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A redshift that is wrong by three

Most galaxies in the largest surveys will never have a spectrum. Their distances are read from five or six magnitudes, and the reading usually lands close. When it fails it does not fail by a little — a nearby galaxy is placed ten billion years away, because two different breaks in two different spectra fell into the same filter.

Assumes Magnitudes and Photometric systems.

15 min read 6 figures Light is the only messengerMeasuring the unreachable

A spectrum gives a galaxy’s redshift almost for certain: find two or three emission or absorption lines, match their pattern to the known rest wavelengths, and the redshift is the ratio, to four or five decimal places. It is also expensive. A spectrum spreads a galaxy’s light over thousands of pixels, and for a faint galaxy each pixel gets very little; the largest spectroscopic surveys have measured tens of millions of redshifts, but the imaging surveys now under way will catalogue billions of galaxies, and for all but a small fraction of them no spectrum will ever be taken.

Their redshifts will be estimated from their brightness in a few broad filters — five or six in the optical, sometimes a few more in the infrared. That is a photometric redshift, and it works because a galaxy’s spectrum is not smooth: it has breaks, places where the flux drops sharply, and as the galaxy is redshifted the breaks move through the filters and change the colours in a predictable way. The estimate is usually good to a few per cent in 1+z1 + z. When it is wrong it is often wrong by a great deal — a galaxy at redshift 0.3 placed at 3, or the reverse — and those failures are not a larger version of the ordinary error; they are a different kind of error, and averaging over many galaxies does not remove them.

Photometric against true redshift for a simulated sample. The redshift fitted from broad-band magnitudes against the true redshift, for 300 simulated galaxies — half old, half young — measured to a signal-to-noise of 10 in their brightest band in six optical bands, and fitted with the same two templates they were made from. 217 fall within 15 per cent in 1 + z of the diagonal, with a scatter of 0.053 in (the fitted z − z)/(1 + z) among them. 83 are catastrophic failures, more than 15 per cent off, and they are not scattered at random: they sit in clumps far from the diagonal, where a galaxy's break was mistaken for the other kind of break. The scatter can be improved by measuring better; the outliers are a different kind of error, and averaging does not remove them.
Fig. 1 The redshift fitted from six broad-band magnitudes against the true redshift, for 300 simulated galaxies — half old and passive, half young and star-forming — measured to a signal-to-noise of 10 in their brightest band and fitted with the same two templates they were made from. Two hundred and seventeen fall within 15 per cent in 1+z1+z of the diagonal, with a scatter of 0.053 among them. Eighty-three are catastrophic failures, and they sit in clumps far from the diagonal, where one kind of break was mistaken for another.

Reading a redshift from a break

The idea is older than the surveys that now depend on it, and it rests on the fact that a magnitude has to say which light it measures: each filter passes a known band of wavelengths, and the band moves across the galaxy’s spectrum as the galaxy is redshifted. In 1962 William Baum measured the brightness of galaxies in nine filters and located the drop in their spectra near 400 nanometres, the 4000 Å break, and read cluster redshifts from where it fell. The break is made by the absorption of metals — ionised calcium and iron — in the atmospheres of old, cool stars, and it is strongest in galaxies whose stars are old; in a passive elliptical the flux just blueward of 400 nanometres is less than half the flux just redward of it.

A second break is at the Lyman limit, 91.2 nanometres, below which hydrogen atoms absorb everything by being ionised. At high redshift a stronger break appears at 121.6 nanometres, the Lyman α wavelength, because every cloud of neutral hydrogen between the galaxy and the observer absorbs at its own redshifted Lyman α, and the absorption lines pile up into a forest that removes much of the light blueward of the galaxy’s own Lyman α. At redshift three the forest removes about half the light just blueward of the break.

A set of broad-band magnitudes is a very coarse spectrum — five or six numbers across a factor of three in wavelength — and what it records best is where a break falls. As a galaxy’s redshift increases, its break moves redward through the filters: the colour across the break is red, the colours on either side are whatever the continuum’s slope makes them, and the filter pair in which the colour jumps says where the break is. That is how a magnitude in a band the source never had — the K-correction — becomes information rather than a nuisance: the change of colour with redshift is the redshift.

