A pre-print this week uses the Milky Way's rotation curve and the vertical acceleration relative to the spiral disk of the Milky Way to compare the predictions of various theories for explaining dark matter phenomena to the data. The novel aspect of the study is its consideration of vertical acceleration data in Section 5 of the paper, rather than relying solely on rotation curve fits, which are more of a wash.
The Theories Compared In The Study
It compares three theories: one modified gravity theory, one bosonic dark matter theory, and one fermionic dark matter theory.
The most well known modified gravity theory is MOND, developed by Israeli Physicist M. Milgrom in the 1983 that does a good job of reproducing galactic rotation curves over a wide range of galaxy types with a single empirically measured parameter, which has a relativistic generalization developed by his colleague, J. Bekenstein, called TeVeS (Tensor-Vector-Scalar with a acronym that is meaningful in Hebrew).
J.W. Moffat at the Perimeter Institute created a similar theory, called SVTG (Scalar-Vector-Tensor Gravity) developed in 2005 that has more parameters but fits a broader range of dark matter phenomena into the galactic cluster scale where MOND fails. The list is not exhaustive.
Yet another gravity modification theory to explain dark matter phenomena is called RGGR (Renormalization Group corrected General Relativity), developed by D.C. Rodrigues and others in 2010, that modifies General Relativity by tweaking the gravitational coupling constant in a manner analogous to renormalization of coupling constants in the Standard Model. RGGR also has more parameters than MOND, but claims to produce tighter fits to galactic rotation curves than leading gravity modfication alternatives to dark matter like MOND and SVTG.
The fermionic dark matter scenario tested models dark matter in the Milky Way as collisionless fermionic fluid with an NFW distribution (the distribution of fermionic thermal relic cold dark matter that one would expect from first principles calculations). This is the "default" dark matter model.
The best summary of the results in the paper's own language is found here: "the RGGR model overestimates the vertical acceleration. The predicted acceleration, despite being able to reproduce adequately the effective force field along the galactic plane, fails to represent the gravitational acceleration along the the vertical axis. Concerning the BEC model, the predicted rotation curve gives a fit quality worse than that derived for the RGGR (or NFW) model but the predicted vertical acceleration agrees better with observation than the RGGR theory."
Unsurprisingly, since it was designed for this purpose, a good fit to the Milky Way's rotation curve was obtained with the RGGR theory (reduced chi squared 1.83) which was superior to either of the dark matter theories. But, its predictions for vertical acceleration of objects in the Milky Way system, a dark matter phenomena that is hard to observe outside the Milky Way and has previously been difficult to measure precisely in the Milky Way, was poor. (No reduced chi square statistic of fit was calculated for this fit). It has one parameter that needs to be adjusted for specific galaxies, but another paper indicates that this parameter is actually close to proportionate to the mass of the galaxy, at least within galaxies of a particular type (e.g. spiral or elliptical) rather than being a truly free parameter that must be tuned on a case by case basis as this study suggested. This is one parameter more than MOND, but one parameter less than an NFW model.
The BES model provided a significantly less good fit to the Milky Way's rotation curve than the RGGR theory (reduced chi squared 4.61), although still profoundly better than a Newtonian approximation to GR model. It also produced what the authors of the study view as a marginally tolerable fit to the observed vertical acceleration of objects in the Milky Way system (reduced chi squared 10.09). But, it has multiple parameters that have to be adjusted to each galaxy considered, limiting its usefulness in making predictions, and it greatly underestimated the total amount of dark matter inferred to be in the Milky Way in dark matter models. It was fit with three parameters: central density, radius and halo mass. This is one parameter more than the NFW model.
The old school collisionless fermionic dark matter model (NFW) was also a decent fit to the Milky Way's rotation curve (reduced chi squared 2.17) and a better fit than the BES model (or RGGR) to the vertical acceleration of objects in the Milky Way (reduced chi squared 5.27). It did, however predict twice as much dark matter as the usually assumed dark matter density, although the total dark matter mass predicted for the Milky Way after fitting the two parameters of an NFW distribution (halo radius and dark matter density) to the Milky Way's rotation curve matched other estimates of the total mass of the Milky Way.
Honestly, just eyeballing the vertical acceleration fit predicted by each of the three theories relative to the data points, I would be hard pressed to tell you that the BES model was superior to the NFW model or visa versa. But, the RGGR model does indeed greatly overestimate vertical acceleration relative to the bosonic and fermionic dark matter halo models. In this regard, the paper notes that: "It is worth mentioning that the relatively high values of the reduced “χ2” associated to the analysis of the vertical acceleration are due only to one or two points whose observational errors were probably underestimated."
