Is Milgrom's Constant Really Constant?

Reductionist physicists always like to derive physical constants, rather than than simply measure them. This paper proposes a way to do so for modified gravity in a MOND-like theory, expanding the theory's domain of applicability to galaxy clusters while sacrificing the universality of Milgrom's constant, a(0). 

It also hints, as Deur explicitly concludes, that MOND-like effects may be influenced by the extent to which a matter distribution is (or is not) spherically symmetric, and provides a mechanism to explain why MOND-like effects arise.

Modified Newtonian Dynamics (MOND) generally resolves the need for dark matter in galaxy rotation curves introducing a single new constant of acceleration a(0). It is well known that increasing a(0) by a factor of a few can alleviate the residual mass discrepancies that MOND leaves in galaxy clusters. 

Within a parameter-free Machian interpretation of MOND, in which a(0) ∼ GM(u)/R(u)^2 arises from the scalar sum of inverse-square distance gravitational mass contributions in the universe, we promote a(0) to a variable influenced by mass external to a locally enclosed region in the spherically symmetric case. 

Instead of a boost of a(0) in terms of gravitational potentials as in EMOND, we show that a boost in terms of this directionless inverse-square field roughly amounts to the boost needed to accommodate the mass discrepancies of MOND in galaxy clusters. We conclude by beginning to generalize the proposed formulation beyond spherical symmetry.

Manuel Uruena Palomo, Juan David Santander, "Machian MOND: a variable a0 in galaxy clusters" arXiv:2608.04894 (August 5, 2026) (published version in International Journal of Modern Physics D).

Another Gravitational Alternative To Dark Matter

Entropic gravity models, which imply that gravity is an emergent law of physics, rather than a fundamental force in its own right, are a very attractive possibility from a reductionist perspective. This paper looks at an entropic gravity theory that would eliminate the need for dark matter and suggests a way to observationally distinguish it from ΛCDM and MOND models. It doesn't work out the cosmological implications of the theory, however, although since it replicates MOND in other contexts, it can be expected to be similar to it in the area of cosmology.

While modified entropy models-such as Barrow, Tsallis, Kaniadakis, Power-law, Logarithmic, and Rényi entropies-have been widely explored in cosmological contexts, their implications on galactic scales remain largely untested. These generalizations of the Bekenstein-Hawking entropy encode quantum gravitational, nonextensive, or fractal spacetime effects and can alter the gravitational entropy-area relation. 

In this paper, we demonstrate that the entropic force framework, when applied to galactic rotation curves and the baryonic mass of galaxy clusters, uniquely selects Tsallis entropy as the specific generalized entropy formulation. We then extend this Tsallis modified gravity to globular clusters to complete the structural hierarchy from galaxies to galaxy clusters to globular clusters and to investigate its behavior as a function of system scale. 

We will show that the nonextensive parameter exhibits no correlation with any of the macroscopic quantities characterizing gravitational systems, such as mass, radius, temperature, or density. Furthermore, it has previously been shown that entropy is not well-defined within the standard thermodynamic approach to gravity. The adoption of nonextensive statistics provides a foundation for entanglement, thereby enabling a consistent definition of entanglement entropy. 

We predict the existence of galaxy clusters with δ=1 (i.e., clusters whose dynamics require no dark matter) analogous to δ=1 systems already observed at galactic and globular cluster scales. This prediction provides a unique observational test to discriminate Tsallis gravity from ΛCDM and MOND. Therefore, for the entropic gravity paradigm to be consistent with observational data across all scales-from globular clusters to galaxies to galaxy clusters-it is inevitably required to be built upon Tsallis entropy.

S. Shamari, A. Sheykhi, "Resolving Galactic and Cluster Dynamics Without Dark Matter: Tsallis Entropy as the Unique Foundation of Emergent Gravity" arXiv:2607.19435 (July 21, 2026).

The introduction to this paper in the body text provides context for the entropic gravity concept, which is follows by subsequent sections that lay out the equations implied by the theory and by its competing explanations of dark matter phenomena in detail:

The quest to understand the fundamental nature of gravity, space, and time has led to profound theoretical innovations, among which the thermodynamic-gravity conjecture stands as a pivotal insight. Originally formulated by Jacobson and later enriched by Padmanabhan’s emergence paradigm, this conjecture posits that gravitational field equations-including those of General Relativity-can be derived from thermodynamic principles applied to spacetime horizons. In this framework, the Bekenstein-Hawking entropy S = A/(4G) plays a central role, linking the geometry of horizons to the statistical mechanics of spacetime microstates. In recent years, however, various quantum-gravitational, nonextensive, and fractal-spacetime considerations have motivated generalizations of this entropy formula, giving rise to modified entropy prescriptions such as those of Barrow, Tsallis, Kaniadakis, Power-law, Logarithmic and Rényi. These extended entropies introduce new parameters that encode departures from standard thermodynamics and may reflect deep-seated quantum or geometric properties of spacetime. 

To date, research on modified entropies has been predominantly cosmology-centric. Studies have explored how such corrections alter the Friedmann equations, influence dark energy models, modify inflationary scenarios, and leave imprints on the cosmic microwave background. 

While these investigations have placed valuable constraints on entropy parameters using large-scale cosmological data, a critical and largely uncharted frontier remains: how do modified entropies manifest on galactic and sub-cosmological scales? 

Galaxies-and the dark matter halos that host them-represent gravitational systems with well-defined effective horizons and rich dynamical observables (rotation curves, velocity dispersion profiles, baryonic mass-velocity relations). Yet, the implications of horizon thermodynamics for such systems have scarcely been explored, leaving open the question of whether galactic kinematics can serve as a new, independent test bed for quantum-gravitational entropy corrections. 

Our aim here is to bridge this gap by developing and applying a framework to test modified entropies through galactic-scale observations. We posit that if horizon thermodynamics underpins gravitational dynamics, then the entropy associated with the boundary of a galaxy or a dark matter halo-be it the Hubble radius of a galaxy group or the radius enclosing a fixed density contrast should govern its equilibrium properties. Using the entropic force scenario proposed by Verlinde [1] we derive modified force laws and mass-velocity relations that depend explicitly on the chosen entropy form. These relations can then be confronted with high-precision galactic data from surveys such as SPARC (for rotation curves), MaNGA (for stellar kinematics), and other spatially resolved kinematic datasets. 

Although the origins of the dark matter problem trace back to the early 1930s with the work of Zwicky and Oort, it only became a hot research topic when Vera Rubin published her observations of galactic rotation curves. Today, after nearly a century of diverse astrophysical observations from the dynamics of galaxy clusters to gravitational lensing studies and baryon acoustic oscillations a consistent picture emerges: approximately 85% of the matter in the universe is non-baryonic. Despite extensive searches, direct detection of dark matter particles remains elusive, motivating serious consideration of alternative ideas [2–7]. 

Many different theories have been proposed to explain this puzzle without non-baryonic dark matter such as Modified Gravity (MOG), Modified Newtonian Dynamics (MOND), Carmelian theory, Cooperstock model, etc [8–12]. 

Deur does not get a mention, although his approach deserves it. References [8-12] are:

[8] J. W. Moffat, Scalar-Tensor-Vector Gravity Theory, J. Cosmol. Astropart. Phys. 2006, 004 (2006). 

[9] M. Milgrom, A Modification of the Newtonian Dynamics as a Possible Alternative to the Hidden Mass Hypothesis, Astrophys. J. 270, 365 (1983). 

