Extended Tully-Fisher relations

Previously I had alluded to some of the major projects I’ve been working on. One has come to fruition and can be found on the arXiv and in the Astrophysical Journal^&. It has taken many years to assemble the data in this paper, during which time the models purporting to explain some of it have evolved considerably while consistently failing to address the real problems they raise. There is a lot to explore, so it will take more than one post.

Here I start with the empirical basis: the stellar mass and baryonic Tully-Fisher relations. The Tully-Fisher relation was originally discovered as a relation between luminosity and linewidth in rotationally supported galaxies – spirals and irregulars. It immediately proved useful as an extragalactic distance indicator. As such, it was instrumental in breaking the impasse in the Hubble constant^* debate (back when it was 50 vs. 100, not 67 vs. 73), and it remains useful in this role.

Physically, the obvious interpretation was that luminosity is a proxy for stellar mass and linewidth^{*^} is a proxy for rotation speed. This is correct. Of the various rotation speeds one can define and measure, the one that works best, in terms of minimizing the scatter in the relation, is the flat rotation speed measured in the outer parts of extended rotation curves. See Stark et al. (2009) and Trachternach et al. (2009) for further examples. The scatter is basically a function of data quality.

On the mass axis, converting measured flux to luminosity to mass is a bit dicier, as we need to know the distance for the first step and the stellar mass-to-light ratio for the second. There is inevitably some intrinsic scatter in the mass-to-light ratio of a stellar population. While I don’t doubt that luminosity is a proxy for stellar mass, improving on it is hard to do: there are many instances in which simply assuming a straight mapping of light to mass can be as effective as applying fancier population models. We might^{^} finally be getting past that, so it is worth discussing a bit.

The procedure to convert starlight into stellar mass involves the construction of stellar population models that use the color(s) or spectral energy distribution of a galaxy to infer the types of stars that make the light. This is a long-argued subject; suffice it to say there are a number of points where it can go wrong. The most obvious is the IMF; the initial spectrum of masses with which stars are born. Most of the light we see from galaxies is produced by its higher mass stars, which are disproportionately bright (there is a steep scaling of stellar luminosity with mass). But most of the mass is locked up in low mass stars that contribute little to the total luminosity. So we are, in effect, using the light of the few to represent the mass of the many. That would go badly wrong if we don’t know the relative mix, i.e., the shape of the IMF. This has been the subject of much research, and over many decades has been narrowed down pretty well. While I hope that this is almost settled, the specter of the IMF lurks as a menace to all stellar mass determinations.

There is a lot else we need to know to build a stellar population model. This includes such essentials as the spectra of individual stars of each and every type and stellar evolution as a function of mass and composition including exotic phases like the asymptotic giant branch. There are a lot of places where this can go badly wrong, and sometimes^{^%} does. So I wouldn’t say we know how to do this perfectly, but we have become pretty good at it.

Converting light to mass suffices to plot the stellar mass Tully-Fisher relation. That accounts for most of the baryonic mass of high mass spirals, but it ignores the mass of the interstellar gas. This can be appreciable in lower mass systems. Indeed, the standard issue dwarf galaxy in the field is more gas than stars:

**Figure 1** from McGaugh et al. (2019): The gas and stellar masses of rotating galaxies. Blue points are galaxies in the SPARC database (Lelli et al. 2016b) and the gas rich galaxies discussed by McGaugh (2012). The location of the Milky Way is noted in red (McGaugh 2016): it is a typical bright spiral. Grey points are the sample of Bradford et al. (2015). The line is the line of equality where M_* = M_g.

With measurements of mass and rotation speed, we can construct the Tully-Fisher relation:

**Figure 4** from McGaugh et al. (2019): The stellar mass (left) and baryonic Tully-Fisher relation (right). Data from Lelli et al. (2016b) and McGaugh (2012) are shown as blue points if both axes are measured with at least 20% accuracy; less accurate data are shown in grey. The latter include cases for which the rotation curve does not extend far enough to measure V_f, in which case the last measure point is used. These cases are systematically offset to lower velocity. Inclination uncertainties and distance errors also contribute to the scatter. The better the data, the tighter the relation. *The location of the Milky Way is noted in red (you are here).*

The stellar mass Tully-Fisher relation is a good correlation by the standards of extragalactic astronomy. The majority of studies in the literature are restricted to massive^% galaxies, mostly those with M_* > 10¹⁰ M_☉ where stars dominate the baryonic mass budget so the omission of gas is not obvious. As we look to lower masses, the relation bends and the scatter increases. That this happens right where gas starts to become important to the mass budget suggests that we’re missing an important component, and voila – a nice, continuous relation that is linear in log space is restored when we plot the baryonic mass Mb = M_*+M_g. Indeed, the data are consistent with a simple power law

M_b = A \, V_f^4

with A = 50 M_☉ km^-4 s⁴. The intercept A has consistently been measured within 10% of this value over the past couple of decades. That this is an integer power law so that the intercept has real physical units is intriguing. That doesn’t happen in most astronomical scaling laws, which are usually more happenstance, like the mass-luminosity relation for main sequence stars.

Why limit ourselves to rotationally supported galaxies? Let’s plots every known type of gravitationally bound extragalactic object, from the smallest ultrafaint dwarfs to the largest clusters of galaxies. Note that I’ve flipped the axes to accommodate the huge dynamic range in baryonic mass, roughly twelve (12) orders of magnitude. This is like having gnats at one end of the scale and blue whales at the other. On that scale, a person is a regular galaxy like the Milky Way.

**Figure 3** from McGaugh et al. (2026): Extended Tully-Fisher relations plotting the flat-equivalent circular velocity of extragalactic systems as a function of stellar mass (top panel) and baryonic mass (bottom panel). Data for rotationally supported galaxies are depicted by circles; squares represent pressure supported systems. The blue circles are galaxies with directly measured distances, V_f from rotation curves, and stellar masses from WISE photometry from Duey et al. (2026, in preparation). Green circles are gas-rich galaxies (M_g > M_*; Stark et al. 2009; Trachternach et al. 2009; Bernstein-Cooper et al. 2014; McNichols et al. 2016; Iorio et al. 2017; Namumba et al. 2025; Xu et al. 2025) not already in Duey et al. (2026). Yellow points are Local Group galaxies, both spirals and dwarfs (McGaugh et al. 2021); gray squares are ultrafaint dwarfs (Lelli et al. 2017). Lensing results for early- and late-type galaxies (Mistele et al. 2024a) are shown as pink squares and magenta circles, respectively. Red squares are clusters of galaxies (Mistele et al. 2025), and purple squares are groups of galaxies (McGaugh et al. 2026). The orange line is the BTFR fit only to rotating galaxies over a more limited range (about three orders of magnitude in baryonic mass, from M_b ~ 4 x 10⁸ to 4 x 10¹¹ M_☉) by McGaugh (2005).

One improvement from twenty years ago, aside from the greater number of objects and the increase in dynamic range, is the accuracy of the mass measurements. I tried a number of prescriptions for the stellar mass-to-light ratio in McGaugh (2005), which resulted in a range of possible slopes. Now we just use the stellar mass from precise population models (Duey et al. 2025) and recover my best estimate from back then. The room to dodge the obvious conclusion about the slope of the relation by complaining about the choice of stellar mass estimator – a popular course of action back then – is gone. Another technical issue we’ve spent a lot of effort working on is how to put all these very different systems on the same scale of V_f. I won’t elaborate on this here: if you’re interested in that level of detail, you can go read the paper and references there in. If we got this wrong, it would add to the scatter in the relation, and/or create offsets between different types of data.

Both of the extended Tully-Fisher relations, that in stellar mass (top panel) and that in baryonic mass (bottom panel, the extended BTFR) are good correlations. That in baryonic mass is clearly better in the sense that it is tighter over a larger dynamic range. From small dwarf galaxies (M_b ~ 5 x 10⁵) to groups of galaxies (5 x 10¹² M_☉), the data are consistent with a single power law (M_b ~ V_f⁴) for all systems with remarkably little scatter. Outside this range, the data for both the lowest and the highest mass systems deviate from a straight line towards higher mass at a given flat velocity. I don’t put much credence in the smallest systems as I think there is little chance that their measured velocity dispersions are representative of their equilibrium gravitational potential. For all practical purposes, our knowledge runs out as we hit the regime of ultrafaint^# dwarfs. The deviations of the most massive systems, clusters of galaxies, are more difficult to dismiss.

Restricting our attention for the moment to the range where a single power law suffices to describe the data, we note that there is not much scatter in the BTFR. Some of it is from random uncertainties; these dominate most studies and lead to a lot more scatter than seen here: these data are very good. We can account for the known observational errors and subtract off their contribution to estimate the intrinsic scatter in the relation. This is the variance of the data from a perfect line. The intrinsic scatter for the best data (the WISE-SPARC sample of Duey et al. 2026) is about 0.11 dex in mass – about what we expect^$ for stellar populations. That doesn’t leave much room for other sources of scatter, so the underlying physical relation has to be very tight indeed: essentially perfect over the range 5 x 10⁵ < M_b < 5 x 10¹² M_☉.

Scatter will also occur if our mass budget is incomplete. We can see this in the transition from the stars-only relation to the BTFR. There is a lot of scatter in the stellar mass Tully-Fisher relation around 10⁷ < M_b < 10⁹ M_☉. Galaxies in this mass range are sometimes star-dominated and sometimes gas-dominated. The gas fraction is all over the place. This shows up as scatter in the stellar mass Tully-Fisher relation. That’s not real; it is a sign that we’ve missed an important mass reservoir. This is cured when we add in the gas mass, which is dominated by atomic gas (HI to spectroscopists and astronomers). That this addition removes the scatter and restores a single power law relation strongly suggests that there are no further substantial reservoirs^** of baryonic material that we’re missing.

This logic applies to other systems as well. Bright spirals do not need much correction because their baryonic mass is dominated by stars. Their stellar mass Tully-Fisher relation is pretty much already their BTFR.

Perhaps this applies to clusters of galaxies as well? There was a huge correction from stars-only to stars plus gas. The gas in this case is the hot, ionized plasma of the intracluster medium (ICM) that belongs to the cluster itself and not any individual galaxy within it. That goes most of the way to close the gap between the stars-only cluster data and the extrapolation of the BTFR fit to individual galaxies, but not all the way. So perhaps we are still missing an important baryonic mass component? It happened before – we didn’t know about the ICM for decades after Zwicky first identified the missing mass problem in clusters – so perhaps there are still more baryons to discover there.

It could also be that the apparent offset occurs because we’ve failed to put clusters on the same V_f scale as galaxies. This is not easy to do, and we’ve spent a lot of time worrying about it. I don’t think this is what’s going on, though it would make my life a lot simpler if it were. Different indicators – dynamics vs. ICM hydrostatics vs. gravitational lensing – can give somewhat different answers, but not in a way that “fixes” the problem: I see no viable path in which the offset turns out to be a simple difference in the way the depth of the gravitational potential is measured. I would love to be wrong here, but I’m not dismissing the offset for clusters as I am for ultrafaint dwarfs (which don’t do lightly).