Similarly, eyeballing the data (all of which are profoundly better than prediction of General Relativity approximated with Newtonian gravity without dark matter), all produce tolerably decent rotation curve fits given that the rotation curve fit parameters were permitted to be adjusted to the specific case to calibrate each of the three theories, and that the data consists of just a single galaxy, rather than an average closeness of fit over many galaxies.
A Null Hypothesis Model Would Have Been Helpful
Unfortunately, the study failed to compare prediction of the default assumption, that the Milky Way's rotation curve and vertical acceleration are governed by conventional Newtonian approximations general relativity unmodified by dark matter of any kind.
While this option would obviously have been wrong and a poor fit for the data, it would better illustrate what each theory adds to this default assumption. By showing the discrepancy between this default assumption and the empirical data, the magnitude of the correction to the Milky Way's rotation curve in each of the theories would be better illustrated, which would allow a reader to better evaluate if the differences in galactic rotation curve fits between the theories is really material.
The comparison to Newtonian vertical accelerations would be even more helpful because while most readers of this paper are familiar with what the default galactic rotation curve looks like, far fewer readers know how much dark matter phenomena impact vertical acceleration in the default case.
Are dark matter halo models outperforming RGGR in this model because RGGR is modifying gravity too much, or too little? The study doesn't tell us.
This Is Not An Unfair Test Of The Particular Models Tested
The implicit conclusion of the study is that collisionless fermionic cold dark matter with an old school NFW profile works best, while bosonic dark matter is a tolerable but inferior alternative, and that modifications to gravity, while tuned to fit one dark matter phenomena, fails to capture the overall picture validly.
The comparison is not entirely unfair.
It is certainly appropriate to see if RGGR, a theory built to fit another parameter that has a theoretical foundation that should be generally applicable, lives up to its promise, and to find it wanting in this regard because it fails to predict Milky Way object vertical acceleration, despite having an adjustable parameter tuned to this particular galaxy's rotation curve.
Likewise, it is hard to dispute that the BES model performs measurably less well than the NFW model in both respects measured in this comparison, despite having a less universal set of parameters.
It Is Premature To Conclude That These Conclusions Can Be Generalized Much
But, it is important not to overstate the conclusions reached either.
All three models secured tolerable fits to galactic rotation curves. Both BES and NFW secured tolerable fits to vertical accelerations.
But, it is easy to imagine that a slightly different bosonic dark matter theory could have performed better with more universal parameters. Indeed, the study itself, notes that it imposed an unrealistic constraint on the BES model that impaired its performance: "our best BEC model has a thin disk much more massive than the thick component. This is certainly a unrealistic result that could be avoided if the condition expressed by eq.13 is relaxed." Rather than addressing this flaw in the model that was compared to a traditional NFW cold dark matter model, the comparison proceeded using this acknowledged straw man version of bosonic dark matter.
And, numerous other studies have concluded that empirically, that an isothermal halo distribution (basically rugby ball shaped), rather than the cuspy NFW halo distribution, is a better fit to observations of multiple different galaxies of different kinds. But, advocates for this approach argue that it can be saved by using models that incorporate the gravitational interaction between baryons and cold dark matter in galaxies.
This Comparison Downplays Some Problems With Dark Matter Models
There is also room to doubt that the analytical comparison method used for the dark matter model in this study accurately reproduces the kind of behavior that would be observed in a N-body cold dark matter simulation.
Also, allowing dark matter halo models, bosonic or fermionic, to be calibrated to best fit the Milky Way rotation curve, rather than universally, sweeps under the rug one of the big weaknesses of these models, which is that dark matter models, in general, do not, in general produce dark matter halos that are best fit to the baryonic matter around them.
One of the stronger arguments for gravity modifications as opposed to dark matter models is that the observed remarkably close link between baryonic matter distributions and dark matter phenomena is a natural feature of gravity modifications, while it is a mere approximate average relationship in dark matter models which feature considerable random halo variation even for otherwise identical baryonic matter distributions.
RGGR Is Unlikely To Be Representative Of Other Gravity Modification Models
The choice of RGGR as a sole representative of modified gravity theories also presents serious uncertainties since it is too new to be well understood. But, more starkly, RGGR is something of a strawman version of modified gravity theories in this comparison. It appears in this paper, because one of the authors of the study, P.L.C. de Oliveria, appears to be one of early investigators of the RGGR theory (and here). (The same author has also investigated unified dark matter-dark energy models also called dark fluid theories involving a Generalized Chaplygin Gas medium defined here, which are similar in substance to theories involving gravitational self-interactions in which the gravitational field warps its own effects such as the theory discussed here.)
While RGGR has some of the best fits to galactic rotation curves of the available theories when it is allowed to have a parameter calibrated to individual galaxies, it has qualitative features that suggest, as this comparison illustrated, that it is not a particularly robust gravity modification theory.
While there is considerable overlap between all non-self-interacting dark matter halo theories which are tweaked only modestly by the specific nature of the dark matter particles modeled, the differences between modified gravity theories are material, because each theory modifies gravity through different kinds of parameters.
