Sunyaev–Zel'dovich Effect

The cosmic microwave background is a nearly perfect $2.725$ K blackbody that has free-streamed to us since the surface of last scattering. On its way, a small fraction of those photons pass through the deep gravitational wells of galaxy clusters, where they meet the intracluster medium (ICM): a diffuse plasma with electron temperatures of tens of millions of kelvin and densities $n_e\sim10^{-3}$ cm $^{-3}$ . The electrons are non-relativistic on average ( $k_BT_e\ll m_ec^2$ ) but far hotter than the $2.7$ K radiation, so inverse-Compton scattering pumps energy from plasma to photons. Because scattering conserves photon number, this does not warm the whole spectrum uniformly — it lifts photons out of the Rayleigh–Jeans tail and deposits them on the Wien side, carving a fixed spectral distortion into the CMB. That is the thermal Sunyaev–Zel'dovich (tSZ) effect. Two small numbers set its size: the optical depth $\tau_e\sim10^{-2}$ (the chance a photon scatters) and the fractional energy boost per scatter $\sim k_BT_e/m_ec^2\sim10^{-2}$ . Their product is the Compton- $y$ parameter, $y\sim10^{-4}$ , which is nothing but the electron pressure integrated along the line of sight — the SZ effect literally weighs the thermal energy of the cluster gas. The distortion is a fractional change $\Delta I/I$ in the CMB brightness, and because a cluster sits inside that same bath at every epoch, the surface-brightness signal is independent of redshift. This makes SZ surveys — SPT, ACT, Planck — nearly mass-selected cluster finders to arbitrary distance, and turns cluster counts into a sharp probe of $\sigma_8$ and $\Omega_m$ . The cluster's bulk motion adds the kinematic SZ effect, and pairing SZ with X-ray emission yields a geometric distance and $H_0$ . This article is the flagship synthesis: what the number $y$ is, what the spectrum $g(x)$ does, and why the crossover at $217$ GHz is a fingerprint.

Key takeaways

The Sunyaev–Zel'dovich effect is the small spectral distortion of the CMB produced when its photons inverse-Compton scatter off the hot ( $T_e\sim10^7$ – $10^8$ K) electrons of the intracluster medium.
The thermal signal is set by the dimensionless Compton- $y$ parameter, the line-of-sight integral of electron pressure $y=\int(k_BT_e/m_ec^2)\,n_e\sigma_T\,\mathrm{d}\ell$ ; rich clusters have $y\sim10^{-4}$ .
The spectrum is a fixed shape $g(x)=x\coth(x/2)-4$ : a temperature decrement below $\sim218$ GHz, a null at $\nu_0\approx217.5$ GHz, and an increment above — and because it is a fractional change it is redshift-independent.
The kinematic SZ effect adds a Doppler shift $\Delta T/T=-\tau_e v_r/c$ from the cluster's peculiar velocity, spectrally a pure temperature shift with no null.
Combining SZ (linear in $n_e$ ) with X-ray brightness (quadratic in $n_e$ ) gives the angular-diameter distance and a ladder-independent $H_0$ ; the integrated $Y_{\rm SZ}$ is a low-scatter cluster mass proxy.
Key idea: The thermal SZ effect is a redshift-independent fractional CMB distortion whose amplitude equals the integrated electron pressure (Compton- $y$ ) and whose spectrum is the fixed function $g(x)=x\coth(x/2)-4$ , crossing zero at $\nu\approx218$ GHz.

Where this lives today

The SZ effect has become industrial cosmology. The South Pole Telescope, the Atacama Cosmology Telescope, and the Planck satellite have assembled catalogues of thousands of clusters selected by their tSZ signature out to $z\gtrsim1.5$ , and because that selection is nearly a clean mass cut, the abundance of clusters versus mass and redshift constrains $\sigma_8$ and $\Omega_m$ and tests structure growth against $\Lambda$ CDM — a probe complementary to the primary CMB. The integrated $Y_{\rm SZ}$ serves as the low-scatter mass proxy that anchors these counts, calibrated by weak lensing. Meanwhile the kinematic SZ effect, once thought hopeless, is now detected statistically through the pairwise-momentum estimator and through velocity-reconstruction cross-correlations, mapping the large-scale peculiar-velocity field, testing gravity, and weighing the diffuse "missing baryons" in the warm–hot intergalactic medium. Next-generation experiments — the Simons Observatory and CMB-S4 — will push tSZ cluster catalogues past $10^5$ objects and turn kSZ into a precision baryon and velocity probe.

