The halo a galaxy drags inward

The argument about the innermost kiloparsec is usually posed as a comparison between two profiles. Simulations of cold dark matter produce haloes whose density rises towards the centre without limit, roughly as the inverse of the radius; the rotation curves of small galaxies say that the density levels off into a core. The simulations that produce the cusp contain only dark matter. The galaxies that show the core contain gas and stars. The comparison is therefore between a halo with no galaxy in it and a halo with one, and the first question is what the galaxy does to its halo simply by being there.

The answer, for a galaxy that forms slowly, is that it pulls the halo inward. Gas that cools and sinks to the centre deepens the potential there, and the dark matter, which feels the potential and nothing else, falls in after it. The effect is called adiabatic contraction, and it works in the opposite direction from the core the observations require.

A quantity that survives slow change #

The calculation rests on a principle that appears wherever a periodic motion is disturbed slowly: if a system’s parameters change on a timescale much longer than its period, certain combinations of its orbital quantities stay fixed. They are called adiabatic invariants. A charged particle spiralling in a slowly strengthening magnetic field keeps the magnetic flux through its orbit constant; a pendulum whose string is slowly shortened keeps the ratio of its energy to its frequency constant. For a particle on a circular orbit in a slowly changing spherical potential, the invariant is its angular momentum, which for a circular orbit is $\sqrt{G\,r\,M(r)}$.

So as long as the dark matter moves on nearly circular orbits and the baryons settle slowly compared with those orbits, the product $r\,M(r)$ — the radius of each dark shell times the total mass inside it — is the same before and after the disc forms. Before, the baryons are spread through the halo in the same proportion as the dark matter, a fraction of a few per cent everywhere. After, they are concentrated in a disc near the centre. A dark shell that started at radius $r_i$ enclosing total mass $M_i(r_i)$ ends at the radius $r_f$ that satisfies

$$r_i\,M_i(r_i) = r_f\left[M_\mathrm{disc}(r_f) + (1-f_b)\,M_i(r_i)\right],$$

since the dark matter inside the shell is still the same dark matter. Where the disc has put more mass inside $r_f$ than there used to be, the shell must move inward to keep the product fixed, and the dark matter becomes denser.

That is the whole calculation, and it is done shell by shell. It was published in 1986 as the answer to exactly the question above: how a galaxy’s formation changes the halo it forms in. The opening figure is its result for a galaxy like the Milky Way. Far outside the disc the halo is untouched — four or five per cent different at eighty kiloparsecs, where the enclosed disc mass is a small fraction of the total. Inside the disc the density at half a scale length is nearly three times what the halo alone would have had, and the logarithmic slope at one kiloparsec steepens from −1.09 to −1.25.

Which galaxies it matters for #

The size of the effect depends on how much of the central mass the disc supplies, and that depends steeply on the galaxy’s mass.

The baryon fraction of galaxies is not constant. Galaxies like the Milky Way have converted a larger share of their halo’s baryons into stars than any others — a few per cent of the halo’s total mass sits in the disc and bulge. Dwarfs have converted far less, a few tenths of a per cent in the faintest, because their shallow potentials let supernovae and stellar winds drive most of their gas away. And the disc of a dwarf is diffuse, with a scale length that is a large fraction of the halo’s own scale radius, so it never dominates the central potential.

The dependence of stellar mass on halo mass is itself a measurement, made by matching the numbers of galaxies of each luminosity to the numbers of haloes of each mass that simulations predict — the census that theory predicts and observations count. It rises steeply from the dwarfs, peaks at about the Milky Way’s halo mass, where a few per cent of the halo is in stars, and falls again towards the clusters, where most of the baryons stay hot. Contraction follows the same curve. It is strongest at the peak and weak on either side, and the galaxies in which it matters most are precisely the ones whose rotation curves are most often used to argue about the inner halo.

The consequence cuts the core problem in two. In dwarfs, where cores are most clearly observed, contraction was never large, so the cusp the simulations predict is close to the profile any other process has to start from. In large disc galaxies contraction is large, so the relevant starting profile is steeper than the simulations say — and any process that makes a core there has to undo the contraction before it can make the core.

A rotation curve whose halo depends on its disc #

The contraction changes the way rotation curves are interpreted, and not only near the centre.

