The mass distribution of dark matter halos that we infer from observations tells us where the dark matter needs to be now. This differs form the mass distribution it had to start, as it gets altered by the process of galaxy formation. It is the primordial distribution that dark matter-only simulations predict most robustly. We* reverse-engineer the collapse of the baryons that make up the visible Galaxy to infer the primordial distribution, which turns out to be… odd.
The Gaia rotation curve and the mass of the Milky Way
As we discussed a couple of years ago, Gaia DR3 data indicate a declining rotation curve for the Milky Way. This decline becomes more steep, nearly Keplerian, in the outskirts of the Milky Way (17 < R < 30 kpc). This is may or may not be consistent with data further out, which gets hard to interpret as the LMC (at 50 kpc) perturbs orbits and the observed motions may not correspond to orbits in dynamical equilibrium. So how much do the data inform us about the gravitational potential?
I am skeptical of the Keplerian portion of this result (as discussed at length at the time) because other galaxies don’t do that. However, I am a big fan of listening to the data, and the people actually doing the work. Taken at face value, the Gaia data show a Keplerian decline with a total mass around 2 x 1011 M☉. If correct, this falsifies MOND.
How does dark matter fare? There is an implicit assumption made by many in the community that any failing of MOND is an automatic win for dark matter. However, it has been my experience that observations that are problematic for MOND are also problematic for dark matter. So let’s check.
Short answer: this is really weird in terms of dark matter. How weird? For starters, most recent non-Gaia dynamical analyses suggest a total mass closer to 1012 M☉, a factor of five higher than the Gaia value. I’m old enough to remember when the accepted mass was 2 x 1012 M☉, an order of magnitude higher. Yet even this larger mass is smaller than suggested by abundance matching recipes, which give more like 4 x 1012 M☉. So somewhere in the range 2 – 40 x 1011 M☉.
The Milky Mass has been adjusted so often, have we finally hit it?
The guy was all over the road. I had to swerve a number of times before I hit him.
Boston Driver’s Handbook (1982 edition)&
If it sounds like we’re all over the map, that’s because we are. It is very hard to constrain the total mass of a dark matter halo. We can’t see it, nor tell where it ends. We infer, indirectly, that the edge is way out beyond the tracers we can see. Heck, even speaking of an “edge” is ill-defined. Theoretically, we expect it to taper off with the density of dark matter falling as ρ ~ r-3, so there is no definitive edge. Somewhat arbitrarily,** we adopt the radius that encloses a density 200 times the average density of the universe as the “virial” radius. This is all completely notional, and it gets worse, as the process of forming a galaxy changes the initial mass distribution. What we observe today is the changed form, not the primordial initial condition for which the notional mass is defined.
Adiabatic compression during galaxy formation
To form a visible galaxy, baryons must dissipate and sink to the center of their parent dark matter halo. This process changes the mass distribution and alters the halo from its primordial state. In effect, the gravity of the sinking baryons drags some dark matter along# with them.
The change to the dark matter halo is often called adiabatic compression. The actual process need not be adiabatic, but that’s how we approximate it. We’ve tested this approximation with detailed numerical simulations, and it works pretty well, at least if you do it right (there are boring debates about technique). What happens makes sense intuitively: the response of the primordial halo to the infall of baryons is to become more dense at the center. While this makes sense physically, it is problematic for LCDM as it takes an NFW halo that is already too dense at the center to be consistent with data and makes it more dense. This has been known forever, so opposing this is one thing feedback is invoked to do, which it may or may not do, depending on how it really works. Even if feedback can really turn a compressed cusp into a core, it is widely to expected to be important only in low mass galaxies where the gravitational potential well isn’t too deep. It isn’t supposed to be all that important in galaxies as massive as the Milky Way, though I’m sure that can change as needed.
There are a variety of challenges to implementing an accurate compression computation, so we usually don’t bother: the standard practice is to assume a halo model and fit it to the data. That will, at best, given a description of the current dark matter halo, not what it started as, which is our closest point of comparison with theory. To give an example of the effect, here is a Milky Way model I built a decade ago:
The change from the green dashed line to the solid green line is the difference compression makes. That’s what happens if a baryon distribution like that of the Milky Way settles in an NFW halo. The inferred mass M200 is lower and the concentration c higher than it originally was – and it is the original version that we should compare to the expectations of LCDM.