[10] F. I. Cooperstock and S. Tieu, Galactic dynamics via general relativity: A compilation and new developments, Int. J. Mod. Phys. A 22, 2293 (2007). 

[11] M. Carmeli, Is Galaxy Dark Matter a Property of Spacetime?, Int. J. Theor. Phys. 37, 2621 (1998). 

[12] S. Behar and M. Carmeli, Derivation of the Tully-Fisher Law from General Relativity Theory: Doubts about the Existence of Halo Dark Matter, Int. J. Theor. Phys. 39, 1397 (2000). 

A parallel line of inquiry emerged from black hole thermodynamics, following the discovery that black holes possess entropy proportional to their horizon area and temperature. Jacobson demonstrated that the Einstein field equations are nothing but an equation of state for spacetime. Verlinde’s entropic gravity framework realizes the idea that gravity itself may be an emergent phenomenon, with spacetime possessing intrinsic thermodynamic properties. In this thermodynamic paradigm, the choice of entropy functional becomes crucial. While Bekenstein-Hawking entropy leads to standard general relativity, any alternative entropy yields modified gravitational dynamics [13–16]. 

In this work, we classify all possible entropy modifications into two types of generalized entropies. We first examine their performance in reproducing galactic rotation curves; in this assessment, only Tsallis entropy succeeds. Next, we investigate their performance in galaxy clusters. We reconstruct a well-known model and again evaluate each modified entropy. Once more, Tsallis entropy emerges successful. We then extend this Tsallis modified gravity to globular clusters to complete the structural hierarchy. After analyzing and plotting the corresponding figures, we turn to the origin of Tsallis entropy and its theoretical implications. 

This paper is structured as follows. Section II critically examines type-II entropies, demonstrating their failure at galactic and cluster scales. In Section III, we derive the Tsallis-modified force law and apply it to galaxy clusters, presenting our observational analysis of 40 clusters. We then extend this framework to globular clusters, analyzing the velocity dispersion profiles of 33 such systems. In Section IV, we discuss the theoretical foundations of nonextensive statistics in gravitational systems, drawing on the work of Chavanis and others, and interpret the physical meaning of the Tsallis parameter δ in terms of dynamical relaxation and hidden constraints. Section V presents our central prediction: the existence of dynamically relaxed galaxy clusters that are observationally dark matter-free, offering a decisive test to distinguish Tsallis gravity from ΛCDM and MOND. Finally, Section VI is devoted to closing remarks. Throughout this paper we set ℏ = c =kB =1.

Another notable new paper on galactic dynamics, which has phenomenological merit, but is probably flaws in its mechanism is:

We investigate whether the observed fine structure and asymmetry of non-averaged galactic rotation curves can be reconstructed directly from the observed HI distribution within the framework of a kinetic gas transport description. Using the observed HI density profiles separately for the approaching (north-eastern) side and the receding (south-western) side of the galaxy NGC~3198, we reconstruct the corresponding rotation curves based on the equation previously derived in Lipovka 2022. It is shown that the reconstructed curves reproduce not only the approximately flat large-scale behaviour of the observed rotation curves, but also their detailed local morphology and asymmetry separately for the north-eastern and south-western sides of the galactic disk. The obtained results indicate that the local structure of galactic rotation curves is closely connected with the local HI distribution and arises naturally as a consequence of kinetic gas transport processes in galactic disks.

Anton A. Lipovka, Anna A. Lipovka "Local morphology and asymmetry of galactic rotation curves in a kinetic gas transport framework: NGC 3198" arXiv:2607.19505 (July 21, 2026).

The abstract of Lipovka 2022 is as follows:

In this paper, I show that generally accepted methods of classical mechanics are not applicable for calculating the outer parts of the rotation curves of galaxies, where an influence of collisions on gas dynamics becomes dominant. In addition, the hydrodynamic approach cannot be used for this purpose due to an extreme rarefaction of the gas. I develop a new approach to describing the gas dynamics in outer regions of galactic disks, where the gas dynamics is determined mainly by collisions. 

Equations (free from restrictions imposed on hydrodynamics) are obtained that describe the dynamics of rarefied gas. The resulting equations relate two quantities: the tangential velocity of the gas as a function of the distance from the center of a galaxy (rotation curve) and the radial distribution of the gas density. It is shown that if the physical properties of the rarefied gas are properly taken into account, then dark matter is not required, and the "nonphysical" (non-Keplerian) rotation curves of the outer parts of the galactic disks are tailwinds that can be described within the framework of conventional gas kinetics. 

To illustrate the correctness of the obtained model, two galaxies with flat rotation curves (NGC7331 and NGC3198) are considered. From the observed rotation curves, using Eq. (14), the radial densities of the gas are calculated. An excellent agreement was obtained between the calculated gas densities and their observed values, which is a serious argument in favor of the developed model. Thus, the non-physical rotation curves of spiral galaxies represent the tailwinds of gas, the dynamics of which is naturally described by the kinetic equation without involving the concept of dark matter. The total masses of two galaxies NGC7331 and NGC3198 have been calculated. The implications for cosmology are discussed.

This theory purports to explain MOND dynamics without either particle dark matter or modified gravity, using interstellar, apparently basically baryonic, gas dynamics. Any theory that works without significant new physics deserves serious attention, but I'm skeptical that it really describes the mechanism of dark matter phenomena. 

A Theoretically Innovative MOG Theory

Canadian physicist John Moffat's MOG modified gravity theory is a long standing tensor, vector, scalar modification of General Relativity (GR). As the link explains:

Scalar–tensor–vector gravity theory, also known as MOdified Gravity (MOG), is based on an action principle and postulates the existence of a vector field, while elevating the three constants of the theory to scalar fields. In the weak-field approximation, STVG produces a Yukawa-like modification of the gravitational force due to a point source. Intuitively, this result can be described as follows: far from a source gravity is stronger than the Newtonian prediction, but at shorter distances, it is counteracted by a repulsive fifth force due to the vector field.

STVG has been used successfully to explain galaxy rotation curves, the mass profiles of galaxy clusters, gravitational lensing in the Bullet Cluster, and cosmological observations without the need for dark matter. On a smaller scale, in the Solar System, STVG predicts no observable deviation from general relativity. The theory may also offer an explanation for the origin of inertia.

Yukawa forces are forces carried by massive mediator bosons (in contrast to the massless mediator boson of electromagnetism, the photon, which has an infinite range as a result), whose range is a function of the mediator mass. 

The most familiar example of a Yukawa force is the nuclear binding force (sometimes called the residual strong force) that holds photons and neutrons in atomic nuclei together, which is mediated by like composite mesons, especially pions (neutral pions have a mass of about 135 MeV, while charged ones have a mass of about 140 MeV) that have an effective range on the order of femtometers, which is similar to the size of an atomic nucleus.

In contrast, GR without a cosmological constant (including Deur's approach to explaining dark matter phenomena as gravitional) is a tensor theory, and GR with a cosmological constant is a tensor-scalar theory. Newtonian gravity is a scalar theory. Several of the main relativistic generalizations of MOND are also tensor, vector, scalar theories.

MOG, while not the subject of as much scholarship as MOND (Israeli physicist Mordehai Milgrom's 1983 non-relativistic toy model modification of Newtonian gravity that does a good job of replicating dark matter phenomena is almost near equilibrium systems of galaxy size or smaller), MOG is still one of the older modified gravity theories, has received considerable investigation from scientists other than its inventor, is relativistic, is more easily generalized to cosmology scale problems, and unlike MOND, models galaxy cluster phenomena often attributed to dark matter more successfully, at the cost of being somewhat less intuitive to understand.