A measured rotation curve is decomposed by modelling the stars’ contribution from their light, the gas’s from its measured distribution, and assigning the rest to the halo. The step that cannot be done from the data is fixing the stars’ mass-to-light ratio, and several very different decompositions fit the same curve — a heavy disc with a light halo, a light disc with a heavy one. One constraint often applied is that the halo should look like a simulated halo. If the simulated halo is the uncontracted one, that constraint favours the wrong decomposition: a real halo that has contracted is denser in the centre than the template, so a fit that forces the template leaves the excess to be absorbed by the disc, and the disc’s mass is overestimated.

The reverse also happens. A decomposition that allows the halo to contract has more dark matter where the disc is, and therefore leaves room for less disc. The contracted fit prefers a lighter disc and the uncontracted one a heavier. The disc’s mass is one of the few properties of a galaxy that can be checked independently — from the vertical motions of its stars, which measure the surface density of the disc through its thickness — and those checks, where they have been made, tend to favour discs somewhat lighter than the heaviest the rotation curves allow.

Compactness matters more than mass #

The contraction is driven by how the disc’s mass is distributed, and two discs of the same mass can produce very different haloes.

The disc’s scale length is set, as the angular momentum of the halo sets it, by how much spin the gas had when it settled. A halo that happened to be spinning slowly produces a compact disc, and a compact disc contracts its halo strongly; a fast-spinning halo produces an extended disc that barely disturbs it. So the inner density of a disc galaxy’s halo is not a property of the halo’s mass alone. It carries the disc’s history, which carries the halo’s spin, and two haloes identical in every property a dark-matter simulation records can end with inner densities differing by a factor of three.

That is one reason the comparison between simulated and observed inner profiles is harder than it looks. A dark-matter-only simulation predicts a distribution of inner densities at each mass, from the scatter in concentration. The galaxies add a second distribution, from the scatter in disc compactness, and the two are correlated through the spin. A sample selected for extended, low-surface-brightness discs — which are the galaxies whose rotation curves are cleanest to interpret — is a sample selected for weak contraction.

The invariant and its assumptions #

The calculation makes two assumptions, and both fail in ways that are now reasonably well understood.

The first is that the dark matter moves on circular orbits. It does not; halo orbits are eccentric, and a particle on an eccentric orbit spends its time over a range of radii, feeling the mass inside its mean distance rather than inside its current one. A modified version of the calculation, fitted to simulations in 2004, conserves the product at an orbit-averaged radius instead, and it gives a weaker contraction than the circular-orbit version — roughly halving the enhancement in the regions that matter for rotation curves. Every figure here uses the circular-orbit version and so shows the largest contraction the slow-settling picture allows.

The second is that the settling is slow. The invariant holds only if the potential changes slowly compared with the orbital periods, and galaxies do not always oblige. If gas falls in as dense clumps, the clumps sink by dynamical friction and heat the dark matter as they go; if gas is expelled suddenly by a burst of supernovae, the potential changes faster than the orbits can follow and the dark matter is left on orbits that are too energetic for the new, shallower potential — it expands. Repeated cycles of inflow and expulsion can pump energy into the dark matter progressively, which is the mechanism by which feedback makes cores. That mechanism is exactly the failure of adiabaticity, and it acts against the contraction.

So contraction and feedback are the same physics — the dark matter’s response to a changing baryonic potential — in two different limits. Slow change pulls the halo in; fast, repeated change pushes it out. Which wins depends on the history of the gas, and the answer differs between galaxies.

A total profile that comes out isothermal #

The clearest evidence that haloes respond to their galaxies comes not from discs but from massive elliptical galaxies, where the baryons are most concentrated of all.

In an elliptical the stars dominate the mass inside the half-light radius and the dark matter dominates outside it. Neither component alone has a simple profile: the stars’ density falls steeply, the dark halo’s more gently. Yet when the total mass profile is measured — combining the deflection of light from background sources, which gives the mass inside the Einstein radius, with the random motions of the stars, which give its run with radius — it comes out remarkably close to a single power law, density falling as the inverse square of radius, over a range where the two components trade places. The same slope is found in galaxy after galaxy, with a scatter of a tenth in the exponent.

