Astronomy & Celestial Mechanics
the sky, computed
Concept

The Sunyaev–Zel'dovich effect — where it appears

The distortion a cluster's hot gas imprints on the microwave background by scattering its photons to higher energy. Because the result is a fraction of a background of fixed brightness, it does not fade with distance, unlike every other way of finding a cluster.

Named by 6 essays across 2 fields — each of them below, with the objects they name alongside it.

Gas outweighs stars 6:1, and the baryons come to 15%. Above: enclosed mass against radius for a 7-keV cluster with a beta-model gas profile, β = 0.65 and a 250-kpc core. The total is from hydrostatic equilibrium — the same equation that holds up a star, with the mass following from the density and temperature gradients and from nothing else — and comes to 9.98·10¹⁴ solar masses inside 2 megaparsecs. The gas, integrated from the same profile, is 1.6·10¹⁴; the stars in all the galaxies are 2.5·10¹³, 6 times less. Most of a cluster's ordinary matter is not in anything anybody would call an object. The two total-mass curves differ by the 15 per cent hydrostatic bias: some of the pressure holding the gas up is turbulence left from the last merger rather than heat, and a mass computed from the thermal pressure alone is low by about that — the assumption that makes the measurement possible is the one that biases it. With the correction, the baryons come to 15 per cent of the total, against the 15.7 per cent the microwave background gives for the universe as a whole. A cluster is large enough to have kept everything it started with, so its own accounting is a cosmological measurement. Below: the same gas seen two ways. X-ray surface brightness falls as (1+z)⁻⁴, so a cluster at redshift one is 16 times fainter per unit sky than the same cluster nearby; the Sunyaev–Zel'dovich distortion of the microwave background does not fall at all, because it is a fraction of a background whose own brightness rises by exactly the same factor. That is why a millimetre survey finds clusters at any distance and an X-ray survey finds the near ones.

The baryons that are not in the galaxies

The stars in a cluster are about a seventh of its ordinary matter. The rest is ten-million-kelvin gas that is invisible optically and dominant in X-rays — and weighing it with hydrostatic equilibrium biases the answer low by exactly the amount the assumption is wrong by.

galaxies · Clusters One cluster moved to z = 1.8: one signal unchanged, the other 57 times fainter. The same cluster of galaxies placed at a series of redshifts, with two ways of detecting it compared. The Sunyaev–Zel'dovich decrement is flat, because it is a fraction of the microwave background and the microwave background has the same surface brightness at every redshift a cluster can sit at — moving the cluster further away shrinks it on the sky but does not make its shadow shallower. The X-ray surface brightness of the identical object falls as (1+z)⁻⁴, the dimming every surface brightness suffers in an expanding universe, and by z = 1.8 it is 57 times below where it started. This is why the cluster surveys that reach the early universe are millimetre surveys: an X-ray telescope's cluster catalogue thins out with distance and a Sunyaev–Zel'dovich catalogue is limited only by how large the cluster looks and by how faint a fractional distortion the instrument can measure. What the effect cannot supply on its own is a distance — the signal that does not know how far away the cluster is also cannot say — so every one of these clusters still needs a redshift measured the ordinary way, from a spectrum of a galaxy inside it.

A shadow that does not get fainter with distance

Every other way of finding a cluster of galaxies gets harder the further away the cluster is. One does not. A cluster's hot gas scatters about one microwave background photon in a hundred to a higher frequency, and because the result is a fraction of a background rather than a flux from a source, the same cluster is exactly as detectable at ten billion light years as at one.

cosmology · Sunyaev zeldovich A path length recovered from two integrals of one cluster. The two line-of-sight integrands through an isothermal β = 0.67 cluster, each normalised to its own centre, against distance along the line of sight in core radii. The upper curve is electron density, which the Compton parameter integrates; the lower is density squared, which the X-ray surface brightness integrates. They are integrals of the same gas along the same line and they weight it differently — the half-width is 1.00 core radii for the linear one and 0.64 for the quadratic, and one core radius either side of the centre holds 51 per cent of the pressure signal against 82 per cent of the X-ray. That difference is the whole method. Two integrals with different powers of one unknown density, down one unknown path, are two equations in two unknowns: y₀ = 1.5·10⁻⁴ and a central X-ray surface brightness of 1.63·10⁻⁵ erg cm⁻² s⁻¹ sr⁻¹ give back a central density of 0.006 cm⁻³ and a physical core radius of 0.250 Mpc. Divide that length by the angular core radius the same cluster subtends, 54.2 arcseconds, and the answer is an angular-diameter distance of 952 Mpc — against the 952 Mpc the cluster was built at, which is the round trip this figure exists to close. Nothing in that chain is calibrated on a Cepheid, a supernova or a parallax. It is a length in centimetres measured against an angle.