Moffat's latest short paper formulates his MOG theory in a manner, that while essentially identical to the original, is easier to apply to cosmology scale questions.

We develop a Stueckelberg gauge-invariant formulation of modified gravity (MOG). 

The massive vector field is made gauge-invariant by introducing a compensating scalar field, without requiring a Higgs field, spontaneous symmetry breaking, or a vacuum expectation value to fix the effective Newtonian gravitational coupling. This separates the gauge-invariant origin of the vector mass from the cosmological evolution of the gravitational coupling. 

The formulation preserves the finite-range vector interaction of MOG, while allowing the effective gravitational coupling to be treated as an independent scalar or scale-dependent quantity. This distinction is important for cosmological tests, since early-universe constraints and late-time large-scale gravitational phenomena need not be tied to a symmetry-breaking vacuum. The Stueckelberg formulation provides a gauge-invariant framework for comparing MOG with nucleosynthesis, cosmic microwave background, large-scale structure, lensing, and distance data.

John W. Moffat, "Stueckelberg Gauge Invariant Formulation of MOG" arXiv:2606.26427 (June 4, 2026).

Another new MOG paper constrains the value of one of that theory's physical constants (to a value inconsistent with the range in the previous literature on the topic):

The scalar-tensor-vector-gravity (STVG), a prototype of modified gravity developed by Moffat, can correctly explain galaxy rotation curves, cluster dynamics, Bullet Cluster phenomena and cosmological data without invoking the observationally elusive general relativistic (GR) dark matter. Further, recent observations of neutron star masses are shown to defy some GR predictions, whereas STVG turns out to be more consistent with those observations. These successes indicate that STVG could be a potential candidate for a new theory of gravity. 

However, an important question concerns the possible range of values of the STVG dimensionless parameter α imposed by various physical scenarios. In the literature, the range 0.03 < α < 2.47 corresponding to different central source masses has been suggested. We show here that the α can be considerably constrained into the range 0 < α < 10^−5 assuming that the updated GPS fluctuation does not exceed the α-dependent correction to the terrestrial Sagnac delay.

R. Kh. Karimov, R. N. Izmailov, K. K. Nandi, "Terrestrial Sagnac delay in scalar-tensor-vector-gravity" arXiv:2606.27033 (June 25, 2026).

A footnote on f(R) gravity

Probably the other modified gravity theory with significant scholarship from multiple astrophysicists that is most often used to explain dark matter phenomena gravitationally is f(R) gravity (the image below is from this link), which like GR with a cosmological constant, and unlike MOG or some relativistic generalizations of MOND, is a tensor-scalar theory. The way f(R) gravity modifies GR is not with an extra vector field, but with a higher order derivative term. The standard Ricci scalar R in the Einstein-Hilbert action is replaced by a general function of R (e.g., R + (alpha)*R^2). Mathematically and dynamically, this higher-order derivative theory is exactly equivalent to standard General Relativity coupled to a single, dynamical scalar field (known as the scalaron), rather than only having the static scalar dark energy field that is equivalent to the cosmological constant.

Like MOG, f(R) gravity has a Yukawa correction to the gravitational potential, which (at least in part, it also has a time and scale dependent gravitational constant) is how it can explain some or all dark matter phenomena without dark matter particles.

The Inferred Milky Way Dark Matter Distribution Isn't Spherical

Measuring matter dynamics outside the plane of spiral galaxies is critical 

Rotations curves of, and gravitational accelerations of, matter in the vicinity of spiral galaxies that is above or below the galactic plane where most of the ordinary matter in these galaxies is found, is critical to distinguishing between competing dark matter particle and gravity or fifth force based explanations of dark matter phenomena (or hybrids of the two paradigms like self-interacting dark matter).

These theories have been formulated and fine tuned to reproduce the dynamics of stars in the plane of spiral galaxies where they are much easier to observe and measure, and good data has been available for many decades. But because good data has not been available for the dynamics of stars outside the galactic plane of spiral galaxies, different models formulated to explain dark matter phenomena differ considerably in what they predict about that.

Measuring matter dynamics outside the plane of spiral galaxies is hard and has only recently become a viable possibility

But until very recently our astrophysical observations provided us with little data and limited accuracy outside the galactic plane of spiral galaxies with various kinds of "telescopes" for a variety of reasons. 

In the case of the Milky Way, the main problems have been that the density of stars to observe outside the galactic plane of the Milky Way is much lower than in or near the thin galactic disk where most of its stars are found, and the complication that as observers who are inside the Milky Way, the vantage point of our observations is obstructed by dense stars in the galactic plane or otherwise non-optimal.

In the case of other galaxies, one of the main problems have been that it is hard to determine if a particular star is in the galactic plane or above (or below) that plane unless we have a close to edge on view of the galaxy, that measuring rotation curves is hard with a true edge on view. Another problem is that the resolution of our view of a galaxy gets worse as the galaxy gets more distant which is especially a concern outside the plane of a spiral galaxy where the density of the stars we are trying to observe is low. And, when looking at another galaxy it is particularly hard to tell if a star outside the main plane of the galaxy is really part of the same gravitationally bound system, or is millions of megaparsecs away from it in the foreground of our observation of that galaxy.

Fortunately, we live in an era where we have an abundance of riches when it comes to astronomy observations, producing a torrent of data from extremely powerful telescopes like the Gaia space observatory (a telescope in orbit around Earth). The data from this space telescope is used by the Gaia collaboration's network of over 400 scientists and engineers funded by the European Space Agency (ESA) to build the most accurate 3D map of the Milky Way ever constructed.

As the paper below explains in its abstract, Gaia's measurement uncertainties are less than 5% for the vertical velocities of stars that it observes in the Milky Way, and are less than 20% for the vertical accelerations that it measures. 

These uncertainties may not seem all that great to someone unfamiliar with the details of galaxy scale astronomy observations. But in that subfield of astronomy,  uncertainties as low as 28% (i.e. 0.1 dex) are the considered good, and relative uncertainties on the order of 50%-100% are common place, so Gaia's measurements are gold standards of precision by comparison.

Comparing models

Both simple cold dark matter models (with their spherically symmetrical NFW dark matter particle halos) and MOND (even in its relativistic generalizations) predict dark matter phenomena are spherically symmetrical, which makes these theories mathematically much more tractable. 

Indeed, coming up with any kind of dark matter particle model without either (1) self-interactions more complex than a simple scalar field, or (2) interactions in excess of ordinary gravitational interactions with ordinary matter, that do not form spherical or nearly spherical dark matter halos, is extremely challenging and could very well be impossible (although I'm not aware of any analytically constructed "no go" theorem to that effect).

But not all explanations of dark matter phenomena predict spherically symmetric effects, and inferred dark matter halo shapes from prior observations have tended to favor non-spherical, rugby ball shaped inferred distributions of dark matter particles (even though theoretically, it has been challenging to come up with dark matter particle theories that reproduce these shapes).

Some gravity based explanations of dark matter phenomena, like the one described by Deur, also propose non-spherical dark matter phenomena in spiral galaxies. In Deur's approach dark matter phenomena arise from non-linear self-interactions within gravitational fields that manifest in, and only in, non-spherical matter distributions like those found in spiral disk disk galaxies. 

In Deur's analysis, in spiral galaxies, the pull of gravity towards the galactic center is stronger than the Newtonian expectation in the direction of rays from the galactic center in the galactic plane (especially at larger radii), while it is weaker than the Newtonian expectation in the vertical direction relative to the galactic plane (an effect which accounts, at least in part, for dark energy phenomena between galaxies).