Two breaks in one filter

The difficulty is that the magnitudes record a break without saying which break it is.

Two galaxies whose spectra break at the same observed wavelength. The model spectra, in flux per unit frequency against observed wavelength, of an old, passive galaxy at redshift 0.3 (its 400 nm break shifted to 520 nm) and a young, star-forming galaxy at redshift 3.3 (its Lyman-α break at 121.6 nm, where intergalactic hydrogen removes 43 per cent of the light, shifted to 523 nm), each scaled to one at 1,000 nm, with six broad bands shaded beneath. Both spectra step down at the same place in the g band, and it is the step that a set of broad-band magnitudes mainly records. The step is made by different physics ten billion years apart — the absorption of an old stellar population's metals in one, the Universe's intervening hydrogen in the other — and the two galaxies differ by a factor of 3.3 in (1 + z).
Fig. 2 The model spectra of an old galaxy at redshift 0.3, its 400 nm break shifted to 520 nm, and a young galaxy at redshift 3.3, its Lyman α break at 121.6 nm — where intergalactic hydrogen removes 43 per cent of the light — shifted to 523 nm, each scaled to one at 1,000 nm, with six broad bands shaded beneath. Both spectra step down at the same place in the g band. The step is made by different physics ten billion years apart: an old stellar population’s metals in one, the Universe’s intervening hydrogen in the other.

The two galaxies differ by a factor of 3.3 in 1+z1+z — in distance, a few hundred megaparsecs against several thousand; in the age of the Universe at the time their light left, nine billion years against two. A few broad magnitudes see a red colour across the g band and a continuum on either side. The continua do differ — the old galaxy is red longward of its break, the young one flat in flux per unit frequency — and with good enough photometry in enough bands the difference is measurable. With five or six bands and modest precision, it often is not.

The same confusion occurs between other pairs. The Balmer break of a young, star-forming galaxy at 365 nanometres can be mistaken for the 4000 Å break of an old galaxy at a slightly different redshift; a dusty star-forming galaxy, reddened by its own dust, can look like an old one. The confusions are between features, not between redshifts, and a feature confused with another at a very different rest wavelength produces a redshift error that is enormous.

Colours that cross

The degeneracy is visible directly in a diagram of two colours, where each kind of galaxy traces a track as its redshift changes.

The colours of two kinds of galaxy as their redshift changes. The track each template follows in a diagram of two colours, u − g against g − r, as its redshift increases: the old galaxy from z = 0 to 1.4, the young one from 0.1 to 4, with points every 0.05 in redshift. The old galaxy's u − g reddens as its 400 nm break moves through the u and g bands; the young galaxy stays blue until its Lyman break reaches the u band near z = 2.5 and then turns sharply red in u − g, which is how Lyman-break galaxies are selected. The tracks come closest where the old galaxy is at z = 0.02 and the young one at z = 3.20, within 0.24 magnitudes of each other in these two colours: a galaxy measured to a few tenths of a magnitude near there could lie on either track, and these two colours alone give no reason to prefer one redshift over another 3.2 units away. The other colours separate them, when they are measured well enough.
Fig. 3 The track each template follows in u−gu - g against g−rg - r as its redshift increases, points every 0.05 in redshift. The young galaxy stays blue until its Lyman break reaches the u band near redshift 2.5, then turns sharply red in u−gu - g — the colour criterion by which Lyman-break galaxies are selected. The tracks come closest where the old galaxy is at redshift 0.02 and the young one at 3.20, within 0.24 magnitudes of each other. In these two colours alone, a galaxy near there could lie on either track.

The Lyman-break technique, developed by Charles Steidel and collaborators in the 1990s, turned this geometry to advantage: galaxies at redshift three are found by their red u−gu - g and blue g−rg - r, and spectroscopic follow-up confirmed that most galaxies selected that way really are at redshift three. It also found that a few per cent are not — low-redshift galaxies and cool stars whose colours land in the same region — and that the contamination depends on how deep the imaging is and how carefully the selection box is drawn. The confusion is between the same two populations that fail photometric redshifts, seen from the side of the selection rather than the side of the estimate.

