A Notable Modified Gravity Theory

Mach's principle is basically that inertia is a product of the combined gravitational pulls of everything in the Universe (although it can be expressed in about a dozen different ways, not all of which are perfectly consistent with each other or observation).  This, in turn, implies that (from the link and also from paper cited below): "Inertial mass is affected by the global distribution of matter."

A new paper tries to derive a relativistic version of a modified gravity theory similar to MOND by incorporating Mach's principle into the theory. This is something that Einstein tried to do until it became clear that this was inconsistent with plain vanilla General Relativity.

The general theory of relativity (GR) was proposed with an aim of incorporating Mach's principle mathematically. Despite early hopes, it became evident that GR did not follow Mach's principle. Over time, multiple researchers attempted to develop gravity theories aligned with Machian idea. Although these theories successfully explained various aspects of Mach's principle, each of these theories possessed its own strengths and weaknesses. 

In this paper, we discuss some of these theories and then try to combine these theories into a single framework that can fully embrace Mach's principle. This new theory, termed Machian Gravity (MG) is a metric-based theory, and can be derived from the action principle, ensuring compliance with all conservation laws. The theory converges to GR at solar system scales, but at larger scales, it diverges from GR and aligns with various modified gravity models proposed to explain dark sectors of the Universe. 

We have tested our theory against multiple observational data. It explains the galactic rotation curve without requiring additional dark matter (DM). The theory also resolves the discrepancy between dynamic mass and photometric mass in galaxy clusters without resorting to DM, but it introduces two additional parameters. It can also explain the expansion history of the Universe without requiring dark components.

Santanu Das (from the Raman Research Institute, Bangalore, India), "Machian Gravity: A mathematical formulation for Mach's Principle" arXiv:2308.04503 (last revised September 7, 2025) (53 pages). 

The theory's success with rotation curves is explored in this preprint (August 31, 2023). The idea was originally proposed in a set of three preprints from the same author in 2012: "Machian gravity and a cosmology without dark matter and dark energy" (May 17, 2012); "Mach's principle and the origin of the quantum phenomenon" (June 4, 2012); Mach Principle and a new theory of gravitation (June 26, 2012).

The introduction to the paper explains that:

Newtonian gravity can provide a very accurate description of gravity, provided the gravitational field is weak, not time-varying and the concerned velocities are much less than the speed of light. It can accurately describe the motions of planets and satellites in the solar system. Einstein formulated GR to provide a complete geometric approach to gravity. GR is designed to follow Newtonian gravity at a large scale. It can explain the perihelion precession of Mercury’s orbit and the bending of light by the Sun, which were never realized before, using Newtonian mechanics. Over the years, numerous predictions of GR, such as the existence of black holes, gravitational waves, etc. have been observed. This makes GR one of the most well-accepted theories of gravity. 

However, the drawbacks of GR come to light when GR is applied on the galactic and cosmological scale. It fails to produce the galactic velocity profiles, provided that calculations are made just considering the visible matter in the galaxy. This led researchers to postulate a new form of weakly interacting matter named dark matter. Earlier it was commonly believed that dark matter (DM) is made up of particles predicted from supersymmetry theory. However, the lack of evidence of these particles from Large Hadron Collider (LHC) strengthens the proposition of other candidates, such as Axions, ultra-light scalar field dark matter, etc.

A further mysterious puzzle is the dark energy (DE) because that requires to produce a repulsive gravitation force. Cosmological constant or Λ-term provides an excellent solution for this. However, as the observations become more precise, multiple inconsistencies come to light. 

There can be two ways to solve the dark sector of the Universe. 

Firstly, we can assume that there is in need some type of matter that does not interact with standard-model particles and acts as dark matter, and we have some form of energy with a negative pressure and provide a dark-energy-like behavior. 

While this can, in need, be the case, the possibility that the GR fails to explain the true nature of gravity in kilo-parsec scale can also not be overlooked. In such a case, we need an alternate theory of gravity that can replicate GR on a relatively smaller scale while deviating from it on a galactic scale. 

Several theories have been proposed in the last decade to explain DM and DE. Empirical theories like Modified Newtonian Dynamics (MOND) can explain the galactic velocity profiles extremely well but violates momentum conservation principles. Therefore, if a mathematically sound theory is developed that can mimic the MOND empirically, then that can explain the dark matter. Bekenstein proposed AQUAdraticLagrangian (AQUAL) to provide a physical ground to MOND. Other theories, such as Modified gravity, Scalar-Tensor-VectorGravity (STVG),Tensor–Vector–Scalar gravity (TeVeS), Massive gravity etc. are also proposed to match the galactic velocity profiles without dark matter. Other higher dimensional theories such as induced matter theory etc. are also proposed by researchers. However, all these theories came from the natural desire to explain the observational data and not build on a solid logical footing. 

Now, let us shift our focus to another aspect of GR. In the early 20th century, Earnest Mach hypothesized that the inertial properties of matter must depend on the distant matters of the Universe. Einstein was intrigued by Mach’s Principle and tried to provide a mathematical construct of it through the GR. He later realized that his field equations imply that a test particle in an otherwise empty Universe has inertial properties, which contradicts Mach’s argument. However, intrigued by the overwhelming success of GR in explaining different observational data, he did not make any further attempt to explain Mach’s principle. 

In view of this, it is worthwhile searching for a theory that implies that matter has inertia only in the presence of other matter. Several theories that abide by Mach’s principle have been postulated in the last century. Among these, the most prominent are Sciama’s vector potential theory, Brans Dicke (BD) theory or the scalar-tensor theory of gravity and Hoyle Narlikar theory etc. Although each of these theories addresses certain aspects of Mach’s principle as discussed in the respective articles, none offers a complete explanation. Thus, only a unified theory that combines these approaches could provide a comprehensive understanding of Mach’s principle in its entirety. 

In this article, we address all the issues described above and propose a theory of gravity based on Mach’s principle. It is based on the following premises. 

• Action principle: The theory should be derived from an action principle to guarantee that the theory does not violate conservation laws. 

• Equivalence principle : Various research groups have tested the Weak Equivalence Principle (WEP) at an exquisite procession. Therefore, any theory must follow the weak equivalence principle. However, the strong equivalence principle has not been tested on a large scale. If the ratio of the inertial mass and the gravitational mass changes over space-time (on a galactic scale or cosmological scale), then that does not violate results from our local measurements. In accordance with Mach’s principle, the inertial properties of matter come from all the distant matter of the Universe. As the matter distribution at different parts of the Universe is different, the theory may not follow the strong equivalence principle. 

