A lens counted instead of measured

Weak lensing is almost always described as a measurement of shapes. A mass between the observer and a background galaxy stretches the galaxy’s image tangentially by a per cent or so, the stretch is buried in the galaxy’s own random ellipticity, and averaging a million galaxies recovers it. That is the shear. But a lens does a second thing to every source behind it, which is to change its brightness — the convergence, an isotropic magnification that changes no shape at all. A magnified source is brighter, and in a survey with a flux limit, brighter sources are more likely to be counted.

That is the basis of a second, much older way to see lensing: count the sources behind a foreground mass and compare with the number elsewhere. It was proposed before weak shear was ever measured, detected first round galaxy clusters, and measured at high significance only when surveys counted hundreds of thousands of quasars behind millions of galaxies. It is noisier than the shear, source for source. It is also free of the one requirement the shear cannot escape — that the source be resolved well enough to have a measurable shape — and its signal carries a sign that can flip, which turns out to be its best defence against being fooled.

Two effects that pull against each other #

A lens that magnifies by a factor $\mu$ does two things to the population of sources behind it. It makes each one $\mu$ times brighter, so sources that were too faint to count are lifted over the survey’s limit. And it stretches the patch of sky behind it, so that a given patch of the observed sky maps onto a smaller patch of the source plane: the sources are spread over $\mu$ times the area, and their density on the sky falls by $\mu$.

Which effect wins depends on how quickly the number of sources grows as the limit goes fainter. If the counts rise steeply — many more sources a little below the limit than above it — then lifting faint ones over the limit adds more than dilution removes, and the lensed patch holds more sources. If the counts rise slowly, the dilution wins, and it holds fewer. The figure shows both happening to one population at once, because quasar counts are steep at bright magnitudes and shallow at faint ones. The same lens raises the count of bright quasars behind it and lowers the count of faint ones.

The slope at which a lens is invisible #

The arithmetic is short. If the number of sources brighter than a flux $S$ grows as $S^{-\alpha}$, then behind a lens the number brighter than $S$ is the number that was brighter than $S/\mu$, divided by $\mu$ for the dilution:

$$\frac{N'(>S)}{N(>S)} = \mu^{\alpha - 1}.$$

Astronomers measure counts per magnitude rather than per unit flux, and the slope $s = d\log_{10} N / dm$ is related to $\alpha$ by $\alpha = 2.5\,s$. So the ratio is $\mu^{2.5s-1}$, and it is exactly one when $s = 0.4$.

The value 0.4 is not special to lensing. It is the slope of counts of sources spread uniformly through a static, transparent, Euclidean space, and a population with that slope has the property that brightening every member by a factor is exactly equivalent to moving the observer closer by the square root of that factor — which leaves the density of sources per unit solid angle unchanged. A lens does to such a population what a step forward would do, and a step forward in a uniform universe changes nothing. Populations whose counts are steeper than that — bright quasars, distant submillimetre galaxies whose counts plunge at bright fluxes — gain from magnification; populations whose counts are shallower, including faint galaxies at the depths where cosmological dimming and the finite age of the universe flatten the counts, lose.

So the magnification is never measured directly. What is measured is $(2.5s-1)$ times it, and the slope $s$ has to be measured from the same survey’s counts before the lens can be weighed at all. In the weak limit, where $\mu \approx 1 + 2\kappa$, the fractional change in the counts is $2(2.5s-1)\kappa$: twice the convergence, scaled by a factor that can be positive, negative or zero.

A signal that changes sign with magnitude #

For quasars the factor runs through zero within the range of a single survey.

The quasar luminosity function has a break: above it the number of quasars falls steeply with luminosity, below it gently. Seen in a flux-limited survey, that break appears as counts that are steep at bright magnitudes and shallow at faint ones, with the slope passing through 0.4 somewhere around twentieth magnitude. Bright quasars therefore cluster round foreground galaxies on the sky, and faint quasars avoid them.

