A history the rotation curve absorbs

When gas cools and settles into a disc at the centre of a dark halo, the dark matter is pulled in after it, and a quantity conserved on slowly changing orbits says by how much. In a large disc galaxy the pull roughly triples the dark matter at the centre. The essay that worked that out ended on the Milky Way, the one galaxy whose halo can be measured from inside: the measurements, it said, are consistent with a contracted halo and with one that has not contracted, and the distinction is a factor of about two in the dark matter near the Sun — exactly the uncertainty that experiments trying to detect dark-matter particles quote.

Worked through, that statement turns out to be true of one kind of calculation and false of another, and the difference between them is the subject here. A halo predicted from cosmology does depend on its history at the Sun. A halo measured from the Galaxy’s rotation does not — and the reason is the same reason a rotation curve cannot divide its speed between a disc and a halo: it measures a total.

One curve, two halos #

Start from what is measured. The Milky Way’s rotation speed, from the motions of gas, stars and masers, is close to 230 kilometres a second and nearly flat between about five and twenty kiloparsecs from the centre, with the Sun at 8.2. The stars and gas contribute part of that speed — how much depends on the mass of the disc, which star counts put at three to six times $10^{10}$ solar masses — and a dark halo has to make up the rest. A decomposition chooses a family of halo profiles and finds the member that, added to the disc and bulge, reproduces the curve.

The two families are fitted separately. One is the Navarro–Frenk–White profile that dark-matter-only simulations produce; the other is the same profile contracted by the disc through the adiabatic invariant, the halo a galaxy formation model would say the Milky Way actually has. Both fits reproduce the band. They are quite different halos: the contracted family needs a much more massive, much less concentrated starting halo to end up with the right curve, because contraction would otherwise put too much mass in the middle, and a concentration of four is low for a halo of that mass in any simulation. That tension is itself a known result. The point here is what the two fits agree on.

Why the history drops out #

At the Sun the two fitted halos differ by about ten per cent. The reason is arithmetic. A rotation curve gives the total mass enclosed within each radius it covers: $v^2 = GM(<r)/r$. Subtract the disc and bulge, which are the same in both fits, and the dark mass enclosed within each radius from five to twenty kiloparsecs is fixed. The local density is the rate at which that enclosed mass grows with radius, divided by the area of the sphere — and both fits are made to match the same enclosed mass over the same range of radii. Whatever the history did to the shape of the halo, a fit that reproduces the curve has to put the same amount of dark matter inside each radius it fits, and so the same density at each.

Inward of the fitted range the constraint loosens, and the first figure shows where. At two kiloparsecs the disc and the bulge supply nearly all of the rotation and the dark matter is a small part of the total, so the curve says little about it; there the contracted halos are 44 per cent denser than the uncontracted one. The history matters exactly where the measurement does not reach.

Predicted instead of measured #

A prediction is a different calculation, and in it the history does not drop out.

Take the halo cosmology predicts for a galaxy of the Milky Way’s mass, and ask what it does when the disc forms. Contraction raises the dark matter at the Sun from 0.31 to as much as 0.51 GeV per cubic centimetre, a factor of 1.65 for circular orbits and 1.4 for the milder eccentric-orbit prescription. That is the factor of about two the earlier essay described, and it is real — for a prediction.

It is also visible in the rotation speeds the three halos would give, 211, 259 and 249 kilometres a second at the Sun, of which only the middle range is close to what is measured. A prediction that gets the rotation wrong has the wrong dark matter at the Sun for the same reason that a fit that gets it right cannot. The factor contraction introduces is the gap between what a galaxy formation model expects and what the Galaxy’s own rotation says, and closing that gap is a test of the model rather than a measurement of the local density.

The uncertainty that remains #

For a measurement fitted to the rotation curve, the uncertainty that matters is the one the fit cannot absorb: the mass of the visible disc.