Deur's analysis is also supported by another key data point that corroborates astronomy observations that infer non-spherical dark matter particle distributions in dark matter particle paradigms. He has observed that the relative proportion of matter in a galaxy that is made of luminous stars to inferred dark matter is strongly correlated in elliptical galaxies, with the extent to which the elliptical galaxy is not perfectly spherical.

New, high quality data shows that the Milky Way's inferred dark matter halo is not spherical 

Gaia has assembled new data with record breaking accuracy and sample sizes on the rotational velocities and accelerations of stars in the Milky Way based upon their polar coordinates (i.e. their distance from the Galactic center and their distance from the plane of the Milky Way spiral disk). This data, was compiled by the Gaia collaboration, and was analyzed and reported in a pre-print released today of an accepted for publication astronomy paper.

The new paper's analysis strongly favors inferred dark matter particle distributions which are not spherically symmetric. Instead, it strongly favors the inference in a dark matter particle paradigm of a flattened disk-like configuration around the ordinary matter of the Milky Way.

The Gaia data generically rules out all dark matter particle explanations of dark matter phenomena, gravity or fifth force based explanations,  and hybrid explanations (like self-interacting dark matter models), that predict spherically symmetric dark matter phenomena effects. 

This is a huge deal because most of the leading explanations of dark matter phenomena are spherically symmetric, and all of those models are now definitively ruled out.

The paper

We derive both the mid-plane and off-plane rotation curves, v(c)(R,z), and the vertical acceleration, a(z)(R,z), of the Milky Way (MW) using Gaia~DR3 data over the ranges of vertical heights z∈(−2,2) kpc and galactocentric distances R∈(8.5,14) kpc where the velocity components are determined with high precision, i.e., with an error <5%. In contrast, the vertical acceleration a(z)(R,z) is dominated by model-dependent systematics, with uncertainties of up to ∼20%. This level of accuracy allows us to place stringent constraints on the geometry of the MW's dark matter (DM) distribution, as the vertical gradients of the gravitational potential attain their maximum within this range of radial and vertical distances corresponding to the characteristic scales of the disk. 

We find that models including the observed stellar components together with a spherical DM halo fail to reproduce both the pronounced variation of v(c)(R,z) with height and the observed behavior of a(z)(R,z). 

In particular, spherical halos with a scale radius of rs∼15 kpc contribute negligibly to the off-plane rotation curve and vertical acceleration in the inner disk, leaving these features primarily determined by the stellar mass distribution. 

Conversely, models in which DM is confined to a flattened, disk-like configuration predict substantial contributions to both v(c)(R,z) and a(z)(R,z), resulting in a markedly better agreement with the data. We conclude that disk-like DM distributions are strongly favored over spherical halo models. 

Forthcoming Gaia data releases will enable even more stringent tests of the geometry and distribution of the MW's DM component.

Francesco Sylos Labini, Roberto Capuzzo-Dolcetta, "Constraining the Geometry of Galactic Dark Matter with Gaia Data Release 3" arXiv:2606.12548 (June 10, 2026) (accepted for publication in The Astrophysical Journal) (emphasis added in abstract).

The body text of the conclusion further explains that:

Our results show that the DM disk model provides a significantly better agreement with the data than the standard Navarro–Frenk–White (NFW) halo profile. 

In particular, spherical halos with characteristic scale radii of order ∼ 10 kpc contribute only marginally to the off-plane rotation curve and to the vertical acceleration within the inner disk, leaving these quantities predominantly determined by the distribution of the stellar mass. As a consequence, halo-based models systematically underestimate the measured vertical accelerations and fail to reproduce the observed decline of the rotation curve at intermediate heights. 

In contrast, models in which the DM is confined to a flattened, disk-like configuration predict substantial contributions to both the radial and vertical components of the gravitational field, leading to a markedly improved agreement with the observed trends of v(c)(R,z) and a(z)(R,z). This improvement is particularly evident at low to intermediate heights (|z| ≲ 2 kpc), where the vertical acceleration inferred from the data cannot be explained by the baryonic components alone. 

The success of the DM disk model arises from its geometry: a flattened mass distribution naturally enhances the vertical component of the gravitational potential without requiring an excessive total mass, and simultaneously reproduces the modest decline of the circular velocity with increasing z. These results strongly suggest that a significant fraction of the MW’s dark matter is distributed in a disk-like structure rather than in a quasi-spherical halo. 

Forthcoming Gaia data releases, offering improved statistics and reduced systematic uncertainties in stellar kinematics, will enable more stringent and spatially extended tests of the geometry of the Galaxy’s dark matter component, potentially allowing one to constrain its vertical and radial scale lengths with unprecedented precision. 

Cold Dark Matter Still Doesn't Work

The missing local baryon problem

Stacy McGaugh at Triton Station explores one of the many bits of empirical evidence, which he calls the missing local baryon problem, that really convincingly disfavors any kind of cold dark matter paradigm.

Basically, he utilizes a proof by contradiction. 

He assumes a standard cold dark matter model, analyzes the data on the share of the mass of galaxies and galaxy clusters that is made up of ordinary baryonic matter (which is about 15.7% in the cold dark matter paradigm), in line for the percentage for the whole universe in that paradigm. Then, he shows how the proportion of baryonic matter gets systemically lower in a very predictable manner as the absolute amount of baryonic matter in a galaxy falls.

The problem is that in the cold dark matter paradigm, galaxy clusters form as galaxies cluster together, and larger galaxies form from the merger of smaller galaxies. But this leaves open the question of how the proportion of baryonic matter in a pair of merged galaxies that form a larger galaxy can be systemically and precisely greater in a merged larger galaxy than it was in any of the smaller galaxies whose merger formed it.

Keep in mind that Standard Model physics demonstrates that in all but ultra-extreme circumstances (which haven't existed since the first few seconds after the Big Bang, at most) the total number of baryons in any system (less the total number of anti-baryons in any system) is constant (which has been experimentally confirmed to extreme precision), and that baryons profoundly outnumber anti-baryons in the universe (on the order of 10^10 to one), so there is no plausible physical mechanism by which new baryons are being created in galaxy mergers.

Indeed, even the proportion of the baryonic mass of the universe of each kind of atomic element, something that can only occur in nuclear fission and nuclear fusion reactions that happen mostly in mature stars, has changes only incrementally from the proportions of those atoms predicted to have been present fifteen minutes after the Big Bang, and even then, in amounts and by mechanisms mostly associated with the nuclear physics of stars, that are reasonably well understood. This strongly reinforces the idea that the new baryons aren't being created in galaxy mergers.

So, the shortfall of baryons in a dark matter particle paradigm, that is present in every system smaller than a galaxy cluster, would have to come from the intergalactic medium (IGM) of cold interstellar gas between galaxies and the circumgalactic medium (CGM) of cold interstellar gas in the dark matter halos of galaxies.

Fig. 1 of McGaugh et al. (2026): Conceptual elements of a galaxy: the stars (yellow/blue) and atomic gas (green) of NGC 6946 (Spitzer 3.6µ and 21 cm data: F. Walter et al. 2008) are shown embedded in an extended dark matter halo (black). The dark matter density decreases continuously with radius so the halo has no hard edge, but for convenience we adopt the common convention that the radius r200 marks the boundary of the dark matter halo and the dividing line between the circumgalactic medium (CGM) and the intergalactic medium (IGM; orange). The stars and atomic gas illustrated here appear within r < 20 kpc while r(200) ≈ 220 kpc (not shown to scale).