The physical picture

Picture a cold CMB photon random-walking through a cloud of blisteringly hot electrons. Each rare collision, on average, kicks the photon to slightly higher energy — the electrons are so much hotter that energy flows from plasma to radiation. Since collisions conserve photon number, the spectrum is not warmed uniformly; instead photons are stolen from the low-frequency Rayleigh–Jeans tail and piled onto the high-frequency Wien tail. Toward a cluster the microwave sky therefore looks slightly cold in the radio band and slightly hot in the sub-millimetre. The first figure fixes the amplitude: it is the product of two tiny numbers, the optical depth and the fractional energy boost, whose product is the Compton- $y$ .

Figure

A cold CMB photon (2.725 K) threads the hot intracluster plasma and is inverse-Compton up-scattered by electrons at T_e ≈ 10⁸ K; the Compton-y amplitude is the product of the optical depth τ_e ≈ 2×10⁻³ and the fractional energy boost k_BT_e/m_ec² ≈ 0.016, giving y ≈ 3×10⁻⁵

The tSZ spectrum is redshift-independent because the CMB is a bath the cluster is immersed in at every epoch: the scattering imprints a fractional distortion $\Delta T/T$ , and as $T_{\rm CMB}\propto(1+z)$ rises with lookback time while the surface brightness dims identically, the fractional signature survives unattenuated. A $z=1.5$ cluster's decrement is as deep in $\Delta T/T$ as a nearby one's — a property no flux-limited survey can match. The second figure is the observable a spectrometer actually measures: the fixed distortion shape, decrement below the null and increment above.

Figure

The thermal SZ brightness distortion of the CMB across frequency: a decrement below the null at ν₀ ≈ 217 GHz (deepest near 128 GHz) and an increment above it peaking near 375 GHz — the fixed shape ΔI ∝ (x⁴eˣ/(eˣ−1)²)(x·coth(x/2) − 4) that separates tSZ from all foregrounds

The core equations

The whole thermal effect is packed into a single line integral: the Compton- $y$ parameter, the electron pressure summed along the sight line in units of the electron rest energy. It is simultaneously the amplitude of the distortion and a direct measure of the gas's thermal energy content.

y=\int\frac{k_BT_e}{m_ec^2}\,n_e\,\sigma_T\,\mathrm{d}\ell \qquad (1)

where: $y$ — dimensionless Compton parameter (integrated electron pressure); $k_B$ — Boltzmann constant; $T_e$ — electron temperature of the ICM; $m_ec^2$ — electron rest energy, $511$ keV; $n_e$ — electron number density; $\sigma_T$ — Thomson cross section, $6.65\times10^{-25}$ cm $^2$ ; $\mathrm{d}\ell$ — path-length element. Read it as the product of optical depth $\tau_e=\int n_e\sigma_T\,\mathrm{d}\ell$ and the mean fractional energy shift per scatter $\sim k_BT_e/m_ec^2$ . The measured distortion is $y$ times a universal frequency shape $g(x)$ . That the shape is fixed — the same for every cluster regardless of temperature or redshift — is what makes tSZ separable from everything else on the sky.

\frac{\Delta T}{T_{\rm CMB}}=y\,g(x),\qquad g(x)=x\coth\!\left(\frac{x}{2}\right)-4,\qquad x=\frac{h\nu}{k_BT_{\rm CMB}} \qquad (2)

where: $\Delta T/T_{\rm CMB}$ — fractional change in CMB brightness temperature; $g(x)$ — universal tSZ spectral function; $x$ — dimensionless frequency; $h$ — Planck constant; $\nu$ — observing frequency; $T_{\rm CMB}=2.725$ K. In the radio limit $g\to-2$ (a decrement); in the sub-mm $g>0$ (an increment). The shape crosses zero at a single frequency — the spectral null — where the decrement turns into the increment. It is fixed by the transcendental root of $g$ , independent of the cluster.

x_0\coth\!\left(\frac{x_0}{2}\right)=4\;\Rightarrow\;x_0\approx3.830,\qquad \nu_0=\frac{x_0k_BT_{\rm CMB}}{h}\approx217.5\ \text{GHz} \qquad (3)

where: $x_0$ — dimensionless frequency where $g(x_0)=0$ ; $\nu_0$ — crossover frequency separating decrement from increment. Observing at $\nu_0$ blinds you to the thermal effect and reveals the kinematic one. In intensity units the distortion carries an extra blackbody weighting; this is the form a bolometer records, with a deepest decrement near $128$ GHz and a peak increment near $370$ GHz.

