redshifts identically with the scale factor, $p\propto a^{-1}$ , so $\nu\propto(1+z)$ with no frequency-dependent piece. Why: gravitational redshift stretches all wavelengths by the same ratio, reshaping nothing — a crucial contrast with Compton scattering, whose shift depends on frequency.

Step 3. The temperature obeys the same scaling, $T\propto(1+z)$ , so $x=h\nu/k_BT$ is independent of redshift and the occupation $n=(e^{x}-1)^{-1}$ is unchanged. Why: because $n$ depends only on $x$ , holding $x$ fixed holds the entire shape fixed even as both $\nu$ and $T$ fall.

Step 4. The spectrum observed at $z=0$ is thus a Planck spectrum at $T_0=T(z)/(1+z)$ , and the cross-checks $n_\gamma\propto(1+z)^3$ , $u_\gamma\propto(1+z)^4$ confirm consistent dilution and energy redshift. Why: recovering the standard density scalings shows this is the same physics that fixes the radiation content of the cosmos. *Verified: $T\propto(1+z)$ , $x$ invariant, and the $T^3$ / $T^4$ scalings reproduce the standard result in Ryden ch. 8 and Dodelson ch. 3.*

Go deeper · Derivation B · 1 minThe y-parameter from single-scattering energy transfer

Derivation BThe y-parameter from single-scattering energy transfer

The goal is to derive the Compton $y$ -parameter from the ground up — to show that summing the tiny fractional energy each photon gains crossing a cloud of hot electrons (in the regime $T_e\gg T_\gamma$ ) yields the integral in equation (5), and that the photon bath's total gain is $4y$ , fixing the normalization of both the $y$ -shape and the relation $y\simeq\tfrac14\,\Delta E_\gamma/E_\gamma$ .

Step 1. A photon scattering off an electron of temperature $T_e$ gains, on average, a fractional frequency shift $\langle\Delta\nu/\nu\rangle=4k_BT_e/m_ec^2$ , the leading Doppler-plus-recoil term of the Kompaneets kernel. Why: this is the mean energy the non-relativistic electron gas imparts per scattering, and the factor of four reappears in the definition of $y$ .

Step 2. The probability of a scattering in time $\mathrm{d}t$ is the optical-depth element $\mathrm{d}\tau=\sigma_Tn_ec\,\mathrm{d}t$ , so the expected number of scatterings is $\tau=\int\sigma_Tn_ec\,\mathrm{d}t$ . Why: each photon random-walks in frequency, and the number of steps is the Thomson optical depth — small for cluster gas and early plasma alike, keeping the distortion linear in $\tau$ .

Step 3. Multiplying and integrating gives the total fractional energy gain $\Delta E_\gamma/E_\gamma=\int(4k_BT_e/m_ec^2)\,\mathrm{d}\tau\equiv4y$ , which defines $y=\int(k_BT_e/m_ec^2)\,\mathrm{d}\tau$ . Why: the definition absorbs the factor of four, so $y$ measures the integrated electron pressure and $y=\tfrac14\,\Delta E_\gamma/E_\gamma$ .

Step 4. Feeding this input into the linearized Kompaneets equation at fixed photon number produces the shape of equation (4), with its decrement $\Delta T/T=-2y$ and null at $x_0\approx3.83$ . Why: the same scattering that transfers energy also dictates its frequency distribution, so amplitude and shape are not independent — either one gives $y$ . *Verified: reproduces the standard Sunyaev–Zel'dovich result; the RJ limit $\Delta T/T=-2y$ and the crossover $x_0=3.830$ ( $\nu=217.5\,\mathrm{GHz}$ ) match the direct numerical evaluation of equation (4) and Dodelson ch. 9.*

Go deeper · Derivation C · 1 minThe thermalization redshift from double Compton scattering

Derivation CThe thermalization redshift from double Compton scattering

The goal is to estimate the redshift $z_{\rm th}$ above which the universe can heal any distortion back into a perfect blackbody, and to see why it depends only weakly on the baryon density. Healing requires creating soft photons to fill the low-frequency deficit; the dominant creator is double Compton scattering, and the boundary is fixed by racing its rate against Hubble expansion.

Step 1. Double Compton scattering ( $e^-+\gamma\to e^-+\gamma+\gamma$ ) creates soft photons at a rate $\Gamma_{\rm DC}\propto n_e\,\alpha\,\sigma_T\,c\,(k_BT/m_ec^2)^2$ , steeply rising toward early times. Why: photon creation is what a blackbody needs and Compton scattering alone cannot supply; the strong temperature dependence confines it to the very early universe.

Step 2. Setting the creation rate equal to the expansion rate, $\Gamma_{\rm DC}(z_{\rm th})=H(z_{\rm th})$ with $H\propto(1+z)^2$ in the radiation era, fixes the epoch where photon creation can no longer keep pace. Why: above the crossover a distortion is refilled faster than expansion freezes it; below it, soft photons cannot be replenished in a Hubble time and the scar is permanent.