• Departure from GR: As GR provides an excellent result in the solar system scale, the proposed theory should follow GR on that scale, and it only deviates from GR at the galactic scale to mimic some of the modified gravity theories proposed by researchers to explain the dark sectors of the Universe. Along with this, the proposed theory should also be able to replicate the behavior of theories like Sciama’s theory or BD theory under the specific circumstances for which they were proposed. 

The paper is organized as follows. 

In the second section I briefly discuss previous developments in gravity theory to explain Mach’s principle. Most of the points covered in this section are generally known positions form various previous research. However, since Mach’s principle is not a mainstream area of study, this discussion is necessary and important for understanding the new insights presented in this article. In some cases I interpret these established ideas from the perspective of this paper, that will help me to built the gravity theory in the later section. 

In the next section, we explain Mach’s principle and discuss the mathematical tools used to formulate the theory. We present the source-free field equations for the theory in the same section. 

The static spherically symmetric solution for the theory in weak field approximation is presented in the fourth section. We show that the solution follows Newtonian gravity and GR at a smaller scale but deviates from it at a large scale. 

Section five presents examples of galactic rotation curves and galaxy cluster mass distributions, demonstrating that the theory yields results in close agreement with observations. 

The source term of the theory has been described in the 6th section. 

In the next section, we provide the cosmological solutions to the MG model. 

The final section is the conclusion and discussion section. We have also added five appendices where we describe the nitty-gritty of the calculations and add multiple illustrations.

This theory is formulated in a 4+1 dimensional space-time. It states that "The momentum in the fifth dimension represents the inertial mass of the particle, which remains constant in any local region." The paper could be more clear regarding which two additional parameters are added to the theory, but appear to be related to a vector field and a scalar field, respectively:

The proposed framework is built upon a five-dimensional metric involving three essential elements: a scalar field ϕ, a vector field Aµ, and an extra dimension x4. (note that these two fields behave as scalar and vector field only if the metric is independent of x4.)

Side commentary on dark matter particle theories 

There are a wealth of papers constraining various versions of dark matter particle theories out there, and I bookmark almost all of them as I encounter them. But synthesizing them into a comprehensive set of constraints on dark matter, in the way that the Particle Data Group does for high energy physics data, is a daunting task.

The bottom line for those papers is that the parameter space of possible dark matter particle candidates is ever more tightly constrained within the context of many dark matter particle proposals, and that there are no positive confirmations of any of them.

A Wide Binary Paper Supporting Non-Newtonian Gravity

Efforts to determine is wide binary star systems show a change in gravitational acceleration at Newtonian accelerations below Milgrom's constant, as predicted by MOND (but not generically by all modified gravity explanations of dark matter phenomena) have had mixed and contradictory results. The latest paper on the subject, a small pilot study using new methods supports the existence of a non-Newtonian gravitational enhancement for wide binaries whose gravitational pull on each other is below Milgrom's constant.

Wide binary tests exclude almost all dark matter particle theories, if they show non-Newtonian gravitational enhancements in weak fields, and also discriminate meaningfully between different gravity based approaches to explain dark matter phenomena if the data allows for sufficiently precise conclusions. 

But competing considerations of data quality (e.g., it is easy to mistake a system with more than two stars for a wide binary system if one of the stars is feint or the angle of observation is poor), and data quantity (to give the observations statistical power), make this astronomy test of weak field gravity challenging to extract convincing results from. 

When 3D relative displacement r and velocity v between the pair in a gravitationally-bound system are precisely measured, the six measured quantities at one phase can allow elliptical orbit solutions at a given gravitational parameter G. Due to degeneracies between orbital-geometric parameters and G, individual Bayesian inferences and their statistical consolidation are needed to infer G as recently suggested by a Bayesian 3D modeling algorithm. 

Here I present a fully general Bayesian algorithm suitable for wide binaries with two (almost) exact sky-projected relative positions (as in the Gaia data release 3) and the other four sufficiently precise quantities. Wide binaries meeting the requirements of the general algorithm to allow for its full potential are rare at present, largely because the measurement uncertainty of the line-of-sight (radial) separation is usually larger than the true separation. 

As a pilot study, the algorithm is applied to 32 Gaia binaries for which precise HARPS radial velocities are available. The value of Γ ≡ log(10)√ G/G(N) (where G(N) is Newton's constant) is −0.002 + 0.012 −0.018 supporting Newton for a combination of 24 binaries with Newtonian acceleration g(N) > 10^−9 ms^−2, while it is Γ = 0.063 + 0.058 − 0.047 or 0.134 + 0.050 − 0.040 for 7 or 8 binaries with g(N) < 10^−9 ms^−2 (depending on one system) showing tension with Newton. The Newtonian ``outlier'' is at the boundary set by the Newtonian escape velocity, but can be consistent with modified gravity. 

The pilot study demonstrates the potential of the algorithm in measuring gravity at low acceleration with future samples of wide binaries.

Kyu-Hyun Chae, "Bayesian Inference of Gravity through Realistic 3D Modeling of Wide Binary Orbits: General Algorithm and a Pilot Study with HARPS Radial Velocities" arXiv:2508.11996 (August 16, 2025).

More Constraints On Primordial Black Holes

A new study, that doesn't rely on the micro-lensing and Hawking radiation exclusions which are the primary methods for constraining primordial black hole frequency, places very strict limitations on the maximum potential abundance of "supermassive" primordial black holes. 

It limits them to less than 0.1% of dark matter in a dark matter particle hypothesis for "supermassive" primordial black holes (i.e. primordial black holes that are 10,000 times more massive than the Sun or more). This had already been ruled out long ago, albeit not quite so strictly. 

The main focus on primordial black holes as a dark matter candidate has been on asteroid sized primordial black holes in the range of 3.5 × 10^−17 to 4 × 10^−12 solar masses (i.e. twelve to seventeen orders of magnitude smaller in mass than the Sun), which by definition cannot arise from stellar collapse. Non-detection of Hawking radiation (which is a net emission for primordial black holes up to about 10^-8 solar masses), and micro-lensing, has largely ruled out larger primordial black holes as a significant component of dark matter (if it exists). 

Also, while the paper frames its constraints in terms of primordial black holes, it would seem to apply to any dark matter candidate in that mass range, such as ordinary black holes and MACHOs (massive compact halo objects).

We present updated constraints on the abundance of primordial black holes (PBHs) dark matter from the high-redshift Lyman-α forest data from MIKE/HIRES experiments. Our analysis leverages an effective field theory (EFT) description of the 1D flux power spectrum, allowing us to analytically predict the Lyman-α fluctuations on quasi-linear scales from first principles. Our EFT-based likelihood enables robust inference across redshifts z = 4.2−5.4 and down to scales of 100 kpc, within previously unexplored regions of parameter space for this dataset. 

