A cluster weighed against the background behind it

The disagreement between counting clusters and reading the microwave background comes down, as the counts showed, to a single number: how much a mass inferred from a cluster’s hot gas underestimates the true one. If the gas-based masses are twenty per cent low, as simulations of the gas suggest, the counts are too few and the growth of structure has been slower than the standard model allows. If they are forty per cent low, the counts are what the model predicts and nothing new is needed. The decision has to be made by a measurement of mass that does not pass through the gas at all.

Lensing is the natural candidate, because light is deflected by total mass whatever it is made of and however it is supported. The lensing of background galaxies has been used for exactly this, and its results have not agreed with one another, for reasons that belong to the galaxies rather than to the clusters: their shapes have to be measured to a per cent, and their redshifts — which set how strongly each one is lensed — come from colours rather than from spectra.

There is another source behind every cluster, and it has none of those problems. The microwave background is a single screen at a single redshift, 1090, whose distance is known from the same data that define the standard model. Its statistical properties are known better than any galaxy population’s. And it is behind every cluster in the sky at every redshift, including the distant ones whose background galaxies are too few and too faint to use. The difficulty is that it is almost featureless on the scales a cluster covers, and the signal it carries is correspondingly faint.

Tens of arcseconds, applied to something smooth #

The deflection a cluster produces depends on how much mass lies inside the radius at which the light passes. For the halo shape simulations give clusters, that mass grows steeply near the centre and then only logarithmically, so the deflection rises to a peak at about the halo’s scale radius and falls away slowly beyond.

Thirty arcseconds is a large deflection by the standards of weak lensing. It is far too small to produce the arcs and multiple images by which the cores of the most massive clusters are weighed, but it is several hundred times the shift that the shear of a background galaxy measures. The reason it cannot simply be seen is the nature of the source. A galaxy has edges; it is displaced, stretched and brightened by lensing in ways that can be measured against its own outline. The microwave background on a scale of an arcminute has no edges at all.

The background’s smoothness on small scales is not an accident of the instruments. Before recombination the photons diffused through the plasma by random-walking from electron to electron, and every fluctuation smaller than the distance they wandered was erased. That diffusion scale corresponds today to a few arcminutes, and below it the primary anisotropies are damped by a factor that grows exponentially with decreasing scale. Over the two or three arcminutes where a cluster’s deflection is largest, the background is to a good approximation a uniform slope: a plane tilted in some direction, rising by about 13 microkelvin per arcminute — the root-mean-square gradient of the measured spectrum on the scales larger than its damping.

Displacing a uniform slope changes the temperature at each point by the slope times the displacement. Light reaching the observer from a point on one side of the cluster was bent inwards on the way, so it left the background at a point a little closer to the cluster’s centre. On the side towards which the gradient rises, that point is colder than the one the line of sight nominally aims at; on the downhill side, it is warmer. The result is the dipole in the opening figure, with its axis set by the background’s gradient and its radial shape set by the cluster’s deflection profile. Multiply 13 microkelvin per arcminute by thirteen arcseconds and the peak is a little under three microkelvin.

The same source behind every cluster #

The geometry of lensing enters through a ratio of distances: how far the source is beyond the lens, as a fraction of how far the source is from the observer. For background galaxies this ratio is a nuisance. A cluster at redshift one, lensing galaxies at redshift one and a half, has a ratio well below a half, and the ratio for each galaxy depends on a redshift that is estimated from its colours. A cluster at redshift 1.5 has almost no background galaxies left that are bright enough to measure.

For the microwave background the ratio is between sixty and ninety-five per cent for every cluster that will ever be catalogued, and it is computed rather than estimated, because the redshift of the source is a single number known to a part in a thousand. The peak deflection is nearly independent of redshift. A survey of clusters selected by their microwave shadow, which does not fade with distance, can therefore be weighed by a lensing signal that does not fade either, with the same source behind every member of the catalogue.

That property matters for more than convenience. The cluster counts are most sensitive to cosmology at high redshift, where the growth of structure has had least time to act and the counts change fastest. Those are exactly the clusters galaxy lensing cannot weigh. A mass scale calibrated on nearby clusters and extrapolated to distant ones assumes that the relation between gas and mass does not evolve, and that assumption is one more place for the disagreement to hide.

Five signals on one patch of sky #

The dipole is not alone on the sky behind a cluster, and its size has to be compared with what else is there.

The thermal signal, the cluster’s own hot gas scattering background photons to higher energy, is four times larger than the lensing at its peak and covers the same region. The instrument’s noise is larger than the lensing in every pixel. The kinetic signal of the cluster’s motion through the background is comparable to it. Taken as numbers, the lensing should be invisible.