Perhaps the extrapolation of the BTFR from individual galaxies to clusters is simply not appropriate. They’re very different kinds of systems, after all. To dig into that, we need some theoretical perspective – why does the observed power law happen? Should we expect different systems to share the same BTFR?

Theory is something I’ve studiously avoided in this post: the possibility that there are baryons that remain to be discovered in clusters can be inferred empirically. All the other data line up, so why not clusters? But unless and until these hypothetical additional baryons are discovered, that’s just one possibility. How likely this possibility seems to be diverges rapidly once we overlay a theoretical preference, which I will leave to future posts. (I did warn it would take more than one.)

^&This paper appears in ApJ volume 1001. The literature has grown quite a bit since I started contributing to it in volume 342. The Astrophysical Journal was founded in 1895. So I’ve been contributing to it for a little over a quarter of its temporal existence, but nearly twice the number of volumes have been published in that shorter time. It’s no wonder none of us can keep up.

^*Indeed, Tully & Fisher’s “preliminary estimate of the Hubble constant is H₀ = 80 km/s/Mpc” remains correct to this day, within the uncertainties (hard to estimate at the time, but roughly ±10 km/s/Mpc).

^{*^}There appears to be an irreducible intrinsic scatter in the linewidth: it is not a perfect proxy for rotation speed. Linewidths are observationally easier to obtain than resolved, extended rotation curves, so the numbers of galaxies in samples using linewidths can be very large without ever approaching the quality provided by resolved interferometric observations. Bigger samples are not necessarily better.

^{^}I emphasize might here because the community seems to have moved towards reporting stellar masses as if we observe these rather than the luminosities and colors/SEDs that the mass estimates are based upon. The latter are data – observed quantities – while stellar masses are a derived quantity that is inevitably model dependent. This doesn’t stop being true just because we decide to invest a lot of faith in our models.

^{*^}The Sloan Digital Sky Survey provides stellar masses based on models that are known to be wrong in the near infrared. Since SDSS itself is entirely optical, one might not notice. If one mixes SDSS data with near-IR data, one will get the wrong answer.

^%This is a classic selection effect. Brighter objects can be seen at a much greater distance than dim ones, so probe a much larger volume. Consequently, their raw numbers always dominate surveys even if their number density is low. Stars are a great example: most of the stars you can see at night are intrinsically luminous: bright stars that are rather far away. Mundane, low mass stars do not stand out even when nearby.

^#This isn’t for lack of observations of ultrafaint dwarfs, it’s the underlying assumptions.

^$No amount of information suffices to perfectly specify the stellar mass that produces an observed luminosity and SED (spectral energy distribution/set of colors), so one always expects at least some intrinsic scatter in the stellar mass-to-light ratio. I’ve seen estimates that range from 0.1 – 0.2 dex for near-IR colors. That’s as good as it can get as there is always some transient population (e.g., AGB stars) that produce an amount of light that depends on the star formation rate some time ago, not what we measure now. Optical colors are worse in the sense of having more intrinsic scatter, as they are more susceptible to the comings and goings of bright but short-lived stars whose numbers fluctuate with the stochastic star formation rate. Finding 0.11 dex intrinstic scatter is pretty much as good as it can get. (By dex we mean the scatter in log space.)

^**We noted this effect in the original BTFR paper to argue that it was unlikely that we were missing substantial amounts of molecular gas (H₂), which was a concern at the time. Flash forward, and we were right: the molecular gas mass is almost always a distant third behind stars and atomic gas in the baryonic mass budgets of individual galaxies. Nowadays, the concern is about the mass of baryons in the circumgalactic medium (CGM). That’s getting ahead of the story, which I’ll save for a future post. For now, it suffices to note that any baryonic mass in the CGM is far beyond the radius where the flat velocity is measured, so is not relevant to the sums here.

The Radial Acceleration Relation to very low accelerations

Flat rotation curves and the Baryonic Tully-Fisher relation (BTFR) both follow from the Radial Acceleration Relation (RAR). In Mistele et al. (2024b) we emphasize the exciting aspects of the former; these follow from the RAR in the Mistele et al. (2024a). It is worth understanding the connection.

First, the basic result:

**Figure 2** from Mistele et al. (2024a). The RAR from weak lensing data (yellow diamonds) is shown together with the binned kinematic RAR from Lelli et al. (2017, gray circles). The solid line is Newtonian gravity without dark matter (g_obs = g_bar). The shaded region at g_bar < 10⁻¹³ m/s² indicates where the isolation criterion may be less reliable according to the estimate by Brouwer et al. (2021). Our results suggest that late type galaxies (LTGs) may be sufficiently isolated down to g_bar ≈ 10⁻¹⁴ m/s². We shade this region where LTGs may still be reliable in a lighter color.

The RAR of weak lensing extends the RAR from kinematics to much lower accelerations. How low we can trust we’ll come back to, but certainly to g_bar ≈ 10⁻¹³ m/s² and probably to g_bar ≈ 10⁻¹⁴ m/s². For the mass of the typical galaxy in the KiDS sample, this corresponds to a radius of 300 kpc and 1.1 Mpc, respectively. Hence our claim that the effective gravitational potentials of isolated galaxies are consistent with rotation curves that remain flat indefinitely far out: a million light years at least, and perhaps a million parsecs.

Note that the kinematic and lensing data overlap at log(g_bar) = -11.5. These independent methods give the same result. Moreover, this region corresponds to the regions in galaxies where atomic gas rather than stars dominates the baryonic mass budget, which minimizes the systematic uncertainty due to stellar population mass estimates. The lensing results still depend on these, but they agree with the gas-dominated portion of the RAR, and merge smoothly into the star-dominated portion of the kinematic data when the same stellar pop models are used for both. To wit: the agreement is really good.

A flat rotation curve projects into the log(g_obs)-log(g_bar) plane as a line with slope 1/2. The data adhere closely to this slope, so I knew as soon as I saw the lensing RAR that the implied rotation curves remained flat indefinitely. How far, in radius, depends on galaxy mass, since for a point mass (a good approximation at radii beyond 100 kpc), g_bar = GM_bar/R². We can split the lensing data into different mass bins, for which the RAR looks like

**Figure 5** from Mistele et al. (2024a). The RAR implied by weak lensing for four baryonic mass bins. The dashed line has the slope a flat rotation curve has when projected into the acceleration plane. That different masses follow the same RAR implies the Baryonic Tully-Fisher relation.

Most dark matter models that I’ve seen or constructed myself predict a mass-dependent shift in the RAR, if they predict a RAR at all (many do not). We see no such shift. But the math is such that the flat rotation speed implied by the slope 1/2 RAR varies with mass in such a way that they only fall on the same RAR, as observed, if there is a Baryonic Tully-Fisher relation with slope 4. So I knew from examination of the above figure that the BTFR was sure to follow, but that’s because I’ve been working on these things for a long time. It isn’t necessarily obvious to everyone else, so it was worth explicitly showing.

Our result differs from the original of Brouwer et al. in two subtle but important ways. The first is that we use stellar population models that are the same as we use for the kinematic data. This self-consistency is important to the continuity of the data. We (especially Jim Schombert) took a deep dive into this, and the models used by Brouwer et al. are consistent with ours for late type (spiral) galaxies (LTGs). However, ours are somewhat heavier^ for early type galaxies (ETGs). That’s part of the reason that they find an offset in the RAR between morphological types and we do not.

Another important difference is the strictness of the isolation criterion. We are trying to ascertain the average gravitational potential of isolated galaxies, those with no big neighbors to compound the lensing signal. Brouwer et al. required that there be no galaxies more than a tenth of the luminosity of the primary within 3 Mpc. That seems reasonable, but we explored lots of variations on both aspects of that limit. It seems to be fine for LTGs, but insufficient for ETGs. That in itself is not surprising, as ETGs are known to be more strongly clustered than LTGs, so it is harder to find isolated examples.

To illustrate this, we show the deviation of the data from the kinematic RAR fit as a function of the isolation criterion:

**Figure 4** from Mistele et al. (2024a). Top: the difference between the radial accelerations inferred from weak lensing and the RAR fitting function, measured in sigmas, as a function of how isolated the lenses are, quantified by Ri_sol. We separately show the result for ETGs (red) and LTGs (blue) as well as for small (triangles with dashed lines) and large accelerations (diamonds with solid lines). LTGs are mostly unaffected by making the isolation criterion stricter. In contrast, ETGs do depend on R_isol, but tend towards with increasing R_isol. Middle and bottom: the accelerations behind these sigma values for R_isol = 3 Mpc/h₇₀ and R_isol = 4 Mpc/h₇₀.

The top panel shows that LTGs do not deviate from the RAR as we vary the radius of isolation. In contrast, ETGs deviate a lot for small R_isol. This is what Brouwer et al. found, and it would be a problem for MOND if LTGs and ETGs genuinely formed different sequences: it would be as if they were both obeying their own version of a similar but distinct MOND-like force law rather than a single universal force law.

That said, the ETGs converge towards the same RAR as the LTGs as we make the isolation criterion more strict. The distinction between ETGs and LTGs that appears to be clear for the R_isol = 3 Mpc/h₇₀ used by Brouwer et al. (middle panel) goes away when R_isol = 4 Mpc/h₇₀ (bottom panel). The random errors grow because fewer galaxies⁺ meet the stricter criterion, but this seems a price well worth paying to be rid of the systematic variation seen in the top panel. This also dictates how far out we can trust the data, which show no clear deviation from the RAR until below the limit g_bar = 10⁻¹⁴ m/s².

Regardless of the underlying theory, the data paint a consistent picture. This can be summarized by three empirical laws of galactic rotation:

Rotation curves become approximately* flat at large radii and remain so indefinitely.
The amplitude of the flat rotation speed scales with the baryonic mass as M_bar ~ V_f⁴ (the BTFR).
The observed centripetal acceleration follows from that predicted by the baryons (the RAR).

These are the galactic analogs of Kepler’s Laws for planetary motion. There is no theory in these statements; their just a description of what the data do. That’s useful, as they provide an empirical touchstone that has to be satisfactorily explained by any theory for it to be considered viable. No dark matter-based theory currently does that.

^The difference is well within the expected variance for stellar population models. We can reproduce their numbers if we treat ETGs as if they were just red LTGs. I don’t know if that’s what they did, but it ain’t right.

⁺For the record, the isolated fraction of the entire sample is 16%: most galaxies have neighbors. As a function of mass, the isolation criterion leaves a fraction of 8%, 18%, 30%, and 42% of LTG lenses and 9%, 14%, and 22% of ETG lenses, respectively, in each mass bin. The fraction of isolated LTGs is generally higher than ETGs, as expected. There is also a trend for the isolation fraction to increase as mass decreases. In part this is real; more luminous galaxies are more clustered. It may also be that it is easier for objects that exceed 10% of the primary mass (really luminosity) to evade detection as the primaries get fainter so 10% of that is harder to reach.