Stars in the same diagram

Galaxies are not the only things in an imaging survey, and the colours of stars add a third population to the confusion.

A star’s colour is set by its temperature — colour is a thermometer — and the coolest stars, M dwarfs and brown dwarfs, are extremely red across the optical, with molecular bands that carve steps into their spectra much as a break does. A cool brown dwarf is faint in every optical band but the reddest, and bright in the near infrared, and at the depth of a modern survey it looks, in a handful of broad magnitudes, very like a galaxy or quasar at redshift six or seven, whose light blueward of Lyman α has been removed entirely by the neutral hydrogen of the early Universe. Both appear only in the reddest bands; both “drop out” of the bluer ones.

The distinction is made partly by shape — a star is a point, a distant galaxy usually slightly extended — and partly by the near-infrared colours, where the molecular absorption of a brown dwarf and the flat continuum of a galaxy differ. Searches for the most distant quasars, which are rare, have to reject brown dwarfs outnumbering them by factors of a hundred or more, and the searches were designed around that rejection. The photometric redshift of a point source is a classification first, and the classes that confuse it are separated by physics that has nothing to do with redshift.

A likelihood with several valleys

A photometric redshift is found by comparing the measured magnitudes with the magnitudes each template would have at each trial redshift and choosing the best match. Done properly, the result is not one number but a likelihood — or, with a prior, a probability — over redshift.

How well each template and redshift fits one galaxy's colours. The goodness of fit, as χ² above its minimum, of the young and the old template at every trial redshift, to the six measured broad-band magnitudes of one simulated young galaxy whose true redshift is 0.32, measured to a signal-to-noise of 10 in its brightest band. The young template's best fit is at z = 1.80, the overall minimum; the old template's is at z = 2.62, 406.4 above it. The fit chose z = 1.80; at the true template and redshift the χ² is 0.5 above the minimum, statistically as good a fit. The lower envelope of the two curves has 3 separate valleys within nine units of the minimum, at z = 0.42, 1.26, 1.80. A photometric redshift is a likelihood over redshift, and when that likelihood has valleys far apart, its error is not a broadening of one answer but a choice between several.
Fig. 4 The goodness of fit of the young and the old template at every trial redshift to the six measured magnitudes of one simulated young galaxy whose true redshift is 0.32. The young template’s best fit is at redshift 1.80, the overall minimum; at the true redshift the fit is only 0.5 worse, statistically as good. The likelihood has three separate valleys within nine units of the minimum, at redshifts 0.42, 1.26 and 1.80, and the noise in six magnitudes was enough to make the wrong one the deepest.

For that galaxy the problem is not a confusion between templates but a feature too weak to locate: a young galaxy at low redshift has no strong break in the optical at all — its Balmer break is shallow and its Lyman break is far in the ultraviolet — and its broad-band colours are nearly flat whatever its redshift. A featureless spectrum gives a redshift with several equally good answers, and a single best-fit value hides that. The surveys that use photometric redshifts for cosmology — measuring the growth of structure through weak lensing and clustering — therefore carry the whole probability distribution for each galaxy, not its peak, and their results depend on those distributions being right in their tails, where the catastrophic failures live.

What better photometry buys

The obvious remedy is to measure the magnitudes more precisely, and it works — up to a point.

The share of catastrophic failures against how well the magnitudes are measured. The fraction of simulated galaxies whose photometric redshift is off by more than 15 per cent in 1 + z, against the signal-to-noise of their brightest band, on a logarithmic axis, fitted from six optical bands (solid) and from the same six with near-infrared J and H added (dashed), 200 galaxies at each point. With optical bands alone the failures fall from 46 per cent at a signal-to-noise of 5 to 3 at 50; adding the near infrared brings them to 42 and 0. Better photometry helps and does not cure it, because the two answers are genuinely similar in colour; a band that sees the spectrum on the far side of both breaks — the near infrared, where an old galaxy at low redshift is bright and a young one at high redshift is faint — separates them.
Fig. 5 The fraction of catastrophic failures against the signal-to-noise of the brightest band, from six optical bands (solid) and with near-infrared J and H added (dashed), 200 simulated galaxies at each point. With optical bands alone the failures fall from 46 per cent at a signal-to-noise of 5 to 3 per cent at 50; with the near infrared added, from 42 to 0. Better photometry helps and does not cure it at the depths that matter, because the faintest galaxies in any survey are always measured at low signal-to-noise.