We derive new bounds on the PBH fraction with respect to the total dark matter fPBH, excluding populations with fPBH≳10^−3 for masses MPBH ∼ 10^4−10^16 M⊙. This offers the leading constraint for PBHs heavier than 10^9 M⊙ and highlights the Lyman-α forest as a uniquely sensitive probe of new physics models that modify the structure formation history of our universe.

Mikhail M. Ivanov, Sokratis Trifinopoulos, "Effective Field Theory Constraints on Primordial Black Holes from the High-Redshift Lyman-α Forest" arXiv:2508.04767 (August 6, 2025).

Another recent study (from August 11, 2025) reaches the same conclusion.

Additional Context

The Ordinary Matter Budget Of The Universe

Most of the ordinary matter in the universe is found in stars (about half) and the intergalactic/interstellar medium (mostly interstellar gas and dust) which is also about half, with planets and asteroids accounting for less than 1% of the total amount of ordinary mass in the universe. 

Stellar-mass black holes (formed from dying stars) account for not more than about 0.1% of the universe's ordinary matter, and supermassive black holes, found at the centers of galaxies, account for not more than about 0.01% of the universe's ordinary matter. 

Contributions to the mass-energy of the universe from photons and neutrinos are also very small (even though both kinds of particles are extremely numerous).

Planets, Asteroids, and Comets

Self-gravity forces planet-like objects of more than 0.5 x 10^21 kg (about one four billionth of the mass of the Sun) and more than 400 km in radius, to become approximately spherical, and this is the lower floor for dwarf planets, regular planets, and planet-sized moons. The mass of the Earth is about 3 x 10^-6 solar masses.

Objects smaller than this (but larger than dust or interstellar gas) tend to form non-spherical asteroids and comets, although some are approximately spherical due to random chance.

Star and Brown Dwarves

As an aside, anything other than a star or a black hole, can't have more than about 1.24% of the mass of the Sun (i.e. 13 Jupiter masses), because then gravity causes unstable nuclear fusion to commence in its core, turning it into a "sub-brown dwarf" although NASA conservatively assumes that planets could be as large as 30 Jupiter masses (about 2.86% of the mass of the Sun). In ideal conditions, a sub-brown dwarf can form at masses as low as one Jupiter mass (about 1/1024th of the mass of the Sun). Sub-brown dwarves and true brown dwarves, which range from 13 to 80 Jupiter masses (i.e. up to about 7.8% of the mass of the Sun) fill a liminal space between true planets with no gravity induced nuclear fusion and the smallest "main-sequence" stars. 

While brown dwarves are an order of magnitude or two heavier than large gas giant planets, like Jupiter and Saturn, they aren't much larger: "most brown dwarfs are slightly larger in volume than Jupiter (15–20%), but are still up to 80 times more massive due to greater density." Jupiter's radius is 11 times that of Earth, and the Sun's radius is 10 times that of Jupiter.

The theoretical maximum mass of a star is on the order of 200 solar masses. Of the billions and billions of stars that astronomers have observed, only 11 of them are potentially more than 150 solar masses, and only 5 of them have an upper end of their two sigma mass range (given the uncertainty of the mass measurement) above 200 solar masses. Only 2 stars have a best fit mass estimate above 200 solar masses, and realistically, given the uncertainty in these mass measurements (which is stated for one of the two and is not stated for the other), a mass of 200 solar masses of less is probably within the two sigma uncertainty range of the observation for both cases (particularly if one considers look elsewhere effects which are significant given the very large number of star masses measured).

The theoretically largest radius star is about 1700 times the radius of the Sun (by comparison, the orbit of Saturn is about 2,048 times the radius of the Sun). The largest radius star ever observed has a radius of 1530 ± 370 times the radius of the Sun.

Thus, any compact object with a mass of more than about 2 * 10^2 solar masses, or a radius more than about 1700 times the radius of the Sun (the Sun has a radius of about 700,000 km) is a supermassive black hole.  

Black Holes

An ordinary stellar collapse black hole has a minimum mass which is more than two times the mass of the Sun, but this minimum mass is a bit under three times the mass of the Sun. This mass, in the non-spinning case is called the Tolman-Oppenheimer-Volkoff limit.  In theory, this threshold mass may vary modestly based upon the spin of the neutron star. The mass limit is 18%-20% higher for a very rapidly spinning neutron star that is on the brink of becoming a black hole. A stellar mass black hole has an event horizon radius  (i.e. Schwarzschild radius radius) of about 6-9 km to 300 km.

The maximum density of anything ever observed in astronomy or high energy physics or nuclear physics is a neutron star/black hole right at the high end of the Tolman-Oppenheimer-Volkoff limit.

Pinning down the exact threshold more precisely is a matter of ongoing astronomy research. The least massive object definitively classified as a black hole has a mass of 3.04 ± 0.06 solar masses. A handful of observations of objects close to the limit have suggested a limit somewhere on the order of 2.01 to 2.9 solar masses.

Between stellar mass black holes (many of which have been indirectly observed) and supermassive black holes at the core of galaxies (many of which have been indirectly observed) are intermediate-mass black holes, which were first observed with gravitational wave telescopes:

An intermediate-mass black hole (IMBH) is a class of black hole with mass in the range of one hundred to one hundred thousand (10^2–10^5) solar masses: significantly higher than stellar black holes but lower than the hundred thousand to more than one billion (10^5–10^9) solar mass supermassive black holes.

An intermediate-mass black hole has an event horizon radius of 300 km to 300,000 km (which is smaller than the radius of the Sun). 

In theory, it would have been possible shortly after the Big Bang and predominantly in the first second after the Big Bang, for matter to be dense enough to form a black hole with less mass than necessary to form an ordinary stellar collapse black hole (even though the density needed to form a black hole increases as the mass which collapses into a black hole gets smaller).  These hypothetical black holes are called primordial black holes. 

But no primordial black holes have ever been observed, despite the fact that they are predicted to emit intense Hawking radiation (a.k.a. Bekenstein-Hawking radiation after Jacob Bekenstein, who died at age 68 in 2015, and Stephen Hawking, who died at age 76 in 2018, who both proposed it) which has never been detected:

Depending on the model, primordial black holes could have initial masses ranging from 10^−8 kg (the so-called Planck relics) to more than thousands of solar masses. However, primordial black holes originally having masses lower than 10^12 kg would not have survived to the present due to Hawking radiation, which causes complete evaporation in a time much shorter than the age of the Universe. . . . Primordial black holes are also good candidates for being the seeds of the supermassive black holes at the center of massive galaxies, as well as of intermediate-mass black holes.

