Flat to a megaparsec, which an isolated halo is not

The Tully–Fisher relation ties a galaxy’s mass to a speed, and the speed it ties it to is a choice. Of all the speeds a rotation curve offers, the relation is tightest for the one at the outer edge of the measured curve — the flat part, read where the hydrogen runs out, typically twenty to forty kiloparsecs from the centre. Read there, the baryonic version of the relation, with the stars and the gas counted together, has almost no scatter beyond what the measurements themselves contribute. It is one of the tightest relations in extragalactic astronomy, and the tightness is at a radius set not by the galaxy’s dynamics but by where its gas happens to stop being detectable.

Beyond the gas nothing orbits that can be measured with a spectrograph. But the mass is still there, and it bends the light of galaxies behind it. Weak gravitational lensing, stacked over tens of thousands of foreground galaxies, measures the average mass profile round them out to a megaparsec and more — the same technique that weighs a galaxy by the light that bends past it, pushed to radii thirty times beyond the hydrogen. In 2024 an analysis of isolated lens galaxies reported something striking: converted into circular speeds, the lensing signal stays flat to the largest radii measured, and the flat speeds lie on the same baryonic Tully–Fisher relation as the rotation curves.

The flat part of a rotation curve was the original evidence for dark matter: a rotation curve that refuses to fall beyond the visible disc says the mass keeps growing where the light has stopped. It was always understood that the curve could not stay flat for ever — a mass growing in proportion to radius without end would be infinite — and that somewhere beyond the gas the halo would run out and the curve would begin to fall. Where that happens is a prediction of the halo model, and until lensing reached these radii it was a prediction nobody could test.

What that would mean depends on what a dark-matter halo predicts at those radii, and this essay draws it. A halo of the kind cosmological simulations produce does not have a flat rotation curve to a megaparsec. Alone, its speed falls to about half its peak there. The neighbours that every real galaxy has add mass that partly refills the curve, and the question of whether the flatness is real becomes a question about how much company an isolated galaxy keeps.

Three predictions beyond the hydrogen #

Take a galaxy like the Milky Way and give it the halo that its stellar mass implies — a halo of about 1.5 × 10¹² solar masses, with the standard density profile found in simulations, whose density falls as the inverse cube of radius at large distances. Its edge, conventionally where the mean density inside is two hundred times the cosmic critical density, is at about 240 kiloparsecs. Its circular speed rises to a peak of about 210 kilometres a second near the region where the gas ends — the region where every rotation curve has been measured — and then turns over.

The turnover is slow at first and then definite. At 300 kiloparsecs, just outside the halo’s nominal edge, the speed is 158 kilometres a second, three-quarters of the peak. At a megaparsec, if the halo is followed beyond its edge with its profile unchanged, it is 111, about half. The decline is gentle compared with the Keplerian fall around a point mass, because the profile keeps adding mass as the logarithm of radius, but it is unmistakable on a plot that runs from ten kiloparsecs to a thousand.

The other two curves are the alternatives. The dashed line is what the lensing analysis reported: a speed that stays where the gas left it. The middle curve adds something every real galaxy has and the halo model leaves out — its neighbours — and it falls between them.

How far an isolated halo falls #

How far the curve falls depends on the halo’s mass, because halo sizes scale with mass. A halo of three hundred billion solar masses — a galaxy a few times fainter than the Milky Way — has its edge at about 140 kiloparsecs and by a megaparsec has fallen to two-fifths of its peak. A halo of six trillion, hosting a giant elliptical, extends to nearly 400 kiloparsecs and has fallen only to two-thirds. The lighter the galaxy, the steeper the predicted decline over the radii lensing measures.

That is the first thing a flat lensing curve would contradict. Stacked lensing averages over many galaxies of similar stellar mass, and the stacks are made separately in mass bins. If every bin stays flat to a megaparsec, then the lightest galaxies — the ones whose haloes should have fallen furthest — are as flat as the heaviest. A halo model predicts not just a decline but a mass-dependent decline, and its absence would be visible in the differences between bins as well as in each one.

A speed nothing has orbited at #

There is something odd about calling the quantity a speed at all. The circular speed at a radius is the speed a particle would need to go round on a circle there, and it is $\sqrt{GM/r}$ with the mass inside that radius; it exists as a number whether or not anything is orbiting. At the edge of the hydrogen, gas really is going round at that speed, completing an orbit in about a billion years, and the galaxy has had time for a dozen of them. At a megaparsec, an orbit at 170 kilometres a second takes 36 billion years. Nothing at that distance has gone round even once. Beyond about 380 kiloparsecs, nothing could have completed an orbit in the entire age of the universe.