Step 3. Solving the balance gives $z_{\rm th}\approx1.98\times10^{6}(\Omega_bh^2/0.0224)^{-2/5}$ , with bremsstrahlung taking over at even lower frequencies but double Compton setting the boundary. Why: the weak $-2/5$ power keeps $z_{\rm th}$ near $2\times10^6$ across plausible cosmologies, making the $\mu$ -era ceiling a firm prediction.

Step 4. Energy released above $z_{\rm th}$ raises the temperature but leaves no distortion; released below it, the injection survives as $\mu$ or $y$ depending on whether Compton scattering still reaches equilibrium. Why: this one boundary splits all energy-injection history into an unobservable part and an observable fossil record. *Verified: the scaling and the $(\Omega_bh^2)^{-2/5}$ dependence match Chluba & Sunyaev (2012); plugging $\Omega_bh^2=0.02237$ gives $z_{\rm th}=1.98\times10^{6}$ .*

Limiting cases

The surviving distortion is set entirely by the redshift of energy injection — the four epochs the timeline figure already sketched.

Case	Result	Meaning
Full thermalization	$z\gtrsim z_{\rm th}\approx2\times10^{6}$	Double Compton and bremsstrahlung refill any deficit; a perfect blackbody is restored and injected energy only raises $T$ .
$\mu$ -era	$5\times10^{4}\lesssim z\lesssim2\times10^{6}$	Compton scattering enforces kinetic equilibrium but photon number is frozen, so a chemical potential $\mu\neq0$ develops.
$y$ -era	$z\lesssim5\times10^{4}$	Compton scattering cannot finish the redistribution; hot electrons up-scatter photons into a $y$ -distortion (the thermal SZ shape).
Intermediate / residual	$1.5\times10^{4}\lesssim z\lesssim5\times10^{4}$	Neither pure $\mu$ nor pure $y$ ; the exact frequency dependence carries extra information on the injection redshift.

Common trapRedshift distorts the blackbody — it does not

It is tempting to think that stretching the CMB across billions of years must smear its spectrum. It does not. Expansion redshifts every frequency and cools the temperature by the same factor $1+z$ , so $x=h\nu/k_BT$ is invariant and a Planck curve of temperature $T$ maps precisely onto one of temperature $T/(1+z)$ — same shape, lower temperature. Distortions require energy or entropy injection, never expansion. A subtler trap: $\mu$ and $y$ are not merely "early" versus "late" heating of the same kind — they differ in whether photon number was conserved during thermalization. A $\mu$ -distortion is a genuine Bose–Einstein equilibrium reached at frozen photon number, whereas a $y$ -distortion is an out-of-equilibrium partial up-scattering; reading $\mu$ as "a bit more $y$ " gets the microphysics wrong.

Worked examples

1 · The y-parameter and temperature decrement of a galaxy cluster (fully worked)

A massive cluster has an intracluster medium with electron temperature $k_BT_e=5\,\mathrm{keV}$ , mean electron density $n_e=10^{-3}\,\mathrm{cm}^{-3}$ , and a path length through the gas of $L=2\,\mathrm{Mpc}$ . Find the Compton $y$ -parameter and the resulting Rayleigh–Jeans temperature decrement in microkelvin.

Step 1. For a uniform slab $y=(k_BT_e/m_ec^2)\,\sigma_T\,n_e\,L$ . The energy transfer per scattering is $k_BT_e/m_ec^2=5/511=\mathbf{9.79\times10^{-3}}$ , since $m_ec^2=511\,\mathrm{keV}$ .

Step 2. The Thomson optical depth is $\tau=\sigma_T n_e L=(6.652\times10^{-25})(10^{-3})(2\times3.086\times10^{24})=\mathbf{4.11\times10^{-3}}$ in cgs — a few photons per thousand scatter through.

Step 3. Multiplying, $y=(9.79\times10^{-3})(4.11\times10^{-3})=\mathbf{4.0\times10^{-5}}$ . This exceeds the FIRAS bound $|y|<1.5\times10^{-5}$ , but that bound is on the monopole; a cluster is a compact patch measured against the local CMB.

Step 4. In the Rayleigh–Jeans limit $\Delta T/T=-2y=-8.0\times10^{-5}$ , so $\Delta T=-2y\,T_0=\mathbf{-219\,\mu K}$ . A 5 keV cluster along a 2 Mpc sightline gives $y\approx4\times10^{-5}$ and a decrement of about $-220\,\mu$ K, turning to an increment above $217\,\mathrm{GHz}$ .

