An emptiness that expands faster

Most of the volume of the universe is empty, or nearly so. A slice through a redshift survey shows galaxies strung along filaments and piled into clusters, and between them regions tens of megaparsecs across with hardly a galaxy in them. Those voids fill about three-quarters of the volume and contain perhaps a tenth of the galaxies, and for a long time they were treated as the background against which structure was drawn — the places where the interesting things were not.

The voids are structures in their own right, and in some ways cleaner ones than the clusters. A cluster is a knot of dark matter, hot gas and galaxies in which every physical process in astrophysics is happening at once, and extracting cosmology from it means modelling all of them. A void is a region where very little is happening at all. Its dynamics are dominated by gravity and by the expansion of the universe, both of which are simple, and it responds to them in a way that is almost linear even when it is deeply underdense. That makes voids a laboratory for the one thing cosmology most wants to test: how gravity grows structure.

A small open universe #

The density profile of a void, averaged over many voids, has a characteristic shape. The centre is emptiest, typically at 15 to 20 per cent of the mean density. The density rises towards the edge and overshoots it, forming a compensating wall of matter slightly denser than average — the filaments and sheets that bound the void — before settling to the mean far away. The figure uses the form fitted to stacked voids in simulations, and its shape is nearly the same for voids of different sizes once distance is measured in units of each void’s radius.

What such a region does under gravity follows from a theorem as old as cosmology. A spherical region of the universe evolves as a universe of its own, with its own mean density, independent of what lies outside it — the cosmological version of the result that a spherical shell exerts no net force on anything inside it. A void is a region with less matter than average, so it is a lower-density universe: it decelerates less, and expands faster, than its surroundings. In the linear theory of structure growth the excess is set by the mass deficit inside each radius, $\Delta(r)$, and the growth rate $f$ of structure:

$$\frac{v}{Hr} = -\frac{f}{3}\,\Delta(r).$$

For the profile drawn, the centre of the void expands 21 per cent faster than the universe around it, and the excess falls towards the edge. The outflow speed — the excess rate times the radius — peaks at about seven-tenths of the void’s radius. Beyond the wall, where the enclosed mass has caught up with the average, the excess vanishes, and just outside it matter is pulled slightly inward towards the dense wall.

Every galaxy in a void is moving away from its centre. Not by much — a few hundred kilometres a second for a void thirty megaparsecs across — but coherently, and with a speed that depends only on the amount of missing mass and on the growth rate. That coherence is what makes voids useful, because the velocities can be seen.

Why voids stop at a fifth of the mean #

An overdense region collapses: its excess gravity decelerates it until it stops expanding and falls in on itself, forming a halo whose density is two hundred times the mean. The same arithmetic run backwards describes a void, and it does not run away.

As a void expands faster than its surroundings, its density falls, but it cannot fall below zero, and it approaches zero ever more slowly. The figure uses Bernardeau’s approximation to the exact spherical solution: the true density contrast against the contrast linear theory would assign. Overdensities curve upward towards collapse; underdensities curve away from the linear line towards −1 but never reach it.

Something happens on the way. Because the inner parts of a void are emptier, they expand faster than the outer parts, and at a linear contrast of −2.72 the inner shells catch up with the outer ones. That is shell crossing: matter piles up at the boundary, the wall becomes sharp, and the void stops evolving as a smooth underdensity and starts behaving as a bubble bounded by a dense rim. At that moment the density inside is 21 per cent of the mean. That is why mature voids everywhere have central densities of about a fifth of the average, whatever their size, and why a void, unlike a cluster, has a natural edge: the shell-crossing wall.

The asymmetry between the two directions explains the shape of the cosmic web. Collapse amplifies small overdensities into compact objects that occupy little volume; expansion smooths underdensities into large, round, nearly empty regions that occupy most of it. The universe is mostly void by volume and mostly cluster and filament by mass, because gravity treats excess and deficit so differently.