So a circular speed at a megaparsec is not a motion. It is a way of quoting the enclosed mass in the units of the Tully–Fisher relation. That matters for interpreting a flat curve, because it means the material out there is not in orbital equilibrium with the galaxy: it is still falling in for the first time, or has turned round once, or belongs to neighbouring systems whose own motions have nothing to do with the lens. A halo’s density profile at those radii is the profile of matter that has not settled, and whatever the lensing measures there is a snapshot of an inflow rather than of a balance. The relation found at the edge of the gas, where things orbit, need not hold where they do not — and if it does hold, the coincidence is the thing to explain.

The relation read further out #

The second thing a flat lensing curve would contradict is the relation itself. Take a sequence of isolated haloes spanning two decades of mass, give each the baryons its mass implies from the statistical matching of galaxy counts to halo counts, and read off the circular speed at three radii. At thirty kiloparsecs the haloes trace a relation between baryonic mass and speed. Read at three hundred kiloparsecs the same galaxies have lower speeds, and at a megaparsec lower again; and because the lighter haloes have fallen further, the relation does not merely shift but changes slope, from 2.7 to 1.7 in logarithmic mass against logarithmic speed.

The slope at thirty kiloparsecs is itself shallower than the measured relation’s slope of about four, which is a separate difficulty for haloes with these baryon fractions and one that many models address with the feedback processes that reshape a galaxy’s centre. The point here is the change with radius. In a halo model the relation has no reason to survive outside the gas: the speed at each radius is set by the halo’s mass inside that radius, and the halo’s mass profile, not the galaxy’s baryons, decides how it changes. A relation tied to the baryons that still holds at a megaparsec is the behaviour of an account in which a single acceleration scale decides where the missing mass appears and the speed far from any mass is set by the baryons alone. In modified dynamics of that kind, the flat speed is $(GM_b a_0)^{1/4}$ at every radius beyond the point where the acceleration drops below $a_0$, with no turnover at all.

Why the relation was tightest where it was measured #

It is worth asking why the relation is so tight at the edge of the gas in the first place, because the answer bears on whether it should survive beyond. In the halo picture the tightness is partly a coincidence of radius. Inside the gas the speed is set by a mixture of baryons and dark matter in proportions that vary from galaxy to galaxy — three different mass models can fit the same curve — and the variations happen to cancel in the outer speed, because a galaxy with more concentrated baryons sits in a halo whose inner profile has responded by contracting. The flat outer speed is close to the halo’s peak speed, and the peak speed tracks the halo’s mass, which tracks the galaxy’s baryons through the statistics of how galaxies form in haloes.

That chain of correlations is real, but it is tied to the peak. A relation that holds at the peak need not hold at three times or thirty times the radius, where the speed depends on how the halo’s mass profile continues, and that continuation depends on the halo’s concentration, its environment and its accretion history rather than on the galaxy at its centre. So in the halo picture the relation is expected to loosen beyond the gas as well as to shift — its scatter should grow with radius, as each galaxy’s neighbours and infall contribute differently.

In modified dynamics it is the other way round. There, the missing mass appears at a fixed acceleration, and beyond the radius where the baryons’ own pull falls below that acceleration the speed is fixed by the baryonic mass alone. It does not loosen with radius, because there is nothing else for it to depend on; and it does not fall, because the effective force declines as $1/r$ rather than $1/r^2$. The edge of the gas is not special at all in that account — merely the last place a spectrograph can see. Lensing therefore separates the two in two ways at once: by whether the curve falls, and by whether the scatter grows. A flat, tight relation to a megaparsec is the modified-dynamics signature; a falling, loosening one is the halo’s.

How much company makes a halo look flat #

The isolated halo is an idealisation, and lensing does not measure it. A lensing signal is the projected mass round the lens, and that includes every other halo whose position is correlated with the lens’s: the satellites inside its own halo, the galaxies of its group, the filament it sits in. Galaxies cluster, and the excess of matter round a typical galaxy at a megaparsec is several times the cosmic mean. Its mass, accumulated out to a given radius, is the mean matter density times the integral of the correlation function — the statistic that measures how much more likely matter is to be found at a given distance from a galaxy than at random — times a factor, the bias, saying how much more clustered this kind of galaxy is than the matter as a whole.

For a typical Milky-Way galaxy the bias is about one, and its neighbours add about 1.7 × 10¹² solar masses inside a megaparsec — more than the halo’s own mass to 240 kiloparsecs. The speed at a megaparsec rises from 111 to 140 kilometres a second. That is a substantial refill, but it is not flatness. To hold the curve at its edge-of-gas speed out to a megaparsec, the neighbourhood would need a bias of more than four — the clustering of a galaxy in a rich group, not of a field galaxy.