When I built this model, I considered several choices for the bulge/bar fraction: something reasonable, something probably too large, and something definitely too small (zero). The model above is the last case of zero bulge/bar. I show it because it is the only case for which the compression procedure worked. If there is a larger central concentration of baryons – i.e., a bulge and/or a bar – then the compression is greater. Too great, in fact: I could not obtain a fit (see also Binney & Piffl and this related discussion).
The calculation of the compression requires knowledge of the primordial halo parameters, which is what one is trying to obtain. So one has to guess an initial state, run the code, check how close it came, then iterate the initial guess. This is computationally expensive, so I was just eyeballing the fit above. Pengfei has done a lot of work to implement a method that iteratively computes the compression and rigorously fits it to data. So we decided to apply it to the newer Gaia DR3 data.
Fitting the Gaia rotation curve with adiabatically compressed halos
We need two inputs here: one, the rotation curve to fit, and two, the baryonic distribution of the Milky Way. The latter is hard to specify given our location within the Milky Way, so there are many different estimates. We tried a dozen.
Another challenge of doing this is deciding which data rotation curve data to fit. We chose to focus on the rotation curve of Jiao et al. (2023) because they made estimates of the systematic as well as random errors. The statistics of Gaia are so good it is practically impossible to fit any equilibrium model to them. There are aspects of the data for which we have to consider non-equilibrium effects (spiral arms, the bar, “snails” from external perturbations) so the usual assumptions are at best an approximation, plus there can always be systematic errors. So the approach is to believe the data, but with the uncertainty estimate of Jiao et al. (2023) that includes systematics.
For a halo model, we started with the boilerplate LCDM NFW halo$. This doesn’t fit the data. Indeed, all attempts to fit NFW halos fail in similar ways for all of the different baryonic mass models we tried. The quasi-Keplerian part of the Gaia rotation curve simply cannot be fit: the NFW halo inevitably requires more mass further out.
Here are a few examples of the NFW fits:
Fig. A.3 from Li et al. (2025). Fits of Galactic circular velocities using the NFW model implementing adiabatic halo contraction using 3 baryonic models. [Another 9 appear in the paper.] Data points with errors are the rotation velocities from Jiao et al. (2023), while open triangles show the data from Eilers et al. (2019), which are not fitted. [The radius ranges from 5 to 30 kpc.] Blue, purple, green and black solid lines correspond to the contributions by the stellar disk, central bar, gas (and dust if any), and compressed dark matter halo, respectively. The total contributions are shown using red solid lines. Black dashed lines are the inferred primordial halos.
LCDM as represented by NFW suffers the same failure mode as seen in MOND (plot at top): both theories overshoot the Gaia rotation curve at R > 17 kpc. This is an example of how data that are problematic for MOND are also problematic for dark matter.
We do have more freedom in the case of dark matter. So we tried a different halo model, Einasto. (For this and many other halo models, see Pengfei’s epic compendium of dark matter halo fits.) Where NFW has two parameters, a concentration c and mass M200, Einasto has a third parameter that modulates the shape of the density profile%. For a very specific choice of this third parameter (α = 0.17), it looks basically the same as NFW. But if we let α be free, then we can obtain a fit. Of all the baryonic models, the RAR model+compressed Einasto fits best:
Fig. 1 from Li et al. (2025). Example of a circular velocity fit using the McGaugh19$$ model for baryonic mass distributions. The purple, blue, and green lines represent the contributions of the bar, disk, and gas components, respectively. The solid and dashed black lines show the current and primordial dark matter halos, respectively. The solid red line indicates the total velocity profile. The black points show the latest Gaia measurements (Jiao et al. 2023), and the gray upward triangles and squares show the terminal velocities from (McClure-Griffiths & Dickey 2007, 2016), and Portail et al. (2017), respectively. The data marked with open symbols were not fit because they do not consider the systematic uncertainties.
So it is possible to obtain a fit considering adiabatic compression. But at what price? The parameters of the best-fit primordial Einasto halo shown above are c = 5.1, M200 = 1.2 x 1011 M☉, and α = 2.75. That’s pretty far from the α = 0.17 expected in LCDM. The mass is lower than low. The concentration is also low. There are expectation values for all these quantities in LCDM, and all of them miss the mark.
Fig. 2 from Li et al. (2025). Halo masses and concentrations of the primordial Galactic halos derived from the Gaia circular velocity fits using 12 baryonic models. The red and blue stars with errors represent the halos with and without adiabatic contraction, respectively. The predicted halo mass-concentration relation within 1 σ from simulations (Dutton & Macciò 2014) is shown as the declining band. The vertical band shows the expected range of the MW halo mass according to the abundance-
matching relation (Moster et al. 2013). The upper and lower limits are set by the highest stellar mass model plus 1 σ and the lowest stellar mass model minus 1 σ, respectively.
