A cusp a black hole would sharpen

The argument about the innermost kiloparsec of dark halos is about whether the dark matter’s density keeps rising towards the centre, as one over the radius in the halos that simulations make, or flattens into a core. Its ending pointed inward, to a question no rotation curve can reach: what happens to the cusp where it meets a central black hole. The answer, if the black hole grew in the right way, is that the cusp becomes something far more extreme than either side of the cusp–core argument — a spike, rising towards the black hole faster than any halo profile does.

The spike has never been seen. Whether it exists depends on the history of the black hole, the stars around it and the galaxy’s mergers, and the reason to care about it is that if it did exist and the dark matter annihilates, it would be among the brightest sources of annihilation products in the sky.

Pulled in on shrinking orbits #

The mechanism is the same conserved quantity that makes a halo contract around a settling disc. A dark-matter particle on an orbit around the centre of a halo has actions — integrals of its motion that stay fixed when the potential it moves in changes slowly compared with its orbital period. Grow a black hole at the centre slowly, by accretion over hundreds of millions of years, and every orbit shrinks while keeping its actions. Particles that were spread through the inner parsec are pulled into a fraction of it, and the density there rises.

Gondolo and Silk worked out the result in 1999 for a cusp falling as $r^{-\gamma}$. Inside the black hole’s sphere of influence — the radius within which its gravity dominates the stars’ and the dark matter’s, $GM/\sigma^2$ with $\sigma$ the stars’ velocity dispersion, about 1.8 parsecs for the Milky Way’s black hole — the dark matter settles into a power law of slope $(9 - 2\gamma)/(4-\gamma)$. For the NFW cusp, $\gamma = 1$, that is seven-thirds. The steepening happens inside about a fifth of the sphere of influence, and between that radius and the black hole’s own vicinity the density climbs by six orders of magnitude.

The scale of the whole construction is the sphere of influence, and it is a measured quantity in only a few galaxies. In the Milky Way the black hole’s mass comes from the orbits of individual stars around it, the mass at the centre that is not stars, and the stars’ velocity dispersion from their motions a few parsecs out; the ratio gives 1.8 parsecs to within ten per cent. In other galaxies the sphere of influence is often smaller than the telescope’s resolution, the black hole’s mass is inferred from a correlation with the dispersion rather than measured, and the black holes that can be weighed directly are biased towards the heavy ones. A spike’s predicted size in another galaxy therefore inherits that bias: the spike scales with the sphere of influence, and the sphere of influence is known only where it is large.

A fifth of the sphere of influence — 0.37 parsecs here — is where the black hole’s potential has become dominant enough that orbits close to it are nearly Keplerian, and the adiabatic calculation applies inside it. Outside, the dark matter feels the stars and the halo as well as the black hole, its orbits shrink less as the black hole grows, and the cusp joins smoothly onto the halo’s own.

A shape that forgets where it started #

The striking part of the formula is how little it depends on what it starts from.

A flat core, a gentle cusp and a steep one all end as spikes with slopes between 2.25 and 2.4. The spike’s shape is set by the black hole’s potential and by the conservation of orbital actions, not by the dark matter’s starting distribution; the starting distribution enters only through the normalisation — how much dark matter there was near the black hole to be pulled in. That makes the spike’s existence a question about the black hole’s growth rather than about the cusp–core argument. Even a cored halo would grow a spike around a black hole that grew slowly at its exact centre.

The conditions are demanding, though. The black hole has to grow slowly compared with the orbital periods around it, from a small seed, at the centre of the halo, and without the halo being disturbed afterwards. Ullio, Zhao and Kamionkowski showed in 2001 that a black hole that formed off-centre and spiralled in, or that grew from a seed already a sizeable fraction of its final mass, makes a much weaker spike or none. The adiabatic spike is the maximal case.

A ceiling set by annihilation #

If the dark matter annihilates, the spike cannot grow without limit. The annihilation rate per particle is proportional to the density, and where the density is high enough the particles are destroyed faster than the spike can be refilled. Over the age of the Galaxy the density is capped at the particle’s mass divided by its annihilation cross-section times the age, and the spike’s centre is a plateau at that ceiling.