RGGR has one universal parameter, and another that must be fit to each galaxy, which controls the exponent of the weak field strength of the Newtonian gravitational field used to determine to "renormalize" the field strength in that galaxy. In this case:
The derived RGGR parameter corresponds to αν = 5.67×10−7, which is about a factor 3.4 higher than that derived from the fit of the rotation curve of NGC 2403 by the authors of reference . They claim that the ν parameter cannot vary from galaxy to galaxy but the α parameter can, contrary to MOND or STVG, which don’t have free parameters varying from one object to another. Despite the fact that the RGGR theory leads to a good fit quality of the rotation curve of the MW, the variation of the energy scale among galaxies is a weak point of this theory.The gravity modification in the toy model MOND regime is triggered at a single, close to universal constant with units of acceleration. It kicks in when the Newtonian approximation of the gravitational field strength falls below the threshold set by this parameter, phasing in the modification using an interpolation function. The lack of fine tuning in MOND to fit particular galaxies (no other theory explains essentially all galaxy rotation curves with just one parameter) distinguishes it from its competition, as does its long track record of making predictions of new phenomena, rather than merely post-dictions of phenomena already observed. Recent efforts to use MOND theory, however, have refined their accuracy by also considering non-luminous particle density in the system and the impact of gravitational fields from other massive bodies outside the measured system.
The SVTG theory, in contrast, modifies gravity in a theory with two almost universal constants, one with units of mass on the order of the galactic scale, and the other with units of length, that have a spread on the order of two to three between different kinds of galaxies or systems. SVTG is relativistic like the MOND generalization TeVeS, but outshines MOND and TeVeS by fitting not just galactic rotation curves, but also galactic cluster data. It has a gravitational field with an added (repulsive) Yukawa potential (which is equivalent to a massive spin-1 fifth force force carrying boson that couples to mass-energy) and with an effective coupling strength and distance range. SVTG has three running constants whose behavior, like the running constant beta functions of the Standard Model are not dependent upon empirically measured parameters.
Notably, Deur's effort to better model graviton-graviton couplings in otherwise standard general relativity, in principle, without introducing new empirically measured constants, like SVTG, introduces a Yukawa term in a weak field approximation, and uses the overall mass of the system as one of two main factors that governs the extent of the gravity modification. But, unlike SVTG, which uses a length factor as a second major determinant of the extent of gravity modification (although it does consider this factor in connection with the skew of the fifth force field), Deur's analysis focuses on a different aspect of the system's geometry - the extent to which it is not spherically symmetric.
MOND, in contrast, is fundamentally spherically symmetric in its design (in addition to failing at cluster scales) and probably share's RGGR's flaws in this respect. (Indeed, an earlier paper concluded just that analyzing the same data.)
SVTG may be less sensitive to geometry than Deur's approach. But, it is not at all obvious that SVTG or Deur's evaluation of graviton self-interactions would have the same failings when it comes to predicting vertical acceleration in the Milky Way as RGGR was revealed to have in this comparison.
I am particularly inclined to think that Deur's approach would outperform RGGR in this respect, because they, while RGGR and MOND appear to be more sensitive to non-spherically symmetric geometries of a system, boosting all weak fields equally, Deur's approach strengthen the radial pull of gravity in a spiral galaxy by weakening the gravitational pull in the direction vertical to the rotating disk of the galaxy to an equal degree. Thus, Deur's model should produce weaker vertical accelerations than either RGGR or MOND in the Milky Way.
The proponents of the RGGR gravity modification theory need to return to the lab and try again. A very basic cold dark matter model is almost as accurate at reproducing galactic rotation curves and far better a reproducing the vertical acceleration seen in the Milky Way.
The superior performance of a particular fermionic dark matter model to a particular bosonic dark matter model in the Milky Way is hardly a definitive test in and of itself. But, the reality is that this corroborates in the Milky Way context what has been clear in a long line of previous studies comparing the two. Fermionic dark matter models are a better match to the dark matter halos that we infer from astronomy observations in a wide variety of contexts than bosonic dark matter models.
But, it is absolutely premature to rule out gravity modification theories in general, relative to dark matter particle theories. The fact that the relatively new and untested RGGR model is flawed when compared to the empirical data from the Milky Way, does not mean that other gravity modification models that have endured the test of time and lack this flaw would also fail.
Also, while this test was a fair way to hypothesis test the RGGR model, it is structured in a way that conceals some of the points that have been identified as critical flaws of cold dark matter theories in other studies.
So, while it is good science to have a new tool available by which to judge the match between observation and reality using precision Milky Way data, it is premature to reach final conclusions about which mechanism best explains dark matter phenomena until more models of the dozens that are in the literature, are considered.