A mean that a few per cent of strays can drag

A galaxy’s redshift can be estimated from its brightness in a handful of broad filters, by finding where a break in its spectrum falls among them. Most of the time this works to a few per cent. A few per cent of the time it fails catastrophically: a young galaxy at a redshift of three, its Lyman break in the blue, is mistaken for an old one at a redshift of 0.3 with its 4000 Å break in the same place, and the estimate is wrong by a factor of ten. The essay on that confusion followed why it happens and what better photometry does about it, and ended on the question of whether surveys of billions of galaxies can know their outliers well enough.

That question has a precise form, because the surveys know exactly what they need. A weak-lensing survey measures the distortion of distant galaxies’ shapes by the matter in front of them, and to turn a distortion into an amount of matter it has to know how far behind the matter the distorted galaxies are. It sorts its galaxies into a few slices of photometric redshift — tomographic bins — and for each slice it needs the distribution of true redshifts, and above all its mean. The requirement the surveys of this decade have set themselves is that each bin’s mean redshift be known to about two thousandths of one plus the redshift.

This essay works out what that requirement does to the outliers. The answer is that it makes them the whole problem.

Why lensing needs a mean and not the galaxies #

A lensing survey never needs any individual galaxy’s redshift. The distortion of one galaxy’s shape is a percent-level effect buried under its intrinsic ellipticity, and it is detected only by averaging over millions; what the survey measures is the average distortion of a population, and what it needs to interpret that is the population’s redshift distribution. Errors on individual galaxies that are symmetric — scattering some a little nearer and some a little further — cancel in the distribution’s mean, and the mean is what matters most, because the lensing efficiency depends on distance smoothly and to first order only through it.

That is why the requirement is stated for the mean and is so stringent. A survey with a billion galaxies has statistical errors on its shear measurements small enough that an error of a per cent in the mean source distance would dominate them. The mean redshift has to be known about ten times better than the photometric redshift of any single galaxy, and it can be, because averaging works for it — except for errors that do not average.

A tail that is invisible and decisive #

Take a bin of galaxies selected to have photometric redshifts near 0.8. Most of them really are near 0.8, scattered by the ordinary errors of the method. Some are not.

Drawn on a linear scale the outliers would be invisible: three per cent of the galaxies spread over a broad bump at a redshift of three, a fortieth of the core’s height. On a logarithmic scale they are a distinct population, and their effect on the mean is not small at all. Each outlier contributes its full displacement — more than two units of redshift — to the average, so three per cent of them move the mean by 0.066. The requirement for a bin at 0.8 is 0.0036. The tail that cannot be seen on a plot of the distribution moves its most important number by eighteen times the tolerance.

That is the arithmetic that sets the priority of photometric-redshift work. The scatter of the core, which dominates every published comparison of photometric with spectroscopic redshifts and every headline figure of merit, hardly affects the mean at all: it is symmetric, and a symmetric error averages away. The outliers are asymmetric by nature — the confusion runs between particular redshifts in particular directions — and an asymmetric error does not average.

How small the outlier fraction must be #

The shift is simply the outlier fraction times the displacement, and that sets a ceiling on how many outliers a bin can carry unknown.

Outliers half a unit of redshift away — confusions between neighbouring features, or galaxies scattered across a bin boundary — must be fewer than three quarters of a per cent of the bin if they are not corrected. Outliers two and a half units away, the Lyman–Balmer confusion, must be fewer than a seventh of a per cent. The best photometric redshifts from optical data alone reach outlier fractions of a few per cent; adding near-infrared bands, which catch the old galaxies’ light on the far side of both breaks, brings that down by a factor of two or three. Both are well above the lines.

So the outliers cannot be made negligible. They have to be measured, and the measurement corrected for. A known fraction of known displacement can be subtracted from the mean as easily as it was added. The requirement then moves from the outlier fraction to the precision with which it is known — the same numbers, applied to an error bar instead of a value.

How many spectra it takes #

The fraction is measured by taking spectra — true redshifts — of a sample of galaxies in each bin, and counting how many of them are outliers.