The plateau has an awkward property for anyone hoping to use the spike to find the particle. The annihilation signal from the spike is dominated by the region just outside the plateau, where the density is highest but not yet capped, and the plateau’s level depends on the particle’s mass and cross-section — the very quantities the signal is meant to measure. A heavier particle gives a higher plateau and a smaller number of brighter annihilations; a lighter one gives a lower plateau and more of them. The spike amplifies whatever signal there is, and couples the amplification to the answer.

Stars wear it down #

The most serious threat to a spike in the Milky Way is not annihilation but the stars that share its volume.

The Galaxy’s central parsec holds millions of stars, more densely packed than anywhere else in the Galaxy, and they exchange energy with each other and with the dark matter in random encounters. The time for those encounters to change a particle’s energy by its own amount is the relaxation time, and at the Milky Way’s sphere of influence it is a few billion years — short compared with the age of the Galaxy. Over several relaxation times the dark matter is heated by the stars and settles into the distribution the stars themselves have around a black hole: a cusp falling as $r^{-3/2}$, much shallower than the adiabatic spike. Merritt and collaborators followed the process in 2007 and found that a spike in the Milky Way, if one formed, would by now be eroded to roughly this profile within the sphere of influence.

The softening is not the same in every galaxy. Relaxation is slow in galaxies whose nuclei are not dense, and in the most massive galaxies the relaxation time at the sphere of influence is far longer than the age of the universe. A spike around a giant elliptical’s black hole could survive the stars — but giant ellipticals are made by mergers, and when two galaxies merge their black holes form a binary that ejects stars and dark matter from the centre as it hardens, scouring out a core of a few times the binary’s mass. Merritt showed in 2002 that this destroys a spike outright. The galaxies most likely to keep a pristine spike are the ones least likely to have one: small, quiet, with black holes that grew in place and nuclei too sparse to relax. Dwarf galaxies and the hypothetical intermediate-mass black holes wandering in the Milky Way’s halo are where the theory points.

The cusp the stars make for themselves #

The profile the stars drive the dark matter towards is not arbitrary; it is the one the stars themselves settle into. Bahcall and Wolf showed in 1976 that a population of stars around a black hole, relaxing by encounters among themselves, reaches a steady state in which a small net flow of stars inward — some of them eventually swallowed or torn apart — is balanced by the energy their encounters exchange, and the density falls as $r^{-7/4}$. A lighter population scattered by a heavier one settles into a shallower $r^{-3/2}$, because it is heated without being able to heat back. Dark-matter particles are the lightest population of all, so they end on the shallower profile.

The Milky Way’s own stellar cusp has been the subject of a long argument of its own, because the old stars near the black hole turned out to have a shallower distribution than Bahcall and Wolf predicted — a deficit of old giants in the inner tenth of a parsec. If the stars have not reached their own relaxed state there, the dark matter’s softening may not have run its full course either, and the spike may be steeper than the relaxed profile in the figure. The same uncertainty about the centre’s history that limits the stellar cusp limits the dark one.

A black hole that leaves its spike behind #

The spike belongs to the black hole only as long as the black hole stays put. When two black holes merge, the gravitational waves they emit carry momentum unevenly, and the merged black hole recoils at hundreds to thousands of kilometres a second, fast enough in some configurations to leave its galaxy altogether. Even a kick too weak to eject it sends it on an oscillation through the nucleus that lasts millions of years, and the dark matter, which was bound to the black hole only within the sphere of influence, is left behind or spread out. A spike that took the black hole’s whole growth to build is erased by a recoil in a few orbits.

So the black holes most likely to have kept a spike are those that never merged with another of comparable mass — which, in a hierarchical universe where galaxies grow by merging, are mostly the small ones. That is the same conclusion as the scouring argument, reached by a different route: the history that makes a black hole large also clears its centre.

Minispikes around the black holes nobody has found #

If spikes survive best around small black holes in quiet surroundings, the natural place to look is around black holes of a few hundred to a hundred thousand solar masses that formed in small halos early on and were later accreted into the Milky Way’s halo without ever reaching its centre. Simulations of galaxy assembly predict that tens to hundreds of such intermediate-mass black holes should be wandering in the halo, each carrying the remnant of the small halo it grew in and, if it grew slowly, a minispike around it.