The expectation for mass and concentration is shown as the bands above. If the primordial halo were anything like what it should be in LCDM, the halo parameters represented by the red stars should be where the bands intersect. They’re nowhere close. The same goes for the shape parameter. The halo should have a density profile like the blue band in the plot below; instead it is more like the red band.
Fig. 3 from Li et al. (2025). Structure of the inferred primordial and current Galactic halos, along with predictions for the cold and warm dark matter. The density profiles are scaled so that there is no need to assume or consider the masses or concentrations for these halos. The gray band indicates the range of the current halos derived from the Gaia velocity fits using the 12 baryonic models, and the red band shows their corresponding primordial halos within 1σ. The blue band presents the simulated halos with cold dark matter only (Dutton & Macciò 2014). The purple band shows the warm dark matter halos (normalized to match the primordial Galactic halo) with a core size spanning from 4.56 kpc (WDM5 in Macciò et al. 2012) to 7.0 kpc, corresponding to a particle mass of 0.05 keV and lower.
So the primordial halo of the Milky Way is pretty odd. From the perspective of LCDM, the mass is too low and the concentration is too low. The inner profile is too flat (a core rather than a cusp) and the outer profile is too steep. This outer steepness is a large part of why the mass comes out so low; there just isn’t a lot of halo out there. The characteristic density ρs is at least in the right ballpark, so aside from the inner slope, the outer slope, the mass, and the concentration, LCDM is doing great.
What if we ignore the naughty bits?
It is really hard for any halo model to fit the steep decline of the Gaia rotation curve at R > 17 kpc. Doing so is what makes the halo mass so small. I’m skeptical about this part of the data, so do things improve if we don’t sweat that part?
Ignoring the data at R > 17 kpc allows the mass to be larger, consistent with other dynamical determinations if not quite with abundance matching. However, the inner parts of the rotation curve still prefer a low density core. That is, something like the warm dark matter halo depicted as the purple band above rather than NFW with its dense central cusp. Or self-interacting dark matter. Or cold dark matter with just-so feedback. Or really anything that obfuscates the need to confront the dangerous question: why does MOND perform better?
*This post is based on the recently published paper by my former student Pengfei Li, who is now faculty at Nanjing University. They have a press release about it.
&A few months after reading this in the Boston Driver’s Handbook, this exact thing happened to me.
**This goes back to BBKS in 1986 when the bedrock assumption was that the universe had Ωm = 1, for which the virial radius was 188 times the critical density. 200 was close enough, and stuck, even though for LCDM the virial radius is more like an overdensity close to 100, which is even further out.
#This is one of many processes that occur in simulations, which are great for examining the statistics of simulated galaxy-like objects but completely useless for modeling individual galaxies in the real universe. There may be similar objects, but one can never say “this galaxy is represented by that simulated thing.” To model a real galaxy requires a customized approach.
$NFW halos consistently perform worse in fitting data than any other halo model, of which there are many. It has been falsified as a viable representation of reality so many times that I can’t recall them all, and yet they remain the go-to model. I think that’s partly thanks to their simplicity – it is mathematically straightforward to implement – and to the fact that is what simulations predict: LCDM halos should look like NFW. People, including scientists, often struggle to differentiate simulation from reality, so we keep flogging the dead horse.
%The density profile of the NFW halo model asymptotes to power laws at both small and large radii: ρ → r-1 as r → 0 and ρ → r-3 as r → ∞. The third parameter of Einasto allows a much wider ranges of shapes.
$$The McGaugh19 model user here is the one with a reasonable bulge/bar. This dense component can be fit in this case because we start with a halo model with a core rather than a cusp (closer to α = 1 than to the α = 0.17 of NFW/LCDM).
so the primordial halo is somehow too light, too fluffy and too steep at once, like a cosmic soufflé that keeps collapsing on both mond and lcdm
Thanks for the clarification. I apologize for my rashness in responses to Francois Hammer.
Really interesting result — especially how the Gaia-induced “odd” halo (low mass, low concentration, cored inside, and over-steep outside) matches what you’d expect if the force law has a slight curvature-suppressed term at large R rather than pure NFW or MOND. A mild scaling like
g ∝ R^(-2) + R^(-1.12)
naturally gives the steep outer decline and the core, while breaking the mass–concentration relation in exactly the way your fits show. Gaia may be pointing to a small, universal deviation in the force-law slope rather than to a specific dark matter profile.