The smaller the black hole, the more rapidly it evaporates due to Hawking radiation. A primordial black hole which initially had the mass of the Sun (2 * 10^30 kg) would now have a mass of something on the order of 10^23 kg (about one 10,000,000th the mass of the Sun) due to Hawking radiation (although accretion of new matter could counteract Hawking radiation and slow down the rate at which a primordial black hole's mass declines).

A hypothetical stable mass primordial black hole has an event horizon radius of at least 24 meters. Evaporating primordial black holes would have a smaller event horizon radius. An asteroid sized black hole would have an event horizon radius of about 0.03 millimeters to 3 meters and would emit significant Hawking radiation.

For black holes formed by stellar mass collapse (about 3 solar masses) or more, the mass loss due to Hawking radiation would be almost completely offset by accretion of mass-energy from its absorption of cosmic background radiation alone, setting aside interstellar dust and other objects that could fall into the black hole. Specifically:

Since the universe contains the cosmic microwave background radiation, in order for the black hole to dissipate, the black hole must have a temperature greater than that of the present-day blackbody radiation of the universe of 2.7 K. The relationship between mass and temperature for Hawking radiation then implies the mass must be less than 0.8% of the mass of the Earth [i.e. about 2.4 * 10^-8 solar masses]. This in turn means any black hole that could dissipate cannot be one created by stellar collapse. Only primordial black holes might be created with this little mass.

The theoretical maximum size of a black hole (with maximal spin) is 2.7 x 10^11 solar masses, and the most massive black hole ever observed has an estimated mass of up to 1 x 10^11 solar masses. The largest theoretically possible black hole has an event horizon radius of about 800 billion (i.e. 800,000,000,000) km.

My Confidence In Various Physics Hypotheses

There are various unresolved questions in physics about which I have an opinion. I'm not 100% sure of any of them, but more sure of some than others.

In this post, I give my subjective probabilities for various possibilities, in numbers rounded to avoid spurious accuracy and to increments not less than 1% (even if the true probability expressed as 1% is a bit less than 0.5%):

1. Dark matter phenomena:

* Dark matter phenomena are explained by general relativity or subtle modifications or quantum gravity, that only discernible in weak gravitational fields: 90%

* Dark matter phenomena are explained by a 5th force or a singlet ultralight dark matter boson: 6%

* Dark matter phenomena are explained by dark matter particles of micro-eV to TeV mass: 3%

* Dark matter phenomena are explained by dark matter particles of greater than TeV mass (including composite dark matter candidates such as MACHOs, primordial black holes, and stable heavy hadrons in addition to heavy fundamental particles): 1%

2. Dark energy phenomena:

* Dark energy phenomena are an emergent result of the same gravitational effects that give rise to dark matter phenomena (and do not violate mass-energy conservation): 60%

* Dark energy phenomena are equivalent to the cosmological constant of general relativity: 15%

* Dark energy phenomena exist and are fundamental and not just a side effect of dark matter phenomena, but dark energy is not a constant: 15%

* Dark energy phenomena are a result of flawed astronomy methods and don't really exist: 10%

3. The Lambda CDM model:

* The Lambda CDM model is deeply flawed (even though it may be a useful crude first order approximation): 95%

* The Lambda CDM model is basically correct (although it may omit some minor factors like neutrino masses): 5%

4. Cosmological inflation:

* Cosmological inflation did not happen: 85%

* Some form of cosmological inflation happened: 15%

5. Quantum gravity:

* Gravity is fundamentally a quantum phenomena involving gravitons in Minkowski space: 65%

* Gravity arises from a discrete or quantum space-time (whether or not it also has gravitons): 15% 

* Gravity is emergent from Standard Model forces: 10%

* Gravity is fundamentally a classical and deterministic phenomena: 10%

6. Universe scale asymmetry:

* The universe is not homogeneous and isotropic at the largest possible scales: 65%

* At the largest possible scales, the universe is homogeneous and isotropic: 35%

7. Maximum density:

* There is no physical constraint on maximum mass-energy density: 65%

* There is a maximum mass-energy density greater than the mass-energy density of a minimum mass stellar black hole (such as a Planck scale limitation): 20%

* There is a maximum mass-energy density close to the mass-energy density of a minimum mass stellar black hole: 15%

8. Supersymmetry:

* There is no version of supersymmetry that exists: 99%

* Some version of supersymmetry exists: 1%

9. String Theory:

* Reality is not fundamentally described by string theory: 98%

* Reality is fundamentally described by string theory: 2%

10. Fundamental fermions:

* The Standard Model includes all of the fundamental particles that are fermions: 95%

* The Standard Model omits up to five fundamental fermions (none of which are additional generations of existing Standard Model fundamental fermions) such as a dark matter particle(s) or right handed neutrinos or supersymmetric partners of Standard Model bosons: 4%

* The Standard Model omits at least one additional generation of Standard Model fermions, and/or omits more than five additional fundamental fermions: 1%

11. Fundamental bosons:

* The Standard Model includes all of the fundamental particles that are bosons other than a possible massless spin-2 graviton: 85%

* The Standard Model omits additional fundamental particles that are bosons beyond a massless spin-2 graviton (e.g. additional Higgs bosons, dark matter bosons, dark matter self-interaction bosons, X17 bosons, bosons involved in neutrino mass generation, bosons involved in cosmological inflation and/or dark energy, fifth force carrying bosons, scalar or vector gravitons, massive gravitons, leptoquarks, supersymmetric partners of Standard Model fermions): 15%

12. Sphalerons:

* Sphaleron interactions are physically possible: 50%

* Sphaleron interactions are not physically possible: 50%

13. Stable heavy hadrons:

* There are no stable or metastable hadrons other than the proton and neutron: 95%

* There are stable or metastable hadrons other than the proton and neutron: 5%

14. Stable heavy elements:

* There are no chemical elements with an atomic number in excess of 118 with a half-life of more than 30 seconds: 65%

* There are chemical elements in "islands of stability" with an atomic number in excess of 118 with a half-life of more than 30 seconds: 35%

15. Neutrino mass:

* Neutrinos have Majorana mass: 10%

* Neutrino mass arises from a see-saw mechanism with one or more heavy right handed neutrinos: 5%

* Neutrino mass arises from some other mechanism not yet widely considered: 85%

16. Sterile neutrinos:

* Right handed sterile neutrinos with the same mass as left handed neutrinos exist: 1%

* One or more sterile neutrinos that oscillate or interact with left handed neutrinos, and masses not identical to left handed neutrinos, exist: 2%

* The three left handed neutrinos of the Standard Model are the only neutrinos that exist: 97%

