A map of galaxies none of which was seen

Every census of galaxies has a floor. A survey detects a galaxy when its light stands clear of the noise, and below that brightness the galaxy is simply absent from the list. At low redshift the floor is so deep that the missing galaxies hardly matter; most of the light is in galaxies bright enough to see. At the redshifts where the first galaxies were forming, when the universe was under a billion years old, the opposite is true. The galaxies are faint, the numbers rise steeply towards fainter ones, and even the deepest images ever taken capture only a fraction of the light that was being emitted.

The sky’s background contains all of it, summed. The far-infrared and optical backgrounds are the integrated glow of every galaxy along every line of sight, detected or not, and the histogram of a map too crowded to separate can recover the number counts of sources nobody can pick out individually. Both of those use broadband light, which arrives from every redshift at once and so has no depth. Line-intensity mapping adds the depth. Map one spectral line — the 21-centimetre line of neutral hydrogen, the rotational lines of carbon monoxide, the 158-micron line of ionised carbon — and the observed frequency of each photon gives its redshift, so a map in many frequency channels is a three-dimensional map of the line’s emission. No galaxy needs to be detected for that map to exist.

A map with no galaxies in it #

The idea is simplest to see by comparing the two ways of observing the same patch of the early universe.

On the left is what a galaxy survey of moderate depth would list: a few dozen objects, scattered across the slice in a pattern so sparse that the structure underneath is barely visible. On the right is the same volume mapped in summed intensity. Every cell contains the light of every galaxy in it, so the map carries the faint galaxies along with the bright ones, and where the matter is dense the cells are brighter. The pattern is noisy — a cell is a sum of a random number of galaxies with random luminosities — but the noise averages down over many cells, and the structure is there statistically even where it cannot be seen by eye.

That statistical character is the point. An intensity map is not a picture to be looked at but a field to be measured, and the measurement is its power spectrum: the variance of its fluctuations as a function of scale. Two physically different things contribute to it, and separating them is what the method is for.

Two terms, two moments #

The intensity in a cell is the sum of the luminosities of the galaxies in it. If galaxies trace the underlying matter with a bias $b$, so that their number density fluctuates by $b$ times the matter’s fractional fluctuation, then the intensity fluctuates on large scales by $b$ times the matter times the mean intensity $\bar I$. The mean intensity is proportional to the first moment of the luminosity function $\Phi(L)$, the total luminosity per unit volume: $\bar I \propto \int L\,\Phi\,dL$. That gives the first term of the power spectrum, which has the shape of the matter’s own clustering.

The second term exists because galaxies are discrete. Even if they were placed at random, with no clustering at all, the sum of a random number of random luminosities would fluctuate from cell to cell. Those fluctuations have no preferred scale — they are white noise in the map, the same at every wavenumber — and their amplitude is set by the second moment of the luminosity function, $\int L^2\,\Phi\,dL$: a field made of a few bright sources is grainier than one made of many faint ones with the same total. Together,

$$\frac{P(k)}{\bar I^2} = b^2 D^2 P_m(k) + \frac{\int L^2\Phi\,dL}{\left(\int L\,\Phi\,dL\right)^2},$$

with $P_m$ the matter power spectrum today and $D$ the growth factor that scales it back to the epoch being mapped.

On large scales, small wavenumber, the clustering term dominates; on small scales the matter power falls and the flat shot term takes over. The crossing, for the parameters drawn, is at a wavenumber of about a third of an $h$ per megaparsec. A map measured on both sides of it measures both terms, and through them two different properties of a population of galaxies of which none has been individually seen: the product $\bar I\,b$ from the clustering, and the ratio of the second moment to the square of the first from the shot noise.

The faint majority and the bright few #

The two moments do not weight the same galaxies, and that is what makes measuring both worth the trouble.

The mean intensity is a sum of luminosities, and for a luminosity function that rises steeply towards faint galaxies it is dominated by galaxies somewhat below the characteristic luminosity $L^*$, with a long tail of contributions from fainter ones. The shot noise is a sum of luminosities squared, which favours the bright end; its contributions peak close to $L^*$ and fall off on both sides. A survey able to detect galaxies only above one and a half times $L^*$ would miss more than nine-tenths of the mean intensity and three-fifths of the shot noise. An intensity map misses neither.

So a single map’s power spectrum, with no galaxy detected, constrains the luminosity function on both sides of any detection limit. The clustering amplitude says how much light there is in total, mostly from faint galaxies. The shot-noise amplitude says how concentrated that light is in bright ones. Combining the two with the galaxies that can be detected individually, in the same volume, pins down the faint-end slope. That slope is the quantity the count of galaxies at the highest redshifts most needs, and which source counts can only extrapolate.