That reversal is the reason the measurement can be trusted. The correlation between foreground galaxies and background quasars could have other causes. Quasars at low redshift sit in the same large-scale structures as galaxies, and would correlate positively at every magnitude; separating the two by redshift with photometric estimates, which a few per cent of strays can drag, never removes that physical overlap completely. Dust in the foreground galaxies’ halos dims and reddens the quasars behind them, which would produce a deficit at every magnitude. Only lensing predicts an excess of bright quasars and a deficit of faint ones from the same foreground, with the crossover at the magnitude where the measured slope passes 0.4. When a survey of two hundred thousand quasars behind thirteen million galaxies was analysed in 2005, the correlation had exactly that shape: positive for the bright, negative for the faint, near zero at the crossover, at eight standard deviations overall. It was the first detection of magnification bias by individual galaxies at a significance comparable to the early shear measurements.

The same data turned out to contain the dust signal too, as a colour dependence: background quasars behind foreground galaxies were slightly redder than the average, by an amount that traced dust out to hundreds of kiloparsecs from the galaxies — far beyond their visible discs. Dust is a contaminant of the magnification measurement and, measured this way, a result in its own right.

How large the change is #

Model a foreground galaxy as an isothermal sphere, the simplest mass profile that gives a flat rotation curve, and its convergence falls as the inverse of the angular distance from it, with an amplitude set by the square of its velocity dispersion.

Round a single large galaxy the change in counts at half an arcminute is a couple of per cent for a steep population and a fraction of a per cent for a shallow one, falling inward-out as the inverse of the distance. Quasars are rare — a few hundred per square degree at most — so a single galaxy has at most one or two behind it within an arcminute, and the per-cent change is invisible in any one field. The measurement is a stack: the counts round millions of foreground galaxies are averaged, and the per-cent excess or deficit emerges from the Poisson noise of millions of counts.

Round a group or a cluster the convergence is larger by the square of the ratio of velocity dispersions.

A group of galaxies with three times the velocity dispersion has nine times the convergence at a given angle, and the count changes reach tens of per cent within an arcminute. That is why the effect was first claimed round clusters, in the 1990s, as deficits of faint galaxies and excesses of bright quasars and radio sources in the fields of massive clusters. The early claims were contentious because cluster fields are crowded and the source populations’ slopes were poorly known, and they are a fair illustration of the method’s weak point: the answer is only as good as the slope of the counts, measured independently.

What a counted source is worth #

The comparison with shear can be made exact for a single source. A shape measurement gives an estimate of the shear with a noise equal to the source’s intrinsic ellipticity scatter, about a quarter per component. A count gives an estimate of the convergence with the Poisson noise of one object, scaled by the factor $2(2.5s-1)$. For a lens whose convergence and shear are equal, as they are for an isothermal sphere, the ratio of what one source contributes to the signal-to-noise by the two routes is $2|2.5s-1|\,\sigma_e$.

For the faint galaxies that a shear survey uses, a count is worth an eighth of a shape: magnification from the same galaxies is roughly eight times weaker. That is the reason it is rarely the headline measurement. But the comparison changes completely for other populations. Bright quasars, whose counts are steep, are worth two-thirds of a shape apiece. Submillimetre galaxies, whose counts at bright fluxes are steeper still, are worth about one. And neither population has a measurable shape at all — quasars are point sources, and submillimetre galaxies are observed with beams far larger than the galaxies. For those sources counting is not the weaker of two methods; it is the only one.

The same holds for the most distant galaxies, at redshifts of three to six, selected by the drop in their spectra blueward of the Lyman limit. Their images are tiny and their shapes, even with space telescopes, are noisy. They are numerous and their counts are well measured, so they serve as a magnification background for lenses at redshifts beyond the reach of shear, where the lenses themselves are the young galaxies and groups whose masses are otherwise hard to weigh.

What counting adds that shapes cannot #

The magnification measures the convergence, and the shear measures something that is blind to one part of it. A uniform sheet of mass in front of a lens changes no shapes, because a uniform convergence stretches nothing; it does change every brightness. That is the content of the sheet of mass that changes nothing but the answer: a shear measurement fixes a cluster’s mass profile only up to the addition of such a sheet, with a compensating rescaling, and the profile it returns can be wrong by tens of per cent. A magnification measurement on the same cluster sees the sheet directly, so the two together pin down what neither can alone.

There is a further, statistical use in large surveys. Magnification correlates the counts of galaxies at high redshift with the matter at lower redshift along the same lines of sight, which adds a term to the clustering measured in any deep survey whether it is wanted or not. A survey that ignores it misreads the two parameters that lensing usually measures as one, and a survey that models it gains an extra probe of the same matter from data it already has. In the deepest galaxy surveys now running, the magnification term in the clustering of the faintest, most distant galaxies is large enough that it has to be included; the counts of those galaxies are therefore part of the lensing measurement whether or not their shapes are.