\frac{\Delta I}{I_0}=y\,\frac{x^4e^x}{(e^x-1)^2}\left[x\coth\!\left(\frac{x}{2}\right)-4\right],\qquad I_0=\frac{2(k_BT_{\rm CMB})^3}{(hc)^2} \qquad (4)

where: $\Delta I$ — change in specific intensity; $I_0$ — characteristic CMB intensity scale; $c$ — speed of light. The factor $x^4e^x/(e^x-1)^2$ converts the temperature distortion $(2)$ to brightness and is why the increment appears larger than the decrement. A cluster in bulk motion Doppler-shifts the photons it scatters, adding a second, spectrally distinct signal: the kinematic SZ effect. It has no null because it is simply a shift of the whole blackbody.

\left.\frac{\Delta T}{T_{\rm CMB}}\right|_{\rm kSZ}=-\,\tau_e\,\frac{v_r}{c} \qquad (5)

where: $\tau_e$ — cluster Thomson optical depth; $v_r$ — line-of-sight peculiar velocity (positive = receding); $c$ — speed of light. A cluster approaching us ( $v_r<0$ ) makes a hot spot; only the scattered fraction $\tau_e$ carries the shift to the observer. Integrating $y$ over the cluster's solid angle gives the total Compton signal $Y_{\rm SZ}$ , proportional to the cluster's total thermal energy — which, through hydrostatic equilibrium, is a tight, low-scatter proxy for mass.

Y_{\rm SZ}\equiv\int y\,\mathrm{d}\Omega=\frac{\sigma_T}{m_ec^2D_A^2}\int P_e\,\mathrm{d}V,\qquad Y_{\rm SZ}\,D_A^2\propto M^{5/3}E(z)^{2/3} \qquad (6)

where: $Y_{\rm SZ}$ — solid-angle integral of $y$ ; $D_A$ — angular-diameter distance; $P_e=n_ek_BT_e$ — electron pressure; $\mathrm{d}V$ — volume element; $E(z)=H(z)/H_0$ . The self-similar scaling makes $Y_{\rm SZ}$ the observable of choice for cluster-count cosmology. Finally, because SZ is linear in density and X-ray brightness is quadratic, their ratio eliminates $n_e$ and fixes the line-of-sight depth in physical units — a purely geometric distance.

D_A\propto\frac{(\Delta T_{\rm SZ})^2}{S_X}\,\frac{1}{T_e^{3/2}\,\theta_c} \qquad (7)

where: $\Delta T_{\rm SZ}$ — central SZ decrement ( $\propto n_eT_e\,\ell$ ); $S_X$ — X-ray surface brightness ( $\propto n_e^2\Lambda(T_e)\,\ell$ ); $\theta_c$ — angular core radius; $\Lambda(T_e)$ — X-ray cooling function. Averaging over many clusters beats down the spherical-symmetry assumption and yields $H_0$ .

Derivation AThe thermal SZ spectral shape from the Kompaneets equation

The goal is to derive the universal tSZ spectral function $g(x)=x\coth(x/2)-4$ directly from the Kompaneets equation, treating a cluster as an optically thin cloud in which each CMB photon undergoes a small number of Compton scatters off electrons far hotter than the radiation. We start from the photon-diffusion equation in frequency space, linearize in the small parameter $y$ , insert the incident Planck spectrum, carry out the frequency derivatives, and read off both the fixed shape and its Rayleigh–Jeans, Wien and crossover limits — the full fingerprint that identifies the effect on the sky.