Read across many frequency channels, the mean intensity becomes a history. Each channel samples a different redshift, so the map’s mean intensity as a function of frequency is the line’s luminosity density as a function of cosmic time — the total emission of every galaxy at each epoch, faint and bright together. For a line whose luminosity is tied to star formation, ionised carbon above all, that history is a history of star formation that includes the galaxies no census reaches. The tie is itself a calibration, measured in nearby galaxies where both the line and the usual tracers of star formation can be observed, and it may change with the metallicity and density of the gas. The mean intensity is also the hardest quantity for a map to measure directly, because a constant added to every pixel is indistinguishable from an instrumental offset or a smooth foreground. In practice it is inferred from the clustering, through the product with the bias, rather than read off the map’s zero level.

How much a catalogue misses #

How large the gap is depends on how steep the faint end is, and at high redshift the faint ends are steep.

The faint-end slopes measured at redshifts of six and beyond, where galaxies are counted in their ultraviolet light, are around −2: steeper than any curve in the figure, so steep that the total light depends on where the luminosity function turns over at the faint end, which no survey has reached. That turnover is a physical question about the first galaxies. It is set by the smallest haloes able to cool their gas and form stars once the ultraviolet background from reionisation began heating it. A census that stops at a tenth of $L^*$ cannot see the turnover. An intensity map includes the light from below it, so it measures the integral that the census can only extrapolate.

This is the same argument Olbers’s paradox began: the total brightness of the sky is a sum over all the sources there are, and the sum depends on how the numbers behave where the sources are too faint to see individually. A dark night sky was first read as a statement about the universe’s size and age. A faint background was later read as a census of the galaxies that make it. An intensity map is the same census split by redshift, one frequency channel at a time.

What it takes to measure each term #

Measuring a power spectrum is a matter of counting independent modes. In a survey of volume $V$ the number of independent Fourier modes in a shell of width $\Delta k$ around $k$ is about $Vk^2\Delta k/(2\pi^2)$, and each mode measures the power with a fractional uncertainty of one. The signal-to-noise ratio of the power in a bin is the square root of the number of modes it contains, degraded by the ratio of the signal to the total power, including the instrument’s own noise.

The two terms are measured in different places. At small wavenumbers there are too few modes in the volume to measure anything well, and the clustering term, though dominant there, is poorly determined. Its best measurement comes near the crossing scale, where it still dominates and the modes are plentiful. The shot term, flat and dominant at large wavenumbers, is measured where there are many modes and is limited mostly by the instrument’s noise. A survey of a few hundred square degrees at redshift six, with noise a few times the shot noise, measures the clustering at a signal-to-noise of over ten per bin over the scales that matter. That is the design target of the surveys now under way.

The instrument’s noise is the tightest requirement. An intensity map needs a sensitivity per voxel comparable to the shot noise of the galaxies being mapped, which for faint lines at high redshift means long integrations with many detectors. And it needs the map’s frequency response to be smooth and stable to a part in thousands, because any structure the instrument imprints in frequency becomes structure along the line of sight in the map.

One product where two numbers were wanted #

The clustering term has a frustrating feature: it measures the product $b\,\bar I$ and not either factor. A brighter population of weakly clustered galaxies and a fainter population of strongly clustered ones give the same large-scale power. Since the mean intensity is the quantity the census of faint galaxies needs, and the bias is a property of the haloes the galaxies live in, the degeneracy hides exactly the number of interest.

The map’s own geometry offers one way to break it. A map built from redshifts is distorted along the line of sight by the galaxies’ peculiar velocities, because infall towards dense regions shifts their emission in frequency. On large scales this stretches the map along the line of sight in a calculable way: the power along the line of sight is enhanced relative to the power across it by a factor that depends on the ratio of the growth rate to the bias. The growth rate is known well from cosmology, so the anisotropy of the clustering measures the bias, and the amplitude then gives the mean intensity. The measurement needs both the line-of-sight and the transverse modes, and so a map with good resolution in both directions.

The second way is to compare the map with something whose bias is known. A catalogue of galaxies in the same volume, detected individually and clustered with a bias measured from their own correlations, cross-correlates with the intensity map with an amplitude proportional to $b\,\bar I\,b_g$. Dividing out $b_g$ leaves the same product as before, so the cross-correlation does not by itself break the degeneracy. It does something more useful at first: it removes everything in the map that is not correlated with galaxies — instrumental noise, foreground emission from the Milky Way, lines from other redshifts. A cross-correlation can detect a faint line under foregrounds thousands of times brighter, because the foregrounds average to zero against the galaxies.

