Concept

Cluster mass function — where it appears

The number of galaxy clusters per unit volume above a given mass. It falls very steeply at the high-mass end, so the count of massive clusters is an exceptionally sensitive measure of how much structure has grown, and correspondingly sensitive to errors in the masses.

Named by 5 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

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

A cluster weighed against the background behind it

Every cluster of galaxies has the same source behind it — the last-scattering surface, at a distance known to a fraction of a per cent — and bends that source's smooth gradient into a dipole of a few microkelvin. The dipole knows nothing about gas or galaxy shapes, so stacked over hundreds of clusters it is a mass scale that no hydrostatic assumption can bias.

cosmology · Sunyaev zeldovich $One cluster weighed two ways, and the redshift where the two swap places. The fractional error on the mass of a single cluster of 3·10¹⁴ solar masses, on a logarithmic axis, against the cluster's redshift. Weighed by the shapes of the galaxies behind it (rising curves), the error is 50 per cent at redshift 0.3 and 750 per cent at 1.5 for a survey of today's depth, 10 galaxies per arcmin²; 26 per cent at redshift 0.3 and 187 per cent at 1.5 for a survey of the next depth, 30 galaxies per arcmin²: the galaxies far enough behind the cluster to be lensed by it run out. Weighed by the dipole it draws on the microwave background (nearly flat curves), the error is 461 per cent at 0.3 and 337 per cent at 1.5 for a map with 10 μK-arcmin of noise; 224 per cent at 0.3 and 153 per cent at 1.5 for a map with 3 μK-arcmin of noise — never good for one cluster, and almost the same at every distance, because the source is at redshift 1090. The curves cross at z = 1.22 (10 galaxies per arcmin², 10 μK-arcmin), z = 0.89 (10 galaxies per arcmin², 3 μK-arcmin), z = 1.77 (30 galaxies per arcmin², 10 μK-arcmin), z = 1.41 (30 galaxies per arcmin², 3 μK-arcmin). Below the crossing a galaxy survey weighs each cluster better; above it, only the background weighs it at all.$

A mass scale checked near and carried far

A cluster can be weighed by the galaxies behind it or by the microwave background behind it, and the two share no systematic error. They do not share a redshift range either: the galaxies run out at about one, the background never does, and a comparison can only be made where both still work.

cosmology · Sunyaev zeldovich

Named alongside it

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

Hydrostatic bias The Sunyaev–Zel'dovich effect Angular-diameter distance Gravitational lensing Hydrostatic mass Integrated y Intracluster medium Matched filter Stacking Surface brightness dimming Baryon fraction Beta model

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