A void is not short of dwarf haloes

A slice through a galaxy survey is mostly empty. The galaxies sit in walls and filaments, and between them are voids thirty or forty megaparsecs across in which a survey that reaches a normal galaxy’s brightness finds almost nothing. The centres of those voids hold, by the measurements of how fast they expand and how they bend light, about a fifth of the average density of matter. The void as an object — its outflow, its stretched appearance in a redshift map, its round average shape — is understood well enough to measure cosmology with. What is inside it is not.

The simplest reading of a void is that it is empty of galaxies because it is empty of matter: a fifth of the matter, so a fifth of the galaxies, and the brightest ones are missing because they are rare anyway. That reading is wrong in an instructive way, and the reason is in how dark matter haloes form. A halo is a gravitationally collapsed clump of dark matter, and every galaxy sits in one. How many haloes of each mass form in a region depends not on the region’s density alone but on how that density compares with the threshold for collapse at each mass scale — and the answer, for a void, is lopsided in a way no argument about proportions predicts.

A threshold crossed at every scale #

The theory that answers it is the excursion-set extension of Press and Schechter’s argument, and its idea is simple enough to state without equations. Take a point in the early universe and ask what the average density contrast is in a sphere round it, extrapolated to today with the linear growth of structure. Start with a sphere so large it holds the whole universe, where the contrast is zero, and shrink it. As the sphere shrinks the contrast wanders — up where it takes in a clump, down where it takes in a gap — and it wanders more widely the smaller the sphere, because small spheres feel the fluctuations that large ones average away. The first sphere, coming down from large scales, whose contrast crosses the collapse threshold of 1.686 is the halo that point belongs to, and its mass is that sphere’s mass.

The wandering is a random walk in the variance $S = \sigma^2(M)$, and a random walk’s first crossing of a barrier has a known distribution. Counting how much mass has its first crossing at each $S$ gives the number of haloes of each mass per unit volume. The unconditional answer is Press and Schechter’s mass function; the conditional one, for a region whose own contrast on its own scale is already known, is the same calculation started from the region’s value instead of from zero — the barrier is then $\delta_c - \delta_0$ away, and the walk has $S - S_0$ of variance left to cover it in.

A void centre is such a region. A sphere five megaparsecs in radius (in units of $h^{-1}$ Mpc throughout) at the middle of a large void, with a density today of a fifth of the mean, corresponds in linear theory to a contrast of about −2.9, obtained by running the spherical-expansion model that stops voids near eighty per cent empty backwards. Its walk begins almost three units below zero, and every halo inside it has to climb from there to 1.686.

The two mass functions part company in the way the figure shows. At the mass of a halo like the Milky Way’s, a trillion solar masses, the void has under two per cent of the average abundance per unit volume, and a region this size would on average hold about a tenth of one such halo. Its absence is guaranteed. At the dwarf end — a billion solar masses, the halo of a faint dwarf galaxy — the void has 38 per cent of the average abundance, more than its share of the matter.

More dwarfs per unit of matter, not fewer #

The comparison worth making is not per unit volume but per unit of matter: how efficiently does a void turn the matter it has into haloes of each mass, compared with an average region?

For every void depth drawn the answer at low mass is above one. A void with a fifth of the mean density turns its matter into billion-solar-mass haloes nearly twice as efficiently as an average place. The curves turn down only at a mass that depends on the void’s depth — near ten billion solar masses for the emptiest, a trillion for a void at half the mean density — and beyond that they fall by orders of magnitude.

The reason the dwarf end rises is not that voids are good at making dwarfs. It is that average regions are good at destroying them. The first crossing is what the theory counts: a point whose walk crosses the barrier at a large mass is counted in the large halo, even if a smaller sphere round it would also have crossed. In the physical universe that is a small halo that formed and was then swallowed by a bigger one. In an average region much of the matter that once sat in dwarf haloes now sits in groups and clusters, having merged. In a void, where large haloes hardly form, the dwarfs have nothing to merge into, and they survive as dwarfs. A void is not a region where small haloes failed to form. It is a region where they were not eaten.

Why the smallest haloes barely notice the void #

The same fact has a name in the language of clustering: halo bias. How much a halo population’s abundance rises or falls with the large-scale density around it is measured by a factor $b$, the halo bias, which the excursion-set theory gives through the peak-background split.