*Some people take “flat” way too seriously in this context. While it is often true that rotation curves look pretty darn flat over an extended radial range, I say approximately flat because we never measure, and can never measure, exactly a slope of dV/dR = 0.000. As a practical matter, we have adopted a variation of < 5% from point to point as a working definition. The scatter in Tully-Fisher naturally goes up if one adopts a weaker criterion; what one gets for the scatter is all about data quality.

Rotation curves: still flat after a million light-years

That rotation curves become flat at large radii is one of the most famous results in extragalactic astronomy. This had been established by Vera Rubin and her collaborators by the late 1970s. There were a few earlier anecdotal cases to this effect, but these seemed like mild curiosities until Rubin showed that the same thing was true over and over again for a hundred spiral galaxies. Flat rotation curves took on the air of a de facto natural law and precipitated the modern dark matter paradigm.

Optical and radio data

Rotation curves shouldn’t be flat. If what we saw was what we got, the rotation curve would reach a peak within the light distribution and decline further out. Perhaps an illustration is in order:

The rotation curve (data points, left) of NGC 6946 (right). The red line shows the expected rotation curve for the detected normal matter, which includes both the stars (yellow, from 2MASS) and atomic gas (blue, from THINGS). This provides a good description of the inner rotation curve but falls short further out. The excess observed rotation leads to the need for dark matter or MOND. Also noted is the extent of the rotation curve measured optically to the effective edge of the stars (Daigle et al. 2006; Epinat et al. 2008) and that measured with radio interferometric observations of the gas (Boomsma et al. 2008).

An obvious question is how far out rotation curves remain flat. In the rotation curves traced with optical observations by Rubin et al., the discrepancy was clear but modest – typically a factor of two in mass. It was possible to imagine that the mass-to-light ratios of stars increased with radius in a systematic way, bending the red line above to match the data out to the edge of the stars. This seemed unlikely, but neither did it seem like a huge ask.

Once one gets to the edge of the stellar distribution, most of the mass has been encompassed, and the rotation curve really should start to decline. Increasing the mass-to-light ratio of the stars ceases to be an option once we run out of stars*. Fortunately, the atomic gas typically extends to larger radii, so provides a tracer further out. Albert Bosma pursued this until there were again enough examples to establish that yes, flat rotation curves were the rule. They extended much further out, well beyond where the mass of the observed stars and gas could explain the data.

How much further out? It depends on the galaxy. A convenient metric is the scale length of the disk, which is a measure of the extent of the light distribution. Some galaxies are bigger than others. The peak of the contribution of the stars to the rotation curve occurs around 2.2 scale lengths. The rotation curve of NGC 6946 extends to about 7 scale lengths, far enough to make the discrepancy clear. For a long time, the record holder was NGC 2403, with a rotation curve that remains flat for 20 scale lengths.

Twenty scale lengths is a long way out. It is observations like this that demanded dark matter halos that are much larger than the galaxies they contain. They also posed a puzzle, since we were still nowhere near finding the edge of the mass distribution. Rotation curves seemed to persist in being flat indefinitely.

Results from gravitational lensing

Weak gravitational lensing provides a statistical technique to probe the gravitational potential of galaxies. Brouwer et al. did pioneering work with data from the KiDS survey, and found that the radial acceleration relation extended to much lower accelerations than probed by the types of kinematic data discussed above. That implies that rotation curves remain flat way far out. How far?

Postdoc Tobias Mistele worked out an elegant technique to improve the analysis of lensing data. His analysis corroborates the findings of Brouwer et al. It also provides the opportunity to push further out.

Weak gravitational lensing is a subtle effect – so subtle that one must coadd thousands of galaxies to get a signal. Beyond that, the limiting effect on the result is how isolated the galaxies are. Lensing is sensitive to all mass; if you go far enough out you start to run into other galaxies whose mass contributes to the signal. So one key is to identify isolated galaxies, and restrict the sample to them. KiDS is large enough to do this. Indeed, Mistele was able to show that while neighbors⁺ were a definite concern for elliptical galaxies, they were much less of a problem for spirals. Consequently, we can trace the implied rotation curve way far out.

How far out? In a new paper, Mistele shows that rotation curves continue way far out. Way way way far out. I mean, damn.

The average rotation curve of isolated galaxies (blue points) inferred from KiDS gravitational lensing data. This remains flat well beyond a million light-years with no end in sight. The width of the figure is the distance between the Milky Way and Andromeda. For comparison, the rotation curve of a single galaxy, UGC 6614, is shown in red. An image of the galaxy is shown to scale centered at the origin. UGC 6614 was selected for this illustration because it has a comparable rotation speed to the KiDS average and because it is one of the largest galaxies known: the red points are already a very extended rotation curve. Image credit: Mistele, Lelli, & McGaugh 2024.

Optical rotation curves typically extend to the edge of the stellar disk. That’s about 8 kpc in the example of NGC 6946 given above. Radio observations of the atomic gas of that galaxy extend to 17 kpc. That fits within the first two tick marks on the graph with the lensing rotation curve.

UGC 6614 is a massive galaxy with a very extended low surface brightness disk. Its rotation curve is traced by radio data to over 60 kpc. It is one of the most extended individual rotation curves known. The statistical lensing data push this out by a factor of ten, and more, with no end in sight. The flat rotation curves found by Rubin and Bosma and everyone else appear to persist indefinitely.

So what does it mean? First, flat rotation curves really are a law of nature, in the same sense of Kepler’s laws of planetary motion. Galaxies don’t obey those planetary rules, they have their own set of rules. This is what nature does.

In terms of dark matter halos, the extent of isolated galaxy rotation curves is surprisingly large. Just as we come to the edge of the stellar disk, then the gas disk, we should eventually hit the edge of the dark matter halo. In principle we can imagine this to be arbitrarily large, but in practice there are other galaxies in the universe so this cannot go one forever.

In the context of LCDM, we now have a pretty good idea of how extended halos should be from abundance matching. A galaxy of the mass of UGC 6614 should live in a halo with a virial radius of about 300 kpc or less. There is some uncertainty in this, of course, but we really should have hit the edge with the lensing data. There should be some sign of it, but we see none.

One complication is the so-called 2-halo term. In addition to the primary dark matter halo that hosts a galaxy, when you get very far out, you run into other halos. Isolated galaxies are selected to avoid this to the extent possible, but eventually there will be some extra mass that causes extra lensing signal that would cause an overestimate of the rotation speed. I’ll forgo a detailed discussion of this for now (see Mistele et al. if you’re eager), but the bottom line is that it would require some unnatural fine-tuning for the 1+2 halo terms to add up to such flat rotation curves. There ought to be a perceptible feature in the transition from the primary halo to the surrounding environment. We don’t see that.

In the context of MOND, a flat rotation curve that persists indefinitely is completely natural. That’s what an isolated galaxy should do. Even in MOND there should be an environmental effect: the mass of everything else in the universe should impose an external field effect that eventually limits the extent of the rotation curve. How this transition happens depends on the density of other galaxies; by selecting isolated galaxies this effect is put off as much as possible. Hopefully it will be detected as the data improve from projects like Euclid.

The primary prediction of MOND is an indefinitely extended rotation curve; the external field effect is a subtle detail. Yet again, that is what we see: MOND gets it right without really trying, and in a way that makes little sense in terms of dark matter. Sometimes I wish MOND had never been invented so we could claim to have discovered something profoundly new, or at least discuss the empirical result without concern that the data would get confused with the theory. MOND predictions keep being corroborated, yet the community persists in ignoring its implications, even in terms of dark matter. It’s gotta be telling us something.

We have a press release about this result, so perhaps you will see it kicking around your news feed.

*We could, of course, invoke dark stars, but that’s just an invisible horse of a different color.

⁺There is a well known correlation between morphology and density such that elliptical galaxies tend to live in the densest environments. This means that they are more likely to have neighbors that interfere with the lensing measurement, so finding that identifying isolated ellipticals with a clean lensing signal is more challenging that finding isolated spirals comes as no surprise. Isolated ellipticals do exist so it is possible, but one has to be very restrictive with the sample.

Recent Developments Concerning the Gravitational Potential of the Milky Way. III. A Closer Look at the RAR Model

I am primarily an extragalactic astronomer – someone who studies galaxies outside our own. Our home Galaxy is a subject in its own right. Naturally, I became curious how the Milky Way appeared in the light of the systematic behaviors we have learned from external galaxies. I first wrote a paper about it in 2008; in the process I realized that I could use the RAR to infer the distribution of stellar mass from the terminal velocities observed in interstellar gas. That’s not necessary in external galaxies, where we can measure the light distribution, but we don’t get a view of the whole Galaxy from our location within it. Still, it wasn’t my field, so it wasn’t until 2015/16 that I did the exercise in detail. Shortly after that, the folks who study the supermassive black hole at the center of the Galaxy provided a very precise constraint on the distance there. That was the one big systematic uncertainty in my own work up to that point, but I had guessed well enough, so it didn’t make a big change. Still, I updated the model to the new distance in 2018, and provided its details on my model page so anyone could use it. Then Gaia data started to pour in, which was overwhelming, but I found I really didn’t need to do any updating: the second data release indicated a declining rotation curve at exactly the rate the model predicted: -1.7 km/s/kpc. So far so good.

I call it the RAR model because it only involves the radial force. All I did was assume that the Milky Way was a typical spiral galaxy that followed the RAR, and ask what the mass distribution of the stars needed to be to match the observed terminal velocities. This is a purely empirical exercise that should work regardless of the underlying cause of the RAR, be it MOND or something else. Of course, MOND is the only theory that explicitly predicted the RAR ahead of time, but we’ve gone to great lengths to establish that the RAR is present empirically whether we know about MOND or not. If we accept that the cause of the RAR is MOND, which is the natural interpretation, then MOND over-predicts the vertical motions by a bit. That may be an important clue, either into how MOND works (it doesn’t necessarily follow the most naive assumption) or how something else might cause the observed MONDian phenomenology, or it could just be another systematic uncertainty of the sort that always plagues astronomy. Here I will focus on the RAR model, highlighting specific radial ranges where the details of the RAR model provide insight that can’t be obtained in other ways.

The RAR Milky Way model was fit to the terminal velocity data (in grey) over the radial range 3 < R < 8 kpc. Everything outside of that range is a prediction. It is not a prediction limited to that skinny blue line, as I have to extrapolate the mass distribution of the Milky Way to arbitrarily large radii. If there is a gradient in the mass-to-light ratio, or even if I guess a little wrong in the extrapolation, it’ll go off at some point. It shouldn’t be far off, as V(R) is mostly fixed by the enclosed mass. Mostly. If there is something else out there, it’ll be higher (like the cyan line including an estimate of the coronal gas in the plot that goes out to 130 kpc). If there is a bit less than the extrapolation, it’ll be lower.

The RAR model Milky Way (blue line) together with the terminal velocities to which it was fit (light grey points), VVV data in the inner 2.2 kpc (dark grey squares), and the Zhou et al. (2023) realization of the Gaia DR3 data. Also shown are the number of stars per bin from Gaia (right axis).