Step 1. Write the Kompaneets equation for the photon occupation number $n$ , $\dfrac{\partial n}{\partial y}=\dfrac{1}{x^2}\dfrac{\partial}{\partial x}\!\left[x^4\!\left(\dfrac{\partial n}{\partial x}+n+n^2\right)\right]$ , in which the Compton- $y$ plays the role of a dimensionless time. Why: repeated small-angle Compton scatters make photons diffuse in energy, and Kompaneets is exactly the Fokker–Planck equation for that diffusion; $y$ measures how far the diffusion has proceeded.

Step 2. Keep only first order in the tiny $y$ , so $\Delta n=y\,\dfrac{1}{x^2}\dfrac{\partial}{\partial x}\!\left[x^4\dfrac{\partial n_0}{\partial x}\right]$ , dropping the $n+n^2$ terms where $n\ll1$ . Why: the cluster is optically thin ( $y\sim10^{-4}$ ), so a single linear term captures the entire observable distortion; the stimulated-scattering terms only enforce Bose statistics deep in the Rayleigh–Jeans regime and do not change the leading shape.

Step 3. Insert the incident Planck spectrum $n_0=(e^x-1)^{-1}$ with derivative $\dfrac{\partial n_0}{\partial x}=-\dfrac{e^x}{(e^x-1)^2}$ , now writing $x=h\nu/k_BT_{\rm CMB}$ since the photons that reach us carry the CMB temperature. Why: the radiation entering the cluster is the unperturbed CMB blackbody, and its curvature in frequency is what the diffusion operator acts on.

Step 4. Carry out the two derivatives and simplify with $\coth(x/2)=(e^x+1)/(e^x-1)$ to obtain $\dfrac{\Delta n}{n_0}=y\,\dfrac{xe^x}{e^x-1}\left[x\coth\!\left(\dfrac{x}{2}\right)-4\right]$ . Why: the bracket is the universal shape; the prefactor $xe^x/(e^x-1)$ is just the conversion between an occupation-number distortion and a brightness-temperature distortion.

Step 5. Convert to temperature to reach the boxed result. Why: observers report $\Delta T/T$ , and using the brightness-temperature relation strips off the prefactor to leave the clean spectral function $g(x)$ .

\frac{\Delta T}{T_{\rm CMB}}=y\left[x\coth\!\left(\frac{x}{2}\right)-4\right]\equiv y\,g(x)

Step 6. Check the limits: as $x\to0$ , $x\coth(x/2)\to2$ so $g\to-2$ (maximum decrement); the root $g(x_0)=0$ sits at $x_0\approx3.83$ ( $\nu_0\approx217.5$ GHz); and for $x\gg1$ , $g\to x-4$ (a growing increment). Why: these three limits are exactly the observed signature — cold in the radio, a null near $218$ GHz, hot in the sub-mm — confirming the derivation reproduces reality.

Go deeper · Derivation B · 1 minAngular-diameter distance from combined SZ and X-ray data

Derivation BAngular-diameter distance from combined SZ and X-ray data

The goal is to show how combining the SZ decrement, which is linear in electron density, with the X-ray surface brightness, which is quadratic in it, eliminates the unknown gas density and delivers the cluster's line-of-sight physical depth in centimetres — which, divided by the measured angular size, yields the angular-diameter distance $D_A$ and hence a Hubble constant independent of the local distance ladder. This is the classic geometric route that made SZ famous before it became a survey tool.

Step 1. The SZ decrement is linear in density, $\Delta T_{\rm SZ}\propto y\propto n_eT_e\,\ell$ , where $\ell$ is the physical path length through the gas. Why: each photon's expected energy gain is proportional to how many electrons it meets, which scales as $n_e\ell$ , times the per-scatter boost $\propto T_e$ .

Step 2. The X-ray surface brightness is quadratic in density, $S_X\propto n_e^2\,\Lambda(T_e)\,\ell$ . Why: X-rays come from thermal bremsstrahlung, a two-body process whose emissivity scales as the number of electron–ion encounters, hence as $n_e^2$ .

Step 3. Form the combination that cancels the density: solving the two relations for $n_e$ and eliminating it gives $\ell\propto\dfrac{(\Delta T_{\rm SZ})^2}{S_X}\,\dfrac{\Lambda(T_e)}{T_e^2}$ , a physical depth in centimetres from observables plus the measured $T_e$ . Why: squaring the linear SZ signal makes it $\propto n_e^2\ell^2$ , and dividing by the quadratic X-ray signal $\propto n_e^2\ell$ leaves one power of $\ell$ with the density gone — the whole trick in one line.