An old argument about quasars beside galaxies #

The effect has a history that predates its use as a measurement, and the history is a warning. In the 1970s and 1980s a number of bright quasars were found unusually close on the sky to nearby galaxies — closer, by the statistics of the time, than chance alignment allowed. The associations were taken by some as evidence that quasars were not at the distances their redshifts implied, but were ejected from the galaxies they sat beside, and that a large part of their redshift was something other than the stretching of the scale between emission and reception. The proposal was debated for two decades.

Magnification bias supplied an explanation that kept the redshifts as distances. A foreground galaxy, or the group or cluster round it, magnifies the quasars behind it, and the brightest quasars, with the steepest counts, are exactly the ones that should appear in excess near foreground mass. The excess predicted for single galaxies was smaller than some of the claimed associations, which led to arguments about whether the galaxies’ halos were more massive than assumed or the statistics overstated; the larger structures round the galaxies, unseen at the time, turned out to carry much of the mass. What settled the matter was not any single association but the magnitude dependence: an ejection model gives no reason for faint quasars to avoid galaxies, and lensing requires it. When surveys large enough to see the deficit of faint quasars arrived, the associations had their cause.

The episode is a reminder of what a sample brighter than the population it came from can do when nobody has modelled why it is bright. A catalogue of the brightest quasars is a catalogue selected in part by the matter in front of them.

The same bias in the strong lenses #

Magnification bias is also why multiply imaged quasars are found as often as they are. A quasar lensed by a galaxy into two or four images is magnified, usually by a factor of several to tens, and for bright quasars with steep counts that magnification lifts many intrinsically faint sources into a bright sample. The fraction of bright quasars that are strongly lensed is therefore several times larger than the fraction of the sky over which strong lensing operates — the optical depth — and surveys of lensed quasars have to correct for it before the number of lenses can say anything about the population of lensing galaxies. The individual systems used to weigh galaxies by the light that bends past them and to time the delay between images are drawn from a sample selected this way, which is why their lensing galaxies are, on average, somewhat more massive and more concentrated than galaxies chosen at random.

Microlensing shows the same physics on a single object rather than a population: a star passing behind another is magnified, and the magnification is measured directly as a brightening in time, with a fold where a caustic is crossed. Weak magnification has no light curve. It has only the counts, and so it can only be read statistically.

Truncated halos, uncertain slopes and dust #

Every figure here uses an isothermal sphere for the lens and a power law or smooth double power law for the counts. Real galaxy halos are truncated, so their convergence falls faster than the inverse of the distance beyond a hundred kiloparsecs, and the signal there comes from the matter round the galaxy rather than from the galaxy itself, which is what the stacking actually measures: the correlation of galaxies with all the mass near them. The slope of the counts is a local derivative of a measured distribution, and the measured distribution is itself affected by incompleteness near the survey limit, by stars misclassified as quasars and by photometric errors that scatter faint sources across the limit — each of which changes the effective slope and so the sign and size of the expected signal. And dust, which the figures ignore, dims the background by an amount comparable to the lensing near galaxies, and has to be separated using colour — the same grains that redden starlight and polarise it in the Milky Way, spread through galaxy halos at a hundredth of the density. The structure of the argument — two effects, a cancellation at one slope, a sign that reverses with magnitude — survives all of these. The numbers depend on every one.

Still open: whether the counts and the shapes agree about the same halos #

Magnification and shear measure the same matter round the same foreground galaxies by entirely different routes: one from the brightness and number of point sources at high redshift, the other from the shapes of resolved galaxies at intermediate redshift. In the standard picture they must agree, once the slopes, the source redshift distributions and the dust are modelled. The surveys now accumulating — imaging over a third of the sky to depths where both are measured with high significance round the same foreground sample — make that comparison possible for the first time at a precision of a few per cent. If the two agree, magnification becomes a routine second channel that tightens every lensing measurement and checks its systematic errors, since the dominant errors of the two are unrelated. If they disagree by more than the models allow, the disagreement will point at something that one of them is missing — most likely the dust, or the blending of faint images that changes both counts and shapes at once — and the counting method, which has been the poor relation of weak lensing for thirty years, will have found the error in the richer one.