Well, a scaling like that would have to work everywhere else as well.
I expect it will take a long time to sort out what is going on with the Gaia determination of the rotation curve in the outer Galaxy.
Totally agree! That’s exactly why the R⁻¹·¹² term is so interesting — it isn’t “tuned” to the Milky Way. The same curvature-suppression shows up independently in molecules, condensed-matter systems, galaxy dynamics, and even folded-protein mutation data. The cross-domain recurrence strongly suggests a small universal geometric effect rather than a galaxy-specific anomaly. If Gaia’s outer rotation curve holds up, it may be pointing straight at the same underlying principle.
Interesting observation. I am working in the modest scaling of g direction, but I can’t wrap my arms around the Gaia numbers – meaning, I would not have predicted this.
Thanks, Jerry — totally understandable! The Gaia numbers are tough to “predict” from any of the usual starting points, whether MOND, ΛCDM, or modest g-rescaling. What caught my attention is that the odd combination Gaia is giving us (steep outside, cored inside, low inferred mass) is exactly the pattern you get when a very small curvature correction nudges the slope away from the idealized asymptotics.
In other words, the correction is small (modest in amplitude), but it disproportionately affects the radial structure of the outer curve. That makes it hard to intuit, but it does seem to line up with the empirical behavior once you look for it.
I’m a bit rusty, but IIRC the declining curve suffer from major systematics,
the milky way has recently discovered major baryonic asymmetries that deviates from the simplifying assumptions of the jeans equation.
Among the many, the most important one probably is that the halo is tilted and triaxial:
https://arxiv.org/abs/2504.07187
the difference between the true circular velocity curve and that inferred from Jeans equations can be as high as 15, but is likely of order 10 for the Milky Way. This is of larger amplitude than the systematics inherent to Jeans equations.
We find that steady-state axisymmetric Jeans modelling becomes less robust at large radii, indicating that particular caution is needed when interpreting the rotation curve inferred in those regions.
https://arxiv.org/abs/2405.19028
https://arxiv.org/abs/2507.23551 most up to date
almost spherically symmetric properties of the Milky Way galaxy at Galactocentric radii above 20 kpc, which is not consistent with the use of axisymmetric Jeans equations.
https://arxiv.org/abs/2503.05877
find that the measured curves deviate from the true curve by 15-20% , as estimated by previous works. Interestingly, there is a large galaxy-to-galaxy variance, and different systematics dominate different galaxies.
Certainly I agree that we have to get past the usual assumptions – this was a key message of mine in https://arxiv.org/abs/1511.06387 – and there are lots of systematics to worry about. So it is good to see others worrying about this, but I think 2507.23551 is exaggerated and I doubt the utility of consulting cosmological simulations as a guide as done in some other papers as such simulations consistently produce rotation curves that are far from realistic. The shape of the halo from GD-1 is interesting but I’ve seen such claims come and go (Sgr required a peculiar triaxial halo at one point; other streams didn’t show precession so MOND was falsified then they did show precession so we stopped talking about it) so I think there remains a great deal to be sorted out.
> implying that commonly neglected terms in the Jeans equations may be non-negligible.
You were ahead of the *curve*! (pun intended)
I agree, but recently the accuracy of the surveys (besides simulations) has significantly improved and IMO along with H3, and DESI DR2 there is a concordance regarding the halo/stars distribution shape
famous h3 study (2022)
https://arxiv.org/abs/2208.04327
DESI dr2 survey of the *exact* same title published yesterday!
https://arxiv.org/html/2512.01350v1
also https://en.wikipedia.org/wiki/Gaia_Sausage
but yes GAIA dr4 will change everything (if not spherex before)
Ah, the age-old problem of using the same word to mean different things. There were places you meant stellar halo where I thought you meant dark matter halo. So yeah, the stellar halo is a broken, tilted mess (quite the convergence of titles there!) As I feared, Gaia is proving to be TMI (too much information)!
I think we can all agree that the classical modeling pipeline is failing here. Axisymmetry, steady-state assumptions, Jeans validity, halo symmetry, tracer equilibrium, and even ΛCDM halo priors all break down in the same ~20–30 kpc band.
Perhaps these aren’t separate puzzles; the “tilt,” the triaxial signatures, the steep outer decline, the broken Jeans inference — they all look like different observational symptoms of the same geometric shift in the potential rather than unrelated systematics. Gaia isn’t giving us noise…it’s giving us structure.