17. Neutrino mass hierarchy:

* The neutrino masses have a "normal" hierarchy: 95%

* The neutrino masses have an "inverted" hierarchy: 5%

18. CP Violation by neutrinos:

* The PMNS matrix exhibits maximal CP violation: 8%

* The PMNS matrix exhibits near maximal CP violation: 85%

* The PMNS matrix exhibits low levels of CP violation: 5%

* The PMNS matrix does not allow for CP violation in neutrino oscillation: 2%

19. Non-standard neutrino interactions:

* There are no non-standard neutrino interactions (i.e. interactions beyond neutrino oscillations and weak force interactions) to be discovered: 90%

* There are some non-standard neutrino interactions: 10%

20. Lepton number and baryon number violation:

* Lepton number and baryon number are always conserved: 50%

* Lepton number and baryon number are only violated in sphaleron interactions: 45%

* Lepton number and baryon number are violated in non-sphaleron interactions (such as neutrinoless double beta decay, proton decay, flavor changing neutral currents, etc.): 5%

21. LP & C:

* The sum of the squares of correctly defined masses of the fundamental particles is equal to the sum of the Higgs vacuum expectation value: 60%

* The sum of the squares of correctly defined masses of the fundamental particles is not equal to the sum of the Higgs vacuum expectation value: 40%

22. Koide's Rule:

* Koide's rule for the masses of charged leptons is true to at least one part per 100,000: 90%

* Koide's rule for the masses of charged leptons is violated by more than one part per 100,000: 10%

23. An extended Koide's rule for quarks:

* The quark masses obey some extended version of Koide's rule: 70%

* The quark masses do not obey some extended version of Koide's rule: 30%

24. The physics desert:

* There are no beyond the Standard Model high energy physics to be discovered between the highest energy scale reached by the Large Hadron Collider (about 10^4 GeV), and energy scales a billion times greater than the highest energy scale reached by the Large Hadron Collider  (about 10^13 GeV): 85%

* There are new high energy physics to be discovered between the highest energy scale reached by the Large Hadron Collider (about 10^4 GeV), and energy scales a billion times greater than the highest energy scale reached by the Large Hadron Collider  (about 10^13 GeV): 15%

25. Planet Nine:

* Planet Nine exists: 65%

* Planet Nine does not exist: 35%

Minimal Gravitational Fields

Gravity is an infinite range force. In isolated circumstances, gravitational pulls from opposite directions can cancel out. But, the vast majority of the time, there is at least some small net gravitational pull in one direction or another.

Stacy McGaugh at Triton Station digs into this observation, in both a Newtonian approximation and MOND, to determine that the minimum gravitational acceleration in deep space in MOND (in light of new data about the percentage of baryons that are in deep space) is about 2% of Milgrom's constant a(0).

This is important in MOND in a way that it isn't in conventional general relativity, because "MOND breaks the strong equivalence principle (but not the weak or Einstein equivalence principle)" with its external field effect.

A Hubble Tension Recap

The Hubble tension has, for whatever reason, been treated as a more serious challenge to the LambdaCDM "standard model of cosmology", which contrary to the statement highlighted below in the abstract, actually has many other serious discrepancies with astronomy observations. A new preprint examines its implications for the model.

Differences in the values of the Hubble constant obtained from the local universe and the early universe have resulted in a significant tension. This tension signifies that our understanding of cosmology (physical processes and/or cosmological data) is incomplete. Some of the suggested solutions include physics of the early Universe. 

In this paper we aim to investigate common features of various early universe solutions to the Hubble constant tension. The physics of the early universe affects the size of the sound horizon which is probed with the Cosmic Microwave Background (CMB) data. Within the standard model, the size of the horizon (within limits of current measurements) is affected by processes that could occur between (approximately) 1 day after the Big Bang and the last scattering instant. We focus on simple extensions incorporating Early Dark Energy (EDE) and show how such a model affects the inferred values of the Hubble constant. We compare this model to LambdaCDM models using MCMC analysis, likelihoods over the parameter space and Bayesian evidence. The MCMC analysis shows that EDE leads to a decrease in the size of the sound horizon that is consistent with H0 = 73.56 km/s/Mpc but we also show that MCMC analysis favours increasing redshift and proportion of EDE. The Bayesian evidence favours our EDE model for very narrow, finely-tuned parameter space. 

The LambdaCDM model used for comparison has good evidence across a wide parameter space. We interpret this as an indication that more sophisticated models are required. We conclude that if the Hubble tension were to be related to the physics of the early universe, EDE could be used as a window to explore conditions of the early universe and extend our understanding of that era.

Gawain Simpson, Krzysztof Bolejko, Stephen Walters, "Beyond LambdaCDM: How the Hubble tension challenges early universe physics" arXiv:2507.08479 (July 11, 2025).

All the GUTs Worth Considering

A fairly short new paper (five pages plus seven pages of footnotes and an appendix) tries to list most or all of the possible Grand Unified Theories a.k.a. GUTs (i.e. theories the unify the three Lie groups of the Standard Model, but not gravity, into a single unified mathematical structure; unified theories that also include gravity are called Theories of Everything a.k.a. TOEs) that could include the Standard Model of Particle Physics, or an extension of it. 

There aren't all that many possibilities that are promising, and several decades of attempts to fit the Standard Model into one in a way that provides useful theoretical insight has not been very fruitful. While this line of inquiry isn't as troubled as supersymmetry (which is a dead man walking) or string theory (which is almost as troubled), it isn't very "hot" either.

Many potential GUTs, including the most minimal SU(5) GUT, would (1) imply violations of baryon number and/or lepton number conservation that aren't observed (e.g. proton decay, flavor changing neutral currents, and neutrinoless double beta decay), (2) lack some fundamental particles that are observed in the Standard Model, or (3) imply the existence of new fundamental particles beyond the Standard Model that haven't been observed (and in some cases, these particles have been ruled out to quite high energies). 

As a general rule, the bigger the Lie group of the unifying GUT, the more likely it is that it will imply far more new fundamental particles than there is any good reason to think that even a many particle dark sector should contain. Theoretical physicists prefer GUTs that imply as minimal an extension of the Standard Model as possible. Moreover, GUTs with certain kinds of new fundamental particles, such as those that imply more than three generations of fundamental Standard Model fermions, are strongly disfavored.

The experimental constraints on baryon number violating and lepton number violating processes (outside sphaleron interactions which are predicted in the Standard Model at extremely high energies but have not been observed) like proton decay, flavor changing neutral currents, and neutrinoless double beta decay are both very strict and very robust (i.e. they have been tested in multiple, independent ways). The exclusions of new fundamental particles are generally up to masses of several hundred to many thousands of GeVs, which is less strict, and the possibility of beyond the Standard Model fundamental particles is also strongly motivated (although not compelled) by the existence of dark matter phenomena. 