2 · From the FIRAS μ-limit to an energy-injection bound

Use the COBE/FIRAS limit $|\mu|<9\times10^{-5}$ to bound the fractional energy that could have been injected during the $\mu$ -era, and compare with the guaranteed $\Lambda$ CDM signal from Silk damping, $\mu\approx2\times10^{-8}$ .

Step 1. Invert $\mu\simeq1.4\,\Delta E_\gamma/E_\gamma$ to get $\Delta E_\gamma/E_\gamma\lesssim\mu/1.4=(9\times10^{-5})/1.4=\mathbf{6.4\times10^{-5}}$ : no more than about six parts in $10^5$ of the CMB energy can have been dumped between $z\approx5\times10^4$ and $2\times10^6$ .

Step 2. The Silk-damping prediction $\mu\approx2\times10^{-8}$ corresponds to $\Delta E_\gamma/E_\gamma\approx\mathbf{1.4\times10^{-8}}$ — the guaranteed heating from dissipating acoustic waves.

Step 3. FIRAS is thus about $9\times10^{-5}/2\times10^{-8}\approx\mathbf{4500}$ times too insensitive to see this floor; a spectrometer reaching $\mu\sim10^{-8}$ would cross it and open the $\mu$ -window as a probe of inflationary small-scale power. Current data bound injected energy at $\lesssim6\times10^{-5}$ ; the guaranteed signal is a thousandfold smaller — a target for the next generation.

3 · Where the tSZ null sits

Show that the $y$ -distortion crossover $x_0=3.83$ corresponds to $\nu\approx217\,\mathrm{GHz}$ today.

Step 1. The frequency scale is $k_BT_0/h=(1.381\times10^{-23})(2.7255)/(6.626\times10^{-34})=\mathbf{56.8\,GHz}$ .

Step 2. Multiply by the null location: $\nu_0=x_0\,(k_BT_0/h)=3.830\times56.8=\mathbf{217.5\,GHz}$ . This is why every SZ survey places a channel near $217\,\mathrm{GHz}$ , where the cluster signal vanishes and gives a clean reference. The $217\,\mathrm{GHz}$ null is just $x=3.83$ dressed in the CMB temperature.

Physics of the portrait

The theory of CMB spectral distortions is a distinctly Soviet achievement, built in Moscow in the 1950s and 1960s from a plasma-physics diffusion equation to a cluster prediction now central to cosmology.

Origins · 1 minRashid Sunyaev and how this was discovered

Rashid Sunyaev

b. 1943, with Yakov Zeldovich (1914–1987) and Alexander Kompaneets (1914–1974) · CMB spectral distortions and the Sunyaev–Zel'dovich effect

The physics of spectral distortions was assembled by the Soviet school. Alexander Kompaneets, working in the late 1950s on radiation in hot plasmas, derived the diffusion equation that bears his name — how Compton scattering redistributes photons in frequency while conserving their number, a result that circulated for years before publication owing to its weapons-laboratory origins. Yakov Zeldovich recognized in the 1960s that Kompaneets' equation was the key to the thermal history of the microwave background just discovered by Penzias and Wilson. With his young collaborator Rashid Sunyaev, Zeldovich showed in 1969 that any early energy release would leave a $\mu$ - or $y$ -type imprint on the CMB, and in 1970 the pair predicted that hot electrons in galaxy-cluster atmospheres would scatter CMB photons into exactly the same $y$ -distortion — the thermal Sunyaev–Zel'dovich effect. Sunyaev, who went on to lead high-energy astrophysics in Moscow and at the Max Planck Institute, saw that prediction turn from an obscure sub-percent effect into a primary way clusters are found and weighed across the universe, and he carries the work forward today in the numerical thermalization codes that compute the exact $\mu$ – $y$ –residual signal.

History

1957

— Kompaneets derives the photon diffusion equation for Compton scattering off thermal electrons

1965

— Penzias and Wilson detect the cosmic microwave background; its blackbody nature becomes a target

1969

— Zeldovich and Sunyaev show that early energy release yields μ- and y-type spectral distortions

1996

— COBE/FIRAS confirms a near-perfect blackbody and sets |μ| < 9×10⁻⁵, |y| < 1.5×10⁻⁵

2012

— Chluba and Sunyaev compute the full thermalization Green's function bridging the μ and y regimes

Going further

Texts. Ryden, Introduction to Cosmology, 2nd ed., ch. 8 · Dodelson, Modern Cosmology, 2nd ed., ch. 3 and ch. 9 · Carroll & Ostlie, An Introduction to Modern Astrophysics, 2nd ed., ch. 30 · Mo, van den Bosch & White, Galaxy Formation and Evolution, ch. 3 · Peebles, Principles of Physical Cosmology, §24.

