A mass scale checked near and carried far

Clusters of galaxies are counted to measure how fast structure has grown, and the counts are only as good as the masses assigned to them. Those masses are read from the hot gas, and the gas has to be calibrated against something that responds to total mass however it is supported. Two such calibrations exist. One uses the shapes of the galaxies behind each cluster, stretched by a per cent as their light passes it. The other uses the dipole the cluster draws on the microwave background, a source with no shapes and no redshift distribution, the same screen behind every cluster in the sky.

The two methods share no systematic error. Galaxy lensing depends on how accurately a per-cent shear is measured on galaxies smeared by the telescope and the atmosphere, and on redshifts estimated from colours. Background lensing depends on how cleanly a cluster’s own thermal and infrared emission is removed from a map, and on the gradient of a screen that was fixed at recombination. If the two agree, the mass scale is settled. That was the hope on which the account of background lensing closed, and it runs into a fact about where each method works. Galaxy lensing weighs a nearby cluster well and a distant one hardly at all, because the galaxies run out. Background lensing weighs every cluster equally badly, and equally at every distance. The comparison can be made only where both still work, and the answer has to be carried from there to everywhere else.

Two instruments pointed at one column #

Both methods respond to the same thing. Lensing deflects light according to the mass projected onto the sky, summed along the line of sight through the cluster, and neither kind of source can tell how that mass is distributed in depth. A cluster elongated towards the observer looks heavier to both than its spherical twin; one elongated across the sky looks lighter to both. This is the largest single source of scatter when a lensing mass is compared with the mass from hot gas: clusters are triaxial, with axis ratios of one half or so, and a cluster’s lensing mass can differ from its spherical mass by twenty per cent depending only on how it happens to be oriented.

In a comparison of two lensing masses that scatter cancels. The galaxies and the background see the same column, so the ratio of the two masses carries no information about orientation at all. The same is true of the cluster’s substructure, of whether its centre has been placed correctly, and of how the mass is shared between the cluster and the filaments feeding it, within the radius both methods measure. What is left in the ratio is the calibration of each method and the noise of each.

Not everything along the line of sight is common. The galaxies sit at redshifts of one or two, and their light picks up deflections from all the structure between them and the observer. The background’s light has crossed the whole universe to redshift 1090 and has picked up more. That uncorrelated structure is noise on each mass, larger for the background, but it is noise rather than a bias and it averages away in a stack. The comparison that the two methods allow is therefore unusually clean: it asks whether two independent instruments give the same mass for the same column of matter, and almost nothing about the cluster itself gets in the way.

The two methods are also not equally useful one cluster at a time. The opening figure shows how unequal. For a nearby cluster in a survey of today’s depth, the galaxies give its mass to a factor of about one and a half; the background gives it to a factor of several. One measurement is a mass with an error bar and the other is a contribution to a stack. That inequality is where the trouble begins, because it reverses with distance.

Galaxies that are no longer behind #

A background galaxy is useful only if it is behind the cluster, and the further behind it is, the more its image is distorted. The efficiency of lensing rises from zero for a galaxy just behind the lens to a maximum for one at a large distance beyond it, through the ratio of distances that sets the critical surface density. The galaxies in a deep image have a distribution of redshifts that peaks below one and has a long tail, and every cluster sees only the part of that distribution that lies beyond it.

The weight that matters is the number of galaxies times the square of their lensing efficiency, because the signal-to-noise of a shear measurement adds in quadrature over galaxies. Both factors fall together as the cluster recedes. Fewer galaxies are behind it, and the ones that are behind are less far behind. By the time a cluster is at redshift 1.3 it has fallen to two per cent of the weight a cluster at 0.3 enjoys, and the error on its mass has grown by a factor of seven. The galaxies have not become fainter or harder to measure; they have simply stopped being behind.

There is a second problem hidden in the same figure. The galaxies’ redshifts are estimated from their colours, and the estimates carry errors of a few per cent in $1+z$, together with a small fraction of catastrophic mistakes that put a galaxy at three times its true redshift or a third of it. For a nearby cluster those errors barely matter: a galaxy believed to be at 1.0 that is really at 0.95 is still far behind a cluster at 0.3, and its efficiency hardly changes. For a cluster at 1.0 the same error decides whether the galaxy is behind the cluster at all. The efficiency of the galaxies that remain is steep in exactly the range where their redshifts are least certain.

A crossing that does not depend on the cluster #

The background suffers neither problem. Its redshift is a number rather than a distribution, and the fraction of the distance it lies beyond any cluster is between sixty and ninety-five per cent for every cluster that will ever be catalogued. The error on a cluster’s mass from background lensing is set by the depth of the map and by the background’s own small-scale fluctuations, and it changes slowly with redshift, a little better at large distances, where a cluster’s dipole is concentrated into a smaller patch of sky.