A minispike would be a point source of annihilation gamma rays with no counterpart at any other wavelength, steady in brightness, with a spectrum set by the particle’s mass. The gamma-ray sky contains about two thousand sources that have not been associated with any known object, most of them presumably pulsars and distant active galaxies too faint to identify. Searches among them for the steady, hard-spectrum point sources minispikes would make have found no clear candidates, which constrains the combination of the number of such black holes, their spikes’ survival and the particle’s annihilation rate. As with the Galactic centre, the constraint on the particle is only as good as the assumption about the spikes.

What the history does to the signal #

The difference between a pristine and a softened spike is not a correction; it is most of the signal.

The annihilation signal goes as the density squared integrated over volume, and a steep spike concentrates almost all of it into the innermost hundredth of a parsec. With the spike confined to a fifth of the sphere of influence — the conservative choice, since the original Gondolo–Silk normalisation extends it over tens of parsecs and raises both numbers by further orders of magnitude — the pristine spike makes the inner ten parsecs nearly five hundred times brighter than a spikeless cusp, and the softened one barely brighter at all. A gamma-ray telescope that sees nothing from the black hole’s position can exclude the pristine case for a thermal particle and say very little about the softened one. Limits on dark matter from the Galactic centre that assume a spike are therefore limits on the spike’s history as much as on the particle, and the published ones state which history they assume.

The Galactic centre is also crowded with ordinary gamma-ray sources — pulsars, cosmic rays striking dense gas, the black hole’s own variable emission — which a spike’s annihilation would have to stand out from. The excess of gamma rays from the inner few kiloparsecs that has been argued about for a decade comes from a much larger region than any spike and is unrelated to it; a spike’s signature would be a point source at the black hole’s position with a spectrum set by the particle’s mass, and the point source that is seen there varies on timescales of hours, which annihilation cannot.

A mass too small for S2 to weigh #

There is a second, purely gravitational test. The star S2 orbits the Galaxy’s black hole every sixteen years, passing within 120 astronomical units, and its orbit has been followed precisely enough to measure the relativistic precession of its pericentre. Any extended mass inside the orbit — a cluster of faint stars, stellar remnants, or a dark-matter spike — would add a precession in the opposite direction, so the measured agreement with relativity limits such a mass to a few thousand solar masses within the orbit.

The conservative spike drawn here puts about 120 solar masses within S2’s apocentre of 0.05 parsecs; a softened one about ten. Both are far below the limit. A spike normalised over tens of parsecs, as in the original calculation, could reach the limit, and so the orbit already excludes the most extreme versions. The ones that survive are invisible to it. The next test is gravitational waves: a stellar-mass compact object spiralling into a black hole of a few hundred thousand solar masses, of the kind a space detector will hear, completes hundreds of thousands of orbits in the detector’s band, and the drag of a dense spike would change its phase measurably over that many — a measurement of the dark matter’s density in exactly the region nothing else reaches.

What the figures leave out #

The spike is drawn as a broken power law: the cusp outside a fifth of the sphere of influence, the adiabatic slope inside, the plateau at the centre. Real calculations of the adiabatic growth give a smooth transition and a spike that turns over at a few Schwarzschild radii, where orbits are captured by the black hole; neither changes the slopes. The heating is drawn as the slope relaxing towards three-halves on the relaxation timescale, rather than by solving the equation that governs it, and published solutions differ in detail by tens of per cent. The black hole’s growth is assumed adiabatic and central, which is the case that maximises the spike.

Still open: whether any black hole has kept its spike #

The adiabatic spike is a clean prediction of a well-understood mechanism, and every effect that removes it — off-centre formation, mergers, stellar heating — is also well understood. What is not known is the history of any particular black hole well enough to say which applied. The Milky Way’s black hole has almost certainly had its spike softened by its stars, if it ever had one. The black holes of dwarf galaxies and the intermediate-mass black holes thought to wander in large halos may have kept theirs, and both are faint, few and hard to find. The first measurement of the dark matter’s density within a parsec of any black hole — from a gravitational-wave inspiral, or from an annihilation point source with the right spectrum — would settle whether the spike is a real object or a maximal case that nature never achieves.