One kpc (i.e. kiloparsec) equals 32,600 light years.

But while this is the only possible solution to the local missing baryon problem in essentially all galaxies (but especially the smaller ones) in the dark matter particle paradigm, there is basically no way to make this work.

Therefore, cold dark matter models are inconsistent with what we observe.

CDM predicts excessive dwarf galaxy masses

Another example demonstrates that in the Local Group that includes the Andromeda galaxy and the Milky Way, one of its minor galaxies should have more mass than its two biggest galaxies and even more mass than the Local Group as a whole, which is contrary to the kinetic dynamics of the system as a whole and contrary to the conservation of matter. As McGaugh explains:

One signature of this misfit is the occurrence of very large V(200) for dwarf galaxies with small V(f). Taken literally, this would mean that some of the smallest dwarf galaxies reside in dark matter halos that outweigh those of giants like the Milky Way. This seems absurd, and it is. For example, by this approach, the dwarf galaxy NGC 3109 residing just outside the Local Group outweighs the Local Group and both its giants, Andromeda and the Milky Way, put together. But it is pretty clear from the local velocity field that the entire Local Group is not orbiting this little dwarf.

Real galaxies rarely have NFW halo distributions 

In dark matter particle paradigms, inferred dark matter halos have a "pseudo-isothermal" distribution, while collisionless cold dark matter must theoretically have, as an inexorable consequence of a very simple statistical mechanics style calculation that applies to dark matter particle with these very simple properties, what is called an NFW distribution, which is a very poor fit to the vast majority of galaxies.

Figure 2 from McGaugh et al. (2026): The observed flat velocity V(f) as it relates to the fitted V(200) for pseudo-isothermal (left panel) and NFW (right panel) halos (Li et al. 2020). Filled points have formal uncertainties < 20% in V(200); open points are less accurate fits. The solid line shows V(f) = V(200). The gray line in the right panel shows Equation (2a) of Katz et al. (2019), which corresponds roughly to f(v) ≈ 1.4.

V(f) is the rotational velocity of a galaxy at about a 65,000 light year radius, V(200) is the velocity of a galaxy at about 715,000 light year radius, and f(v) is equal to V(200)/V(f). 

The bottom line is that pseudo-isothermal dark matter halo distributions are a decent fit to what is observed with f(v) approximately equal to 1 and little scatter in the data (and scatter mostly associated with data points that have high uncertainties), while an NFW dark matter halo distribution has f(v) approximately equal to 1.4 with a great deal of scatter in the data.

This is a problem for the dark matter paradigm because coming up with a dark matter candidate with properties the naturally form pseudo-isothermal halos (for a candidate that isn't excluded by other data) is a challenging enterprise. Indeed, pseudo-isothermal dark matter halo density distributions aren't even theoretically stable.

CDM predicts the wrong slope for the Tully-Fischer scaling law

In a cold dark matter paradigm, the baryonic Tully-Fischer relationship (which roughly speaking related galaxy size to the speed of its flat rotation) has a slope of four when the observed relationship has a slope of three. 

When your power law exponent is a power of three rather than a predicted power of four, you have a seriously flawed functional form for your model.

Gravity based solutions compared

Toy-model MOND has challenges of its own (especially in galaxy clusters, although the intra-cluster medium of cold interstellar gas that was recently estimated makes the discrepancy smaller), but it is much more descriptive of the data, and predictive, than the cold dark matter paradigm. It even fits clusters reasonably well also with a tweak to just one of its parameters, rather than to the model as a whole. 

Deur chalks up the different gravitational behavior of galaxy clusters and galaxies to the different geometries of the mass distributions involved.

The Limits Of Post-Newtonian Approximations

Recognizing that non-perturbative GR effects can be important in circumstances conventionally considered to be non-relativistic, is a big step forward. Deur hit on this a long time ago, and this paper, independently, and indeed without citing to Deur, reaches the same conclusion. This could provide an answer to key unsolved problems in astrophysics, especially dark matter and dark energy phenomena.

Post-Newtonian theory is considered a reliable effective expansion of General Relativity in the weak-field and slow-motion limit. We argue that such a belief is misplaced. 

In generic many-body relativistic dynamics, the absence of globally conserved charges in the region of interest and non-integrability can drive strong sensitivity to angular-momentum exchange across inhomogeneous curvature, invalidating naive power counting in an effective theory expansion. 

Building on general lessons from effective field theory, we derive an explicit breakdown criterion that delineates when post-Newtonian truncations become unreliable despite small local potentials and velocities. This supplies a controlled systematic for weak-field mass inference, relevant to the dark matter puzzle in astrophysics and cosmology.

Marco Galoppo, Giorgio Torrieri, "When Weak Fields Arent Weak: Post-Newtonian effective theory and the Dark Matter Puzzle"  arXiv:2605.13557 (May 13, 2026) (Honorable mention, Gravity Research Foundation essay competition 2026).

Some Astrophysics Quick Hits

Many good theories start from the foundation of patterns in the empirical data.

The standard theory of galaxy formation predicts that all galaxies should contain dark matter, yet a handful of recently discovered galaxies appear to lack it, challenging our understanding of galaxy formation. We investigate whether such dark-matter deficient objects can be identified from their baryonic properties alone, analogously to the radial-acceleration relation, which tightly links baryon and dark matter distributions in spiral galaxies. 

Using a sample of ultra-diffuse and dwarf spheroidal galaxies -- systems whose baryonic properties resemble those of the confirmed dark-matter-deficient galaxies -- we systematically search for a formula to predict baryonic fractions from stellar mass, effective radius, distance to the host, and the host's baryonic mass. We find that baryonic fraction correlates most strongly with the gravitational acceleration expected from baryons alone, a(bar), or equivalently, with mean surface brightness, following an approximately a(bar)^−1 dependence. This scaling resembles the radial-acceleration relation but differs in functional form and applies to a different galaxy population. 

Strikingly, the dark-matter-deficient galaxies occupy the extreme end of the correlation. This suggests that they result from standard formation processes operating at unusual intensities rather than from exotic mechanisms. Importantly, the correlation predicts that all ultra-diffuse galaxies brighter than approximately 25 mag arcsec^−2 in the g-band should have very low dark matter content, offering a straightforward observational criterion for identifying these rare objects.

Michal Bílek, "A correlation predicting galaxies without dark matter" arXiv:2605.11070 (May 11, 2026).

Footnote: "cosmic noon" (i.e. the middle of the age of the universe measured in years) is sometimes operationally defined as "0.5 < z < 3".

Deur's approach to gravity provides a mechanism that explains the seeming observational preference for planar structures over spherical ones at galactic and larger scales.

An update of the evidence that radio galaxies and clusters of galaxies are more common than average near the plane of the de Vaucouleurs Local Supercluster shows that in the distance range 100 to 200Mpc objects whose positions are correlated with the plane of the Local Supercluster include galaxies that are exceptionally luminous at two microns, radio galaxies, and clusters of galaxies. There can be little doubt about this property of cosmic structure. I also argue for detection of this correlation for the galaxies at 400Mpc distance that are exceptionally luminous at two microns. 

It will be interesting to learn whether these results are expected in the standard cosmology.

