A line from the wrong redshift lands at the wrong distance

A line-intensity map turns frequency into distance, and it inherits from every measurement of the sky’s summed light the property that it adds up everything along the line of sight. Every photon arriving at 250 gigahertz is assumed to have been emitted by ionised carbon at its rest frequency of 1,900.5 gigahertz, and so to have travelled from redshift 6.6, when the universe was 800 million years old. That assumption is what lets a stack of frequency channels become a three-dimensional map of the early universe. It is also false for some of the photons. A galaxy at redshift 0.84 emits carbon monoxide in its fourth rotational transition at 461 gigahertz, which reaches the telescope at 250. Nothing about the photon says which line it came from, and the map places it at redshift 6.6 with the rest.

These interlopers are not a small correction. The carbon monoxide lines from galaxies at redshifts below three are, line for line, often comparable to or brighter than ionised carbon from the reionisation era, because the galaxies are so much closer. A map made for one line contains several, superposed, each from a different epoch of cosmic history. The same summing of every galaxy that makes intensity mapping powerful makes it unable to refuse light it was not looking for.

What rescues it is geometry. The interloper light is placed at the wrong distance, and the wrong distance distorts its structure in a specific, predictable way.

Which lines land where #

The first step is the bookkeeping: which lines, from which redshifts, fall in a given band.

Carbon monoxide’s rotational transitions are regularly spaced, at very nearly integer multiples of 115.27 gigahertz, so at any observed frequency there is a series of interlopers at a series of redshifts, one per transition. At 250 gigahertz the lowest is CO(3–2) from redshift 0.38, four billion years ago, and the highest that matters is CO(7–6) from redshift 2.2. Galaxies at those redshifts are forming stars vigorously, and their molecular gas is abundant. The band that makes ionised carbon from the reionisation era accessible is the band where the late universe’s carbon monoxide is brightest.

Each interloper redshift is fixed by the observed frequency and the rest frequency. The interlopers therefore move through redshift together as the observing frequency changes, and a broad band contains a continuum of them. A single channel holds one redshift slice of each line.

Why the nearer lines are so bright #

That a line from a few billion years ago should rival one from the reionisation era is not obvious, and the reason is partly the galaxies and partly the geometry of the map.

The galaxies at redshifts of one to three are the ones that formed most of the universe’s stars. The cosmic rate of star formation peaked around redshift two, ten times higher per unit volume than today and perhaps several times higher than at redshift six, and their molecular gas — the reservoir carbon monoxide traces — was correspondingly abundant. Per unit volume, the late universe at those redshifts was simply emitting more in its lines than the early universe.

The geometry compounds it. The intensity a line produces at a given observed frequency is the luminosity per unit volume times the comoving volume that fits into a unit of frequency and solid angle, divided by the area over which the light is spread. For a nearer emitter that volume per unit frequency is different, and the light is spread over a smaller sphere. The count of sources needed to make a background depends on exactly this bookkeeping. Worked through for carbon monoxide against ionised carbon, it leaves the nearer lines, per unit of their own luminosity density, brighter in the map than the distant one. The result, in most models, is interloper power comparable to or exceeding the target’s — uncertain by a factor of several because the carbon-monoxide luminosity density at those redshifts is itself uncertain by that much.

Placed too far away, magnified and flattened #

The map assigns every photon a position by converting its direction on the sky into a transverse distance, using the comoving distance to the target’s redshift, and its frequency into a distance along the line of sight, using the comoving distance per unit frequency at that redshift. An interloper photon is assigned both from the target’s redshift rather than its own.

Across the sky, a structure of true size $r$ at the interloper’s distance $\chi_i$ subtends an angle $r/\chi_i$, which the map converts back into a size $r\,\chi_t/\chi_i$ at the target’s distance. Since the interloper is much nearer, $\chi_i < \chi_t$, and its structure is magnified by $1/\alpha_\perp$, where $\alpha_\perp = \chi_i/\chi_t$. Along the line of sight, a frequency interval corresponds to a comoving length proportional to $(1+z)/H(z)$, which is larger at the interloper’s redshift relative to the target’s than the transverse distances suggest. The map assigns the target’s length per unit frequency, so the interloper’s structure is compressed along the line of sight by a factor $\alpha_\parallel = (1+z_i)H(z_t)/\left((1+z_t)H(z_i)\right)$.

For CO(4–3) the transverse factor is 0.35 and the line-of-sight factor 1.76. A cluster of galaxies that is round at redshift 0.84 appears in the map nearly three times wider across the sky than it is, and nearly half as deep along the line of sight: a pancake facing the observer. The lower the interloper’s redshift, the more extreme the mismatch, because the ratio of distances is smaller.