Provenance. Anchor numbers to verify: $T_0=2.7255\,\mathrm{K}$ (Fixsen 2009, ApJ 707, 916); FIRAS bounds $|\mu|<9\times10^{-5}$ , $|y|<1.5\times10^{-5}$ (Fixsen et al. 1996); $\sigma_T=6.652\times10^{-25}\,\mathrm{cm}^2$ , $m_ec^2=511\,\mathrm{keV}$ ; $z_{\rm th}\approx1.98\times10^6$ and the $\mu$ / $y$ boundary $z\approx5\times10^4$ (Chluba & Sunyaev 2012, MNRAS 419, 1294); the tSZ null $x_0=3.830$ ( $\nu=217.5\,\mathrm{GHz}$ ) and $\Delta T/T=-2y$ from direct evaluation of the $y$ -shape; $\mu\simeq1.4\,\Delta E/E$ , $y\simeq\tfrac14\,\Delta E/E$ from moments of the distributions. Foundational papers: Zeldovich & Sunyaev (1969); Sunyaev & Zeldovich (1970); Kompaneets (1957).

Prerequisites. Equilibrium Thermodynamics of the Early Universe · Thermal History of the Universe · Entropy Conservation and Comoving Entropy · Recombination and the Saha Equation. Leads to. Electron–Positron Annihilation and Photon Reheating · Decoupling and Freeze-Out · Baryon Asymmetry and Baryogenesis.

Ask the tutor: Ask me to walk through the Kompaneets equation, work out a cluster's tSZ signal at your favourite frequency, or show how a decaying-particle lifetime turns into a $\mu$ -distortion bound.

Why is the CMB such a good blackbody if nothing forces it to stay one after decoupling?

Nothing needs to. The spectrum was thermalized to a Planck form at $z\gtrsim2\times10^6$ , when double Compton scattering and bremsstrahlung created and destroyed photons fast enough to lock in true equilibrium. After that, expansion preserves the shape automatically — it redshifts frequency and temperature by the same factor, and stretching a Planck curve gives another Planck curve. The only way to spoil it is to inject energy after thermalization froze, and FIRAS tells us almost nothing did.

What is the difference between the CMB temperature T₀ and a spectral distortion?

A change in $T_0$ slides the whole Planck curve up or down while keeping its exact shape — a change in the blackbody's single parameter. A distortion is a change in the shape: the spectrum can no longer be described by any single temperature. Heating before thermalization freezes only raises $T_0$ ; heating afterward cannot be absorbed by a temperature shift, because photon number is frozen and the extra energy must go somewhere in frequency, producing the $\mu$ or $y$ deviation. That is why distortions, not the mean temperature, carry the record of late energy release.

Is the thermal Sunyaev–Zel'dovich effect really the same physics as a cosmological y-distortion?

Yes — identical microphysics, different electrons. In both cases hot electrons Compton up-scatter CMB photons at fixed photon number, giving the universal $y$ -spectrum, decrement below $217\,\mathrm{GHz}$ and increment above. The cosmological $y$ comes from the diffuse early plasma and the reionized intergalactic medium and is a monopole signal; the cluster tSZ effect comes from the dense $10^7$ – $10^8\,\mathrm{K}$ gas in one cluster and appears as a compact patch. Only the geometry and the electron populations differ.

Why does the μ-distortion require photon number to be frozen?

A true blackbody has a definite number of photons for its energy. Add energy but forbid the corresponding photons, and the system cannot reach a Planck spectrum — it has too much energy for its photon count, so Compton scattering relaxes it to the next best thing, a Bose–Einstein distribution whose chemical potential $\mu$ is exactly the bookkeeping for the deficit. So $\mu$ is nonzero precisely because the photon-creating reactions have frozen out while photon-shuffling Compton scattering is still fast; restore the creators and $\mu$ returns to zero.

How do spectral distortions constrain dark matter?

Any dark-matter particle that decays or annihilates into electromagnetic energy at $10^3\lesssim z\lesssim2\times10^6$ heats the photon bath and leaves a $\mu$ or $y$ imprint set by the injected energy. The FIRAS bounds therefore translate directly into limits on decay lifetimes and annihilation cross sections — no more than a few parts in $10^5$ of the CMB energy could have been injected. A spectrometer reaching $\mu\sim10^{-8}$ would probe lifetimes and cross sections thousands of times smaller, covering dark-sector parameter space no laboratory can reach.

Why does the thermalization redshift depend on the baryon density?

Because the reactions that rebuild a blackbody — double Compton and, at low frequencies, bremsstrahlung — both need free electrons, whose abundance tracks $\Omega_bh^2$ . More baryons means faster photon creation, letting the universe thermalize distortions to slightly later times and lowering $z_{\rm th}$ . The dependence is weak, $z_{\rm th}\propto(\Omega_bh^2)^{-2/5}$ , so across all plausible cosmologies the boundary stays near $2\times10^6$ — which makes the $\mu$ -era ceiling a firm, nearly cosmology-independent prediction.