A heavier disc supplies more of the rotation at the Sun and leaves less for the dark matter, and the range of disc masses that star counts allow moves the fitted local density by thirty per cent. That is three times the effect of the halo’s history. The disc’s mass is uncertain because most of it is in faint, low-mass stars that are counted only nearby and extrapolated, and because the conversion from light to mass depends on a mass function that is steep at the faint end. The decomposition therefore converts an uncertainty in the census of red dwarfs into an uncertainty in dark matter, at a rate of about one per cent of local density for every two per cent of disc mass.

There is also the question of the halo’s shape. A spherical halo is assumed throughout, and a flattened one — as halos in simulations with discs tend to be — puts more dark matter in the plane, raising the local density by tens of per cent at the same rotation curve. The rotation curve measures the mass in the plane; the shape decides how much of the halo lies outside it.

A measurement that needs no history at all #

There is a second way to measure the local density, and it is the one the experiments increasingly rely on, because it makes no assumption about the halo’s history or shape.

Stars near the Sun oscillate up and down through the disc, and the amplitude of their oscillation at a given speed measures the vertical pull, which measures the mass within a slab around the plane. Kuijken and Gilmore made the measurement from the velocities and heights of a few hundred stars in 1989, and the Gaia satellite has turned it into a measurement from millions. Close to the plane the pull is dominated by the disc’s stars and gas; a kiloparsec away the disc’s contribution has saturated, because there is almost no disc that high, and the pull keeps growing only because the dark matter is still there. The slope of the surface density against height, far from the plane, is twice the local dark density.

The measurement is older than the dark-matter problem itself. Jan Oort made it in 1932 from the stars he could count and their vertical speeds, and found that the pull near the Sun required about twice the mass the visible stars supplied — the first claim of mass that does not shine in the Galaxy, a year before Zwicky’s in the Coma cluster. Oort’s excess turned out to be mostly an underestimate of faint stars and gas rather than a dark disc, and the modern measurement, made with far better data, finds the disc’s own mass close to what is counted and the dark matter as a smooth component that only becomes visible above a kiloparsec. The method was the same; what changed was the census it is compared against.

The method’s weakness is the same disc, entering the other way. The dark matter is found as the excess over the visible matter, so an error in the census of stars and gas — six solar masses per square parsec is typical — is an error of 0.08 GeV per cubic centimetre in the result. The measurements of the last decade cluster around 0.4 to 0.6, consistent with the rotation-curve values and more precise than them, and neither depends on whether the halo contracted.

What the number does to a detection experiment #

The experiments that look for dark-matter particles colliding with nuclei in underground detectors do not measure the local density; they assume it. An experiment counts recoils, or the absence of them, and the expected rate is the number of particles passing through the detector times the probability that one scatters. The first factor is the local density divided by the particle’s mass, times the speed at which the particles arrive. So every published limit on the scattering cross-section is a limit on the product of the cross-section and the local density, and is quoted at a conventional density of 0.3 GeV per cubic centimetre, chosen decades ago when the measurements were cruder.

If the true local density is 0.45, every published limit is stronger than stated by a factor of one and a half; if it is 0.3, they are as stated. That factor is small compared with the orders of magnitude the experiments have covered, and large compared with the improvements each new generation claims. The convention is deliberate — it lets experiments be compared with one another without each choosing its own astrophysics — and it means that the local density’s uncertainty is carried, unstated, by every comparison between a detection claim and a limit.

The speeds matter as much. The standard assumption is a smooth distribution, a Maxwellian with a dispersion set by the rotation speed, truncated at the Galaxy’s escape speed. Gaia’s survey of stellar motions has shown that much of the inner halo’s stars arrived in one merger about ten billion years ago, on strongly radial orbits, and if the dark matter that came with them shares their motions, the speed distribution at the Sun is not Maxwellian. For particles light enough that only the fastest can make a detectable recoil, the tail of that distribution decides the rate, and the tail is the part the merger reshaped most.