What saves it is its shape. The thermal and kinetic signals are roughly symmetric about the cluster’s centre: their contribution to a dipole along any particular axis averages to nearly zero. The lensing dipole has a specific orientation, and that orientation is set by something measured independently — the background gradient, estimated from the map on scales of five to twenty arcminutes, where the cluster contributes little and the primary anisotropy is strong. An estimator that projects the map onto the expected dipole, oriented along each cluster’s own measured gradient, keeps the lensing and rejects most of everything else.

Rejecting most is not rejecting all. A cluster’s hot gas is not perfectly symmetric — it is elongated, it has substructure, and any part of it that happens to line up with the background gradient leaks into the estimate. The leak is suppressed by measuring the gradient from a map in which the thermal signal has been removed by its spectrum: the thermal effect has a characteristic shape in frequency, a decrement below 217 gigahertz and an increment above — a null whose position moves slightly with the gas temperature — and a combination of frequencies can null it while keeping the background. Infrared emission from dusty galaxies inside the cluster contaminates in the same way and is removed the same way, less cleanly. The kinetic signal has exactly the spectrum of the background itself and cannot be removed by frequency at all; it can only be averaged down, since a cluster’s motion along the line of sight is uncorrelated with the direction of the gradient behind it.

The noise that is the background itself #

The measurement’s floor is set by two kinds of noise, and the second is less obvious than the first. Instrument noise is white, and deeper maps reduce it. But the background is not exactly a uniform slope over the region where the dipole is spread; its own fluctuations on arcminute scales, damped but not zero, are the same size as the signal on the scales where the signal lives. The dipole has to be found against the structure of the screen it is drawn on.

No single cluster is detected. The noise curve shows why the measurement is always a stack: the signal from one cluster is below one standard deviation in any map that exists. But the stack is coherent, because each cluster’s dipole is lined up with its own gradient and the estimator knows which way that is. Ten clusters do not add ten random dipoles; they add ten copies of the same pattern, rotated into a common frame, and the signal-to-noise grows as the square root of their number.

The depth of the map matters less than might be expected. Halving the noise from ten to five microkelvin-arcminutes halves the number of clusters needed rather than quartering it, because at five microkelvin-arcminutes the background’s own fluctuations are already a large part of the noise and no improvement in the instrument reduces them. What reduces them is information about the background from somewhere else: its polarisation, which is lensed in the same way and carries none of the cluster’s own thermal or infrared emission, gives an independent and cleaner estimate of the same deflection, though at present a noisier one.

The numbers in the figure leave out several things, and all of them push the same way. A real cluster’s gradient is not the root-mean-square value but a draw from a distribution around it, and clusters behind weak gradients contribute little. The thermal and infrared residuals add noise. The cluster positions are uncertain by a fraction of an arcminute, which smears the dipole. So the curves are a floor rather than a forecast, and an analysis needs perhaps twice the clusters the figure says.

A cluster on a flat patch of sky weighs nothing #

One feature of the method has no counterpart in galaxy lensing, and it decides how a stack should be built. The signal is the product of two things, the cluster’s deflection and the background’s gradient behind it, and only the first belongs to the cluster. The second is a draw from the sky.

The gradient at a random point is a two-dimensional vector whose components are Gaussian, so its magnitude follows a Rayleigh distribution with a root-mean-square of 13 microkelvin per arcminute. About one cluster in eleven sits on a patch where the gradient is below 4 microkelvin per arcminute — a third of the typical value — and its lensing signal is reduced in the same proportion. A cluster that happens to lie on a turning point of the background, a local hot or cold spot where the gradient passes through zero, produces no dipole at all, whatever its mass. It is not hidden by noise. It is genuinely invisible to this method, because there is no slope for it to displace.

The signal-to-noise each cluster contributes goes as the square of its gradient, so the information in a stack is distributed very unequally. Weighting the Rayleigh distribution by the gradient squared, half of the total information comes from clusters whose gradient exceeds 1.3 times the root-mean-square value, and those are only a fifth of the sample. A stack of a thousand clusters is, for this purpose, a stack of about two hundred well-placed ones plus eight hundred that add a little each.

This has two consequences. The optimal estimator weights each cluster by its own measured gradient rather than averaging them equally, which is what the analyses do; an equal-weight stack keeps only the fraction $\pi/4$ of the information an optimally weighted one has, which is the same as discarding a fifth of the clusters. And the gradient has to be measured without the cluster contaminating it. If the slope is estimated from a region that includes the cluster’s own dipole, the dipole is partly absorbed into the slope and the inferred mass is biased low. The standard remedy is to estimate the gradient only from scales larger than the dipole, where the damped background has its power and the cluster has little, and the cost is that the gradient is then known less precisely. The trade between the bias from including small scales and the noise from excluding them sets where the filter is cut, and it is one of the choices that differ between analyses of the same data.