The faintest galaxies are the problem because they are most of the galaxies. A survey’s number counts rise steeply towards its detection limit, so at any depth the majority of the catalogue is within a magnitude or so of the limit, measured at a signal-to-noise of five or ten. The catastrophic outlier fraction of a survey is set by its faintest galaxies, however well its brighter ones are measured, and it is those galaxies that cosmological analyses most need, because they are the most distant and the most numerous.

The band on the far side of both breaks

What separates the two confused populations is the part of the spectrum where they differ most, and that is longward of both breaks — in the near infrared.

Photometric against true redshift for a simulated sample. The redshift fitted from broad-band magnitudes against the true redshift, for 300 simulated galaxies — half old, half young — measured to a signal-to-noise of 10 in their brightest band, with near-infrared J and H added to the six optical bands, and fitted with the same two templates they were made from. 244 fall within 15 per cent in 1 + z of the diagonal, with a scatter of 0.059 in (the fitted z − z)/(1 + z) among them. 56 are catastrophic failures, more than 15 per cent off, and they are not scattered at random: they sit in clumps far from the diagonal, where a galaxy's break was mistaken for the other kind of break. The scatter can be improved by measuring better; the outliers are a different kind of error, and averaging does not remove them.
Fig. 6 The same simulation with near-infrared J and H bands added to the six optical ones, at a signal-to-noise of 10. Two hundred and forty-four galaxies now fall within 15 per cent of the diagonal and 56 are catastrophic failures, against 83 without the near infrared. The remaining failures are mostly featureless young galaxies at low redshift, whose spectra give no break to find in any band.

An old galaxy at redshift 0.3 is bright in the near infrared: its old stars emit most of their light there, and the near-infrared magnitudes rise steeply above the optical ones. A young galaxy at redshift 3.3 is faint there, its flat ultraviolet continuum redshifted into the near infrared with nothing to brighten it. The near-infrared colour therefore separates exactly the pair the optical colours confuse. That is the scientific case for pairing the large optical surveys with near-infrared imaging — from space, where the sky is dark in the infrared — and it is why the photometric redshifts of the deepest surveys are computed from a dozen or more bands spanning the optical and near infrared together.

A galaxy at redshift sixteen that was at five

The sharpest recent illustration of a catastrophic failure came from the first images of the James Webb Space Telescope in 2022, and it involved exactly the confusion drawn here, moved to longer wavelengths.

Webb’s near-infrared camera images the sky in a dozen broad bands between one and five microns, where the Lyman break of a galaxy at redshift ten or more falls. Within weeks of the first data release several candidates were reported with photometric redshifts above twelve, one of them near sixteen — galaxies apparently existing two hundred and fifty million years after the big bang, earlier and brighter than models of galaxy formation allowed. The candidate near sixteen was one of the most discussed objects of the year.

A spectrum taken months later put it at redshift 4.9. Its apparent Lyman break was a different feature: a strong break from dust-reddened light and bright emission lines falling in a way that mimicked the drop-out of a much more distant galaxy. The likelihood from its photometry had a second valley at low redshift that the first analyses had judged less probable, and the spectrum chose the other one. A redshift wrong by eleven, from a dozen bands measured by the most sensitive infrared telescope ever built, is the same kind of error as a redshift wrong by three from six optical bands: a feature correctly located and wrongly identified.

The episode was not a failure of the method so much as a reminder of what the likelihood contains. Many of the other high-redshift candidates were confirmed spectroscopically, some at redshifts above thirteen, and the early Universe does appear to hold more bright galaxies than expected. But every photometric candidate in that regime now carries its full probability distribution, and the second valleys are reported rather than discarded.