Finding a void is a choice #

A cluster announces itself — a peak in the galaxy density with hot gas at its centre — but a void has to be defined, and the definition is part of the measurement. The most widely used method starts from the galaxies themselves. Each galaxy is given the region of space closer to it than to any other galaxy, a Voronoi cell, whose volume is large where galaxies are sparse. Cells are then merged, starting from the largest, into basins that flow downhill in density to a common minimum, the way a landscape’s catchments are found by following water downhill — a watershed. Each basin is a void; its centre is the volume-weighted centre of its cells, and its radius is that of the sphere of the same volume.

The method has no free parameters, which is its virtue, but it responds to everything in the galaxy distribution, including its sparseness. A survey that samples galaxies thinly finds more small spurious voids; one that stops at a bright magnitude limit sees different voids from one that goes deep. Voids found in different tracers — bright galaxies, faint galaxies, quasars — have different profiles because the tracers sit differently in the web. The stacking results quoted here hold for voids found consistently in one tracer and compared with simulations processed in exactly the same way, and a comparison that skips that step compares two different kinds of object.

A void stretched by its own outflow #

A redshift survey measures a galaxy’s distance by its redshift, and the redshift includes the galaxy’s own motion as well as the expansion. Around a cluster, galaxies are falling in, and the effect is to squash the cluster’s surroundings along the line of sight on large scales: galaxies on the near side, moving away from the observer towards the cluster, appear further away, and those on the far side appear nearer. Around a void everything runs in reverse.

Galaxies on the far side of a void are moving away from its centre — away from the observer — and so appear further away than they are; those on the near side are moving towards the observer and appear nearer. Both are displaced away from the centre along the line of sight, and the void in the redshift map is stretched in that direction. Averaged over many voids, which in real space are round on average, the stretch appears as an elongation of the stacked void along the line of sight, a few per cent in the figure.

The calculation behind the figure has a subtlety worth stating. The distortion is not only a change in apparent density at each point but a change of where each point appears, and a model that evaluates the density at the redshift-space position without mapping it back to where the galaxy really was loses the elongation entirely and gets the sign of the effect wrong at the void’s edge. Models that skipped that mapping disagreed with simulations near the void’s edge; the analyses that map every point back through the outflow before evaluating the profile agree with them, and that step is now standard.

A growth rate with the bias taken out #

The stretch is proportional to the growth rate $f$, which is what makes it a measurement.

The part of the distortion that depends on the angle to the line of sight — the quadrupole of the cross-correlation between void centres and galaxies — is, in linear theory, the growth rate times a function of the void’s own profile: $(2f/3)$ times the difference between the density at a radius and the mean density inside it. The profile is measured from the same data, averaged over directions, so the quadrupole’s height gives $f$ directly. In general relativity with the measured matter density, $f$ is about 0.75 near redshift 0.5; theories that modify gravity on large scales change it by amounts comparable to the spread drawn.

Measurements of the growth rate from redshift-space distortions around all galaxies — the Kaiser effect, mapped over whole surveys — carry a persistent difficulty: galaxies are biased tracers of matter, clustering more strongly than the matter does by a factor that must be modelled, and the distortion measures $f$ divided by that bias. Around voids the difficulty largely disappears. The voids are found in the galaxy distribution itself, so the profile $\xi(r)$ that enters the quadrupole is the galaxy profile, and the velocities that produce the distortion are those of the galaxies. In the linear model the bias cancels out of the ratio, and what remains is $f$.

Voids also test gravity where modifications are expected to show. The theories that change gravity on cosmological scales while agreeing with the solar system’s tests usually include a screening mechanism that suppresses the modification in dense environments. The emptiest places in the universe are where screening is weakest and the modification, if any, is largest. A growth rate measured around voids that differed from one measured around clusters would be the signature of exactly such a theory. The measurements made so far, from surveys of a million galaxies, find $f$ around voids consistent with general relativity, with uncertainties of five to ten per cent.

A round average and a wrong cosmology #

A single void is irregular, but a stack of thousands of voids must be spherical on average, because the universe has no preferred direction. That makes stacked voids a standard shape, and a standard shape measures geometry.

To draw a map from a redshift survey, redshifts and angles have to be converted to distances, and the conversion assumes a cosmology. Along the line of sight the conversion depends on the expansion rate $H(z)$; across it, on the angular-diameter distance. If the assumed cosmology is wrong, the two conversions are wrong by different factors, and a sphere in the sky becomes an ellipsoid on the map. This is the Alcock–Paczyński test, and it needs no standard ruler — only an object known to be round. A matter density wrong by 0.06 distorts the stacked void by 3.5 per cent at redshift one, which surveys of hundreds of thousands of voids can measure.