From 8 to 19 kpc, the Gaia data as realized by Zhao et al. fall bang on the model. They evince exactly the slowly declining rotation curve that was predicted. That’s pretty good for an extrapolation from R < 8 kpc. I’m not aware of any other model that did this well in advance of the observation. Indeed, I can’t think of a way to even make a prediction with a dark matter model. I’ve tried this – a lot – and it is as easy to come up with a model whose rotation curve is rising as one that is falling. There’s nothing in the dark matter paradigm that is predictive at this level of detail.

Beyond R > 19 kpc, the match of the model and Zhou et al. realization of the data is not perfect. It is still pretty damn good by astronomical standards, and better than the Keplerian dotted line. Cosmologists would be wetting themselves with excitement if they could come this close to predicting anything. Heck, they’re known to do that even when they’re obviously wrong*.

If the difference between the outermost data and the blue line is correct, then all it means is that we have to tweak the model to have a bit less mass than assumed in the extrapolation. I call it a tweak because it would be exactly that: a small change to an assumption I was obliged to make in order to do the calculation. I could have assumed something else, and almost did: there is discussion in the literature that the disk of the Milky Way is truncated at 20 kpc. I considered using a mass model with such a feature, but one can’t make it a sharp edge as that introduces numerical artifacts when solving the Poisson equation numerically, as this procedure depends on derivatives that blow up when they encounter sharp features. Presumably the physical truncation isn’t unphysically sharp anyway, rather being a transition to a steeper exponential decline as we sometimes see in other galaxies. However, despite indications of such an effect, there wasn’t enough data to constrain it in a way useful for my model. So rather than introduce a bunch of extra, unconstrained freedom into the model, I made a straight extrapolation from what I had all the way to infinity in the full knowledge that this had to be wrong at some level. Perhaps we’ve found that level.

That said, I’m happy with the agreement of the data with the model as is. The data become very sparse where there is even a hint of disagreement. Where there are thousands of stars per bin in the well-fit portion of the rotation curve, there are only tens per bin outside 20 kpc. When the numbers get that small, one has to start to worry that there are not enough independent samples of phase space. A sizeable fraction of those tens of stars could be part of the same stellar stream, which would bias the results to that particular unrepresentative orbit. I don’t know if that’s the case, which is the point: it is just one of the many potential systematic uncertainties that are not represented in the formal error bars. Missing those last five points by two sigma is as likely to be an indication that the error bars have been underestimated as it is to be an indication that the model is inadequate. Trying to account for this sort of thing is why the error bars of Jiao et al. are so much bigger than the formal uncertainties in the three realization papers.

That’s the outer regions. The place where the RAR model disagrees the most with the Gaia data is from 5 < R < 8 kpc, which is in the range where it was fit! So what’s going on there?

Again, the data disagree with the data. The stellar data from Gaia disagree with the terminal velocity data from interstellar gas at high significance. The RAR model was fit to the latter, so it must per force disagree with the former. It is tempting to dismiss one or the other as wrong, but do they really disagree?

Adapted from Fig. 4 of McGaugh (2019). Grey points are the first and fourth quadrant terminal velocity data to which the model (blue line) was matched. The red squares are the stellar rotation curve estimated with Gaia DR2 (DR3 is indistinguishable). The black squares are the stellar rotation curve after adjustment to be consistent with a mass profile that includes spiral arms. This adjustment for self-consistency remedies the apparent discrepancy between gas and stellar data.

In order to build the model depicted above, I chose to split the difference between the first and fourth quadrant terminal velocity data. I fit them separately in McGaugh (2016) where I made the additional point that the apparent difference between the two quadrants is what we expect from an m=2 mode – i.e., a galaxy with spiral arms. That means these velocities are not exactly circular as commonly assumed, and as I must per force assume to build the model. So I split the difference above in the full knowledge that this is not the exact circular velocity curve of the Galaxy, it’s just the best I can do at present. This is another example of the systematic uncertainties we encounter: the difference between the first and fourth quadrant is real and is telling us that the galaxy is not azimuthally symmetric – as anyone can tell by looking at any spiral galaxy, but is a detail we’d like to ignore so we can talk about disk+dark matter halo models in the convenient limit of axisymmetry.

Though not perfect – no model is – the RAR model Milky Way is a lot better than models that ignore spiral structure entirely, which is basically all of them. The standard procedure assumes an exponential disk and some form of dark matter halo. Allowance is usually made for a central bulge component, but it is relatively rare to bother to include the interstellar gas, much less consider deviations from a pure exponential disk. Having adopted the approximation of an exponential disk, one inevitably get a smooth rotation curve like the dashed line below:

Fig. 1 from McGaugh (2019). Red points are the binned fourth quadrant molecular hydrogen terminal velocities to which the model (blue line) has been fit. The dotted lines shows the corresponding Newtonian rotation curve of the baryons. The dashed line is the model of Bovy & Rix (2013) built assuming an exponential disk. The inset shows residuals of the models from the data. The exponential model does not and cannot fit these data.

The common assumption of exponential disk precludes the possibility of fitting the bumps and wiggles observed in the terminal velocities. These occur because of deviations from a pure exponential profile caused by features like spiral arms. By making this assumption, the variations in mass due to spiral arms is artificially smoothed over. They are not there by assumption, and there is no way to recover them in a dark matter fit that doesn’t know about the RAR.

Depending on what one is trying to accomplish, an exponential model may suffice. The Bovy & Rix model shown above is perfectly reasonable for what they were trying to do, which involved the vertical motions of stars, not the bumps and wiggles in the rotation curve. I would say that the result they obtain is in reasonable agreement with the rotation curve, given what they were doing and in full knowledge that we can’t expect to hit every error bar of every datum of every sort. But for the benefit of the chi-square enthusiasts who are concerned about missing a few data points at large radii, the reduced chi-squared of the Bovy & Rix model is 14.35 while that of the RAR model is 0.6. A good fit is around 1, so the RAR model is a good fit while the smooth exponential is terrible – as one can see by eye in the residual inset: the smooth exponential model gets the overall amplitude about right, but hits none of the data. That’s the starting point for every dark matter model that assumes an exponential disk; even if they do a marginally better job of fitting the alleged Keplerian downturn, they’re still a lot worse if we consider the terminal velocity data, the details of which are usually ignored.

If instead we pay attention the details of the terminal velocity data, we discover that the broad features seen there in are pretty much what we expect for the kinematic signatures of photometrically known spiral arms. That is, the mass density variations inferred by fitting the RAR correspond to spiral arms that are independently known from star counts. We’ve discussed this before.

Spiral structure in the Milky Way (left) as traced by HII regions and Giant Molecular Clouds (GMCs). These correspond to bumps in the surface density profile inferred from kinematics with the RAR (right).

If we accept that the bumps and wiggles in the terminal velocities are tracers of bumps and wiggles in the stellar mass profiles, as seen in external galaxies, then we can return to examining the apparent discrepancy between them and the stellar rotation curve from Gaia. The latter follow from an application of the Jeans equation, which helps us sort out the circular motion from the mildly eccentric orbits of many stars. It includes a term that depends on the gradient of the density profile of the stars that trace the gravitational potential. If we assume an exponential disk, then that term is easily calculated. It is slowly and smoothly varying, and has little impact on the outcome. One can explore variations of the assumed scale length of the disk, and these likewise have little impact, leading us to infer that we don’t need to worry about it. The trouble with this inference is that it is predicated on the assumption of a smooth exponential disk. We are implicitly assuming that there are no bumps and wiggles.

The bumps and wiggles are explicitly part of the RAR model. Consequently, the gradient term in the Jeans equation has a modest but important impact on the result. Applying it to the Gaia data, I get the black points:

*The red squares are the Gaia DR2 data. The black squares are the same data after including in the Jeans equation the effect of variations in the tracer gradient. This term dominates the uncertainties.*

The velocities of the Gaia data in the range illustrated all go up. This systematic effect reconciles the apparent discrepancy between the stellar and gas rotation curves. The red points are highly discrepant from the gray points, but the black points are not. All it took was to drop the assumption of a smooth exponential profile and calculate the density gradient numerically from the data. This difference has a more pronounced impact on rotation curve fits than any of the differences between the various realizations of the Gaia DR3 data – hence my cavalier attitude towards their error bars. Those are not the important uncertainties.

Indeed, I caution that we still don’t know what the effective circular velocity of the potential is. I’ve made my best guess by splitting the difference between the first and fourth quadrant terminal velocity data, but I’ve surely not got it perfectly right. One might view the difference between the quadrants as the level at which the perfect quantity is practically unknowable. I don’t think it is quite that bad, but I hope I have at least given the reader some flavor for some of the hidden systematic uncertainties that we struggle with in astronomy.

It gets worse! At small radii, there is good reason to be wary of the extent to which terminal velocities represent circular motion. Our Galaxy hosts a strong bar, as artistically depicted here:

*Artist’s rendition of the Milky Way. Image credit: NASA/JPL-Caltech*.

Bars are a rich topic in their own right. They are supported by non-circular orbits that maintain their pattern. Consequently, one does not expect gas in the region where the bar is to be on circular orbits. It is not entirely clear how long the bar in our Galaxy is, but it is at least 3 kpc – which is why I have not attempted to fit data interior to that. I do, however, have to account for the mass in that region. So I built a model based on the observed light distribution. It’s a nifty bit of math to work out the equivalent circular velocity corresponding to a triaxial bar structure, so having done it once I’ve not been keen to do it again. This fixes the shape of the rotation curve in the inner region, though the amplitude may shift up and down with the mass-to-light ratio of the stars, which dominate the gravitational potential at small radii. This deserves its own close up:

Colored points are terminal velocities from Marasco et al. (2017), from both molecular (red) and atomic (green) gas. Light gray circles are from Sofue (2020). These are plotted assuming they represent circular motions, which they do not. Dark grey squares are the equivalent circular velocity inferred from stars in the VVV survey. The black line is the Newtonian mass model for the central bar and disk, and the blue line is the corresponding RAR model as seen above.

Here is another place where the terminal velocities disagree with the stellar data. This time, it is because the terminal velocities do not trace circular motion. If we assume they do, then we get what is depicted above, and for many years, that was thought to be the Galactic rotation curve, complete with a pronounced classical bulge. Many decades later, we know the center of the Galaxy is not dominated by a bulge but rather a bar, with concominant non-circular motions – motions that have been observed in the stars and carefully used to reconstruct the equivalent circular velocity curve by Portail et al. (2017). This is exactly what we need to compare to the RAR model.

Note that 2008, when the bar model was constructed, predates 2017 (or the 2016 appearance of the preprint). While it would have been fair to tweak the model as the data improved, this did not prove necessary. The RAR model effectively predicted the inner rotation curve a priori. That’s a considerably more impressive feat than getting the outer slope right, but the model manages both sans effort.