The bias is one where a halo population traces the matter exactly, above one where it exaggerates the matter’s contrasts and below one where it softens them. Rare, massive haloes are strongly biased: they form only where a small-scale peak happens to sit on top of a large-scale one, so a slight rise in the large-scale density multiplies their number. Common, low-mass haloes are anti-biased: they form from peaks so typical that a large-scale density barely tilts the odds, while the merging that removes them happens preferentially in the dense places. At a billion solar masses the bias is about a half, so a region with a tenth less matter than average has only about five per cent fewer such haloes — and per unit of its matter, more.

Galaxy surveys measure the bias of galaxies of each luminosity from their clustering, and for bright galaxies it is well above one, as the theory expects for the massive haloes they occupy. For the faintest galaxies the measured bias is still close to one and does not fall towards a half. That is the same discrepancy seen from the other side: the faint galaxies are clustered as if they were in haloes more massive than the dwarf haloes that dominate the count, which is what would happen if most dwarf haloes — the anti-biased ones, the ones that fill voids — held no galaxy at all.

A census that dates the haloes #

Whether a dwarf halo holds a galaxy depends less on its mass than on when it formed. Before about redshift six the gas between the galaxies was neutral and cold, and any halo heavy enough to let its gas cool by atomic hydrogen could collect it and form stars. Reionisation heated that gas to about ten thousand kelvin, and gas that hot is not held by a halo whose circular speed is below about twenty or thirty kilometres a second: it stays spread out, or is boiled off the halo it was in. A halo that reached a few billion solar masses before reionisation could make stars; one that reached it afterwards, and grew no further, could not. The threshold is soft and moves with time, but its existence is not in doubt.

The figure applies the same excursion-set arithmetic at earlier times, when the barrier the walk must cross was higher because structure had had less time to grow. Everywhere, only a small part of today’s dwarf haloes existed at redshift six. At the void centre the part is six times smaller still, because the void’s own density was lagging behind from the start: its haloes have further to climb, so each one crosses later. The void’s dwarf haloes are, overwhelmingly, haloes that formed after the gas around them had been heated — the ones least able to make a galaxy.

That is the second, independent reason a void is empty of dwarf galaxies. The first was that it is short of the massive haloes that make bright ones. The second is that its small haloes are young in the one sense that matters for whether they light: they arrived after the window closed.

How many should be there #

The arithmetic can be turned into a count for a single sphere.

Taken at face value, a five-megaparsec sphere at the centre of a deep void contains about four hundred and fifty haloes above a billion solar masses and a hundred and forty above three billion. If each held a galaxy, the sphere would be full of dwarfs. Surveys of the nearest voids, deep enough to reach galaxies a thousand times fainter than the Milky Way, find a handful. The count of haloes that had formed by redshift six, at the same threshold, is about two; the count above a hundred billion solar masses, massive enough to have captured and held gas whenever they formed, is two or three. The observed handful sits between these numbers, and closer to the second pair than the first.

The comparison is known as the void phenomenon, and it was once counted among the problems of the standard cosmology. It no longer is, for the reason the figures make plain. The halo count at the void centre is not in dispute and is not particularly small; what is small is the number of haloes that had the chance to make stars, and that number is set by a threshold in mass and time that the theory of haloes alone cannot supply. The count of dwarfs in voids is one of the cleanest measurements of where that threshold lies, because a void removes the complication that makes it hard to measure anywhere else: the merging of dwarfs into bigger galaxies, which in dense regions destroys the evidence.

The nearest void, seen from its edge #

The void that can be searched most deeply is the one next door. The Milky Way and its neighbours sit on the edge of the Local Void, a region that begins a few megaparsecs away and extends for tens of megaparsecs, much of it hidden behind the dust of the Galactic plane. Its expansion is felt directly: the peculiar velocities of the nearby galaxies show the whole Local Group being pushed away from it at a couple of hundred kilometres a second, one of the larger single terms in the budget for the Local Group’s motion. A void is a place that pushes, and this one pushes on the observer.

Inside it, the galaxies found so far can be counted on two hands. They include dwarfs a million times fainter than the Milky Way, found because they are close, and at least one of them is among the most gas-rich and least evolved galaxies known — a disc of hydrogen with a scattering of young stars, as if it had begun forming stars only recently. The volume searched is not large and the census is incomplete where the Galaxy’s dust hides the sky, but the contrast with the prediction is already stark: a sphere of that size in the void’s interior should hold, by the counts drawn above, a hundred or more dark haloes heavy enough to have held a galaxy had they formed in time. The nearest void is the strongest single case that most of them are dark.