P. J. E. Peebles, "The Extended Plane of the Local Supercluster" arXiv:2605.11184 (May 11, 2026).

We study how constraints on the abundance of ultralight axions (ULAs) from cosmic microwave background (CMB) data depend on their nonlinear modelling. We focus on the axion mass range 10^−25 ≤ m/eV ≤ 10^−23, where the axion Jeans scale falls in the quasi-linear regime probed by CMB lensing, making constraints highly sensitive to the choice of nonlinear prescription. 

We show that the inferred constraints depend significantly on the choice of nonlinear model, which must therefore be treated carefully. Performing Markov Chain Monte Carlo (MCMC) analyses with Planck 2018, ACT DR6 and DESI DR2 BAO data, we find naive nonlinear modelling of non-cold matter can produce an artificial preference for a subdominant ULA dark matter component with mass m ≈ 10^−24 eV. This arises from a lensing-like enhancement of the CMB power spectrum.

Lauren Gaughan, Anne M. Green, Adam Moss, "Ultra-light axion constraints from Planck and ACT: the role of nonlinear modelling" arXiv:2605.12054 (May 12, 2026).

Noting this paper for future reference. Here is the chart from it that is most interesting to me:

We present results from the Big Mysteries Survey, a large-scale survey conducted through the American Physical Society's Physics Magazine on foundational and controversial topics in contemporary physics. The survey provides a snapshot of physicists' views on issues in cosmology, black-hole physics, quantum mechanics, quantum gravity, and anthropic coincidences. A central finding is that several positions often described publicly as field-wide ``consensus'' views are, in practice, supported by much narrower majorities or by pluralities rather than majorities.

Niayesh Afshordi, Phil Halper, Matteo Rini, Michael Schirber, "Big Mysteries Survey: Physicists' Views on Cosmology, Black Holes, Quantum Mechanics, and Quantum Gravity" arXiv:2605.11058 (May 11, 2026).

Does a(0) Evolve Over Time?

The radical acceleration relation (RAR) which is implied by MOND but isn't necessary caused by MOND, holds true for all low-z observations (i.e. nearby galaxies). But this study concludes that while the RAR still holds in intermediate age galaxies (i.e. those that are farther away), that Milgrom's constant a(0) for these galaxies has a numerical value that is a factor of two greater than what it is for low-z galaxies.

The radial acceleration relation (RAR) is a tight empirical correlation between the observed radial acceleration (a_tot) and the baryonic radial acceleration (a_bar) measured across galaxy radii: these two accelerations start to deviate significantly from each other below a characteristic acceleration scale, a0. So far, observational studies of the RAR have predominantly focused on galaxies in the local Universe, leaving its evolution with cosmic time largely unexplored. 

Using high signal-to-noise data from the MUSE Hubble Ultra Deep Field survey, we investigate the RAR with a sample of 79 star-forming galaxies (complete above M* >10^8.8 Msun) at intermediate redshifts (0.33 < z <1.44). We estimate the observed intrinsic acceleration and the baryonic acceleration from a disk-halo decomposition that incorporates stellar, gas, and dark matter components, with corrections for pressure support, using 3D forward modelling. 

We find a RAR in our intermediate-z sample offset from the local relation, with a higher characteristic acceleration scale, a0(z~1) = 2.38+/-0.1* 10^-10 m/s^2, and a larger intrinsic scatter (~0.17 dex). Dividing the sample into redshift bins and refitting the RAR in each bin, we find a characteristic acceleration scale that systematically increases with z. Parametrizing the z-dependence as a0(z)= a0(0) + a1 * z, we obtain a1 = 1.59 +/- 0.1 * 10^-10 m/s^2, providing evidence for a z-evolution. 

We find similar results using various dark matter halo profiles as well as the Modified Newtonian Dynamics framework in our 3D forward modelling. Our results show that the RAR persists at intermediate redshift, with statistically significant redshift evolution of the characteristic acceleration, pointing to a possible evolution of the baryon-missing mass connection over cosmic time.

B. I. Ciocan, N. F. Bouché, J. Fensch, D. Krajnović, J. Freundlich, H. Desmond, B. Famaey, R. Techi, "MUSE-DARK III: The evolution of the radial acceleration relation at intermediate redshifts" arXiv:2604.22613 (April 24, 2026) (Accepted in A&A).

For reference z=0.33 is about 3.7 to 3.8 billion years ago, z=1 is about 7.7 to 8 billion years ago, and z=1.44 is about 9 to 10 billion years ago. The universe is about 13.8 billion years old. A variation of 0.17 dex is about ± 48%. The intrinsic scatter in the recent time SPARC galaxy sample is about ± 8% (0.034 dex), which is about is small as possible given the precision of the astronomy instrumentation involved. Milgrom's constant is about a(0) ≈ 1.2 × 10^−10 m/s^2.

Ciocan (2026), above, and the cluster data, both point to something very like MOND, except that a(0) evolves under certain circumstances to higher values. 

Missing baryonic matter (i.e. matter made up of ordinary atoms) is, at least, a partial explanation and one that could evolve other time. Indeed, it should evolve over time, because over time more baryonic matter ends up in stars, which are easy for astronomers to see, rather than interstellar gas and dust, which are hard for astronomers to see (and hence often called "missing" when it isn't seen and couldn't be seen even if it was there with current instrumentation). Still, missing baryonic matter may not be the entire explanation, because the magnitude of the change in a(0) may not be big enough, and changes in the naively measured value of Milgrom's constant shouldn't be very uniform since some galaxies are forming starts more actively than others (although this may be reflected in the greater dispersion of Milgrom's constant measurements in older samples).

Deur (who bibliography is linked in the sidebar) argues that the missing piece for cluster scale phenomena is the geometry of the mass distributions, by an appealing analogy to similar phenomena in QCD (which is attractive theoretically because in many respects gravity behaves like QCD squared). (QCD stands for quantum chromodynamics which is the Standard Model theory of the strong force that holds hadrons together and indirectly through hadron mediated forces accounts for the nuclear binding force that binds atomic nuclei together.)

Stacy McGaugh at Triton Station has another post about MOND v. dark matter particles (DM) and why the evidence favors something like MOND but the sociology of astrophysics favors dark matter particles.

The search for a final explanation of dark matter phenomena continues, and while toy-model MOND isn't the final solution, it does a remarkably good job over a very wide range of masses. McGaugh is surely right that the final solution looks a lot more like MOND than it does like most DM models, because for DM to describe the universe we see, we need a theory that explains how DM particles consistently form in a way entirely predicted by baryonic mass distribution, which contrary to protests that it has, it hasn't.

Even if a(0) changes over time, it provides a vastly smaller degree of freedom in how galaxy dynamics can vary than DM, especially if the variation is systemic between galaxies and galaxy clusters, or between galaxies over billions of years of time, and not just random.

The Hubble Tension Is Real

The Hubble constant is a measurement of the expansion of the universe, sometimes attributed to a cosmological constant in General Relativity (and the source of more than two-thirds of the mass-energy of the universe in conventional cosmology). Except, it appears that the Hubble constant isn't quite constant. So the explanation must be more complicated than a simple cosmological constant.

The Hubble tension isn't huge in relative terms, 10% over measurements more than ten billion years removed from each other. 

But it is highly statistically significant at the five sigma plus level, and isn't a simple methodological artifact of late time Hubble constant measurements (although it could be a methodological artifact of model dependent cosmic microwave background radiation measurements).

Context. The direct empirical determination of the local value of the Hubble constant (H(0)) has markedly advanced thanks to improved instrumentation, measurement techniques, and distance estimators. However, combining determinations from different estimators is nontrivial due to their correlated calibrations and different analysis methodologies.