So the two error curves cross. For a cluster of three hundred trillion solar masses, a galaxy survey with ten usable galaxies per square arcminute and a background map with ten microkelvin-arcminutes of noise give equal errors at redshift 1.22. Below that, each cluster’s galaxy-lensing mass is the better one; above it, the background’s is, and soon it is the only one. What is less obvious is how the crossing depends on the cluster.

A heavier cluster is weighed better by both methods, and by almost exactly the same factor. Its deflection and its shear both grow with the mass, both are measured over a region that grows with the cluster’s size, and the ratio of the two errors comes out independent of the mass to the precision drawn. The crossing is therefore a property of the two kinds of source, not of what is in front of them: of where the galaxy distribution ends and how deep and how smooth the microwave map is. A survey can state its crossing redshift before it has found a single cluster.

That makes the crossing a clean divide in a catalogue. Every cluster on one side of it, light or heavy, is weighed better by galaxies; every cluster on the other side, light or heavy, is weighed better by the background. A survey selected by the clusters’ microwave shadows, which do not fade with distance, finds a substantial fraction of its members beyond that divide.

Where the comparison is actually made #

The crossing is where the two errors are equal, and it would be natural to expect the comparison to be made near it, where both methods are working at comparable strength. That expectation is wrong, and the reason is the size of the background’s error.

The quantity compared is the ratio of the two masses. Its error, for one cluster, is the two fractional errors added in quadrature, and below the crossing that sum is dominated by the background’s term, which is several hundred per cent. A cluster at redshift 0.3 and a cluster at 0.9 contribute almost equally to the comparison, because what limits both is the background’s noise, which hardly depends on the distance. The galaxies’ error, small for the first and larger for the second, barely enters. So the information about the ratio is spread over the redshifts in proportion to the number of clusters there, as long as the galaxies’ error stays smaller than the background’s.

The comparison is therefore made on the bulk of the catalogue, from redshift 0.2 to about one, and not at the crossing. Half its information comes from below redshift 0.63 and nine-tenths from below 1.02. The clusters beyond the crossing are the ones whose masses the comparison is meant to make trustworthy, and they contribute almost nothing to it, because each of them now has one precise-enough mass and one useless one.

This is the shape of the whole procedure. The background’s mass scale is checked against the galaxies’ on the clusters where both exist. Then, on the clusters where only the background’s exists, it is used alone, on the understanding that a method calibrated at redshift 0.6 is still calibrated at 1.6. The clusters that matter most for the growth of structure, the distant ones where the counts are most sensitive to it, are weighed by a scale checked on other clusters.

A calibration checked in slices #

A single ratio averaged over all redshifts says whether the two methods agree on average. It cannot say whether either has a calibration that drifts with distance, and drift is the failure that would matter, because it is what would corrupt the extrapolation. Checking for it means dividing the clusters into slices of redshift and asking whether the ratio is the same in each.

The figure has two lessons, and the second is the less welcome. The first is that the comparison is limited by the background’s noise, not the galaxies’. With today’s maps the ratio is known in each slice to five per cent at best, larger than the error the galaxies’ own calibration carries. Only with a map three times deeper do a few slices, between redshift 0.4 and 0.8, measure the ratio more precisely than the galaxy side’s floor — which is the condition for the comparison to test the galaxies rather than merely to be consistent with them.

The second is that the galaxy side’s floor rises with redshift, and for the reason already given. A mean error of 0.02 in the galaxies’ redshifts changes the efficiency of the galaxies behind a cluster at 0.2 by a little over one per cent, and the efficiency of those behind a cluster at one by four. The slices in which the galaxies are least certain are the slices in which the background is about to take over. A drift found there could belong to either method, and the comparison by itself cannot say which.

Deeper maps move the boundary nearer #

The crossing is set by two surveys, and improving either moves it. A denser galaxy survey moves it outward, by giving each cluster more galaxies behind it; a deeper microwave map moves it inward, by weighing every cluster better at every distance.

Neither kind of improvement is free of diminishing returns. A twentyfold deeper map weighs a cluster seven and a half times better, not twenty, because once the instrument’s noise is low the background’s own fluctuations on arcminute scales are the noise — damped by the photons’ diffusion before recombination, but not to zero — and they are part of the sky. A denser galaxy survey has to go fainter, and its faint galaxies are smaller, harder to measure and more likely to have their redshifts wrong.