No dark matter model can make an equivalent boast. Indeed, it is not obvious how to do this at all; usually people just make a crude assumption with some convenient approximation like the Hernquist potential and call it a day without bothering to fit the inner data. The obvious prediction for a dark matter model overshoots the inner rotation curve, as there is no room for the cusp predicted in cold dark matter halos – stars dominate the central potential. One can of course invoke feedback to fix this, but it is a post hoc kludge rather than a prediction, and one that isn’t supposed to apply in galaxies as massive as the Milky Way. Unless it needs to, of course.

So, lets’s see – the RAR model Milky Way reconciles the tension between stellar and interstellar velocity data, indicates density bumps that are in the right location to correspond to actual spiral arms, matches the effective circular velocity curve determined for stars in the Galactic bar, correctly predicted the slope of the rotation curve outside the solar circle out to at least 19 kpc, and is consistent with the bulk of the data at much larger radii. That’s a pretty successful model. Some realizations of the Gaia DR3 data are a bit lower than predicted, but others are not. Hopefully our knowledge of the outer rotation curve will continue to improve. Maybe the day will come when the data have improved to the point where the model needs to be tweaked a little bit, but it is not this day.

*To give one example, the BICEP II experiment infamously claimed in March of 2014 to have detected the Inflationary signal of primordial gravitational waves in their polarization data. They held a huge press conference to announce the result in clear anticipation of earning a Nobel prize. They did this before releasing the science paper, much less hearing back from a referee. When they did release the science paper, it was immediately obvious on inspection that they had incorrectly estimated the dust foreground. Their signal was just that – excess foreground emission. I could see that in a quick glance at the relevant figure as soon as the paper was made available. Literally – I picked it up, scanned through it, saw the relevant figure, and could immediately spot where they had gone wrong. And yet this huge group of scientists all signed their name to the submitted paper and hyped it as the cosmic “discovery of the century”. Pfft.

Wide Binary Weirdness

My last post about the Milky Way was intended to be a brief introduction to our home galaxy in order to motivate the topic of binary stars. There’s too much interesting to say about the Milky Way as a galaxy, so I never got past that. Even now I feel the urge to say more, like with this extended rotation curve that I included in my contribution to the proceedings of IAU 379.

The RAR-based model rotation curve of the Milky Way extrapolated to large radii (note the switch to a logarithmic scale at 20 kpc!) for comparison to the halo stars of Bird et al (2022) and the globular clusters of Watkins et al (2019). The location of the solar system is noted by the red circle.

But instead I want to talk about data for binary stars from the Gaia mission. Gaia has been mapping the positions and proper motions of stars in the local neighborhood with unprecedented accuracy. These can be used to measure distances via trigonometric parallax, and speeds along the sky. The latter once seemed impossible to obtain in numbers with much precision; thanks to Gaia such data now outnumber radial (line of sight) velocities of comparable accuracy from spectra. That is a mind-boggling statement to anyone who has worked in the field; for all of my career (and that of any living astronomer), radial velocities have vastly outnumbered comparably well-measured proper motions. Gaia has flipped that forever-reality upside down in a few short years. It’s third data release was in June of 2022; this provides enough information to identify binary stars, and we’ve had enough time to start (and I do mean start) sorting through the data.

OK, why are binary stars interesting to the missing mass (really the acceleration discrepancy) problem? In principle, they allow us to distinguish between dark matter and modified gravity theories like MOND. If galactic mass discrepancies are caused by a diffuse distribution of dark matter, gravity is normal, and binary stars should orbit each other as Newton predicts, no matter their separation: the dark matter is too diffuse to have an impact on such comparatively tiny scales. If instead the force law changes at some critical scale, then the orbital speeds of widely separated binary pairs that exceed this scale should get a boost relative to the Newtonian case.

The test is easy to visualize for a single binary system. Imagine two stars orbiting one another. When they’re close, they orbits as Newton predicts. This is, after all, how we got Newtonian gravity – as an explanation for Kepler’s Laws or planetary motion. Ours is a lonely star, not a binary, but that makes no difference to gravity: Jupiter (or any other planet) is an adequate stand-in. Newton’s universal law of gravity (with tiny tweaks from Einstein) is valid as far out in the solar system as we’ve been able to probe. For scale, Pluto is about 40 AU out (where Earth, by definition, is 1 AU from the sun).

Let’s start with a pair of stars orbiting at a distance that is comfortably in the Newtonian regime, say with a separation of 40 AU. If we know the mass of the stars, we can calculate what their orbital speed will be. Now imagine gradually separating the stars so they are farther and farther apart. For any new separation s, we can predict what the new orbital speed will be. According to Newton, this will decline in a Keplerian fashion, v ~ 1/√s. This will continue indefinitely if Newton remains forever the law of the land. If instead the force law changes at some critical scale s_c, then we would expect to see a change when the separation exceeds that scale. Same binary pair, same mass, but relatively faster speed – a faster speed that on galaxy scales leads to the inference of dark matter. In essence, we want to check if binary stars also have flat rotation curves if we look far enough out.

We have long known that simply changing the force law at some length scale s_c does not work. In MOND, the critical scale is an acceleration, a₀. This will map to a different s_c for binary stars of different masses. For the sun, the critical acceleration scale is reached at s_c ≈ 7000 AU ≈ 0.034 parsecs (pc), about a tenth of a light-year. That’s a lot bigger than the solar system (40 AU) but rather smaller than the distance to the next star (1.3 pc = 4.25 light-years). So it is conceivable that there are wide binaries in the solar neighborhood for which this test can be made – pairs of stars with separations large enough to probe the MOND regime without being so far apart that they inevitably get broken up by random interactions with unrelated stars.

Gaia is great for identifying binaries, and space is big. There are thousands of wide binaries within 200 pc of the sun where Gaia can obtain excellent measurements. That’s not a big piece of the galaxy – it is a patch roughly the size of the red circle in the rotation curve plot above – but it is still a heck of a lot of stars. A signal should emerge, and a number of papers have now appeared that attempt this exercise. And ooooo-buddy, am I confused. Frequent readers will have noticed that it has been a long time between posts. There are lots of reasons for this, but a big one is that every time I think I understand what is going on here, another paper appears with a different result.

OK, first, what do we expect? Conventionally, binaries should show Keplerian behavior whatever their separation. Dark matter is not dense enough locally to have any perceptible impact. In MOND, one might expect an effect analogous to the flattening of rotation curves, hence higher velocities than predicted by Newton. And that’s correct, but it isn’t quite that simple.

In MOND, there is the External Field Effect (EFE) in which the acceleration from distant sources can matter to the motion of a local system. This violates the strong but not the weak Equivalence Principle. In MOND, all accelerative tugs matter, whereas conventionally only local effects matter.

This is important here, as we live in a relatively high acceleration neighborhood that is close to a₀. The acceleration the sun feels towards the Galactic center is about 1.8 a₀. This applies to all the stars in the solar neighborhood, so even if one finds a binary pair that is widely separated enough for the force of one star on another to be less than a₀, they both feel the 1.8 a₀ of the greater Galaxy. A lot of math intervenes, with the net effect being that the predicted boost over Newton is less than it would have been in the absence of this effect. There is still a boost, but its predicted amplitude is less than one might naively hope.

*The location of the solar system along the radial acceleration relation is roughly (g_bar, g_obs) = (1.2, 1.8)* a₀*. At this acceleration, the effects of MOND are just beginning to appear, and the external field of the Galaxy can affect local binary stars.*

One of the first papers to address this is Hernandez et al (2022). They found a boost in speed that looks like MOND but is not MOND. Rather, it is consistent with the larger speed that is predicted by MOND in the absence of the EFE. This implies that the radial acceleration relation depicted above is absolute, and somehow more fundamental than MOND. This would require a new theory that is very similar to MOND but lacks the EFE, which seems necessary in other situations. Weird.

A thorough study has independently been made by Pittordis & Sutherland (2023). I heard a talk by them over Zoom that motivated the previous post to set the stage for this one. They identify a huge sample of over 73,000 wide binaries within 300 pc of the sun. Contrary to Hernandez et al., they find no boost at all. The motions of binaries appear to remain perfectly Keplerian. There is no hint of MOND-like effects. Different.

OK, so that is pretty strong evidence against MOND, as Indranil Banik was describing to me at the IAU meeting in Potsdam, which is why I knew to tune in for the talk by Pittordis. But before I could write this post, yet another paper appeared. This preprint by Kyu-Hyun Chae splits the difference. It finds a clear excess over the Newtonian expectation that is formally highly significant. It is also about right for what is expected in MOND with the EFE, in particular with the AQUAL flavor of MOND developed by Bekenstein & Milgrom (1984).

So we have one estimate that is MOND-like but too much for MOND, one estimate that is straight-laced Newton, and one estimate that is so MOND that it can start to discern flavors of MOND.

I really don’t know what to make of all this. The test is clearly a lot more complicated than I made it sound. One does not get to play God with a single binary pair; one instead has to infer from populations of binaries of different mass stars whether a statistical excess in orbital velocity occurs at wide separations. This is challenging for lots of reasons.

For example, we need to know the mass of each star in each binary. This can be gauged by the mass-luminosity relation – how bright a main sequence star is depends on its mass – but this must be calibrated by binary stars. OK, so, it should be safe to use close binaries that are nowhere near the MOND limit, but it can still be challenging to get this right for completely mundane, traditional astronomical reasons. It remains challenging to confidently infer the properties of impossibly distant physical objects that we can never hope to visit, much less subject to laboratory scrutiny.

Another complication is the orientation and eccentricity of orbits. The plane of the orbit of each binary pair will be inclined to our line of sight so that the velocity we measure is only a portion of the full velocity. We do not have any way to know what the inclination of any one wide binary is; it is hard enough to identify them and get a relative velocity on the plane of the sky. So we have to resort to statistical estimates. The same goes for the eccentricities of the orbits: not all orbits are circles; indeed, most are not. The orbital speed depends on where an object is along its elliptical orbit, as Kepler taught us. So yet again we must make some statistical inference about the distribution of eccentricities. These kinds of estimates are both doable and subject to going badly wrong.

The net effect is that we wind up looking at distributions of relative velocities, and trying to perceive whether there is an excess high-velocity tail over and above the Newtonian expectation. This is far enough from my expertise that I do not feel qualified to judge between the works cited above. It takes time to sort these things out, and hopefully we can all come to agreement on what it is that we’re seeing. Right now, we’re not all seeing eye-to-eye.

There is a whole session devoted to this topic at the upcoming meeting on MOND. The primary protagonists will be there, so hopefully some progress can be made. At least it should be entertaining.

Take it where?

I had written most of the post below the line before an exchange with a senior colleague who accused me of asking us to abandon General Relativity (GR). Anyone who read the last post knows that this is the opposite of true. So how does this happen?

Much of the field is mired in bad ideas that seemed like good ideas in the 1980s. There has been some progress, but the idea that MOND is an abandonment of GR I recognize as a misconception from that time. It arose because the initial MOND hypothesis suggested modifying the law of inertia without showing a clear path to how this might be consistent with GR. GR was built on the Equivalence Principle (EP), the equivalence¹ of gravitational charge with inertial mass. The original MOND hypothesis directly contradicted that, so it was a fair concern in 1983. It was not by 1984². I was still an undergraduate then, so I don’t know the sociology, but I get the impression that most of the community wrote MOND off at this point and never gave it further thought.