From that perspective, both MOND and ΛCDM struggle for the same fundamental reason: they assume the wrong asymptotic slope of gravity in exactly the regime where Gaia is now giving us the clearest data.
A small correction at large radii appears to resolve all of these features at once.
My “not quite working” theory is that momentum is a function of the total system mass, so when the budget for momentum is exhausted on a galactic scale, energy is radiated rather than vectored. (Hence the robust nature of the Tully-Fisher Relationship.) My problem is the Microscope Eotvos results. The publish accuracy all but rules-out any swing in inertial capacity. I have to either ignore it, find a flaw in it, or find another stump to crawl onto.
“Taken at face value, the Gaia data show a Keplerian decline with a total mass around 2 x 1011 M☉. If correct, this falsifies MOND.”
Falsifies MOND? That seems hard to do when we don’t yet have a theoretical basis identified. Perhaps it could falsify MOND as a description of a local or real gravitational field. We’ll have to see the data says, but I think MOND still has a leg to stand on either way.
Let’s not forget about the cosmological coincidence of the MOND acceleration. A galactic coincidence shouldn’t seem unreasonable either, as long as we keep an open mind about the underlying theory.
There could be a very good reason we don’t need to modify Newton for observations in our own galaxy.
Maybe as we learn more about our own acceleration MOND will start to make more sense.
Some pretends that they have just seen dark matter directly for the first time, not through gravity. Tomonori Totani, “20 GeV halo-like excess of the Galactic diffuse emission and implications for dark matter annihilation,” Journal of Cosmology and Astroparticle Physics (IOPscience): November 26, 2025, doi:10.1088/1475-7516/2025/11/080.
Yeah, I saw that. Might even write about it if I get the chance. A similar claim was made for data from the same (Fermi gamma ray) telescope before, and fell flat. In this case, as in that one, the Fermi team did not itself make the claim. Large teams are not in the habit of leaving Nobel prizes for others to find in their data.
The fact that many of the DM halo experts think that a failure of MOND is a success for DM halos might indicate that they are somewhat aware of the importance of MOND.
According to Professor Milgrom, “Roughly speaking, predictions made by MOND theories can be divided into primary ones, those that follow, within a margin of minor latitude, from only the basic tenets of MOND, and secondary ones, which do not. The former are then shared by all MOND theories (defined as those that embody the basic tenets), while the latter can differ greatly among MOND theories, and are much better suited for distinguishing observationally between theories. I thus also warn against viewing predictions of the two workhorse MOND MG theories, AQUAL and QUMOND, as absolute predictions of MOND.”
“MOND as manifestation of modified inertia”, 2023, https://arxiv.org/pdf/2310.14334
Professor Milgrom thinks that my basic FUNDAMOND theory (based on the monster group & the 6 pariah groups) is completely wrong. Put 3 artificial satellites into approximate orbit around the planet Mercury — this might somewhat clarify the FUNDAMOND problem.
Satellites would be helpful. Some flavors of theories predict strong effects near Lagrange points, where the gradient of the potential goes to zero. And an obvious place to test trajectories is the far outer solar system, though one would have to be patient about getting there.
A weird thing about the sociology of dark matter: while every failure of MOND is seen as a success of dark matter, so is each of its successes: “oh, we explain that too.” Not really, but they ignore anyone who calls them on it. But even if they did, so what? Why should we prefer a theory that only persists by mimicking its rival by making ad hoc adjustments after the fact?
Dear Prof. McGaugh,
Was it ever attempted to involve the rotation velocity of stars in MOND interpolation function?
I mean, instead of e.g. f(a) = 1/(1 + a/a0) something like f(a,υ) = 1/[g(υ) + 1 + a/a0]
Thank you in advance!
That sounds like eMOND: https://arxiv.org/abs/1207.6232
Yes the data could be wrong. But they all seem to go down. Could be the warp too. But it seems EFE-y to me. At 20 kpc we’re at something like gbar=3×10^-11 m/s^2. That’s 10x larger than the local group gext but the decline also isn’t fully Keplerian
Indeed – there are all sorts of things that could go amiss out there.
Is it possible ro rule out Renzo’s rule from the Gaia data? That is, the Milky Way has an overdensity in the region surveyed by Gaia and this skews the results? Can this be excluded from other observations? I don’t know how easily we can figure out how dense a region from the forest is from within that region.