The direction of the trend is the interesting part. The microwave surveys now being built will reach a few microkelvin-arcminutes over half the sky, and with them the crossing comes in to redshift one or below for today’s galaxy surveys. The two methods’ overlap does not grow as the background is measured better; it shifts, and more of the catalogue falls to the side where only the background weighs the clusters. The galaxy surveys of the next decade, at thirty galaxies per square arcminute or more, push the crossing back out. What decides the range in which the comparison can be made is not either survey but the relative pace of the two.

The assumption the hand-over makes #

Carrying a calibration from one set of clusters to another rests on one assumption: that whatever error the background’s mass scale has, it has the same error at every redshift. That is not automatic.

The largest risk is the cluster’s own emission. The thermal signal of the hot gas is several times larger than the lensing dipole and has to be removed by its spectrum — a decrement below 217 gigahertz and an increment above — and any residual that happens to line up with the background’s gradient leaks into the lensing estimate. A cluster’s pressure, and so its thermal signal at a given mass, changes with redshift, and so does its angular size relative to the beam. If the leak is a few per cent at redshift 0.5, there is no reason it should be the same few per cent at 1.5. The infrared emission of the cluster’s star-forming galaxies, which is also removed imperfectly, rises steeply with redshift as star formation does. Both effects could make the background’s mass scale drift in exactly the range where it is used alone.

The way to check this does not involve the galaxies. The background can also be lensed in polarisation, and the polarised background carries none of the cluster’s thermal emission and very little of its infrared emission. A lensing mass from the polarisation alone is noisier but has a different, much smaller leak. If the masses from temperature and from polarisation agree at every redshift, the temperature estimator’s leak is not drifting. That is an internal test, made within the background’s own data, and it is what turns the hand-over from an assumption into a measurement.

The galaxy side has an equivalent check in principle: the ratio of the shear behind a cluster at two different source redshifts depends only on distances, not on the cluster’s mass, and testing it tests the redshift estimates directly. Both checks matter, because the comparison of the two methods can only say that they disagree, not which of them moved.

The same move, with candles #

The structure is familiar from a different measurement entirely. The expansion rate is measured with supernovae far away, whose brightness is calibrated against stars nearby — Cepheids, or the tip of the red-giant branch — in the handful of galaxies close enough to contain both. The overlap is a narrow range of distances; the supernovae are then used alone out to distances where no star can be resolved; and the whole result rests on the supernovae being the same objects at both ends of the range.

The cluster masses repeat that pattern almost exactly. The galaxies play the nearby calibrator, precise for the clusters they can see and blind beyond them. The background plays the far indicator, noisy for each object but usable at every distance. The comparison is made where both are available, and its result is carried outward on the assumption that the far indicator does not change its behaviour with distance. The supernova calibration has been argued over for exactly that assumption, and the cluster calibration will be argued over for it too.

There is one difference. A supernova’s calibration has to be carried across a range of distance in which no other method overlaps it. The background’s has a second, independent internal check — temperature against polarisation — that works at every redshift. The far indicator here can test its own stability without a nearby calibrator, which is more than the supernovae have.

What the figures leave out #

Each curve comes from an idealised survey. The clusters are all one concentration, centred exactly, with no scatter between their masses and their signals. The galaxies’ redshift distribution is a smooth function with one median, their intrinsic alignments with the tidal field are ignored, their shapes have a single noise per component and are measured without error in calibration except where the floor says so, and no correction is made for the shear approaching a significant fraction of unity near the centre. The background’s gradient is its root-mean-square value for every cluster, not the spread of values that leaves some clusters on nearly flat patches of sky. The thermal and infrared residuals are assumed removed. The cluster count comes from a fitted halo abundance on 40 per cent of the sky and a fixed conversion between mass definitions.

None of these moves the shape of the argument, and most move the numbers in the same direction: real analyses need more clusters than the figures say for a given precision, and the crossing shifts by perhaps a tenth in redshift for plausible changes in the galaxy distribution. What the figures cannot show at all is the drift the hand-over depends on, because a drift is exactly what an idealised calculation does not contain.

Still open: whether the background’s calibration is the same at every redshift #

The comparison of the two lensing masses is about to become possible at the few-per-cent level, on tens of thousands of clusters from the microwave surveys now running and the galaxy surveys now imaging the same sky. Where the two overlap, out to redshift one or a little beyond, it will say whether two methods with nothing in common give the same mass. Beyond that, the background’s masses will be used alone for the clusters that count most for the growth of structure. Whether its calibration holds out there depends on how the cluster’s own emission leaks into the estimate as clusters change with cosmic time, and the test that decides it — lensing masses from the polarised background, compared with those from temperature, slice by slice in redshift — needs maps deep enough in polarisation to weigh a stack of distant clusters on their own.