I guess this is why I still encounter people with this attitude, that someone is trying to rob them of GR. It’s feels like we’re always starting at square one, like there has been zero progress in forty years. I hope it isn’t that bad, but I admit my patience is wearing thin.

*I’m trying to help you. Don’t waste you’re entire career chasing phantoms.*

What MOND does ask us to abandon is the Strong Equivalence Principle. Not the Weak EP, nor even the Einstein EP. Just the Strong EP. That’s a much more limited ask that abandoning all of GR. Indeed, all flavors of EP are subject to experimental test. The Weak EP has been repeatedly validated, but there is nothing about MOND that implies platinum would fall differently from titanium. Experimental tests of the Strong EP are less favorable.

I understand that MOND seems impossible. It also keeps having its predictions come true. This combination is what makes it important. The history of science is chock full of ideas that were initially rejected as impossible or absurd, going all the way back to heliocentrism. The greater the cognitive dissonance, the more important the result.

Continuing the previous discussion of UT, where do we go from here? If we accept that maybe we have all these problems in cosmology because we’re piling on auxiliary hypotheses to continue to be able to approximate UT with FLRW, what now?

I don’t know.

It’s hard to accept that we don’t understand something we thought we understood. Scientists hate revisiting issues that seem settled. Feels like a waste of time. It also feels like a waste of time continuing to add epicycles to a zombie theory, be it LCDM or MOND or the phoenix universe or tired light or whatever fantasy reality you favor. So, painful as it may be, one has find a little humility to step back and take account of what we know empirically independent of the interpretive veneer of theory.

As I’ve said before, I think we do know that the universe is expanding and passed through an early hot phase that bequeathed us the primordial abundances of the light elements (BBN) and the relic radiation field that we observe as the cosmic microwave background (CMB). There’s a lot more to it than that, and I’m not going to attempt to recite it all here.

Still, to give one pertinent example, BBN only works if the expansion rate is as expected during the epoch of radiation domination. So whatever is going on has to converge to that early on. This is hardly surprising for UT since it was stipulated to contain GR in the relevant limit, but we don’t actually know how it does so until we work out what UT is – a tall order that we can’t expect to accomplish overnight, or even over the course of many decades without a critical mass of scientists thinking about it (and not being vilified by other scientists for doing so).

Another example is that the cosmological principle – that the universe is homogeneous and isotropic – is observed to be true in the CMB. The temperature is the same all over the sky to one part in 100,000. That’s isotropy. The temperature is tightly coupled to the density, so if the temperature is the same everywhere, so is the density. That’s homogeneity. So both of the assumptions made by the cosmological principle are corroborated by observations of the CMB.

The cosmological principle is extremely useful for solving the equations of GR as applied to the whole universe. If the universe has a uniform density on average, then the solution is straightforward (though it is rather tedious to work through to the Friedmann equation). If the universe is not homogeneous and isotropic, then it becomes a nightmare to solve the equations. One needs to know where everything was for all of time.

Starting from the uniform condition of the CMB, it is straightforward to show that the assumption of homogeneity and isotropy should persist on large scales up to the present day. “Small” things like galaxies go nonlinear and collapse, but huge volumes containing billions of galaxies should remain in the linear regime and these small-scale variations average out. One cubic Gigaparsec will have the same average density as the next as the next, so the cosmological principle continues to hold today.

Anyone spot the rub? I said homogeneity and isotropy should persist. This statement assumes GR. Perhaps it doesn’t hold in UT?

This aspect of cosmology is so deeply embedded in everything that we do in the field that it was only recently that I realized it might not hold absolutely – and I’ve been actively contemplating such a possibility for a long time. Shouldn’t have taken me so long. Felten (1984) realized right away that a MONDian universe would depart from isotropy by late times. I read that paper long ago but didn’t grasp the significance of that statement. I did absorb that in the absence of a cosmological constant (which no one believed in at the time), the universe would inevitably recollapse, regardless of what the density was. This seems like an elegant solution to the flatness/coincidence problem that obsessed cosmologists at the time. There is no special value of the mass density that provides an over/under line demarcating eternal expansion from eventual recollapse, so there is no coincidence problem. All naive MOND cosmologies share the same ultimate fate, so it doesn’t matter what we observe for the mass density.

MOND departs from isotropy for the same reason it forms structure fast: it is inherently non-linear. As well as predicting that big galaxies would form by z=10, Sanders (1998) correctly anticipated the size of the largest structures collapsing today (things like the local supercluster Laniakea) and the scale of homogeneity (a few hundred Mpc if there is a cosmological constant). Pretty much everyone who looked into it came to similar conclusions.

But MOND and cosmology, as we know it in the absence of UT, are incompatible. Where LCDM encompasses both cosmology and the dynamics of bound systems (dark matter halos³), MOND addresses the dynamics of low acceleration systems (the most common examples being individual galaxies) but says nothing about cosmology. So how do we proceed?

For starters, we have to admit our ignorance. From there, one has to assume some expanding background – that much is well established – and ask what happens to particles responding to a MONDian force-law in this background, starting from the very nearly uniform initial condition indicated by the CMB. From that simple starting point, it turns out one can get a long way without knowing the details of the cosmic expansion history or the metric that so obsess cosmologists. These are interesting things, to be sure, but they are aspects of UT we don’t know and can manage without to some finite extent.

For one, the thermal history of the universe is pretty much the same with or without dark matter, with or without a cosmological constant. Without dark matter, structure can’t get going until after thermal decoupling (when the matter is free to diverge thermally from the temperature of the background radiation). After that happens, around z = 200, the baryons suddenly find themselves in the low acceleration regime, newly free to respond to the nonlinear force of MOND, and structure starts forming fast, with the consequences previously elaborated.

But what about the expansion history? The geometry? The big questions of cosmology?

Again, I don’t know. MOND is a dynamical theory that extends Newton. It doesn’t address these questions. Hence the need for UT.

I’ve encountered people who refuse to acknowledge⁴ that MOND gets predictions like z=10 galaxies right without a proper theory for cosmology. That attitude puts the cart before the horse. One doesn’t look for UT unless well motivated. That one is able to correctly predict 25 years in advance something that comes as a huge surprise to cosmologists today is the motivation. Indeed, the degree of surprise and the longevity of the prediction amplify the motivation: if this doesn’t get your attention, what possibly could?

There is no guarantee that our first attempt at UT (or our second or third or fourth) will work out. It is possible that in the search for UT, one comes up with a theory that fails to do what was successfully predicted by the more primitive theory. That just lets you know you’ve taken a wrong turn. It does not mean that a correct UT doesn’t exist, or that the initial prediction was some impossible fluke.

One candidate theory for UT is bimetric MOND. This appears to justify the assumptions made by Sanders’s early work, and provide a basis for a relativistic theory that leads to rapid structure formation. Whether it can also fit the acoustic power spectrum of the CMB as well as LCDM and AeST has yet to be seen. These things take time and effort. What they really need is a critical mass of people working on the problem – a community that enjoys the support of other scientists and funding institutions like NSF. Until we have that⁵, progress will remain grudgingly slow.

¹The equivalence of gravitational charge and inertial mass means that the m in F=GMm/d² is identically the same as the m in F=ma. Modified gravity changes the former; modified inertia the latter.

²Bekenstein & Milgrom (1984) showed how a modification of Newtonian gravity could avoid the non-conservation issues suffered by the original hypothesis of modified inertia. They also outlined a path towards a generally covariant theory that Bekenstein pursued for the rest of his life. That he never managed to obtain a completely satisfactory version is often cited as evidence that it can’t be done, since he was widely acknowledged as one of the smartest people in the field. One wonders why he persisted if, as these detractors would have us believe, the smart thing to do was not even try.

³The data for galaxies do not look like the dark matter halos predicted by LCDM.

⁴I have entirely lost patience with this attitude. If a phenomena is correctly predicted in advance in the literature, we are obliged as scientists to take it seriously⁺. Pretending that it is not meaningful in the absence of UT is just an avoidance strategy: an excuse to ignore inconvenient facts.

⁺I’ve heard eminent scientists describe MOND’s predictive ability as “magic.” This also seems like an avoidance strategy. I, for one, do not believe in magic. That it works as well as it does – that it works at all – must be telling us something about the natural world, not the supernatural.

⁵There does exist a large and active community of astroparticle physicists trying to come up with theories for what the dark matter could be. That’s good: that’s what needs to happen, and we should exhaust all possibilities. We should do the same for new dynamical theories.

Are there credible deviations from the baryonic Tully-Fisher relation?

There is a rule of thumb in scientific publication that if a title is posed a question, the answer is no.

It sucks being so far ahead of the field that I get to watch people repeat the mistakes I made (or almost made) and warned against long ago. There have been persistent claims of deviations of one sort or another from the Baryonic Tully-Fisher relation (BTFR). So far, these have all been obviously wrong, for reasons we’ve discussed before. It all boils down to data quality. The credibility of data is important, especially in astronomy.

Here is a plot of the BTFR for all the data I have ready at hand, both for gas rich galaxies and the SPARC sample:

*Baryonic mass (stars plus gas) as a function of the rotation speed measured at the outermost detected radius.*

A relation is clear in the plot above, but it’s a mess. There’s lots of scatter, especially at low mass. There is also a systematic tendency for low mass galaxies to fall to the left of the main relation, appearing to rotate too slowly for their mass.

There is no quality control in the plot above. I have thrown all the mud at the wall. Let’s now do some quality control. The plotted quantities are the baryonic mass and the flat rotation speed. We haven’t actually measured the flat rotation speed in all these cases. For some, we’ve simply taken the last measured point. This was an issue we explicitly pointed out in Stark et al (2009):

Fig. 1 from Stark et al (2009): Examples of rotation curves (Swaters et al. 2009) that do and do not satisfy the flatness criterion. The rotation curve of UGC 4173 (top) rises continuously and does not meet the flatness criterion. UGC 5721 (center) is an ideal case with clear flattening of the rotational velocity. UGC 4499 marginally satisfies the flatness criterion.

If we include a galaxy like UGC 4173, we expect it will be offset to the low velocity side because we haven’t measured the flat rotation speed. We’ve merely taken that last point and hoped it is close enough. Sometimes it is, depending on your tolerance for systematic errors. But the plain fact is that we haven’t measured the flat rotation speed in this case. We don’t even know if it has one; it is only empirical experience with other examples that lead us to expect it to flatten if we manage to observe further out.

For our purpose here, it is as if we hadn’t measured this galaxy at all. So let’s not pretend like we have, and restrict the plot to galaxies for which the flat velocity is measured:

*The same as the first plot, restricted to galaxies for which the flat rotation speed has been measured.*

The scatter in the BTFR decreases dramatically when we exclude the galaxies for which we haven’t measured the relevant quantities. This is a simple matter of data quality. We’re no longer pretending to have measured a quantity that we haven’t measured.