This is the crux, and it is why the measurement is hard to read. Lensing analyses that look for flatness select isolated lenses precisely to suppress the neighbours’ contribution, by requiring that no brighter galaxy lie within some projected radius and redshift interval. A selection that works makes the signal approach the isolated curve, which falls. A selection that leaks admits lenses that are really members of groups, whose signal is enhanced by exactly the kind of company that flattens the curve. And a lens that is the brightest member of its group passes any isolation criterion based on brighter neighbours.

The mass a flat curve needs #

In terms of mass the discrepancy is starker than in terms of speed, because speed goes as the square root. A flat curve needs an enclosed mass growing in proportion to radius — the profile of an isothermal sphere, whose density falls as the inverse square of radius rather than the inverse cube. At a megaparsec, a curve flat at 211 kilometres a second needs ten trillion solar masses inside. The halo, followed beyond its edge, holds 2.8 trillion; a typical neighbourhood adds 1.7. The remaining factor of two is the mass that the reported lensing signal would be finding, if it is what it appears to be.

That also says where to look for a mistake. The conversion from a lensing signal to a circular speed assumes the mass distribution is spherical and that the measured excess surface density can be deprojected into a three-dimensional enclosed mass with a particular profile. The neighbours’ mass is not spherical — it lies along filaments — and in projection a filament seen end-on can add a great deal of surface density at large radius. The stacking mixes galaxies of different masses, which flattens a stack of declining curves if the heavier galaxies, whose curves fall more slowly, dominate the signal at large radii. And the lensing signal at a megaparsec is small, measured from tiny distortions averaged over enormous numbers of background galaxies, with systematic errors in their shapes and redshifts that grow in relative importance as the signal falls.

Other ways of weighing the outer halo #

Lensing is not the only measurement that reaches beyond the gas, and the others give a consistency check. The Milky Way’s own halo can be weighed at a hundred kiloparsecs and beyond from the motions of its satellite galaxies and of its most distant globular clusters, and the number of those clusters, which scales with the mass of the halo they sit in, gives an independent estimate of the whole. The answers — about a trillion solar masses inside two hundred kiloparsecs — sit comfortably with a halo of the kind drawn here, and the circular speed they imply at that radius is already below the speed at the edge of the disc.

Further out, the Milky Way and Andromeda are approaching each other at about 110 kilometres a second from a distance of 770 kiloparsecs, and the time they have had to turn round weighs the pair: a few trillion solar masses in total, again in line with two haloes of the standard kind and their surroundings. Both measurements are of one galaxy or one pair, with all the uncertainty that carries, but neither sees the mass growing in proportion to radius out to a megaparsec. A flat lensing signal round isolated field galaxies would therefore be at odds not only with simulated haloes but with the timing of the Local Group — unless the Local Group is not a fair sample of the galaxies in the stacks, which, as a pair of large spirals with a dozen satellites, it may not be.

What the figures leave out #

The haloes here are spherical, with a fixed concentration scaling, followed beyond their nominal edge with their inner profile unchanged; a real halo’s density beyond about the edge steepens and then gives way to infalling matter, which these curves do not model. The baryons are a point mass outside thirty kiloparsecs, which overstates the speed inside it. The neighbours’ mass is the linear-theory mean excess for a galaxy of the given bias, in three dimensions; lensing measures a projected quantity, and the one-halo contribution of satellites inside the lens’s own halo, and the nonlinear clustering at a few hundred kiloparsecs, are not included. The speeds of the alternatives are drawn as flat at the halo’s own edge-of-gas speed, which in this model is higher than the measured relation gives for a galaxy of this baryonic mass. None of these changes the shape of the argument: a lone halo falls, typical neighbours partly refill it, and flatness needs more company than a field galaxy keeps.

Still open: whether the flatness survives a stricter isolation #

The test is stated by the curves. If the reported flatness is the neighbours’, a stricter isolation criterion — no comparable galaxy within two or three megaparsecs, in projection and in redshift — should make the stacked signal fall towards the isolated halo’s curve, and it should fall faster for the lighter lenses. If the flatness persists for lenses isolated well beyond the range at which their neighbours contribute, and if it persists equally at every stellar mass, then the mass round isolated galaxies grows linearly with radius to a megaparsec, which no halo of the standard kind does, and the falling rotation curve recently reported inside the Milky Way would sit oddly against flat lensing curves outside other galaxies.

Imaging surveys covering a third of the sky, with source redshifts measured precisely enough to place the lenses in three dimensions, will provide samples large enough to make the isolation strict and still measure the signal at a megaparsec. The other route is to model the neighbours rather than avoid them: measure the lensing round all galaxies, measure the clustering of the same galaxies, and fit both together, as cosmological analyses of galaxy–galaxy lensing already do. If the two approaches agree, the halo has been weighed beyond its gas. If they disagree in the direction of the reported flatness, the relation that is tightest at the edge of the hydrogen will have been found to carry on into a region where nothing orbits — and that would be a statement about gravity, not about galaxies.