Calibrating what cannot be measured directly

A photometric redshift method has to be tested against galaxies whose redshifts are known from spectra, and here the problem turns on itself. The galaxies for which spectra are easiest to obtain are the bright ones and the ones with strong emission lines; the faint, featureless galaxies whose photometric redshifts fail most often are exactly those for which spectra are hardest to get. The calibration sample is biased against the failures it is meant to measure, and the outlier fraction measured on it underestimates the outlier fraction of the survey.

Two strategies address this. One is to obtain spectra for a carefully chosen sample that spans the whole range of colours in the survey — every region of colour space, weighted by how many survey galaxies live there — so that no kind of galaxy is left uncalibrated, even if it takes many hours of telescope time per spectrum. The other uses the clustering of galaxies: 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 individual redshifts. Neither is complete, and the residual uncertainty in the redshift distribution of the faintest galaxies is one of the leading systematic errors in the cosmological measurements of the current generation of surveys.

Clues that are not colours

Broad-band magnitudes are not the only information an image holds about a galaxy’s distance, and the others fail in different ways from the colours, which is what makes them useful.

A galaxy’s apparent size and surface brightness change with redshift in ways its colours do not. Surface brightness is the one photometric quantity distance does not touch in a static space, and in an expanding one it falls as (1+z)4(1+z)^4 — a factor of about three at redshift 0.3 and more than three hundred at redshift 3.3. An old galaxy at redshift 0.3 that happens to have the colours of a young one at 3.3 is typically far larger on the sky and of far higher surface brightness than any galaxy at 3.3 could be, and a prior on size and surface brightness at each redshift suppresses the confusion. A galaxy’s brightness itself is a clue of the same kind: a galaxy placed at redshift three by its colours must be enormously luminous if it is bright, and very bright galaxies at redshift three are rare.

Combining these with the colours is what modern methods do, either through explicit priors or by training a machine-learning estimator on galaxies with spectroscopic redshifts, using every measured property at once. Each additional clue is useful exactly to the extent that its failures are uncorrelated with the colours’, and size, surface brightness and luminosity are informative because the physics that sets them has nothing to do with where a break falls in a filter.

What the simulation simplifies

There are two templates and the galaxies are made from them. Real galaxies span a continuum of star-formation histories, dust contents and metallicities, and a real template set — or a machine-learning method trained on spectroscopic redshifts — has to cover that range; mismatch between the templates and the galaxies adds failures the simulation does not have.

The bands are top-hats standing in for u, g, r, i, z and y and for the near-infrared J and H, and the noise model gives every band a fixed fractional error scaled by how faint the galaxy is in it relative to its brightest band. Real surveys have different depths in different bands, and the u band, which is crucial for the Lyman break, is usually the shallowest.

The priors are flat. A real analysis weights each redshift by how many galaxies of that brightness are expected there, which suppresses some failures — a bright galaxy is unlikely to be at redshift three — and can create others if the prior is wrong.

And the intergalactic absorption is a smooth average. The Lyman α forest varies from one sight line to the next, and at the highest redshifts its fluctuations are large enough to move the break by a detectable amount.

Still open: redshifts for ten billion galaxies

The surveys of the next decade will measure positions, shapes and colours for several billion galaxies and spectra for a few tens of millions. The cosmology they are built for — the expansion history, the growth of structure, the sum of the neutrino masses — needs the redshift distribution of the photometric galaxies known to a fraction of a per cent in its mean, and the catastrophic outliers, a few per cent of the catalogue, can shift that mean by more. Whether the combination of deeper near-infrared data, larger and more representative spectroscopic samples, and clustering-based calibration can pin down the outlier population to the required precision — or whether it will remain the limiting systematic error of photometric cosmology — is not yet known, and the answer depends most on the galaxies about which the least is known: the faint, the featureless and the dusty, whose breaks the filters cannot find.