That is how intensity mapping first became a measurement rather than a proposal. The 21-centimetre signal from galaxies within a few billion years was first detected by cross-correlating radio maps with optical galaxy redshift surveys of the same volume, years before any radio map alone could separate it from its foregrounds. The same strategy is the plan for the high-redshift lines: cross-correlate the line maps with the few galaxies that can be detected at those redshifts, or with maps of other lines tracing the same galaxies, and measure the signal there first.

Voxels, haloes, and the scale in between #

A real map has finite resolution. Its pixels on the sky are set by the telescope’s beam, typically arcminutes, and its channels in frequency by the spectrometer’s resolution. At redshift six an arcminute is about two comoving megaparsecs, and a spectral resolving power of a few hundred corresponds to a few megaparsecs along the line of sight. The map’s voxels are therefore a few megaparsecs on a side, each holding many galaxies, and the power spectrum is measured only on scales larger than a voxel; the shot noise is the variance of the voxel sums, not of individual galaxies.

Between the two terms of the figure lies a third, which the simple model omits. Galaxies are not placed independently even on small scales: several share each dark-matter halo, and the ones in a massive halo are bright and close together. Their correlation within haloes adds power on the scales of haloes, a megaparsec or so, on top of both the large-scale clustering and the shot noise. In galaxy surveys this is the one-halo term. In an intensity map it blurs the crossing between the two terms the method relies on, and it carries information of its own about how line luminosity depends on halo mass. Separating it from the shot noise needs resolution finer than a halo, which most current instruments at high redshift do not have.

The lines being mapped #

Several lines are being mapped, each with its own reason and its own difficulties.

The 21-centimetre line of neutral hydrogen is the oldest target and the one with the most mature instruments. In the late universe, after reionisation, most neutral hydrogen sits in galaxies, and a 21-centimetre intensity map traces them the way a galaxy survey would, at a fraction of the cost and with no galaxy detected. Its first detections came in cross-correlation, comparing radio maps with optical galaxy surveys of the same volume, and dedicated arrays have since measured the signal’s power directly in the nearer universe. The 21-centimetre line is weak per atom but hydrogen is abundant, and its great difficulty is foreground synchrotron emission from the Milky Way, thousands of times brighter, which has to be removed by its smoothness in frequency.

Carbon monoxide traces molecular gas and so the fuel for star formation. Its rotational lines from galaxies at redshifts of two to three fall in the centimetre and millimetre bands, and the first dedicated surveys have set limits on their power and are approaching a detection. Ionised carbon at 158 microns is the brightest cooling line of star-forming gas, and from redshift six it falls at a few hundred gigahertz, where several instruments are now beginning to map it. Lyman-alpha and hydrogen-alpha, mapped from space in the optical and near-infrared, trace star formation and the ionised gas around it.

Each line also brings light from other redshifts into the same frequency channels, from other lines that happen to land there. A map of ionised carbon from redshift six contains carbon monoxide from redshifts below three, mapped at the wrong distance, and separating the two is a problem with its own geometry.

What the figures assume #

The galaxies are drawn from one Schechter luminosity function, with every galaxy emitting the line in proportion to a single luminosity, and a single bias relating them to the matter. Real emitters have a distribution of line luminosity at fixed halo mass, which adds to the shot noise and makes it a moment of a joint distribution rather than of one function. The matter power spectrum is the smooth linear shape with no acoustic features, scaled by the growth factor and evaluated on scales where the linear approximation holds; at large wavenumbers non-linear clustering adds power that the figures leave out. The map simulation is a projection of a thin slab drawn with a two-dimensional field, enough to show the principle but not a forecast for any survey. And the mode-counting estimate assumes a cubic volume with uniform noise, ignoring the loss of modes along the line of sight to foreground removal, which in practice is the largest single loss.

Still open: whether the faint galaxies reionised the universe #

The question line-intensity mapping is best placed to answer is one no galaxy survey can. Something reionised the universe’s hydrogen between redshifts of about fifteen and six, and the leading suspects are the stars in faint galaxies, numerous enough to make up in number what they lack individually. Whether they supplied enough ionising photons depends on the integral of the luminosity function to its faint end — exactly the integral the census cannot reach and the mean intensity measures.

The maps that would answer it are being made, in ionised carbon, in the 21-centimetre line during reionisation itself, and in Lyman-alpha. Whether their first measurements, mostly of shot noise and of the clustering at the largest accessible scales, will be precise enough to fix the total light of the faint population is not yet known. That depends on noise levels that have not been reached and foreground removals that have not been demonstrated at the required depth. The method’s promise is a census of every galaxy without counting one. Its first results will say whether that census can be taken through the foregrounds that sit in front of it.