Step 4. Convert depth to distance: assuming the line-of-sight depth equals the transverse extent, $D_A=\ell/\theta_c$ from the imaged angular core size $\theta_c$ , and $H_0$ follows from $D_A$ and the redshift. Why: a physical length measured in cm together with the angle it subtends is precisely an angular-diameter distance; the sphericity assumption introduces per-cluster scatter that averages down over a sample.

Limiting cases

The non-relativistic, single-scattering, diffusion approximation governs the standard SZ formulae; departures appear in well-defined regimes.

Case	Result	Meaning
Rayleigh–Jeans limit ( $x\to0$ )	$g(x)\xrightarrow{x\to0}-2$	Maximum fractional decrement $\Delta T/T_{\rm CMB}=-2y$ ; radio/microwave observations see a deficit.
Wien limit ( $x\gg1$ )	$g(x)\xrightarrow{x\to\infty}x-4$	Positive increment growing with frequency; the sub-mm/far-IR sees an excess.
Crossover null	$x_0\coth(x_0/2)=4\Rightarrow x_0\approx3.830,\ \nu_0\approx217.5$ GHz	tSZ vanishes at first order; the ideal band to isolate the kinematic SZ effect.
Relativistic corrections ( $k_BT_e\gtrsim5$ keV)	$g(x)\to g(x)[1+\theta_e\delta_1(x)+\cdots],\ \theta_e=k_BT_e/m_ec^2$	For the hottest clusters the null shifts up by a few GHz and the shape distorts at the percent level.
Kinematic vs thermal	$\Delta T/T=-\tau_e v_r/c$ (no null) vs $y\,g(x)$	The kSZ is a pure temperature shift; the two separate cleanly at $\nu_0$ .

Common trapThe SZ effect just heats the CMB to a higher temperature

It does not. Inverse-Compton scattering conserves photon number, so it cannot warm the blackbody uniformly — it can only rearrange photons within the spectrum, removing them from the Rayleigh–Jeans tail and adding them to the Wien tail. That is why the signal is a decrement at low frequency and an increment at high frequency with a null in between, not a global temperature rise. Two further traps ride along with this one. First, the surface-brightness decrement $\Delta T/T=y\,g(x)$ is redshift-independent, but the integrated flux through a fixed beam still falls as $D_A^{-2}$ because a distant cluster subtends a smaller angle — so $Y_{\rm SZ}\propto D_A^{-2}$ even though $y$ does not. Second, "the electrons are non-relativistic" is only true on average; the hottest clusters ( $k_BT_e\sim10$ keV) need percent-level relativistic corrections that shift the null upward by a few GHz.

Worked examples

1 · Compton-y and the millikelvin decrement of a rich cluster (fully worked)

A massive cluster has central electron density $n_e=3\times10^{-3}$ cm $^{-3}$ , temperature $k_BT_e=9$ keV, and an effective path length $\ell=1.5$ Mpc through the core. Find the fractional energy boost, the optical depth, the central Compton- $y$ , and the Rayleigh–Jeans temperature decrement.

Step 1. Fractional energy boost per scatter: $\dfrac{k_BT_e}{m_ec^2}=\dfrac{9\ \text{keV}}{511\ \text{keV}}=\mathbf{1.76\times10^{-2}}$ . The electrons are hot but comfortably non-relativistic, so leading-order tSZ applies (with small relativistic corrections at this temperature).

Step 2. Optical depth, using $1$ Mpc $=3.086\times10^{24}$ cm so $\ell=4.63\times10^{24}$ cm: $\tau_e=n_e\sigma_T\ell=(3\times10^{-3})(6.652\times10^{-25})(4.63\times10^{24})=\mathbf{9.2\times10^{-3}}$ . About one photon in a hundred scatters — the cluster is optically thin, as assumed.

Step 3. Compton- $y$ is the product: $y=\tau_e\cdot\dfrac{k_BT_e}{m_ec^2}=(9.2\times10^{-3})(1.76\times10^{-2})=\mathbf{1.6\times10^{-4}}$ . This sits at the upper end of the observed range, appropriate for one of the richest clusters.

