A void dated by the dwarfs that lit in time

A sphere five megaparsecs across at the centre of a deep void holds, by the excursion-set theory of halo formation, about a hundred and forty dark matter haloes heavy enough to have held a dwarf galaxy, and the deepest surveys of real voids find a handful of galaxies in such a volume. The reconciliation is a threshold in time as well as in mass. A small halo could collect gas and make stars only while the gas between the galaxies was cold. Once the first stars and quasars had ionised it and heated it to ten thousand kelvin, a halo too shallow to hold gas that warm could no longer begin, and the haloes that had not crossed the mass threshold by then stayed dark. Counted that way, the void’s sphere held about two lit haloes when the heating arrived.

That count treated the heating as one moment for the whole universe. It was not. Reionisation was a matter of bubbles, opened by galaxies that formed first in the dense places, growing outward and meeting. The last regions the bubbles reached were the ones with the fewest galaxies of their own: the centres of the large voids. A void’s gas was heated by radiation that arrived from outside, late, and the date it arrived is a local fact about that void. Ask how the count depends on that date and the void’s dwarfs turn into a clock.

A threshold that is also a date #

The calculation behind the figure is the one that counted the haloes in the first place, with the date left free. For each redshift, the excursion-set theory gives the number of haloes above a given mass that had formed from a region’s matter, starting the random walk of density from the region’s own lagging value. A void centre at a fifth of the mean density today began with a linear density contrast nearly three units below zero, and every halo inside it had further to climb to the collapse barrier than a halo anywhere else. Raising the barrier, as looking back to earlier times does, makes the climb harder still.

The number that comes out is the count of haloes that were heavy enough, early enough. It is not quite the count of galaxies today, because some of those early haloes have since merged with one another; following them forward, about three in five survive as separate systems. But it is the count of separate places where star formation began before the void’s gas was heated, and it is what the date controls.

At the void centre that count is a handful for any plausible date and changes rapidly with it. Heated at redshift 5.3 — the latest the absorption of quasar light allows for the end of reionisation anywhere — the sphere would have four lit haloes. Heated at 7.7, around the middle of reionisation by the measurement from the microwave background’s polarisation, it would have one in five chance of any. Between those two dates, about four hundred million years apart, the count changes twentyfold.

The average sphere’s count changes too, but less: under sevenfold across the same interval. Both curves fall because earlier dates leave less time for haloes to cross the threshold, and the void’s falls faster. That difference is the whole of the argument, and it has a familiar cause.

What the heating does to a small halo #

The threshold is a statement about temperature and depth. Gas in a halo is held if the halo’s gravity can confine it against its own pressure, and the measure of that is the ratio of the gas’s sound speed to the halo’s circular speed. Before reionisation the intergalactic gas had cooled with the expansion to a few tens of kelvin, and its sound speed was a fraction of a kilometre a second: any halo deep enough to let the gas cool further by radiating could pull it in. Ionisation heated it to about ten thousand kelvin at a stroke, a sound speed near fifteen kilometres a second, and from then on a halo with a circular speed below twenty or thirty kilometres a second could not compress it.

The transition was not quite instantaneous for the haloes either. Gas already settled in a halo’s centre before the heating arrived was dense and could shield itself from the radiation for a while; gas still falling in was heated on the way and stopped. A halo just above the threshold kept forming stars from what it had; one just below lost its gas over a few hundred million years, boiled off by the heating and by its own young stars. That is why the threshold is soft, and why its value at the heating date depends on what the halo was doing when the radiation arrived. The quasar spectra that show the forest of absorption thinning to nothing by redshift six, and the dark troughs that persist in some of them a little later, are measurements of when the heating had finished in the regions those quasars happen to light, not of when it reached any particular void.

What matters for the count is that the heating is a one-way gate in time. A halo that crossed the threshold before it began, in a void or anywhere else, had already begun. A halo that crossed afterwards found the gas around it too warm to gather and had to grow much heavier before it could start.

Why the emptiest count is the steepest #

The haloes that had crossed the threshold by redshift six were rare everywhere: two per cent of the average region’s matter, and much less at a void centre. Rare objects sit in the tail of a distribution, and a tail is steep. A small change in how far the walk has to go — a change in the date, which moves the barrier, or a change in the region’s starting density — moves the count by a large factor, because what is being counted is the far end of a Gaussian.

The emptier the region, the further into the tail its early haloes are, and the more sharply their number depends on when it is counted. A void at a tenth of the mean density more than quadruples its count of early-lit haloes for each unit of delay. The places that hold the fewest dwarfs are the most sensitive clocks.