Aims. Using covariance weighting and leveraging community expertise, we have constructed a rigorous and transparent “Distance Network” to find a consensus value and uncertainty for the locally measured Hubble constant.

Methods. Experts across all relevant distance measurement domains were invited to critically review the available datasets spanning parallaxes, detached eclipsing binaries, masers, Cepheids, the tip of the red giant branch, Miras, carbon-rich asymptotic giant branch stars, Type Ia (SNe Ia) and Type II supernovae, surface brightness fluctuations, the fundamental plane, and Tully–Fisher relations. Before any calculations, the group voted for first-rank indicators to define a “baseline” Distance Network. Other indicators were included to assess the robustness and sensitivity of the results. We provide open-source software and data products to support full transparency and future extensions of this effort.

Results. Our key findings are as follows: (1) The local H(0) is robustly determined, with first-rank indicators internally consistent within their uncertainties. (2) A covariance-weighted combination yields a relative uncertainty of 1.1% (baseline) or 0.9% (all estimators). (3) The contribution from SNe Ia is consistent across compilations of optical or NIR magnitudes. (4) Removing either Cepheids or the tip of the red giant branch has a minimal effect on the central value of H0. (5) Replacing SNe Ia with galaxy-based indicators changes H(0) by less than 0.1 km s^−1 Mpc^−1 while doubling its uncertainty. (6) The baseline result is H(0) = 73.50 ± 0.81 km s^−1 Mpc^−1, 7.1σ from the early Universe plus ΛCDM result 67.24 ± 0.35 km s^−1 Mpc^−1 and 5.0σ from BBN+BAO within a flat ΛCDM DESI DR2 (68.51 ± 0.58 km s^−1 Mpc^−1).

Conclusions. A networked approach, such as the one presented here, is invaluable for enabling further progress in Hubble constant measurements, as it provides the much needed advances in accuracy and precision without overreliance on any single method, sample, or group.

H0DN Collaboration, "The Local Distance Network: A community consensus report on the measurement of the Hubble constant at ∼1% precision" 708 A&A, A166 (April 10, 2026).

Worth noting that in Deur's approach, there is no cosmological constant, and that the apparent cosmological constant varies over time, and is expected to increase as galaxy and cluster structure increase somewhat over time. And, in Deur's approach, galaxy formation comes earlier than in ΛCDM.

A Decent Modified Gravity Candidate

This modified gravity proposal explains galactic rotation curves without dark matter, it's relativistic, and its key parameter beyond general relativity is determined on a very consistent basis from data from nine different galaxies. It bears some general similarities to other modified gravity proposals that do the same thing. 

The author's conjecture that the reason we don't have a workable quantum gravity theory is that the standard equations of general relativity that we're trying to quantize aren't quite right also seems plausible.

This candidate isn't as mature as some of the competing modified gravity proposals, so it hasn't be tested against the cosmic microwave background, galaxy formation rates, in non-spiral galaxies, and in galaxy clusters yet. But its a promising proposal that deserves further attention.

A modification of the Einstein-Hilbert Lagrangian by introducing a coupling between the Weyl tensor and the stress-energy tensor was proposed to explain flat galactic rotation curves without the exotic (non-baryonic) dark matter (DM). The proposed coupling constant was previously determined by fitting the rotational velocities of the Milky Way and M31 modeled with constant density, yielding the same coupling constant for both. In this work, we have modified the formalism for a variable density by modeling the galactic systems with realistic, spherically symmetric and radially varying density profiles for the baryonic matter and this analysis is applied to seven edge-on spiral galaxies of the local cluster and the Milky Way.

Asghar Qadir, Ashmal Shahid, Noraiz Tahir, "The Galactic Halo Rotation by Weyl Incorporated Gravity" arXiv:2604.01643 (April 2, 2026) (Arabian Journal of Mathematics (2026)).

The introduction to the paper is also encouraging, although some of the summary of the criticisms of MOND are overstated. The explicit treatment of the effect of the gravitational field, similar to the approach of Deur, is particularly notable. It says:

One of the most striking observations in galactic dynamics is the discrepancy between the predicted and observed rotational velocities of galaxies. According to the standard theories of gravity, the rotational velocity of the galaxies should decrease sharply at large radii where visible matter becomes sparse. However, observations of their rotation curves remain nearly flat out to very large distances [11–13]. Other dynamic considerations had already led Zwicky [14] to propose the existence of DM, but this evidence was much stronger. Rubin’s investigation was extended to galactic clusters [15, 16] providing yet stronger evidence. The observations of the cosmic microwave background (CMB) had already provided minimum and maximum values for baryonic matter in the Universe according to the standard model of particle physics (SMpp). The observations required a value well beyond the limit of the baryonic matter [17]. This has led to various suggestions for exotic (non-baryonic) DM, but there is no direct evidence for any of the proposed candidates. Nevertheless, CMB observations also indicate that ≃ 5% of the Universe should be made up of baryons (the usual protons and neutrons), but observations of the luminous parts of the galaxies show only half of these baryons, this is the “missing baryon problem”. Since the baryons are dark, we call this the baryonic DM. It is proposed that a significant fraction of this baryonic DM is present in the galactic halos [18–23]. 

For the non-baryonic DM an alternative suggestion was that the standard law of gravity should be modified instead of looking for other forms of matter. The first such suggestion came from Milgrom [24], who proposed the modification of Newton’s law by inserting a Yukawa-like term to damp gravity at large distances, Modified Newtonian Dynamics (MOND). It was not able to explain the dynamics at different scales, especially of single galaxies and clusters, or provide for the formation of structure in the early Universe [25–31]. Most of all, the damping term was totally ad-hoc and was embedded in an obsolete Newtonian framework, which could not be converted to General Relativity (GR) [29]. Apart from galactic dynamics, arguably the most outstanding problem of fundamental physics is the incompatibility of GR and Quantum Theory. In particular, the Renormalization Group Equation of ’t Hooft and Veltman demonstrated that Dirac quantization of GR produced a non-renormalizable theory [32], leading to the well-known “Quantum Gravity (QG)” problem. To avoid these separate problems various ad-hoc modifications of GR involving arbitrarily many new parameters have been proposed [24, 33–36]. 

Qadir and Lee took the view that there must be a sound physical basis for any modification of the highly successful GR, and it must be minimal, i.e., it should remain geometric and involve only one free parameter to explain the discrepancies of galactic dynamics at all scales. Further, it should also provide a base for solving the QG problem. In 2019, they proposed an explicit interaction term between matter and the gravitational field λT.C.T, where λ is a new coupling constant, T is the stress-energy tensor, and C is the Weyl tensor, which represents a pure gravitational field [1]. This idea was inspired by the Feynman vertex representing a similar explicit interaction between the source (an electron) and the electrodynamic field in Quantum Electrodynamics (QED), Aµjµ. In QED the electron is given by a spin-half spinor, which comes twice over in the current jµ and the field, Aµ, which appears singly, while in QG the source would be represented by the rank-two tensor T, which comes twice, and the gravitational field by the rank-four tensor C, which comes once. As QED with this source term is renormalizable, it can be hoped that so would this modified QG. It was called Modified Relativistic Dynamics (MORD). 