The test around voids has a complication that is also its strength. The outflow stretches the voids too, so the geometric distortion and the dynamical one both appear as ellipticity, and they have to be fitted together. They are separable because they depend differently on distance from the centre: the geometric distortion rescales the whole profile, while the dynamical one follows the shape of $\xi(r) - \Delta(r)$, peaking near the void’s edge. Fitting both gives the growth rate and the expansion history from the same stack, and the constraints from voids on the matter density and the equation of state of dark energy are now comparable to those from the acoustic scale measured in galaxies, and independent of it.

What the surveys have measured #

The largest spectroscopic surveys of the last decade, mapping a million or so galaxies out to redshift 0.7, yield void catalogues of several thousand voids with radii from about twenty to sixty megaparsecs. Stacked, their redshift-space elongation has been measured with enough precision to give the growth rate to about ten per cent and, combined with the Alcock–Paczyński distortion of the same stacks, the ratio of the transverse to the line-of-sight distance scales to a few per cent. Both agree with general relativity and the standard cosmology, and the geometric result is one of the more precise single measurements of that ratio at those redshifts, because the voids’ round average is a cleaner reference shape than anything a galaxy cluster offers.

The next generation of surveys, mapping tens of millions of galaxies to redshift two, will find hundreds of thousands of voids. At that scale the statistical errors on the void growth rate fall below those of the galaxy-clustering measurements, and the systematic questions — how voids are defined, how well the linear model fits the edge, how the galaxies’ own random motions are handled — become the limit. Whether voids then become the most precise test of gravity on cosmological scales, or a cross-check on the others, depends on how well those questions are answered.

What the model leaves out #

The drawing uses linear theory for the velocities, which is accurate for the outflow in most of a void but fails near the dense wall and inside the filaments that cross many voids; there, galaxies have random velocities of a few hundred kilometres a second that smear the distortion, and the fits add a dispersion term for them. It treats void centres as fixed points, when in practice they are found by algorithms that search the redshift-space galaxy distribution, and a void found in redshift space is already distorted by the outflow it is meant to measure; the modern analyses reconstruct the real-space positions of the galaxies before finding the voids, removing most of that circularity.

It also treats every void as a sphere with the average profile. Real voids are irregular, contain sub-voids and thin filaments, and differ in profile with size: small voids sit inside larger overdense regions and are surrounded by high walls, while large voids are embedded in underdense surroundings and have almost no wall. Stacking averages over that variety, which is why stacks behave simply; it also means that a result from voids is a statement about the average, and that the choice of which voids to stack is part of the measurement.

Other things voids weigh #

Voids also bend light. A void’s deficit of mass acts as a weak diverging lens, and the same weak-lensing measurements that map dark matter around galaxies detect the demagnification of background galaxies behind stacked voids, giving the void’s mass profile directly rather than through the galaxies inside it. And because photons crossing a void while it expands and the gravitational potential decays gain a little energy, the microwave background is very slightly colder in the directions of large voids — the integrated Sachs–Wolfe effect, which in a universe with dark energy is a direct signature of the acceleration that was not supposed to be there. Both effects are weak and have been detected only statistically, by stacking many voids, and the size of the second has been the subject of some tension, with the largest voids appearing to leave a colder imprint than the standard model predicts.

Still open: how empty the emptiest places are #

The one quantity no stack measures well is the density at the very centre of the largest voids, because there are almost no galaxies there to measure it with. Whether the centres hold small, faint galaxies that surveys miss, a diffuse web of gas, or nearly nothing at all bears on galaxy formation — how small a dark matter halo can be and still make a galaxy — and on the shape of the profile the growth-rate measurement depends on. The deepest surveys are now finding dwarf galaxies inside nearby voids, far fewer than in denser regions and with more gas and younger stars, as if they had formed late and slowly. The emptiness that expands faster than everything around it turns out to be a place where galaxies can still be found, and how many there are is the next thing it will be asked to weigh.