This is the same arithmetic that makes the count of massive clusters so sensitive to the amplitude of the matter fluctuations. The massive clusters are a tail of the halo distribution at the top end of mass; the early dwarfs in a void are a tail at the low end of density and time. In both, a parameter that moves the barrier by a few per cent moves the count by tens of per cent, and in both the steepness is what makes the count a measurement and what makes it fragile. A few per cent of error in anything that moves the barrier — the threshold mass, the void’s density — is a large error in the inferred date.

A late date against a low density #

A void centre was heated late, and a late date gives more of its haloes time to cross the threshold. Its low density makes its haloes cross late. The two delays push in opposite directions, and how far the first undoes the second decides whether a void’s dwarf count is suppressed at all.

Simulations of reionisation that follow the radiation from the galaxies that made it place the centres of large voids among the last regions to be ionised, by a fraction of a unit of redshift to about one after the average. A delay of that size closes the whole gap for a void at half the mean density, so that per unit of matter it lit as many dwarfs as anywhere else. For a void at a fifth of the mean density it shrinks the deficit from a factor of twelve to a factor of about four, and for the deepest voids it barely helps.

So the void phenomenon — the shortage of dwarf galaxies in voids that was once counted among the problems of the standard cosmology — has two parts that the theory of haloes keeps separate. The shallower parts of voids should not be short of lit dwarfs at all, once their late heating is taken into account. The deep centres should be, by factors that depend steeply on how late they were heated. A survey that counts dwarfs as a function of depth within voids measures the race between the two delays, and the depth at which the deficit sets in is a statement about the timing of reionisation inside voids that no other measurement can make.

A count that weighs two things at once #

The count depends on two things: when the gas was heated, and how heavy a halo had to be by then. The threshold mass of 3.2 billion solar masses is a convenience — roughly the mass whose gas, once cooled by atomic hydrogen, can collapse to make stars — and it is not known to better than a factor of a few.

A later date and a higher threshold give the same count. In the plane of the two, every measured count is a line, and a quarter of a decade of uncertainty in the threshold is a full unit of uncertainty in the date. The count alone measures one combination.

Worse, the threshold is not expected to be the same everywhere. A halo holds its gas against the heating radiation partly by gravity and partly by shielding: dense gas absorbs the radiation in its outer layers and keeps its interior cold. The gas in a void is thinner than average, so it shields less, and a halo there should need to be heavier to keep enough of its gas cold to form stars. Against that, the radiation reaching a void centre has crossed more neutral gas and is weaker when it arrives. The direction of the net effect is not settled, and it has the same form as a shift in the date. A void that appears to have been heated early may have been heated late with a stricter threshold, and the count cannot tell the two apart.

The dwarfs that lit are not the smallest #

One natural hope is that the masses of the lit galaxies would break the degeneracy. A later date and a higher threshold light different haloes, and the masses of dwarf galaxies can be measured from the widths of their 21-centimetre hydrogen lines, which give the speed at which the gas rotates.

The figure answers it, and the answer is no. The haloes lit before their gas was heated are not today’s smallest dwarfs. A halo that crossed three billion solar masses at redshift six has grown about eighty-fold since, through the accretion and mergers that continue in a void as they do everywhere, if more slowly. The lit haloes are, today, a hundred to five hundred billion solar masses, rotating at eighty to a hundred and thirty kilometres a second — the void’s larger galaxies, not its faintest. And the curve for the degenerate history, later and stricter, falls exactly on the curve for the earlier, laxer one. The progenitor odds of a halo depend on its own mass, and in this theory the two histories give the same odds at every mass. The masses follow the same combination of date and threshold as the number, and add nothing to it.

The fainter dwarfs that surveys of voids actually find — gas-rich, blue and still forming stars, with rotation speeds of a few tens of kilometres a second — are therefore not what the early count is about. They are haloes that grew massive enough after the heating to collect warm gas anyway, by a separate and later route whose threshold depends on the present state of the gas rather than on the date of its heating. A census of a void that does not separate the two routes mixes a date-sensitive count with a date-blind one.

Old stars as the separating mark #

What separates them is the age of the stars. A galaxy that began forming stars before its gas was heated carries stars from that time — formed more than twelve and a half billion years ago — whatever it has done since. A galaxy that began afterwards cannot. Close enough to be resolved into individual stars, a galaxy’s history can be read from the colours and brightnesses of its red giants and the faint end of its main sequence, which together date when its stars were made.