A dark-matter-only halo added to the observed stars does not produce that. The sum has a visible kink where the components cross. A halo that has contracted in response to the stars fills in the kink: the extra dark matter pulled into the region where the stars dominate is just what makes the total smooth. The near-isothermal total has been called a conspiracy between baryons and dark matter, and contraction is the most natural explanation for it — though the fits that reproduce it require a contraction weaker than the circular-orbit calculation gives, closer to the orbit-averaged version.

The clusters complicate the picture from the other side. In the centres of massive clusters, where a giant elliptical sits at the bottom of the potential, lensing and stellar dynamics together have measured dark-matter profiles shallower than the uncontracted prediction rather than steeper — the opposite of contraction. The proposed explanation is the same as for cores in dwarfs, operating at a different scale: the central galaxy was assembled by mergers, and each merger’s infalling galaxies transferred energy to the dark matter by dynamical friction as they sank. Slow settling contracts; lumpy, rapid assembly expands. A cluster’s centre and a disc galaxy’s centre were built differently and ended with different responses.

What a core costs after contraction #

If a large disc galaxy is to have a core, the feedback that makes it must first undo the contraction, and the energy bill can be computed.

The raw numbers need interpretation. A supernova releases about $10^{51}$ erg of kinetic energy, but almost all of it goes into heating and moving gas, and only a small fraction ends up as work done on the dark matter through the changing potential — a per cent or less, in simulations that follow it. The dwarf’s bill of a thousand supernovae’ worth, at a coupling of one per cent, is a hundred thousand supernovae, which a dwarf with a hundred million solar masses of stars has had over its lifetime. The large galaxy’s contracted bill is a hundred times larger, and it would have to be delivered into a region already dominated by a massive disc whose own gravity holds the dark matter in.

That is the quantitative form of a qualitative expectation: feedback can make cores in dwarfs and struggles to make them in galaxies like the Milky Way. The simulations that produce cores find them at intermediate masses, peaking at a stellar-to-halo mass ratio of about a few tenths of a per cent, and not in the most massive discs. The contraction calculation says why the massive end is difficult: the galaxy has already dragged its halo deeper than the simulations of dark matter alone ever put it.

What the picture leaves out #

The figures use a spherical approximation for the disc’s enclosed mass, which is accurate far from the disc and overestimates the pull at radii comparable to its scale length, where a flattened distribution pulls less in the radial direction than a sphere of the same mass. They ignore the bulge, which in many large galaxies is more compact than the disc and contracts the inner halo more strongly still. And they ignore the gas that never cooled — the hot halo of a large galaxy, which holds a substantial fraction of its baryons and stays spread through the dark matter.

Whether a given change counts as slow is a matter of timescales that can be estimated. An orbit at five kiloparsecs in a galaxy like the Milky Way takes about a hundred and fifty million years; its disc was assembled over several billion. That is slow by a factor of tens, and the invariant should hold well. At the centre of a dwarf an orbit at half a kiloparsec takes perhaps fifty million years, while the bursts of star formation that drive gas in and out of it last ten to a hundred million years. That is not slow at all, and it is exactly the regime in which the dark matter’s response becomes irreversible — each cycle leaving the orbits slightly more energetic than before. The same galaxy can be adiabatic in its outskirts and violently non-adiabatic at its centre.

They also give one galaxy’s history. A real galaxy’s disc formed over billions of years, while the halo itself was still accreting, merging and changing shape, and the invariant applies to each slow stage separately. The contraction computed here is the cumulative effect of a disc that appeared all at once but slowly — an idealisation that simulations with full gas physics have both confirmed in direction and revised in size.

Still open: what the Milky Way’s own halo did #

The one galaxy in which the inner halo can be measured directly, rather than inferred from a rotation curve, is the Milky Way seen from inside. The motions of stars near the Sun and towards the centre constrain the total mass inside each radius, the stars’ own mass is counted, and the difference is the dark matter. The measurements are consistent with a halo that has contracted, and consistent with one that has not, and the distinction is a factor of about two in the dark matter near the Sun — which is exactly the uncertainty in the local dark-matter density that direct-detection experiments quote. Whether the Galaxy’s halo was pulled in by its disc, pushed out by its early bursts of star formation, or both in turn, is recorded in the orbits of its oldest stars, and the surveys now mapping those orbits in six dimensions are the measurement that could decide it.