It is also the one where the surveys are most likely to improve. The faintest galaxies are found either as resolved stars — a scattering of red giants at a common distance, picked out in deep imaging — or as clouds of neutral hydrogen detected by radio surveys with no reference to starlight at all. The second method is the one that can find a halo that holds gas and almost no stars, the object the late-formation argument predicts in numbers, and the radio surveys now under way will reach the Local Void’s interior to masses of gas well below those of the known members.

What the theory leaves out #

The excursion-set theory is an approximation, and a known one. Against simulations it overpredicts the abundance of small haloes by tens of per cent and underpredicts the rarest massive ones; the refinements that fix this, using an ellipsoidal rather than a spherical collapse, change the numbers but not the shapes, and the figures draw ratios of void to average, which the approximation gets closer to right than it gets either alone. It treats the void centre as a sphere with one density; a real void centre is a thin web of sheets and tendrils at a range of densities, and dwarf haloes form preferentially along those tendrils. The counts drawn are for a smooth sphere and should be read as an order of magnitude.

The count also stops at the halo. Whether a halo holds a galaxy depends on its full history — when it crossed the threshold, whether it was ever bigger and was stripped, what radiation field it saw — and on physics the figures represent only by one line at redshift six. The line is an idealisation of a transition that took hundreds of millions of years and happened at different times in different places. Voids, being far from the first sources of ionising light, were probably reionised late, which would give their small haloes more time to form stars before being heated. That effect runs against the one drawn, and how much it offsets it depends on the geometry of reionisation — which is itself a question of when the bubbles met, and in voids they met last.

And the theory is written for cold dark matter. A dark matter particle light enough to stream out of the smallest fluctuations — the same mechanism by which the neutrinos leave a measurable hole in the growth of structure, on much larger scales — or one that interacts with itself, would remove small haloes everywhere; the void’s excess of dwarf haloes per unit mass would be erased at masses below the cut-off, and the count at a void centre would be low for a reason that has nothing to do with star formation. The two explanations predict different things elsewhere — in the Lyman-α forest, in the substructure of lensing galaxies, in the faint end of the galaxy mass function — and void counts are only one of the tests that must agree.

What lives in the few that did light #

The galaxies that are found in voids are not a random sample of dwarfs. They are, on average, bluer — on the star-forming side of the colour gap almost without exception — more gas-rich and forming stars at a higher rate per unit of stellar mass than galaxies of the same mass in walls and filaments. Some are almost pure gas, with a few per cent of their mass in stars. Those properties are what the late-formation argument predicts: haloes that collapsed recently, in a region where nothing strips their gas by ram pressure and no larger neighbour heats it, and which are still converting their gas slowly into stars. A galaxy that runs out of gas in a fraction of its lifetime needs feeding from outside, and in a void the feeding is thin and steady rather than interrupted.

The same argument predicts something harder to see: haloes that formed late and never lit, holding gas that is ionised, warm and too diffuse to cool, detectable only through the faint absorption it imprints on the light of more distant quasars, or through the Lyman-α glow it emits where the ionising background strikes it. Whether that gas is there, and how much, is one of the targets of the next generation of integral-field spectrographs, which can map the faint emission of intergalactic hydrogen across regions the size of a void.

A test of the halo count as much as of the galaxies #

The argument cuts both ways. If the star-formation threshold were known independently — from simulations of the first galaxies, or from the faint end of the luminosity function in dense regions where it can be measured by counting satellites — then the void count would test the halo abundance itself: how many billion-solar-mass haloes the dark matter actually makes. It is one of the few places where that number can be tested at all, because everywhere else the dwarf haloes are either inside bigger ones, where they are tidally stripped, or too faint to count. A void is the one environment where a dwarf halo is left alone. The difference between how many haloes it holds and how many galaxies is the product of two unknowns, and the cosmology fixes the first more firmly than the astrophysics fixes the second — which is why the void phenomenon is now read as a statement about galaxies rather than about dark matter.

Still open: whether the threshold depends on where the halo lives #

The threshold below which a halo stays dark has been treated here as one mass at one time. It is probably not. In a void the ionising background arrived late and the gas was thinner; in a filament, the background arrived early and the gas was denser and could self-shield against it. A threshold that depends on environment would change the void’s dwarf count by a factor that is not small, and it would make the count at a void centre a measurement of the reionisation history of the void rather than of the universe. Deep surveys of nearby voids are beginning to find their faintest members — gas-rich dwarfs detected in the 21-centimetre line with few stars at all — and whether those dwarfs are too many, too few or just right for a threshold that moves with environment is the next thing the emptiest places will be asked.