The parallel with galaxy lensing is instructive. There, the source is a population of galaxies with random intrinsic shapes, and each galaxy contributes the same small information whatever its orientation. Here the source is one smooth screen whose local slope varies from cluster to cluster, and the information arrives in lumps determined by where the clusters happen to fall on it. The clusters do not choose where they sit on the background; the background chose, at recombination, long before any of them formed.

What a stack can say about the gas #

The question the stack was built to answer is whether the masses the counts were read against are twenty per cent low or forty per cent low.

The answer is a matter of numbers. A hundred clusters measure the amplitude to about twenty per cent, which is the width of the whole disagreement: such a stack cannot choose. A thousand clusters measure it to seven per cent, which separates the two hypotheses at about three standard deviations. Ten thousand would separate them at eleven, before the systematic errors begin to dominate.

The first measurements were at the smaller end of that range. Detections of lensing of the background by clusters were first reported in 2015, from stacks of several hundred clusters at about three standard deviations. Planck used its own lensing of the background to calibrate the masses of its cluster catalogue and found the ratio of lensing to hydrostatic mass consistent with one, with an uncertainty of about twenty per cent — compatible with no bias at all, with the simulated twenty per cent and with the forty per cent the counts needed. The measurement had arrived and could not yet decide.

What has changed since is the number of clusters. Microwave surveys with arcminute resolution over thousands of square degrees now catalogue clusters by the thousand, and the deep maps they come from contain the lensing signal of every one. The ten-thousand-cluster stack in the figure is not hypothetical; it is the size of the catalogues the current generation of surveys is producing, measured against maps of the depth the noise curves assume.

Where the answer could still go wrong #

The lensing signal answers the question of total mass without any assumption about the gas, but it is not free of assumptions.

The first is the shape of the mass. The stack measures a dipole profile, and converting it to a mass means fitting a halo model — here, the profile simulations give, with a concentration that is itself uncertain. A cluster that is more concentrated than assumed produces a larger deflection at small radii and a smaller one at large radii, and the fitted mass depends on which radii carry the weight. The effect is a few per cent, comparable to the statistical error of the largest stacks.

The second is everything along the line of sight. The deflection is produced by all the mass between the observer and the last-scattering surface, and the matter correlated with each cluster — the filaments feeding it, the neighbouring haloes — adds to the signal at large radii. This is the same problem galaxy lensing faces, and the same models of correlated structure correct for it, but it means the mass that comes out is not exactly the cluster’s.

The third is the selection. Clusters are selected by their thermal signal, and a cluster that happens to have a larger signal than its mass would predict is more likely to be in the catalogue. The selected sample is therefore biased towards clusters whose gas over-represents their mass, which is the quantity being measured. Correcting for that needs a model of the scatter in the relation between signal and mass, and the correction is largest for clusters near the detection threshold, which are the majority.

None of these is as large as the twenty-percentage-point difference between the two hypotheses. All of them are larger than the statistical error of a ten-thousand-cluster stack.

A source with no properties of its own #

The general point is about what makes a calibration trustworthy. Every mass measurement that has been brought to bear on the cluster counts so far has had a scale of its own. Hydrostatic masses have the gas’s support; galaxy lensing has the shapes’ calibration and the redshifts’ accuracy; the cluster’s velocity dispersion has the galaxies’ orbits. Each of those scales has to be calibrated in turn, and the chain ends somewhere uncertain.

The microwave background as a lensing source is close to the end of every chain. Its redshift is a number, not a distribution. Its statistics are Gaussian to the precision anyone has been able to test, and their power spectrum is measured. It has no intrinsic shapes to be aligned with the cluster, no colours to be confused with redshifts, no population that evolves. What it lacks is signal: the price of a source with no properties of its own is that it shows almost nothing of what happens in front of it, and the measurement has to be built from hundreds or thousands of clusters added in exactly the right orientation.

There is a pleasing symmetry in using the background in this way. The disagreement began as a comparison between the background’s prediction for the cluster counts and what the counts found. The background is now being used to weigh the clusters it made the prediction about — as the screen behind them rather than as the source of the forecast — and the two roles are entirely separate: the prediction came from the acoustic peaks at degree scales, the weighing comes from the damped arcminute scales where those peaks are long gone.

Still open: whether the stacks agree with the galaxy lensing #

The lensing of the background by clusters has reached the precision at which it can decide between a twenty and a forty per cent hydrostatic bias, and it has not yet been compared in detail with galaxy lensing for the same clusters. Those two measurements of total mass share no systematic errors: one depends on galaxy shapes and redshifts, the other on the background’s gradient and the cleaning of the cluster’s own emission. If they agree, the mass scale of the cluster counts is set, and whether the counts are too few becomes a question about cosmology rather than about calibration. If they disagree, one of the two has a scale of its own after all — and the overlap between the clusters that galaxy lensing can weigh, mostly below redshift one, and those the background can weigh, at every redshift, is where the comparison has to be made.