Step 4. Temperature decrement in the Rayleigh–Jeans band, where $g\to-2$ : $\Delta T=-2y\,T_{\rm CMB}=-2(1.6\times10^{-4})(2.725\ \text{K})=\mathbf{-0.89\ mK}$ . A sub-millikelvin decrement — routinely mapped by SPT and ACT at $150$ GHz. Result: y ≈ 1.6×10⁻⁴, giving a Rayleigh–Jeans decrement of about −0.9 mK.

2 · The spectral null and a kinematic hot spot

(a) Confirm the tSZ null frequency from $x_0=3.830$ . (b) The same cluster ( $\tau_e=9.2\times10^{-3}$ ) moves toward us at $v_r=-1000$ km s $^{-1}$ ; find its kinematic SZ signal.

Step 1. The natural frequency unit is $\dfrac{k_BT_{\rm CMB}}{h}=\dfrac{(1.381\times10^{-23})(2.725)}{6.626\times10^{-34}}=\mathbf{56.8\ GHz}$ . This is the frequency at which $x=1$ .

Step 2. The null sits at $\nu_0=x_0\times56.8\ \text{GHz}=3.830\times56.8=\mathbf{217.5\ GHz}$ — exactly the band where multi-frequency experiments place a channel to zero out the thermal effect.

Step 3. Kinematic signal: with $v_r/c=(-1000)/(2.998\times10^5)=-3.34\times10^{-3}$ , $\dfrac{\Delta T}{T_{\rm CMB}}=-\tau_e\dfrac{v_r}{c}=-(9.2\times10^{-3})(-3.34\times10^{-3})=+3.1\times10^{-5}$ , so $\Delta T=+3.1\times10^{-5}(2.725\ \text{K})=\mathbf{+84\ \mu K}$ . An approaching cluster is a hot spot.

Step 4. Compare magnitudes: the tSZ decrement was $\sim-900\ \mu$ K while the kSZ is only $\sim+84\ \mu$ K — an order of magnitude smaller, with no null. This is why kSZ is best hunted at $\nu_0$ where tSZ disappears, and why single-cluster kSZ detections are so demanding. Result: ν₀ ≈ 217.5 GHz; the kSZ hot spot is about +84 μK, ten times weaker than the thermal decrement.

3 · Redshift independence, and the flux caveat

Take the cluster of Example 1 ( $y=1.6\times10^{-4}$ ) and place it first at $z=0.2$ , then at $z=1.0$ . Compare the central decrement and the integrated flux.

Step 1. Central surface-brightness decrement: $\Delta T/T=y\,g(x)$ depends only on the local pressure, not on distance. In the Rayleigh–Jeans band it is $\mathbf{-3.2\times10^{-4}}$ at both redshifts — the $z=1$ cluster is exactly as easy to detect as the nearby one.

Step 2. Integrated Compton signal: $Y_{\rm SZ}=\int y\,\mathrm{d}\Omega\propto D_A^{-2}$ , because a fixed physical cluster subtends a smaller solid angle when farther away. Using standard $\Lambda$ CDM values $D_A(0.2)\approx670$ Mpc and $D_A(1.0)\approx1700$ Mpc.

Step 3. The ratio of integrated fluxes is $\left(\dfrac{670}{1700}\right)^2\approx\mathbf{0.16}$ : the distant cluster delivers about six times less total flux through a fixed beam, even though its central decrement is identical. Result: the decrement is redshift-independent (−3.2×10⁻⁴ at both z), but the integrated flux falls by ≈6× from z = 0.2 to z = 1.0.

Physics of the portrait

The SZ effect is one of the few cosmological probes named for a still-active line of theory: predicted in Moscow at the turn of the 1970s, ignored as undetectable for a decade, then turned into one of the sharpest tools in cosmology once the detectors caught up.