The method has been applied to the faintest satellites of the Milky Way, the ultra-faint dwarfs, and the result is one of the clearest fossil records of reionisation there is: most of them formed nearly all their stars more than twelve billion years ago and then stopped, as a halo below the threshold would if its gas was heated around redshift six. They are the Milky Way’s early-lit haloes, frozen at the moment their gas was taken.

The same measurement in a void would count early-lit galaxies directly, rather than inferring them from a total. The Local Void, whose push on the Local Group is felt in the motions of the nearby galaxies, begins a few megaparsecs away, within reach of the telescopes that resolve individual red giants. Its known members can be counted on two hands, and whether any of them holds an ancient population, and how large a share of its stars that population is, are questions that space telescopes can answer for the nearest of them. A void galaxy with no stars older than ten billion years is one that lit after its gas was heated. A void with several galaxies holding ancient stars was heated late, or had a lax threshold — the degeneracy remains, but now it is a degeneracy in a population whose meaning is not in doubt.

How many voids a date would need #

The count in one void sphere is a few galaxies, and a few is a number with a large Poisson error.

The emptiest spheres are the most sensitive per galaxy and the least informative per sphere: their counts depend most steeply on the date, but there are too few galaxies in them for the steepness to make up for it. A survey that wants a date for void centres in general would count galaxies with ancient stars in a few dozen spheres, at a range of depths, and fit the dependence of the count on depth — which is itself sensitive to the date, as the race between the two delays showed.

The figure leaves out the error that would dominate a real measurement: the uncertainty in each sphere’s own density. A sphere’s count depends steeply on its depth, and the depth of a void centre is measured from the galaxies around it, with an error of several per cent of the mean density. Through the steepness of the tail, that is an error of tens of per cent in the expected count, and so of a few tenths in the date.

The sphere is not a void #

Every number here comes from an idealisation, and the idealisations should be named. The excursion-set theory overpredicts small haloes by tens of per cent against simulations; the figures draw ratios and slopes, which it gets closer to right, but the absolute counts are an order of magnitude statement. A void centre is a web of thin sheets and filaments, not a uniform sphere, and small haloes form along those structures preferentially; voids themselves come in every size up to the scale at which the universe becomes smooth, and the five-megaparsec sphere is one choice among many. The theory assumes cold dark matter: a particle that streams out of the smallest fluctuations would remove small haloes everywhere and lower every count for a reason with no date in it. Heating is treated as instantaneous for each sphere, when in fact it took tens of millions of years to cross a void. The threshold is a single mass, when it is really a soft transition depending on the halo’s history. And the later route by which haloes light after the heating is left out of the count entirely, as it should be for an argument about the date, but it is in every real census.

None of these changes the direction of the result. The early-lit count at a void centre is a steep function of the local date, steeper than anywhere else, and degenerate with the threshold in a way that the halo masses cannot break. What the idealisations change is the conversion from a count to a date, and that conversion is where a real measurement would spend most of its effort.

A history written locally #

Reionisation is usually described by its global history: the fraction of the universe’s hydrogen ionised at each redshift, a curve rising from zero to one between redshift fifteen and five or six. That curve is an average over a process that was entirely local — each region was neutral, then for a short time partly ionised, then ionised — and the variation in the local date from place to place is the geometry of the bubbles, which the global curve does not record.

The 21-centimetre line of the neutral gas can in principle map that geometry directly, by imaging the neutral regions as they shrink, and experiments are being built to try. But the map is a snapshot at the time the light left, and it records the bubbles, not the regions that would become voids. The dwarfs in a void are a different kind of record: a census made today of the consequences of a date the void itself experienced. They are not a sharp record, because a count is a noisy instrument and the threshold muddies it. But they are a record of the date in a particular place, held by the galaxies that live there, and they are the only one of their kind.

Still open: whether the void dwarfs hold ancient stars #

The early-lit galaxies in a void centre should be the larger ones, rotating at around a hundred kilometres a second, and they should carry stars older than twelve and a half billion years. The fainter, gas-rich dwarfs that void surveys find should mostly not. Neither statement has been tested on a single void. The resolved stellar populations of the few galaxies inside the Local Void are within reach of the deepest imaging, and the census of void galaxies at larger distances — with ages estimated from their integrated spectra rather than from individual stars — is growing with each wide spectroscopic survey. A void whose larger galaxies all hold ancient stars, at a density where the theory expects none, would say the void was heated late or the threshold was low there. Which of the two is a question the threshold’s dependence on environment, still unsettled, would then have to answer.