Counting is subject to binomial statistics. A bin with three per cent outliers two and a half units away needs its outlier fraction known to about a seventh of a per cent, and that takes about fourteen thousand spectra in that bin — provided the spectroscopic sample is a fair one, drawn from the bin with no preference, and that every spectrum yields a redshift. A survey with five tomographic bins and a range of galaxy types needs tens to hundreds of thousands of such spectra, of galaxies at the survey’s faint limit, where each spectrum takes hours on the largest telescopes.

The figure also shows the lever. The spectra needed scale as the outlier fraction itself, so halving the outliers halves the calibration burden. That is why the investment in near-infrared photometry, which puts a band where the source had light that the optical bands miss, is worth more to a lensing survey than its modest improvement in the scatter suggests: it reduces the population that is expensive to count.

A calibration that misses what it calibrates #

The assumption of a fair sample fails in a specific way.

A spectrum yields a redshift when it contains a recognisable feature — an emission line, a strong absorption break — in the wavelength range the spectrograph covers. The outliers are, by construction, the galaxies whose features fall in awkward places. A galaxy at a redshift of three has its strongest optical emission lines shifted into the infrared, beyond most spectrographs’ reach, and its Lyman break into the blue, where faint galaxies are faintest. Spectra of outliers fail more often than spectra of the core.

When they do, the outliers are under-counted among the successful spectra, and the correction based on them is too small. If outliers’ spectra succeed half as often as the core’s, the corrected mean is still off by nine times the requirement. The bias does not shrink with more spectra; it is a property of which spectra succeed. A calibration sample’s failures fall preferentially on exactly the galaxies it most needs, which is the deepest reason the redshift distributions of the faintest galaxies remain among the leading systematic errors in lensing cosmology.

The methods that address it work around the spectra rather than through them. Galaxies at the same redshift cluster together on the sky, so the photometric galaxies’ cross-correlation with a spectroscopic sample, slice by slice in redshift, measures how many of them lie in each slice without needing their own redshifts — and it works for galaxies whose spectra would fail, because it needs only their positions. Its own limitation is that the clustering of the two samples depends on what kinds of galaxies they contain — a bias of galaxies relative to matter that differs between the faint photometric galaxies and the bright spectroscopic ones and changes with redshift — which has to be modelled, or cancelled by comparing ratios of correlations rather than the correlations themselves.

The outliers lens more than the galaxies they are mistaken for #

A wrong mean redshift matters to lensing through geometry. The shear a galaxy shows depends on how far behind the lensing mass it is, through the ratio of the lens–source distance to the observer–source distance.

An outlier at a redshift of three lies far behind every lensing structure the survey studies, and it is lensed almost as strongly as any source can be. A galaxy at 0.8 is lensed by mass at 0.4 with about half the efficiency of a galaxy at infinity; the outlier at three, with about eighty per cent. So three per cent of outliers raise the bin’s average lensing efficiency for nearby mass by a couple of per cent, and for mass near the bin’s own redshift — which the core galaxies barely see — they supply most of the signal.

That is the mechanism by which an error in redshift becomes an error in cosmology. A survey that assumed its bin held only the core would see shears a few per cent larger than its model of the core predicts, and would read them as a few per cent more matter.

An outlier fraction read as an amplitude #

The size of the misreading follows directly.

The lensing signal rises almost in proportion to the outlier fraction: two per cent for three per cent of outliers. Lensing measures the amplitude of the matter fluctuations through the size of the signal — a combination of the matter density and the amplitude that lensing measures as one — and an uncorrected two per cent in the signal is a two per cent error in that combination. The disagreement between lensing surveys and the microwave background’s prediction, which has occupied the field for a decade, is a few per cent in the same combination.

The direction is worth noticing. Uncorrected outliers at high redshift make the lensing signal larger, pushing the inferred amplitude up, while the surveys find it lower than the microwave background predicts. So outliers left in the bins would, if anything, have hidden part of the disagreement rather than created it — unless the corrections applied for them were too large, which is the opposite failure, and the one an over-eager calibration would produce.

Strays in both directions #

The Lyman–Balmer confusion runs both ways. Young galaxies at high redshift are mistaken for old ones at low redshift, and old galaxies at low redshift — less often, because their breaks are stronger — are mistaken for young ones at high redshift. A low-redshift bin therefore receives high-redshift interlopers, which raise its mean and its lensing efficiency, while a high-redshift bin receives low-redshift interlopers, which lower both.