Origins · 1 minsunyaev–zeldovich effect and how this was discovered

sunyaev–zeldovich effect

CMB distortion ΔT/T = y·g(x), predicted 1969–1972 · high-energy astrophysics and cosmology

The effect carries two names because it came from a two-generation collaboration. Yakov Borisovich Zel'dovich (1914–1987) was already a titan of Soviet science — a founder of detonation theory, a leader of the nuclear program, and by the 1960s a driving force in relativistic astrophysics — when he turned to the microwave background. Rashid Alievich Sunyaev (born 1943), then his young student, would go on to co-author the Sunyaev–Zel'dovich reionization signature, the theory of accretion-disk spectra, and much of modern X-ray astronomy. Around 1969–1970 they realized that the hot electrons known to fill galaxy clusters must Compton-scatter CMB photons, and that Abram Kompaneets's equation for photon diffusion in a hot plasma predicted a precise, redshift-independent distortion. Their 1972 paper laid out the decrement, the increment, the $217$ GHz null, and the astonishing claim that the signal does not dim with distance. For years it seemed a curiosity: the decrements are sub-millikelvin, and early attempts through the 1970s were marginal. Firm detections toward rich clusters arrived in the 1980s (Birkinshaw and collaborators), interferometers such as OVRO and BIMA matured the technique in the 1990s, and the wide-field cameras of the South Pole Telescope, ACT and Planck finally realized Sunyaev and Zel'dovich's vision — catalogues of thousands of clusters selected by their SZ shadow.

History

1969

— Zel'dovich and Sunyaev predict a spectral distortion of the CMB by hot cluster electrons

1970

— the Kompaneets-based y-distortion and its universal spectral shape are worked out

1972

— Sunyaev and Zel'dovich publish the definitive treatment, including the 217 GHz null

1984

— Birkinshaw and collaborators secure the first firm cluster decrements

2009

— the South Pole Telescope discovers the first clusters found by their SZ signal alone

Going further

Texts. Sunyaev & Zel'dovich, Comments Astrophys. Space Phys. 4, 173 (1972) · Birkinshaw, Physics Reports 310, 97 (1999), "The Sunyaev–Zel'dovich effect" · Carlstrom, Holder & Reese, ARA&A 40, 643 (2002) · Carroll & Ostlie, An Introduction to Modern Astrophysics, 2nd ed., §27 · Dodelson, Modern Cosmology, 2nd ed., Ch. 9 · Mo, van den Bosch & White, Galaxy Formation and Evolution, §6.7 · Ryden, Introduction to Cosmology, 2nd ed., Ch. 8.

Provenance. Definitions and formulae follow Sunyaev & Zel'dovich (1972), Birkinshaw (1999), and Carlstrom, Holder & Reese (2002), with textbook treatments in Dodelson Ch. 9 and Ryden Ch. 8. Standard constants: $\sigma_T=6.652\times10^{-25}$ cm $^2$ , $m_ec^2=511$ keV, $T_{\rm CMB}=2.725$ K, crossover $x_0=3.830$ ( $\nu_0\approx217.5$ GHz), $k_BT_{\rm CMB}/h=56.8$ GHz. A verifier should recompute $g(x)=x\coth(x/2)-4$ and confirm $g(0)=-2$ and the root $x_0=3.830$ ; reproduce $y\approx1.6\times10^{-4}$ , $\tau_e\approx9\times10^{-3}$ , and the $-0.9$ mK decrement from the Example 1 parameters; and confirm the kSZ hot spot $\approx+84\ \mu$ K.

Prerequisites. Cosmic Microwave Background · Compton and Inverse-Compton Scattering · Intracluster Medium and X-ray Clusters · Kompaneets Equation. Leads to. Cluster Cosmology and Halo Mass Function · Peculiar Velocity Surveys · Supermassive Black Holes.

Ask the tutor: Ask me to walk through the Kompaneets derivation of $g(x)$ line by line, or to work a Compton- $y$ , null-frequency, or kinematic-SZ estimate for a cluster of your choosing.

Why is the SZ decrement redshift-independent but the flux is not?

The decrement $\Delta T/T=y\,g(x)$ is a fractional distortion of the CMB, and the cluster is bathed in that same CMB at every epoch. As the background heats as $(1+z)$ and dims in surface brightness identically, the fractional imprint survives untouched, so $y$ — set only by the local electron pressure — is what you measure regardless of distance. The total flux is different: it is $y$ integrated over the cluster's solid angle, $Y_{\rm SZ}\propto D_A^{-2}$ , and a fixed physical cluster subtends a smaller angle when farther away. So a distant cluster is just as easy to detect per beam but delivers less integrated signal — the redshift independence lives in the surface brightness, not the flux.

Is the SZ effect the same thing as the CMB dipole from our own motion?