There are still some outliers as there are still things that can go wrong. Inclinations are a challenge for some galaxies, as are distances determinations. Remember that Tully-Fisher was first employed as a distance indicator. If we look at the plot above from that perspective, the outliers have obviously been assigned the wrong distance, and we would assign a new one by putting them on the relation. That, in a nutshell, is how astronomical distance indicators work.

If we restrict the data to those with accurate measurements, we get

*Same as the plot above, restricted to galaxies for which the quantities measured on both axes have been measured to an accuracy of 20% or better.*

Now the outliers are gone. They were outliers because they had crappy data. This is completely unsurprising. Some astronomical data are always crappy. You plot crap against crap, you get crap. If, on the other hand, you look at the high quality data, you get a high quality correlation. Even then, you can never be sure that you’ve excluded all the crap, as there are often unknown unknowns – systematic errors you don’t know about and can’t control for.

We have done the exercise of varying the tolerance limits on data quality many times. We have shown that the scatter varies as expected with data quality. If we consider high quality data, we find a small scatter in the BTFR. If we consider low quality data, we get to plot more points, but the scatter goes up. You can see this by eye above. We can quantify this, and have. The amount of scatter varies as expected with the size of the uncertainties. Bigger errors, bigger scatter. Smaller errors, smaller scatter. This shouldn’t be hard to understand.

So why do people – many of them good scientists – keep screwing this up?

There are several answers. One is that measuring the flat rotation speed is hard. We have only done it for a couple hundred galaxies. This seems like a tiny number in the era of the Sloan Digitial Sky Survey, which enables any newbie to assemble a sample of tens of thousands of galaxies… with photometric data. It doesn’t provide any kinematic data. Measuring the stellar mass with the photometric data doesn’t do one bit of good for this problem if you don’t have the kinematic axis to plot against. Consequently, it doesn’t matter how big such a sample is.

Other measurements often provide a proxy measurement that seems like it ought to be close enough to use. If not the flat rotation speed, maybe you have a line width or a maximum speed or V_2.2 or the hybrid S_0.5 or some other metric. That’s fine, so long as you recognize you’re plotting something different so should expect to get something different – not the BTFR. Again, we’ve shown that the flat rotation speed is the measure that minimizes the scatter; if you utilize some other measure you’re gonna get more scatter. That may be useful for some purposes, but it only tells you about what you measured. It doesn’t tell you anything about the scatter in the BTFR constructed with the flat rotation speed if you didn’t measure the flat rotation speed.

Another possibility is that there exist galaxies that fall off the BTFR that we haven’t observed yet. It is a big universe, after all. This is a known unknown unknown: we know that we don’t know if there are non-conforming galaxies. If the relation is indeed absolute, then we never can find any, but never can we know that they don’t exist, only that we haven’t yet found any credible examples.

I’ve addressed the possibility of nonconforming galaxies elsewhere, so all I’ll say here is that I have spent my entire career seeking out the extremes in galaxy properties. Many times I have specifically sought out galaxies that should deviate from the BTFR for some clear reason, only to be surprised when they fall bang on the BTFR. Over and over and over again. It makes me wonder how Vera Rubin felt when her observations kept turning up flat rotation curves. Shouldn’t happen, but it does – over and over and over again. So far, I haven’t found any credible deviations from the BTFR, nor have I seen credible cases provided by others – just repeated failures of quality control.

Finally, an underlying issue is often – not always, but often – an obsession with salvaging the dark matter paradigm. That’s hard to do if you acknowledge that the observed BTFR – its slope, normalization, lack of scale length residuals, negligible intrinsic scatter; indeed, the very quantities that define it, were anticipated and explicitly predicted by MOND and only MOND. It is easy to confirm the dark matter paradigm if you never acknowledge this to be a problem. Often, people redefine the terms of the issue in some manner that is more tractable from the perspective of dark matter. From that perspective, neither the “cold” baryonic mass nor the flat rotation speed have any special meaning, so why even consider them? That is the road to MONDness.

Astronomical Acceleration Scales

A quick note to put the acceleration discrepancy in perspective.

The acceleration discrepancy, as Bekenstein called it, more commonly called the missing mass or dark matter problem, is the deviation of dynamics from those of Newton and Einstein. The quantity D is the amplitude of the discrepancy, basically the ratio of total mass to that which is visible. The need for dark matter – the discrepancy – only manifests at very low accelerations, of order 10^-10 m/s/s. That’s one part in 10¹¹ of what you feel standing on the Earth.

MDacc_wclusters_uptomergingBH — The mass discrepancy as a function of acceleration. There is no discrepancy (D=1) at high acceleration: everything is normal in the solar system and at the highest accelerations probed. The discrepancy only manifests at very low accelerations.

Astronomical data span enormous, indeed, astronomical, ranges. This is why astronomers so frequently use logarithmic plots. The abscissa in the plot above spans 25 orders of magnitude, from the lowest accelerations measured in the outskirts of galaxies to the highest conceivable on the surface of a neutron star on the brink of collapse into a black hole. If we put this on a linear scale, you’d see one point (the highest) and all the rest would be crammed into x=0.

Galileo established that the we live in a regime where the acceleration due to gravity is effectively constant; g = 9.8 m/s/s. This suffices to describe the trajectories of projectiles (like baseballs) familiar to everyday experience. At least is suffices to describe the gravity; air resistance plays a non-negligible role as well. But you don’t need Newton’s Universal Law of Gravity; you just need to know everything experiences a downward acceleration of one gee.

As we move to higher altitude and on into space, this ceases to suffice. As Newton taught us, the strength of the gravitational attraction between two bodies decreases as the distance between them increases. The constant acceleration recognized by Galileo was a special case of a more general phenomenon. The surface of the Earth is a [very nearly] constant distance from its center, so gee is [very nearly] constant. Get off the Earth, and that changes.

In the plot above, the acceleration we experience here on the surface of the Earth lands pretty much in the middle of the range known to astronomical observation. This is normal to us. The orbits of the planets in the solar system stretch to lower accelerations: the surface gravity of the Earth exceeds the centripetal force it takes to keep Earth in its orbit around the sun. This decreases outward in the solar system, with Neptune experiencing less than 10^-5 m/s/s in its orbit.

We understand the gravity in the solar system extraordinarily well. We’ve been watching the planets orbit for ages. The inner planets, in particular, are so well known that subtle effects have been known for ages. Most famous is the tiny excess precession of the perihelion of the orbit of Mercury, first noted by Le Verrier in 1859 but not satisfactorily* explained until Einstein applied General Relativity to the problem in 1916.

The solar system probes many decades of acceleration accurately, but there are many decades of phenomena beyond the reach of the solar system, both to higher and lower accelerations. Two objects orbiting one another intensely enough for the energy loss due to the emission of gravitational waves to have a measurable effect on their orbit are the two neutron stars that compose the binary pulsar of Hulse & Taylor. Their orbit is highly eccentric, pulling an acceleration of about 270 m/s/s at periastron (closest passage). The gravitational dynamics of the system are extraordinarily well understood, and Hulse & Taylor were awarded the 1993 Nobel prize in physics for this observation that indirectly corroborated the existence of gravitational waves.

ghostbusters-20090702101358857 — The mass-energy tensor was dancing a monster jig as the fabric of space-time was rent asunder, I can tell you!

Direct detection of gravitational waves was first achieved by LIGO in 2015 (the 2017 Nobel prize). The source of these waves was the merger of a binary pair of black holes, a calamity so intense that it converted the equivalent of 3 solar masses into the energy carried away as gravitational waves. Imagine two 30 solar mass black holes orbiting each other a few hundred km apart 75 times per second just before merging – that equates to a centripetal acceleration of nearly 10¹¹ m/s/s.

We seem to understand gravity well in this regime.

The highest acceleration illustrated in the figure above is the maximum surface gravity of a neutron star, which is just a hair under 10¹³ m/s/s. Anything more than this collapses to a black hole. The surface of a neutron star is not a place that suffers large mountains to exist, even if by “large” you mean “ant sized.” Good luck walking around in an exoskeleton there! Micron scale crustal adjustments correspond to monster starquakes.

High-end gravitational accelerations are 20 orders of magnitude removed from where the acceleration discrepancy appears. Dark matter is a problem restricted to the regime of tiny accelerations, of order 1 Angstrom/s/s. That isn’t much, but it is roughly what holds a star in its orbit within a galaxy. Sometimes less.

Galaxies show a large and clear acceleration discrepancy. The mob of black points is the radial acceleration relation, compressed to fit on the same graph with the high acceleration phenomena. Whatever happens, happens suddenly at this specific scale.

I also show clusters of galaxies, which follow a similar but offset acceleration relation. The discrepancy sets in a littler earlier for them (and with more scatter, but that may simply be a matter of lower precision). This offset from galaxies is a small matter on the scale considered here, but it is a serious one if we seek to modify dynamics at a universal acceleration scale. Depending on how one chooses to look at this aspect of the problem, the data for clusters are either tantalizingly close to the [far superior] data for galaxies, or they are impossibly far removed. Regardless of which attitude proves to be less incorrect, it is clear that the missing mass phenomena is restricted to low accelerations. Everything is normal until we reach the lowest decade or two of accelerations probed by current astronomical data – and extragalactic data are the only data that test gravity in this regime.

We have no other data that probe the very low acceleration regime. The lowest acceleration probe we have with solar system accuracy is from the Pioneer spacecraft. These suffer an anomalous acceleration whose source was debated for many years. Was it some subtle asymmetry in the photon pressure due thermal radiation from the spacecraft? Or new physics?

Though the effect is tiny (it is shown in the graph above, but can you see it?), it would be enormous for a MOND effect. MOND asymptotes to Newton at high accelerations. Despite the many AU Pioneer has put between itself and home, it is still in a regime 4 orders of magnitude above where MOND effects kick in. This would only be perceptible if the asymptotic approach to the Newtonian regime were incredibly slow. So slow, in fact, that it should be perceptible in the highly accurate data for the inner planets. Nowadays, the hypothesis of asymmetric photon pressure is widely accepted, which just goes to show how hard it is to construct experiments to test MOND. Not only do you have to get far enough away from the sun to probe the MOND regime (about a tenth of a light-year), but you have to control for how hard itty-bitty photons push on your projectile.

That said, it’d still be great experiment. Send a bunch of test particles out of the solar system at high speed on a variety of ballistic trajectories. They needn’t be much more than bullets with beacons to track them by. It would take a heck of a rocket to get them going fast enough to return an answer within a lifetime, but rocket scientists love a challenge to go real fast.

*Le Verrier suggested that the effect could be due to a new planet, dubbed Vulcan, that orbited the sun interior to the orbit of Mercury. In the half century prior to Einstein settling the issue, there were many claims to detect this Victorian form of dark matter.

RAR fits to individual galaxies

The radial acceleration relation connects what we see in visible mass with what we get in galaxy dynamics. This is true in a statistical sense, with remarkably little scatter. The SPARC data are consistent with a single, universal force law in galaxies. One that appears to be sourced by the baryons alone.