Why other galaxies cannot do the same #

The Milky Way is the one galaxy where the history drops out of the measurement at the Sun, for two reasons that no other galaxy shares. Its rotation curve is measured at the position where the density is wanted, and its vertical pull can be measured there too, from the motions of individual stars. In any other galaxy only the rotation curve is available, and only as a projection on the sky, and the dark matter of interest is usually in the innermost kiloparsec, inward of where the disc stops dominating. That is precisely the region the first figure shows the history deciding.

So the debate about whether halos have cusps or cores at their centres is a debate conducted entirely in the region where contraction, feedback and the disc’s mass-to-light ratio cannot be separated by the curve. A cored halo fitted to a dwarf galaxy may be a cusp that feedback flattened, a cusp that never contracted because the dwarf’s disc was too light, or a cusp read through the wrong stellar mass. In the Milky Way the local measurement escapes the problem by being made far enough out; for the centres of galaxies there is no such escape.

The bar’s verdict on the inner halo #

Inward of the fitted range the Milky Way offers one more handle, and it is dynamical rather than kinematic. The Galaxy has a bar about five kiloparsecs long, rotating as a rigid pattern, and a bar rotating through a dense dark halo loses angular momentum to it by dynamical friction and slows down. The more dark matter there is in the inner few kiloparsecs, the faster the slowing. The bar’s present pattern speed, measured from the motions of stars at its ends and from where its gas collects, is moderately fast, which argues for an inner halo that is not very dense — less dense than the strongly contracted halo predicts, and closer to one that contracted mildly or not at all. The argument depends on how old the bar is and how fast it started, and both are uncertain, but it is a constraint from the one quantity in the inner Galaxy that the dark matter acts on directly.

Where the history does matter #

The history is not irrelevant; it has moved to where the experiments that look for annihilation rather than collision are looking. Gamma-ray searches for dark matter annihilating in the inner Galaxy depend on the density squared, integrated towards the centre, where the first figure shows the contracted and uncontracted halos parting company. A factor of 1.4 in density at two kiloparsecs is a factor of two in the annihilation signal from there, and inward of that the difference grows. An excess of gamma rays from the inner Galaxy found in the Fermi telescope’s data has been argued for over a decade as a possible annihilation signal, and the halo’s inner profile — set by contraction, by the bar, and by whatever feedback did — is one of the largest uncertainties in what it would imply. The same history that drops out of the local density is the one that most needs to be known for the centre.

The Galaxy measured from inside it has given the inner rotation curve from the motions of stars in the bulge and bar, and fits that extend inward to two or three kiloparsecs are beginning to constrain the inner halo directly. They find the inner dark matter consistent with a halo that contracted modestly and then had some of the contraction undone, which is what simulations with both cooling and feedback produce — and also consistent, within the uncertainty of the bar’s mass, with no contraction at all.

What the decomposition leaves out #

The rotation curve is idealised as flat at exactly 230 kilometres a second; the measured one declines slightly beyond fifteen kiloparsecs, and the decline is one of the inputs that fixes the halo’s total mass. The halo is spherical, the disc is a single exponential and the bulge a single sphere, where the real Galaxy has a bar, a thick disc and a gas layer with its own scale height. The contraction prescriptions are the two standard ones; simulations with feedback give responses between none and the circular-orbit case, and some give expansion. None of these changes the central point, which follows from the rotation curve fixing the enclosed mass: within the radii the curve measures, the halo’s history cannot change the dark matter a fit puts there.

Still open: whether the measured local density and the predicted one agree #

The measurements from the vertical pull and the rotation curve give a local density near 0.4 to 0.5 GeV per cubic centimetre. The predictions from cosmological simulations of Milky Way–like galaxies give a range that depends on contraction, on the halo’s shape and on the halo’s mass, which is itself uncertain by a factor of two. Whether the Galaxy’s halo is what cosmology expects for a galaxy of its mass, given its disc, is a test of the whole picture of how discs form inside halos — and it is the comparison of a measured number that does not depend on history with a predicted one that does. The particle physicists who need the local density can take the measurement; the cosmologists who want the history have to take the comparison.