This is the same distortion that underlies the Alcock–Paczyński test, turned to a different purpose. There, a structure known to be statistically round, the acoustic scale in the galaxy distribution, is measured along and across the line of sight. If the assumed cosmology is wrong, its distances are wrong, and the structure appears flattened or stretched; the flattening measures the error. Here the cosmology is right and the redshift is wrong, by a factor of several, and the flattening is enormous rather than a few per cent.

Round where it is, elongated in the map #

The distortion is clearest in the power spectrum, drawn in the plane of wavenumbers across the sky and along the line of sight.

The target’s power, isotropic in its own coordinates, is round in the plane of wavenumbers. Its contours are quarter-circles because the power at a given wavenumber does not depend on direction. The interloper’s power is also isotropic where it really is, but the map has rescaled its coordinates differently in the two directions, and in Fourier space a stretch becomes a compression. Its power is pushed to small transverse wavenumbers by the transverse magnification, and out to large line-of-sight wavenumbers by the line-of-sight compression. Its contours become ellipses elongated along the $k_\parallel$ axis by $\alpha_\parallel/\alpha_\perp$, a factor of five for CO(4–3). Its amplitude is also changed, by the volume factor $1/(\alpha_\perp^2\alpha_\parallel)$, since the map spreads the interloper’s power over a volume larger than the one it came from.

The shape difference is what makes the two separable. If the interloper’s power were simply added to the target’s with the same shape, no analysis of one map could tell how much was which. Because the shapes differ, a fit to the full two-dimensional power spectrum, with the target’s shape and each interloper’s distorted shape as templates, can measure the amplitude of each. The fit uses only geometry and the known redshifts of the lines; it needs no model of how bright any of them is.

A quadrupole as a census of impostors #

The two-dimensional power spectrum is usually summarised by its multipoles: its average over all directions, the monopole, and the part that varies as the second Legendre polynomial of the angle to the line of sight, the quadrupole.

A signal with no preferred direction has no quadrupole. An interloper with its power pushed towards the line-of-sight axis has a large positive one, approaching the theoretical maximum of five for power concentrated entirely along the line of sight. The ratios in the figure, between one and two at the scales that matter, are far larger than anything the target line produces by itself. A measured quadrupole is therefore, to a first approximation, a census of interloper power, and the size of the quadrupole relative to the monopole says what fraction of the map’s fluctuations came from the wrong redshift.

The approximation has one important exception. The target line also has a quadrupole, from the redshift-space distortion that peculiar velocities imprint on any map made from redshifts: infall towards dense regions compresses structure along the line of sight on large scales and boosts the power there. That quadrupole is positive too, but much smaller — a ratio of a few tenths for strongly biased emitters — and its shape as a function of wavenumber is calculable from the growth rate. Separating the two is a fit with several templates rather than a single number, and the precision of the separation depends on how many modes the map contains along and across the line of sight.

Masking the interlopers one can see #

Geometry is not the only weapon. Unlike the faint target galaxies, many interloper galaxies are near enough to be detected individually in other surveys — optical and infrared catalogues of galaxies at redshifts below three, with known positions and redshifts. A voxel of the map that contains a bright known interloper galaxy can be masked, discarded from the analysis, at the cost of the target signal in that voxel.

Masking is efficient against the shot noise, because the shot noise is dominated by the brightest galaxies, and the brightest galaxies are rare. Removing the voxels that hold the few most luminous interlopers cuts half their shot noise at the cost of a fraction of a per cent of the map, and ninety per cent at the cost of two per cent. Masking is much less effective against the interloper’s clustered power, which is carried by the many faint galaxies that no external catalogue lists. That part has to be removed by its shape. The two methods are complementary in the same way the two terms of the power spectrum are: masking attacks the second moment of the interloper’s luminosity function, geometry the first.

Masking has a cost beyond the lost voxels. A mask with holes at the positions of galaxies is itself correlated with structure, since the galaxies cluster, and the holes imprint a pattern on the map’s power spectrum that has to be modelled. In practice the masks are kept to a few per cent of the volume and their effect is computed by applying the same mask to simulated maps.

Foregrounds that are smooth and interlopers that are not #

Intensity maps have a second kind of contamination, and the contrast between the two explains why interlopers are the harder problem. The continuum foregrounds — the microwave background, thermal dust in the Milky Way and in distant galaxies, synchrotron emission from cosmic-ray electrons — are thousands of times brighter than any line signal, but each is smooth in frequency. A pixel’s foreground spectrum is a slowly varying function, a power law or a modified blackbody, and it can be fitted and subtracted pixel by pixel, keeping only the structure that varies rapidly with frequency. The line signal varies rapidly, because each channel samples a different slice of structure, and survives the subtraction. That is the strategy that makes a 21-centimetre signal recoverable from beneath a Galactic foreground, and the same separation by spectral smoothness is how dust’s polarised glow is handled in maps of the microwave background.