In the early days of GUT theories, a much sought after GUT property was that the three Standard Model forces unify at high enough energies in a manner that echos electroweak unification theory (which was one of the very attractive features of supersymmetry theory). But this has also been elusive. 

The Standard Model beta functions of the three Standard Model forces (electromagnetism, the weak force, and the strong force), which govern how the strength of these forces change with energy scale, extrapolated to arbitrarily high energy scales, based upon data all of the way up to the energy scales that can be reached by the Large Hadron Collider a.k.a. LHC (the highest energy scale high energy physics experiment every conducted), never unify. So, if a GUT the unifies the three Standard Model forces exists is some high energy scale, this must be due to new physics at energy scales above those that can be experimentally probed so far that is outside the domain of applicability of the Standard Model. 

Basically, given the energy scales that have already been reached by the LHC, energies at which the three Standard Model force could possibly unify haven't been present anywhere in the universe since some fraction of a second elapsed after the Big Bang. Of course, it is entirely possible that the three Standard Model forces simply don't unify at any energy scale that has ever existed or ever could exist.

Under a reasonable set of ab-initio assumptions, we define and chart the atlas of simple gauge theories with families of fermions whose masses are forbidden by gauge invariance. We propose a compass to navigate the atlas based on counting degrees of freedom. When searching for Grand-unification Theories with three matter generations, the free energy singles out the SU(5) Georgi-Glashow model as the minimal one, closely followed by SO(10) with spinorial matter. The atlas also defines the dryland of grand-unifiable gauge extensions of the standard model. We further provide examples relevant for gauge dual completions of the standard model as well as extensions by an additional SU(N) gauge symmetry.

Giacomo Cacciapaglia, Aldo Deandrea, Konstantinos Kollias, Francesco Sannino, "Grand-unification Theory Atlas: Standard Model and Beyond" arXiv:2507.06368 July 8, 2025).

The final paragraph of the conclusion of the main paper also enumerates some limitations on this paper serving as a truly comprehensive list of possibilities:

We have not considered yet scalar fields, as their mass cannot be prevented by any symmetry. Including spontaneous symmetry breaking of the gauge symmetry and generation of Yukawa couplings could imprint further constraints on the atlas, providing a phenomenological compass to navigate us towards the optimal high-energy theory. In our analysis, asymptotic freedom plays a crucial role in counting the degrees of freedom of each theory.

Non-Linear Cosmology Dynamics

Assuming the data has a Gaussian distribution (i.e. is distributed in a "normal" probability curve) is often reasonable, since this is what happens when data comes from independent simple percentage probability events. And, it is a convenient assumption when it works, because mathematically it is much easier to work with Gaussian distributions than most other probability distributions. But, sometimes reality is more complicated than that and this assumption isn't reasonable. 

The supernova data used to characterize dark energy phenomena isn't Gaussian. 

Trivially, this means that statistical uncertainty estimates based upon Gaussian distributions overestimate the statistical significance of observations in the fat tailed t-distribution. 

Non-trivially, this means that the underlying physics of dark matter phenomena are more mathematically complex than something like Newtonian gravity (often assumed for astronomy purposes as a reasonable approximation of general relativity) or a simple cosmological constant. Simple cosmology models don't match the data. 

This paper estimates dark energy parameters for more complex dark energy models that can fit the data.

Type Ia supernovae have provided fundamental observational data in the discovery of the late acceleration of the expansion of the Universe in cosmology. However, this analysis has relied on the assumption of a Gaussian distribution for the data, a hypothesis that can be challenged with the increasing volume and precision of available supernova data. 

In this work, we rigorously assess this Gaussianity hypothesis and analyze its impact on parameter estimation for dark energy cosmological models. We utilize the Pantheon+ dataset and perform a comprehensive statistical, analysis including the Lilliefors and Jarque-Bera tests, to assess the normality of both the data and model residuals. 

We find that the Gaussianity assumption is untenable and that the redshift distribution is more accurately described by a t-distribution, as indicated by the Kolmogorov Smirnov test. Parameters are estimated for a model incorporating a nonlinear cosmological interaction for the dark sector. The free parameters are estimated using multiple methods, and bootstrap confidence intervals are constructed for them.

Fabiola Arevalo, Luis Firinguetti, Marcos Peña, "On the Gaussian Assumption in the Estimation of Parameters for Dark Energy Models" arXiv:2507.05468 (July 7, 2025).

A New Relativistic Generalization Of MOND (And More)

This six page article is just a conference paper summary of a much more involved modified gravity theory and its implications. The abstract is silent on how well it handles galaxy cluster physics, which deviate (in a quite systemic way) from simple toy-model MOND theories, or the Hubble tension.

We propose an alternative scalar-tensor theory based on the Khronon scalar field labeling a family of space-like three-dimensional hypersurfaces. This theory leads to modified Newtonian dynamics (MOND) at galactic scales for stationary systems, recovers GR plus a cosmological constant in the strong field regime, and is in agreement with the standard cosmological model and the observed cosmic microwave background anisotropies.

Luc Blanchet, Constantinos Skordis, "Khronon-Tensor theory reproducing MOND and the cosmological model" arXiv:2507.00912 (July 1, 2025) (Contribution to the 2025 Gravitation session of the 59th Rencontres de Moriond).

A fuller explanation of the theory can be found here.

Another lengthy paper by P. S. Bhupal Dev et al., examines the constraints dark matter-neutrino interactions which are very strict.

We present a comprehensive analysis of the interactions of neutrinos with the dark sector within the simplified model framework. We first derive the exact analytic formulas for the differential scattering cross sections of neutrinos with scalar, fermion, and vector dark matter (DM) for light dark sector models with mediators of different types. We then implement the full catalog of constraints on the parameter space of the neutrino-DM and neutrino-mediator couplings and masses, including cosmological and astrophysical bounds coming from Big Bang Nucleosynthesis, Cosmic Microwave Background, DM and neutrino self-interactions, DM collisional damping, and astrophysical neutrino sources, as well as laboratory constraints from 3-body meson decays and invisible Z decays. 

We find that most of the benchmarks in the DM mass-coupling plane adopted in previous studies to get an observable neutrino-DM interaction effect are actually ruled out by a combination of the above-mentioned constraints, especially the laboratory ones which are robust against astrophysical uncertainties and independent of the cosmological history. 