Where this goes next

Four spectra, and 3.9 magnitudes between them at the same redshift. The K-correction — the magnitude that has to be added to compare a redshifted object with a nearby one through the same filter — against redshift, for four power-law spectra. At zero redshift every correction is zero by construction. Beyond that they diverge, because a filter at a fixed observed wavelength samples a different part of the source's own spectrum at every redshift, and how much flux is there depends on the spectrum. A flat spectrum needs no correction at all at any redshift, and the two extremes drawn differ by 3.9 magnitudes by z = 1.2. The circularity is the point: applying the correction requires the spectrum, and the spectrum is what a magnitude is being used to constrain. The dashed line is the way out — observe in a band chosen so that it lands on the rest-frame band of interest, and the spectral term cancels, leaving only the bandwidth stretch. A magnitude in a band the source never had A filter passes light at a fixed observed wavelength, and a redshifted source emitted that light at a shorter one. Comparing a distant galaxy with a nearby one through the same filter therefore compares two different parts of two spectra — and the correction between them needs the spectrum, which is what the magnitude was going to be used to find out. B − V against temperature, computed and measured. B − V against effective temperature. The curve is the colour index of a blackbody, obtained by integrating Planck's law against B and V response functions and subtracting a constant so that the index is exactly zero at 9600 K — the convention that an A0V star has every colour index zero, which is a choice and not a measurement. The 15 points are the main sequence as it is actually measured, and they do not lie on the curve: at the Sun's temperature the blackbody gives 0.446 where the sky gives 0.653, and at M0V 0.995 against 1.40. The model is too blue almost everywhere, and least wrong near 9600 K — which is why the zero point is put where it is. A colour index is a difference of two magnitudes, so it is a difference of two integrals, and changing either filter changes the number. A magnitude has to say which light The same star is a different magnitude in every filter, and a colour index is a difference of two conventions rather than a property of the star. Both are integrals of a spectrum against a piece of glass, and the zero point is a choice somebody made in 1953. Fingers 3.1 times long, a large scale squashed to 0.92, and a test of gravity. Left: a 360-megaparsec slice of a clustered universe, as the galaxies actually sit. Right: the same galaxies as a redshift survey records them, with the line of sight up the page. Nothing has moved sideways, because an angle is an angle; every displacement is along the line of sight, because that coordinate came from a redshift and a redshift is the expansion plus whatever the galaxy is doing on its own. Two effects, opposite in sign and separated by scale. Inside a cluster the motions are virial and random, 720 kilometres a second of them, which at H₀ = 67.36 is 11 megaparsecs of smearing on an object a few across: the clusters become fingers 3.1 times longer than they are wide, all pointing at the observer, which is the one structure in cosmology that is definitely not real. On the scale of a supercluster the motions are coherent — everything is falling in, so the far side is approaching and the near side receding — and the structure is compressed rather than stretched, to 0.92 of its true extent here, measured on cluster centroids so the fingers have already averaged away. Below: why that compression is worth having. Its amplitude is the rate at which structure is currently growing, and the growth rate is Ωₘ(z) raised to a power that general relativity fixes at about 0.55. A theory of gravity that differs from general relativity on cosmological scales while matching every solar-system test changes that exponent and nothing else, and the two curves drawn — γ = 0.55 and γ = 0.68 — differ by only 4 per cent at redshift a half, against error bars of 12 per cent on the points beside them. The worst systematic in a redshift survey is the measurement — and it is a hard one, because a quarter of a change in the exponent that governs how gravity assembles structure moves the observable by less than the width of the curve it is drawn on. What the picture cannot show is the degeneracy that limits it: what is measured is fσ₈, a product, and separating the growth rate from the amplitude of clustering needs something else entirely. A map that is not of positions One axis of every redshift survey is not a distance but a velocity, and the difference is not noise. Inside a cluster it smears the galaxies into a finger pointing at the observer; on the scale of a supercluster it compresses the structure — and the amount of that compression is a test of gravity.

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Balmer breakBroad-band photometryCatastrophic outlierK-correctionLikelihoodLyman breakPhotometric redshiftSpectral energy distribution