No, though both are Doppler-related. The CMB dipole is our motion through the radiation, imprinting a $\sim3.4$ mK pattern on the whole sky. The kinematic SZ is a cluster's motion, and it only appears because the cluster's electrons scatter a fraction $\tau_e$ of the photons toward us, giving $\Delta T/T=-\tau_e v_r/c$ localized on that cluster. The suppression by $\tau_e\sim10^{-2}$ is exactly why kSZ signals are tens of microkelvin while the global dipole is millikelvin. The thermal SZ is different again: it comes from the random thermal motions of the electrons, not any bulk drift, and carries the distinctive $g(x)$ shape with a null.

If the electrons lose energy to the photons, why do we say the CMB is distorted rather than the gas cooled?

Both happen, but the numbers are wildly lopsided. Each scatter transfers a fraction $\sim k_BT_e/m_ec^2\sim10^{-2}$ of the electron's energy to a photon, and only a fraction $\tau_e\sim10^{-2}$ of photons scatter at all, so the energy the cluster loses to the CMB is utterly negligible compared with its vast thermal reservoir — the gas is not measurably cooled on any relevant timescale. From the photons' side, though, that same tiny transfer is the only thing happening to them, so it is entirely visible as a $y\sim10^{-4}$ distortion. We describe the effect from the photons' point of view because that is where the observable lives.

What fixes the 217 GHz null — is it a property of each cluster?

No, and that is the beauty of it. The null sits wherever the universal function $g(x)=x\coth(x/2)-4$ crosses zero, at $x_0\approx3.83$ , which corresponds to $\nu_0\approx217.5$ GHz for the fixed CMB temperature. It does not depend on the cluster's density, temperature, size, or redshift — every thermal SZ source in the Universe shares the same crossover (up to small relativistic shifts for the hottest gas). That universality is what makes the null so useful: place a detector there and the thermal effect vanishes, leaving the kinematic effect and letting you calibrate the separation of tSZ from primary CMB and foregrounds.

How do observers separate tSZ from primary CMB anisotropies, dust, and synchrotron?

By the fixed spectral shape. Primary CMB anisotropies have the spectrum of a temperature fluctuation (flat in $\Delta T$ , no null); galactic dust rises steeply into the sub-mm; synchrotron falls with frequency. The tSZ alone has the specific $g(x)$ form that is negative below $217$ GHz, exactly zero there, and positive above. Observing the same patch at several frequencies (say $95$ , $150$ , and $220$ GHz for SPT/ACT, or Planck's nine bands) lets a linear combination isolate the component that changes sign at the null and matches $g(x)$ . This multi-frequency "component separation" is what turns a jumble of overlapping signals into a clean Compton- $y$ map.

When do relativistic corrections matter, and what do they do?

They matter for the hottest clusters, $k_BT_e\gtrsim5$ keV, where the expansion parameter $\theta_e=k_BT_e/m_ec^2$ reaches a few percent. The corrections multiply the spectrum by a series $g(x)[1+\theta_e\delta_1(x)+\theta_e^2\delta_2(x)+\cdots]$ , which slightly reduces the decrement, reshapes the increment, and — most importantly — shifts the null upward by a few GHz (from $\sim217$ to $\sim219$ – $220$ GHz for a $10$ keV cluster). Ignoring them biases temperature and $Y$ estimates at the percent level, which matters for precision cluster cosmology, so modern analyses include the relativistic tSZ (rSZ) terms and can in principle use them as an independent thermometer of the gas.

Why is the integrated Compton-Y such a good mass proxy?

Because $Y_{\rm SZ}$ measures the cluster's total thermal energy, $\int P_e\,\mathrm{d}V$ , and for gas in hydrostatic equilibrium in the cluster's potential well that energy is tied directly to mass by the virial relation. The self-similar model predicts $Y_{\rm SZ}D_A^2\propto M^{5/3}E(z)^{2/3}$ with remarkably low intrinsic scatter — lower than X-ray luminosity, which is $\propto n_e^2$ and so is easily biased by dense cool cores and mergers. Since $Y$ is a pressure integral, it is insensitive to the messy central gas physics that plagues other proxies. That robustness is why SZ cluster cosmology reports masses through $Y$ , cross-calibrated with weak lensing to pin down the normalization.