The two errors have opposite signs, and in a survey that combines its bins they partly cancel in the overall amplitude. They do not cancel in the cross-correlations between bins, which are what lensing tomography uses to separate the growth of structure at different epochs: an interloper population makes a high-redshift bin correlate with nearby mass more than it should, mimicking a change in how fast structure grew. The same outliers that move one number up and another down can leave the survey’s headline amplitude roughly right and its measurement of growth wrong — a failure that a survey checking only the amplitude would not see.

Why the brightest galaxies are no help #

There is a tempting shortcut: calibrate on bright galaxies, whose spectra are easy, and extrapolate. It fails for a reason the earlier figures contain. The outliers are a property of the faint, featureless galaxies; bright galaxies at the same photometric redshift have clean breaks and good spectra and few outliers. A calibration on bright galaxies measures an outlier fraction lower than the survey’s and corrects too little.

The same problem appears in every chain of calibrations in astronomy, wherever a brightness is a distance only if something is known about the objects it is applied to: a relation calibrated on easy objects is applied to hard ones on the assumption that nothing changes between them. Here the assumption fails in a measurable way, because the outliers are precisely the galaxies that differ from the bright ones in the property that matters.

The redshift is one error among several #

The outliers’ bias has to be weighed against the survey’s other systematic errors, which are of comparable size and partly degenerate with it. The calibration of galaxy shapes — how much a measured ellipticity underestimates the true shear for faint, small, noisy galaxies — enters the lensing amplitude multiplicatively, at the per cent level, just as the redshift error does. The suppression of small-scale power by the gas that galaxies push out of their haloes lowers the signal on small scales by a similar amount. And the intrinsic alignment of galaxies by the tidal fields they formed in adds a spurious correlation whose size depends on the redshift distribution too, since galaxies close together in true redshift align with each other.

All of these are marginalised over in a modern lensing analysis, with priors from simulations and from calibration data, and the redshift calibration is usually the one with the widest prior. That is why the outliers matter beyond their own effect: a redshift distribution known poorly lets the analysis absorb part of every other systematic error into it, and the cosmological result inherits the combined uncertainty.

What the arithmetic assumes #

Outliers are a single population. The figures place all outliers at one displacement. Real outliers have several origins — the Lyman–Balmer confusion, confusions between emission-line galaxies at different redshifts, blends of two galaxies at different redshifts — each with its own displacement, and the mean shift is the sum.

The core is symmetric. A skewed core moves the mean too, by an amount comparable to the outliers’ when the skew is a few per cent of the scatter. Real cores are skewed by the same template mismatches that produce outliers.

The requirement is a round number. Two thousandths of one plus the redshift is the order of magnitude the planned surveys quote; their exact requirements depend on how many bins they use and on how much the other systematic errors have been reduced, and they are traded against each other in the error budget.

Outliers are assumed to be galaxies. Some of the catastrophic failures in a photometric catalogue are not galaxies at the wrong redshift but stars mistaken for galaxies, or galaxies blended with a neighbour so that the colours belong to neither. Blends in particular grow with survey depth, because the faintest galaxies overlap on the sky more often, and a blend’s photometric redshift can be anything between its two components’. They add a population of outliers whose displacement has no fixed value.

The lensing efficiency is for a single lens redshift. A survey integrates over all lenses, and the outliers’ effect on the measured amplitude depends on how the bins are combined; the figure’s two per cent is representative, not exact.

Still open: whether the faint end can be calibrated at all #

The redshift calibration of lensing surveys has turned out to be a problem about a few per cent of galaxies — the catastrophic outliers — rather than about the precision of the method for the rest. Measuring them directly needs tens of thousands of spectra per bin, of exactly the galaxies whose spectra fail most, and the failure biases the correction by more than the requirement allows.

Three routes are being followed at once: deeper spectroscopy of a complete sample in small fields, with near-infrared spectrographs that catch the lines of high-redshift galaxies; clustering cross-correlations over the whole survey area, which need no spectra of the faint galaxies themselves; and better photometry, including space-based near-infrared imaging, to shrink the outlier population before it has to be counted. Whether they converge to two thousandths, or whether the redshift calibration of the faintest galaxies remains the limit on what a survey of billions of galaxies can say about the growth of structure, is the question the coming surveys will answer in their first years — and their headline cosmology will depend on the answer.