This was not expected with dark matter. Indeed, it would be hard to imagine a less natural result. We can only salvage the dark matter picture by tweaking it to make it mimic its chief rival. This is not a healthy situation for a theory.

On the other hand, if these results really do indicate the action of a single universal force law, then it should be possible to fit each individual galaxy. This has been done many times before, with surprisingly positive results. Does it work for the entirety of SPARC?

For the impatient, the answer is yes. Graduate student Pengfei Li has addressed this issue in a paper in press at A&A. There are some inevitable goofballs; this is astronomy after all. But by and large, it works much better than I expected – the goof rate is only about 10%, and the worst goofs are for the worst data.

Fig. 1 from the paper gives the example of NGC 2841. This case has been historically problematic for MOND, but a good fit falls out of the Bayesian MCMC procedure employed. We marginalize over the nuisance parameters (distance and inclination) in addition to the stellar mass-to-light ratio of disk and bulge. These come out a tad high in this case, but everything is within the uncertainties. A long standing historical problem is easily solved by application of Bayesian statistics.

NGC2841_RAR_MCMC — RAR fit (equivalent to a MOND fit) to NGC 2841. The rotation curve and components of the mass model are shown at top left, with the fit parameters at top right. The fit is also shown in terms of acceleration (bottom left) and where the galaxy falls on the RAR (bottom right).

Another example is provided by the low surface brightness (LSB) dwarf galaxy IC 2574. Note that like all LSB galaxies, it lies at the low acceleration end of the RAR. This is what attracted my attention to the problem a long time ago: the mass discrepancy is large everywhere, so conventionally dark matter dominates. And yet, the luminous matter tells you everything you need to know to predict the rotation curve. This makes no physical sense whatsoever: it is as if the baryonic tail wags the dark matter dog.

IC2574_RAR_MCMC — RAR fit for IC 2574, with panels as in the figure above.

In this case, the mass-to-light ratio of the stars comes out a bit low. LSB galaxies like IC 2574 are gas rich; the stellar mass is pretty much an afterthought to the fitting process. That’s good: there is very little freedom; the rotation curve has to follow almost directly from the observed gas distribution. If it doesn’t, there’s nothing to be done to fix it. But it is also bad: since the stars contribute little to the total mass budget, their mass-to-light ratio is not well constrained by the fit – changing it a lot makes little overall difference. This renders the formal uncertainty on the mass-to-light ratio highly dubious. The quoted number is correct for the data as presented, but it does not reflect the inevitable systematic errors that afflict astronomical observations in a variety of subtle ways. In this case, a small change in the innermost velocity measurements (as happens in the THINGS data) could change the mass-to-light ratio by a huge factor (and well outside the stated error) without doing squat to the overall fit.

We can address statistically how [un]reasonable the required fit parameters are. Short answer: they’re pretty darn reasonable. Here is the distribution of 3.6 micron band mass-to-light ratios.

MLdisk_RAR_MCMC — Histogram of best-fit stellar mass-to-light ratios for the disk components of SPARC galaxies. The red dashed line illustrates the typical value expected from stellar population models.

From a stellar population perspective, we expect roughly constant mass-to-light ratios in the near-infrared, with some scatter. The fits to the rotation curves give just that. There is no guarantee that this should work out. It could be a meaningless fit parameter with no connection to stellar astrophysics. Instead, it reproduces the normalization, color dependence, and scatter expected from completely independent stellar population models.

The stellar mass-to-light ratio is practically inaccessible in the context of dark matter fits to rotation curves, as it is horribly degenerate with the parameters of the dark matter halo. That MOND returns reasonable mass-to-light ratios is one of those important details that keeps me wondering. It seems like there must be something to it.

Unsurprisingly, once we fit the mass-to-light ratio and the nuisance parameters, the scatter in the RAR itself practically vanishes. It does not entirely go away, as we fit only one mass-to-light ratio per galaxy (two in the handful of cases with a bulge). The scatter in the individual velocity measurements has been minimized, but some remains. The amount that remains is tiny (0.06 dex) and consistent with what we’d expect from measurement errors and mild asymmetries (non-circular motions).

RAR_MCMC — The radial acceleration relation with optimized parameters.

For those unfamiliar with extragalactic astronomy, it is common for “correlations” to be weak and have enormous intrinsic scatter. Early versions of the Tully-Fisher relation were considered spooky-tight with a mere 0.4 mag. of scatter. In the RAR we have a relation as near to perfect as we’re likely to get. The data are consistent with a single, universal force law – at least in the radial direction in rotating galaxies.

That’s a strong statement. It is hard to understand in the context of dark matter. If you think you do, you are not thinking clearly.

So how strong is this statement? Very. We tried fits allowing additional freedom. None is necessary. One can of course introduce more parameters, but we find that no more are needed. The bare minimum is the mass-to-light ratio (plus the nuisance parameters of distance and inclination); these entirely suffice to describe the data. Allowing more freedom does not meaningfully improve the fits.

For example, I have often seen it asserted that MOND fits require variation in the acceleration constant of the theory. If this were true, I would have zero interest in the theory. So we checked.

Here we learn something important about the role of priors in Bayesian fits. If we allow the critical acceleration g_† to vary from galaxy to galaxy with a flat prior, it does indeed do so: it flops around all over the place. Aha! So g_† is not constant! MOND is falsified!

gdagger_MCMC — Best fit values of the critical acceleration in each galaxy for a flat prior (light blue) and a Gaussian prior (dark blue). The best-fit value is so consistent in the latter case that the inset is necessary to see the distribution at all. Note the switch to a linear scale and the very narrow window.

Well, no. Flat priors are often problematic, as they have no physical motivation. By allowing for a wide variation in g_†, one is inviting covariance with other parameters. As g_† goes wild, so too does the mass-to-light ratio. This wrecks the stellar mass Tully-Fisher relation by introducing a lot of unnecessary variation in the mass-to-light ratio: luminosity correlates nicely with rotation speed, but stellar mass picks up a lot of extraneous scatter. Worse, all this variation in both g_† and the mass-to-light ratio does very little to improve the fits. It does a tiny bit – χ² gets infinitesimally better, so the fitting program takes it. But the improvement is not statistically meaningful.

In contrast, with a Gaussian prior, we get essentially the same fits, but with practically zero variation in g_†. wee The reduced χ² actually gets a bit worse thanks to the extra, unnecessary, degree of freedom. This demonstrates that for these data, g_† is consistent with a single, universal value. For whatever reason it may occur physically, this number is in the data.

We have made the SPARC data public, so anyone who wants to reproduce these results may easily do so. Just mind your priors, and don’t take every individual error bar too seriously. There is a long tail to high χ² that persists for any type of model. If you get a bad fit with the RAR, you will almost certainly get a bad fit with your favorite dark matter halo model as well. This is astronomy, fergodssake.

One Law to Rule Them All

One Law to rule them all, One Law to guide them,
One Law to form them all and in the dark halo bind them.

Galaxies appear to obey a single universal effective force law.

Early indications of this have been around for some time. It has become particularly clear in our work using near-infrared surface photometry to trace the stellar mass distribution of late type galaxies (SPARC). It takes a while to wrap our heads around the implications.

The observed phenomenology constitutes a new law of nature. One Law to rule all galaxies.

The Astrophysical Journal just published our long and thorough investigation of this issue eponymously titled One Law to Rule Them All: The Radial Acceleration Relation of Galaxies. It includes this movie showing the build-up of the radial acceleration relation in the data.

So far, the ubiquitous effective force law had only been clearly demonstrated in rotating galaxies. Federico Lelli and Marcel Pawlowski went to great lengths to also include pressure supported galaxies, from giant ellipticals to dwarf spheroidals. They appear to follow the same effective force law as rotating galaxies.

rar_todo_raronly — The Radial Acceleration Relation defined by rotating late type galaxies (blue points) is also obeyed by early type galaxies, regardless of whether they be fast rotators (orange points) or pressure supported slow rotators (red point) or dark matter dominated dwarf spheroidal satellite galaxies (grey and green points).

This is not a fluke of a few special galaxies. It involves galaxies of all known morphological types spanning an enormous range in mass, size, and surface brightness. I have spent the last twenty years adding new data for all varieties of galaxy types to this relation in the expectation that it would break. Instead it has become stronger and clearer.

Understanding the observed relation is one of the pre-eminent challenges in modern physics. Once we exclude metaphysical nonsense like multiverses, it is arguably the most important unsolved problem. Why does this happen?

The usual ad hoc interpretation of rotation curves in terms of dark matter does nothing to anticipate the observed phenomenology, which is in fact quite troubling from this perspective as it requires excessive fine-tuning. This has been known (if widely ignored) for a while, but doesn’t preclude the more rabid advocates of dark matter from asserting that it all comes about naturally. Lets not mince words here: claims that the radial acceleration relation occurs naturally with dark matter are pure, unadulterated bullshit fueled by confirmation bias and cognitive dissonance. Perhaps dark matter is the root cause, but there is nothing natural about it.

The natural explanation of a single effective force law is that it is caused by a truly universal force law.

So far, the theory that comes closest to explaining these data is MOND. Milgrom, understandably enough, argues that these data require MOND. He has a valid point. It is a good argument, but does it suffice to overcome the other problems MOND faces? These are not as great as widely portrayed, but they aren’t exactly negligible, either. I tried to look at the problem from both perspectives in this review for the Canadian Journal of Physics. [Being able to see things from both sides is an essential skill if one is to be objective, an important value in science that seems disturbingly absent in its modern practice.]

MOND anticipates an asymptotic slope of 1/2 at low acceleration (g_obs ~ g_bar^1/2). In the figure above, the data for the faintest (“ultrafaint”) dwarf spheroidals show a flattening in the empirical law at low accelerations that is not predicted by MOND. Perhaps the underlying force law is subtly different from pure MOND? On the other hand, weak lensing observations show that the MOND slope extrapolates well to much lower accelerations.

It is possible that the data for ultrafaint dwarfs are in some cases misleading. Are these objects in dynamical equilibrium (a prerequisite for analysis)? Are they even dwarf galaxies? Some of the ultrafaints are not clearly distinct objects in the sense of dSph satellites like Crater 2: it is not clear that all of them deserve the status of “dwarf galaxy.” Some are little more than a handful of stars that occupy a similar cell in phase space – perhaps they are fragmentary structures in the Galactic stellar halo? Or the rump end of dissolving satellites? This is anticipated to occur in both ΛCDM and MOND. If so, their velocity dispersions probably tell us more about their disruption history than their gravitational potential, in which case their location in the plot is misleading.

Detailed questions like these are the subject of much current research. For now, lets take a step back and appreciate the data for what they say, irrespective of the underlying theoretical reason for it. We’re looking at a new law of nature! How cool is that?

Ash nazg durbatulûk, ash nazg gimbatul, ash nazg thrakatulûk, agh burzum-ishi krimpatul.

Triton Station

A Blog About the Science and Sociology of Cosmology and Dark Matter

Category: Laws of Nature