Previously, MORD was tested by checking whether a single value of the new coupling constant λ could reproduce the rotational velocity at the outer rim of two galaxies, the Milky Way and M31, incorporating only the baryonic DM and not any postulated non-baryonic DM, by assuming a simple-minded model in which both the galaxies were represented as a constant density sphere with a peak density of the baryonic matter from the core to the edge of the galaxy [2, 3]. In that study, a single value of λ was indeed found to fit the rotational velocity values for both galaxies. This approach was inherently limited, i.e., it neglected the radial variation of the galactic halo density, and treated the galaxies as idealized, uniform objects. The aim of this paper is to completely modify the previous formalism of a constant density case to a variable density case, where ρ′= 0, and take the next step forward by generalizing the baryonic DM component to spherically symmetric, radially varying density profiles for the galactic halos of eight spiral galaxies [4–10, 37, 38]. 

This extension is conceptually significant because it will allow us later to test whether the universality of λ persists under physically motivated halo structures at different radii, rather than only at the rim. By moving from a toy model to a realistic halo description, we not only refine the numerical estimate of λ but also provide a more robust and physically meaningful assessment of MORD across multiple spiral galaxies. We stress that while more realistic baryonic distributions, such as double exponential stellar disks combined with bulge components, are commonly used to model luminous matter, these structures are intrinsically non-spherical, and would require extension of the formalism to two or more variables. 

The plan of the paper is as follows: in Section 2, we will briefly explain the Weyl modified Einstein field equations for varying spherically symmetric density profiles and demonstrate how the value of the coupling constant λ is obtained for the Milky Way galactic halo. In Section 3, we will use the analysis for seven other spiral galaxies. Finally, in Section 4, the obtained results will be discussed.

The key formulas are as follows:

The Weyl-modified Einstein-Hilbert Lagrangian is [1] 

L =√−g (R − 2Λ − kT + λC(αµβν)T^(αβ)T^(µν)), (1) 

where √−g is the determinant of the metric, k = 8πG/c^4 is the coupling constant for matter, where G is the Newton’s gravitational constant, c is the speed of light. This leads to the Weyl incorporated Einstein field equation (WIFE) 

R(µν) − 1/2g(µν)R + g(µν)Λ =  kT(µν) + λI(µν), (2) 

[Ed. For comparison the unmodified Einstein field equation is as follows:

 

So, the only modification is the addition of the λI(µν) term on the RHS.]

where I(µν) is the interaction term given by 

I(µν)= 

1/4(−g(αβ)g(ρµ)g(σν)−g(ρσ)g(αµ)g(βν)−g(ασ)g(ρµ)g(βν)−g(ρβ)g(αµ)g(σν))□(T^(αβ)T^(ρσ))

+1/6 (g(αβ)g(ρσ)−g(ρβ)g(ασ))(g(µν)□ −∇(µ)∇(ν)) T(αβ)T(ρσ). (3) 

The paper sums up its findings in the conclusion:

The very first modified gravity approach to explain the flat rotational curves of galaxies without invoking DM is MOND [24]. However, MOND’s phenomenological success comes with challenges in covariant formulation and cluster-scale dynamics, motivating alternative modifications rooted in GR. Furthermore, a major challenge to MOND has emerged from the analysis of wide binary stars in Gaia DR3. A comprehensive study by Ref. [59] found that the relative velocities of widely separated binaries (2−30 kAU) are inconsistent with the MOND prediction, which expects a ≈ 20% enhancement over Newtonian gravity due to the external field effect. Their analysis, which rigorously modeled the Galactic external field and population uncertainties, excluded MOND at a statistical significance of 16σ in favour of Newtonian dynamics [59, 60]. These challenges motivate the exploration of alternative single-parameter modified gravity theories rooted in a covariant framework. 

Qadir and Lee’s MORD [1] proposed a modification to the standard Lagrangian by incorporating an interaction coupling constant λ with a term involving the Weyl tensor and the stress-energy tensor, expressed as λC(µνρπ)T^(µν)T^(ρπ) whose purpose was to see whether a single unique value of λ can account for the rotational velocity curves of galaxies by replacing the exotic DM by what we now feel should be called Weyl Incorporated Gravity (WIG), solving the outstanding DM problem with a single new parameter. This was tested using a simplistic constant density model which did have just the one value of the coupling for all galaxies considered [2, 3]. 

In the present work, we have modeled the galactic halos of eight spiral galaxies, modifying the previous framework for a variable density case to estimate the value of the coupling constant λ. For this purpose we adopt three widely used density profiles, the Navarrow-Frenk-White (NFW), Moore, and Burkert models, normally used for all DM in the halos, but here used only for the baryonic DM to test the robustness of the proposal by verifying that the choice of model makes no difference to the results [40-50]. 

We find that a tiny range, λ = (6.9546 ± 0.00012) × 10^−18 km^2s^4kg^2, consistently reproduces the observed halo rotational velocities at r = 100 kpc. 

For the Milky Way, the fitted rotational velocities span v(rot) ≃ 153–159 km s^−1, compared to the observed value 150 ± 10 km s^−1, with an enclosed halo mass M(h)(≤ 100 kpc) ≃ 1.0 × 10^12 M⊙. 

For M31, we obtain v(rot) ≃ 230–232 km s^−1 versus the observed 225 ± 10 km s^−1, corresponding to a halo mass M(h) ≃ 1.4 × 10^12 M⊙. 

In the case of M33, the modeled velocities v(rot) ≃ 121–122 km s^−1 agree with the observed 120 ± 5 km s^−1, yielding M(h) ≃ 3.2 × 10^11 M⊙. 

For M81 and M82, the fitted velocities lie in the ranges 256-257 km s^−1 and 251–253 km s^−1, respectively, consistent with the observed values 250 ± 15km s^−1 and 250 ± 18 km s^−1, with inferred halo masses M(h) ≃ 1.3 × 10^12 M⊙ and 1.0 × 10^11 M⊙. 

Similarly, for NGC 5128, NGC 4594, and M90, the modeled rotational velocities at 100 kpc fall within the observed ranges reported in the literature, with corresponding halo masses M(h) ≃ 4.4 × 10^12 M⊙, 6.3 × 10^13 M⊙, and 3.5 × 10^13 M⊙, respectively. It is clearly seen that these mass and velocity estimates are consistent with the observed values (see Refs. [40, 44], and Table 1). 

Before closing the paper, note that we have used spherical symmetry to explain the dynamics of the galactic halo to estimate the value of λ. However, to make a more realistic model of the galaxy, we should take into account the vertical component of the velocity, which is missing in the present geometry. The hope is that one can fit the complete rotational velocity curve and get a more robust model. Indeed, the chosen metric would change, and we may need to consider the Kerr geometry, or a slow rotation approximation of it [61], to incorporate the vertical component, as it accounts for the angular momentum effects [62, 63]. This will be addressed separately later. 

As previously discussed, the problem of DM and QG may share a common origin [1]. Addressing observational issues related to DM could provide insights into resolving fundamental difficulties in QG. Instead of assuming an indirect interaction, we have introduced a direct nonlinear coupling between matter and gravity, analogous to the interaction between electromagnetic sources and the electromagnetic field. The modified Lagrangian, given in eq. (1), represents the minimal extension of the Einstein-Hilbert Lagrangian and is proposed as a potential solution to both problems with a single additional parameter. Naturally, the feasibility of this approach must first be tested against the DM problem before seeing if our WIG fits on the messier head of QG. 

As a further conjecture of my own, their coupling constant could be parsed out suggestively into λ = X x Λ^2/G^2 where Λ is the cosmological constant, G is Newton's constant, and X is a dimensionless coupling constant with a value on the order of 10^48. 

You could also have λ = X x Λ/G^2 with a much small constant (on the order of 10^-4) that has dimensions of m^2.