To illustrate the consequences of our new results, we take the galactic supernova neutrinos in the MeV energy range as a concrete example and highlight the difficulties in finding any observable effect of neutrino-DM interactions. 

Finally, we identify new benchmark points potentially promising for future observational prospects of the attenuation of the galactic supernova neutrino flux and comment on their implications for the detection prospects in future large-volume neutrino experiments such as JUNO, Hyper-K, and DUNE. We also comment on the ultraviolet-embedding of the effective neutrino-DM couplings.

Missing Baryon Problem Solved

Connor et al. assess that about 3/4 of all baryons are in the intergalactic medium (IGM), give or take 10% – the side bars illustrate the range of uncertainty. Many of the remaining baryons are in other forms of space plasma associated with but not in galaxies: the intracluster medium (ICM) of rich clusters, the intragroup medium (IGroupM) of smaller groups, and the circumgalactic medium (CGM) associated with individual galaxies. All the stars in all the galaxies add up to less than 10%, and the cold (non-ionized) atomic and molecular gas in galaxies comprise about 1% of the baryons.

For a long time at least half to a third of the ordinary atoms that other observations and Big Bang Nucleosynthesis predicted exist hadn't been found. (And, to be clear, this "missing baryon problem" was separate and distinct from the "dark matter" problem.)

Now, they are all accounted for. The missing ones were in the intergalactic medium (i.e. in the deep space between galaxies at a density of about one hydrogen atom per cubic meter). 

Stacy McGaugh at his blog Triton Station explains how this happened. The money chart is above.

The Latest Wide Binary Paper Shows Newtonian Behavior

The Wide Binary analysis debate continues. This paper doesn't see MOND-like behavior in wide binaries.

Of the 44 pairs observed with HARPS, 27% show sign of multiplicity or are not suitable for the test, and 32 bona-fide WBs survive our selection. Their projected separation s is up to 14 kAU, or 0.06 parsec. We determine distances, eccentricities and position angles to reproduce the velocity differences according to Newton's law, finding reasonable solutions for all WBs but one, and with some systems possibly too near pericenter and/or at too high inclination. Our (limited) number of WBs does not show obvious trends with separation or acceleration and is consistent with Newtonian dynamics. We are collecting a larger sample of this kind to robustly assess these results.

From R. Saglia (2025).

Conformal Gravity As A Dark Matter Alternative

One proposed modification of General Relativity that has been proposed to explain dark matter phenomena is Conformal Gravity, which basically preserves angles in transformations adding an additional symmetry to GR.

A new paper suggests that Conformal Gravity fails in elliptical galaxies.

As an alternative gravitational theory to General Relativity (GR), the Conformal Gravity (CG) has recently been successfully verified by observations of Type Ia supernovae (SN Ia) and the rotation curves of spiral galaxies. 

The observations of galaxies only pertain to the non-relativistic form of gravity. In this context, within the framework of the Newtonian theory of gravity (the non-relativistic form of GR), dark matter is postulated to account for the observations. On the other hand, the non-relativistic form of CG predicts an additional potential: besides the Newtonian potential, there is a so-called linear potential term, characterized by the parameter γ∗, as an alternative to dark matter in Newtonian gravity. 

To test CG in its non-relativistic form, much work has been done by fitting the predictions to the observations of circular velocity (rotation curves) for spiral galaxies. 

In this paper, we test CG with the observations from elliptical galaxies. Instead of the circular velocities for spiral galaxies, we use the velocity dispersion for elliptical galaxies. By replacing the Newtonian potential with that predicted by non-relativistic form of CG in Hamiltonian, we directly extend the Jeans equation derived in Newtonian theory to that for CG. By comparing the results derived from the ellipticals with that from spirals, we find that the extra potential predicted by CG is not sufficient to account for the observations of ellipticals. Furthermore, we discover a strong correlation between γ∗ and the stellar mass M∗ in dwarf spheroidal galaxies. This finding implies that the variation in γ∗ violates a fundamental prediction of Conformal Gravity (CG), which posits that γ∗ should be a universal constant.

Li-Xue Yue, Da-Ming Chen, "Test of conformal gravity as an alternative to dark matter from the observations of elliptical galaxies" arXiv:2506.03955 (June 4, 2025).

The Pros And Cons Of MOND

In my view, this analysis is too critical and misses key achievements of MOND-derived theories in the cosmology realm (while doing to little to compare MOND to the competition). But it is still a notable article.

Modified Newtonian Dynamics (MOND) is an alternative to the dark matter hypothesis that attempts to explain the "missing gravity" problem in astrophysics and cosmology through a modification to objects' dynamics. Since its conception in 1983, MOND has had a chequered history. Some phenomena difficult to understand in standard cosmology MOND explains remarkably well, most notably galaxies' radial dynamics encapsulated in the Radial Acceleration Relation. But for others it falls flat -- mass discrepancies in clusters are not fully accounted for, the Solar System imposes a constraint on the shape of the MOND modification seemingly incompatible with that from galaxies, and non-radial motions are poorly predicted. An experiment that promised to be decisive -- the wide binary test -- has produced mainly confusion. This article summarises the good, the bad and the ugly of MOND's observational existence. I argue that despite its imperfections it does possess ongoing relevance: there may yet be crucial insight to be gleaned from it.

Harry Desmond, "Modified Newtonian Dynamics: Observational Successes and Failures" arXiv:2505.21638 (May 27, 2025) (8 pages) (invited contribution to the 2025 Gravitation session of the 59th Rencontres de Moriond).

BBN Tensions And The LambdaCDM Model

Stacy McGaugh's latest post at his Triton Station blog explains why Big Bang Nucleosynthesis (BBN) poses a challenge to the LambdaCDM Standard Model of Cosmology. 

Basically, BBN favors a lower primordial baryon density as of the time of nucleosynthesis, while the cosmic microwave background (CMB) astronomy data when interpreted in light of the LambdaCDM model in a model-dependent way, favors a primordial baryon density as of the time of nucleosynthesis that is two times higher.

But, there are lots of technical issues that make the 4-5 sigma discrepancy less obviously an irreconcilable conflict than it might otherwise seem to be.

Another Dark Matter Particle Model

Overview: An Improvement But Worse Than MOND

Another day, another dark matter particle model. 

This time, a spin-0 massive scalar boson and a spin-1 massive vector boson. Unsurprisingly with the additional degree of freedom that two bosonic dark matter particles can provide relative to single dark matter particle models, it can fit the data a little better than most one parameter dark matter particle models.

The authors "fix the vector boson mass µV = 9 × 10^−26 eV across all galaxies[.]" They allow "the scalar boson mass to vary in the range µS ∈ [10^−10, 10^−16] eV." [Ed. I have converted the MeV units used in the paper for the scalar boson mass to eV units for ease of comparison.]