Interlopers are line emission too. They vary from channel to channel exactly as the target does, because they are structure seen through a spectral line, and a smoothness filter that removes continuum leaves them untouched. In frequency they look exactly like signal. The only differences are the ones this essay has traced: their structure has the wrong shape when placed at the target’s distance, and their brightest members can sometimes be identified in other catalogues. A map can be cleaned of its continuum foregrounds by its spectrum alone. It can be cleaned of its interlopers only by its geometry and by outside information.

The same confusion, one galaxy at a time #

The problem is older than intensity mapping. Any survey that identifies galaxies by a single emission line faces it galaxy by galaxy. A large survey searching for Lyman-alpha emitters at redshifts of two to three and a half, to map the acoustic scale in that era, finds a single emission line in each spectrum and has to decide whether it is Lyman-alpha from a distant galaxy or the oxygen doublet at 372.7 nanometres from a nearby one, which falls at the same wavelength from redshifts below half. Misclassifying even a few per cent of the nearby galaxies as distant ones puts a population of impostors into the distant map, at distances wrong by a factor of several, with exactly the distortion described here.

The individual survey can attack the problem galaxy by galaxy, because it detects each source. A nearby galaxy emitting oxygen is usually bright in its continuum light, so its line has a small equivalent width, while a distant Lyman-alpha emitter is faint in continuum and its line stands out. A cut on equivalent width removes most impostors, and the residual contamination is measured from the distortion it leaves in the clustering. Intensity mapping has no individual sources to cut, only the summed light, and so it relies entirely on the statistical route that individual surveys use as a check. The individual surveys’ experience with that route — fitting the distorted clustering of impostors whose fraction they also know from the cuts — is the best calibration of how well it works.

Cross-correlation, again #

The cleanest separation of all uses a second line. Ionised carbon and, say, the oxygen line at 88 microns from the same galaxies at redshift six fall at different observed frequencies, and the interlopers in each map come from different redshifts. Cross-correlating the two maps, channel pairs matched to the same target redshift, keeps only the emission that is at the target’s redshift in both. The interlopers in one map are uncorrelated with those in the other, because they come from different volumes of the late universe, and they average away.

The same logic lies behind cross-correlating a line map with a catalogue of galaxies at the target redshift, which first brought intensity mapping to a detection in the nearer universe. At redshift six such catalogues are sparse, but even a few hundred galaxies with spectroscopic redshifts in the volume of a map give a cross-correlation that no interloper and no foreground can fake.

The price is that the cross-correlation measures a product of the two lines’ signals, so it says nothing about either alone without further assumptions. It is a detection and a check, and the auto-correlation of each map, cleaned of interlopers by geometry and masking, remains the measurement. The cross-correlation’s job is to confirm that whatever the cleaning left is at the right redshift.

What the figures assume #

The interlopers’ structure is the same linear matter power spectrum as the target’s, scaled by a bias and a growth factor, so the distortion is the only thing distinguishing them in the figures. Real interlopers have their own non-linear clustering, their own redshift-space distortions in their own coordinates — which the map then rescales again — and their own shot noise, all of which enter the full model. The distortion factors are computed for a single observed frequency; a real channel has a width, across which the interloper redshift changes slightly, smearing the template. The masking calculation assumes every interloper galaxy above a luminosity cut is known, at its true position, and occupies one voxel; incomplete external catalogues mask less, and bright galaxies whose emission spans several voxels cost more.

Still open: how bright the impostors are #

The geometric separation works whatever the interlopers’ brightness, but how well it works depends on it. A map in which interlopers carry ten times the target’s power needs a far more precise fit than one in which they carry a tenth, because the target is then a small residual in a large anisotropic signal, and any error in the interloper templates leaks into it. How bright the carbon-monoxide lines are at redshifts of one to three, in the aggregate and including faint galaxies, is uncertain by factors of several. It is constrained by the same kind of measurement in reverse: intensity maps of carbon monoxide at its own redshift, which are only now approaching a detection.

The first maps of ionised carbon from the reionisation era will therefore have to measure their interlopers as well as their target, and the quadrupoles they find will say how much of what they see came from the wrong redshift. Whether the target survives that subtraction with enough precision to say how bright the faint galaxies of the reionisation era were depends on numbers nobody yet has. One consequence is certain: a map of the early universe made in one spectral line is also a map of the late universe in several others. The method that refuses no light is also, unavoidably, a census of the light it was not looking for.