A length in centimetres, measured against an angle

A cluster's hot gas offers two line integrals of the same electrons — one linear in density, one quadratic. Two equations in two unknowns give back the path length in centimetres, and a length divided by the angle it subtends is a distance with no rung of any ladder beneath it.

cosmology · Sunyaev zeldovich A cluster moving at +500 km/s, and the frequency where only the motion is left. The two distortions one cluster imprints on the microwave background, against observing frequency, scaled to the largest excursion drawn. The thermal effect is from the random motion of electrons at 8 keV, with a central Compton parameter of 10⁻⁴; the kinematic effect is from the bulk motion of the same gas at 500 km/s along the line of sight, positive meaning receding, through an optical depth of 0.00639, which is the Compton parameter divided by kT/mₑc². A bulk velocity shifts every scattered photon by a common Doppler factor, and a blackbody shifted by a common factor is a blackbody at another temperature — so the kinematic distortion has exactly the shape of a temperature change, ΔT/T = −τv/c, which here is −29.0 µK at every frequency. In intensity that shape is the derivative of the Planck spectrum, and its largest value falls at 217.5 GHz, the same frequency at which the thermal distortion crosses zero, 217.5 GHz: both conditions reduce to x coth(x/2) = 4. At 150 GHz the thermal decrement is −260 µK, so the motion is 11.2 per cent of it there and all of the signal at the null. What the kinematic spectrum cannot be told apart from is the primary microwave background itself, which is also a temperature change with this shape — so the frequency that isolates the velocity from the gas is no help at all against the sky behind it.

A velocity that has the colour of the sky

A cluster moving through the microwave background shifts the light it scatters by a common Doppler factor, which leaves a spectrum shaped exactly like a change of temperature. That shape is loudest precisely where the hot gas falls silent — and it is the one shape the background itself already has.

cosmology · Sunyaev zeldovich At 15 keV the decrement is 9 per cent shallower and the null has moved to 224.4 GHz. The thermal Sunyaev–Zel'dovich distortion at one fixed Compton parameter, 10⁻⁴, computed with the relativistic kinetic equation expanded to second order in kTₑ/mₑc² for gas at 5, 10, 15 keV, against the non-relativistic shape that is the same for every temperature. All are scaled to the non-relativistic curve's largest excursion. Heating the gas at fixed y does two things to the spectrum. The decrement becomes shallower — by 9.3 per cent at its deepest point for 15 keV — and the increment becomes lower and broader, by 17.3 per cent at its peak, because fast electrons scatter photons over a wider spread of frequencies than slow ones and some of the boost is carried to frequencies above the drawn range. The crossing moves up, from 217.5 GHz to 224.4 GHz. A cluster's temperature is therefore written into the shape of its distortion and not only its amplitude, which is what a thermometer needs; and a Compton parameter read off one frequency with the non-relativistic shape is biased low by the drawn amount, which is what a mass estimate does not need. The expansion is good to well under a per cent below 15 keV; above 20 it has to be replaced by the exact integral.

A null that moves with the temperature

The frequency at which a cluster's hot gas vanishes from the microwave sky was derived for slow electrons. The electrons in a massive cluster move at a quarter of the speed of light, the null drifts half a gigahertz per keV, and what is left at the old frequency reads as a velocity as large as the ones being sought.

cosmology · Sunyaev zeldovich 1020 clusters or 372, and the survey cannot say which parameter moved. The number of clusters per unit redshift a survey finds above an integrated Compton signal of 8·10⁻⁵ arcmin² over 6 per cent of the sky, computed from the Sheth–Tormen halo count grown by the linear growth factor, the comoving volume in each redshift slice, and the calibrated relation between signal and mass. Each curve is one pair of assumptions: the amplitude of structure, σ₈, and the hydrostatic mass bias, 1 − b, which says how far the X-ray masses the relation was calibrated on fall below the true masses. With σ₈ = 0.811 and 1 − b = 0.8 the survey finds 1020; σ₈ = 0.811 with 1 − b = 0.6 gives 372; σ₈ = 0.75 with 1 − b = 0.8 gives 548. The counts fall at low redshift because there is little volume, and at high redshift because massive halos have not yet formed, and the peak sits near z = 0.24. Lowering 1 − b pushes the threshold onto more massive and rarer halos; lowering σ₈ makes every halo rarer. The two lower curves differ in total by 47 per cent and in normalised shape by at most 2 per cent of the peak. A survey that assumed 1 − b = 0.8 would read the 372 clusters of the curve with 1 − b = 0.6 as σ₈ = 0.716, with a redshift distribution that differs from it by at most 12 per cent of the peak — which is the only handle the survey has on the difference, and it is smaller than the counting noise in any redshift bin holding fewer than about 67 clusters. A total count cannot distinguish a universe with less structure from a survey that has misjudged its masses, and it is that degeneracy, not the counting, that has been argued about since the first large catalogue.

Too few clusters, or a scale that reads light

A catalogue selected on the microwave shadow is, past redshift one half, very nearly a catalogue of everything above a fixed mass — so its count by redshift is the growth of structure read almost directly. Almost, because the mass behind the threshold comes from a calibration, and a scale that reads twenty per cent light is indistinguishable from a universe with less in it.

cosmology · Sunyaev zeldovich

Named alongside it

The objects these essays reach for when they reach for this one.

Intracluster mediumAngular-diameter distanceCluster mass functionCompton y parameterHydrostatic biasThomson scatteringBeta modelBremsstrahlungCosmic microwave backgroundHydrostatic massIntegrated yInverse Compton scattering

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