The vector boson mass is (as is typical of ultralight bosonic dark matter models) of the same order of magnitude as the mass-energy of a typical graviton (for which there is also an obvious theoretical basis for gravitons to vary in mass-energy), suggesting a convergence towards the predictions of a gravitationally based explanation for dark matter phenomena with properties similar to a massless tensor (spin-2) graviton.

The second scalar bosonic dark matter particle found in dark matter sub-halos, however, has less of a clear analog for that mass scale, although the behavior of a scalar boson and a tensor boson have a lot of similarities. The authors of the paper note that:"The origin of this second DM source is unknown, which somehow points to a limitation of the model."

The model doesn't explain why the mass of the scalar bosonic dark matter candidate varies in average mass by a factor of a million from one galaxy to another, despite the fact that all of the bosonic dark matter of both types is assumed to be in its ground state in every galaxy, which is necessary for it to produce the assumed halos shapes. 

To be charitable, however, the number of sub-halos times the mass of each sub-halo could be addressed by varying the number of sub-halos rather than the mass of the dark matter particles in each sub-halo. 

But this would create a different problem. The size of each dark matter sub-halo in the model is basically a function of the mass of the scalar bosonic dark matter candidate (related to its reduced Compton wave length), but not all dark matter sub-halos that are inferred from rotation dynamics and gravitational lensing are the same size.

With three parameters: a fixed vector dark matter particle mass, a scalar dark matter particle mass which can be fit to the data on a galaxy by galaxy basis, and a factor to scale the total amount of dark matter to the total galaxy mass on a case by case basis, with a basically fixed and well-motivated formula for the halo and sub-halo shapes, this model it does almost as well at fitting a galactic rotation curve as a much simpler single fixed parameter MOND model with no parameters that vary from galaxy to galaxy, as shown in the figures below from the paper, although eyeballing it (I've seen the MOND fits to the rotation curves of many similar galaxies many times), it doesn't look like quite as tight a fit.

This model can't predict the total mass of the galaxy from the luminous matter distribution in the way that MOND does, without also resorting to the Tully-Fischer relationship to which MOND is equivalent. And, this model doesn't provide any theoretical explanation for the systemic variation in the mass-luminosity ratio from one galaxy to the next that MOND does.

Given the additional two degrees of freedom in this two type bosonic dark matter model, it's Chi-square fit should have a Chi-square fit two better than a MOND fit, which would be very tight indeed, instead of being sightly worse.

This model also, at least point, has been tested in a much narrower domain of applicability than MOND. It has basically only been tested in selected spiral galaxies in the SPARC sample, while MOND has been fit to essentially all galaxies of all shapes from smallest to largest. MOND doesn't work quite right in galaxy clusters, but this model hasn't been tested in galaxy clusters, so it provides nothing to compare to there. And, while simple extensions of MOND have been fit neatly to the cosmic microwave radiation background (CMB), and there are single particle type dark matter models that have been fit to the CMB, I have not yet seen a two particle type dark matter fit to the CMB and I'm not even really sure who that would work when one of the dark matter particle types has a mass that varies by a factor of a million from one galaxy to the next.

This is an improvement over models that took fifteen or so free parameters to fit galactic rotation curves as well as MOND, that I blogged about quite a few years ago, which still wasn't really any more predictive than this model. But, it is also definitely a work in progress that has multiple problems to solve before it is an attractive dark matter particle model that fits all of the data well.

The Paper

The introduction to the paper explains the model more fully:

Observations of galaxy and galaxy cluster rotation curves reveal a striking deviation from classical expectations. Instead of exhibiting a Keplerian decline, the measured velocities remain unexpectedly flat, extending far beyond the visible boundaries of galaxies. This persistent flatness, commonly known as the Rotation Curves (RC) problem, constitutes a critical argument in favor of non-baryonic dark matter (DM). Various theories have been proposed to explain these anomalies. Some authors have suggested modifications to Newtonian gravity, while others advocate for the existence of invisible, non-interacting DM. The early observations of the Coma cluster together with more precise measurements of galaxy RC during the 1970s reinforced the DM hypothesis. Freeman’s model of spherical halos introduced the concept of a linearly increasing mass function, and subsequent studies have mapped DM halos exceeding quantitatively the observable galactic regions. 

The possibility that galactic halos could be composed of bosonic DM has also been investigated in several works. In particular, models incorporating an ultralight axion-like particle have attracted much attention, as they naturally give rise to DM halos modeled as Newtonian Bose-Einstein Condensates. The Scalar Field Dark Matter model, which is consistent with the ΛCDM paradigm, predicts large-scale phenomena that align with linear-order perturbations. By employing ground state solutions of the Schrödinger-Poisson (SP) system—the only stable configuration where all bosonic particles reside in the lowest energy state—these models successfully reproduce the observed RC. Stability analyses further confirm that while the ground state is robust against gravitational perturbations, excited state configurations remain inherently unstable. 

Models in which a bosonic field minimally coupled to gravity acts as a source of DM are directly linked with bosonic stars. These can be primarily classified into scalar boson stars (BS) and Proca stars (PS), which are localized, regular, horizonless solutions modeled by massive, free or self-interacting, complex scalar and vector fields bound by gravity, respectively. Recent advances have expanded this framework to include ProcaHiggs stars (PHS), where complex vectors interact with real scalars to yield richer dynamics. Moreover, investigations into multi-field configurations have led to the development of multi-state boson stars and ℓ-bosonic stars, thereby broadening the spectrum of viable models. 

Ed. Proca theories were initially devised as massive photon theories. They don't actually describe the behavior of photons well, but does provide the propagators for massive spin-1 bosons, which include the W and Z bosons of the Standard Model.

The vector boson dark matter candidate is essentially a sterile Z boson (i.e. one that unlike the Z boson doesn't interact via the weak force) that is 35 orders of magnitude less massive than a Z boson. The scalar bosonic dark matter candidate is a massive spin-0 boson, much like a sterile Higgs boson (i.e. one that doesn't interact via any non-gravitational force), but 21 to 27 orders of magnitude less massive than the Standard Model Higgs boson. The vector dark matter candidate is about 9 to 15 orders of magnitude less massive than the scalar dark matter candidate.

The primary dark matter halo component has to be less massive than the subhalo dark matter component, because the size of the Bose-Einstein condensate boson star distribution of a less massive bosonic dark matter candidate is large enough to extend across the entire galaxy, while the size of the Bose-Einstein condensate boson start distribution of a more massive bosonic dark matter candidate is only large enough to extend across a dark matter subhalo which is much smaller than a galaxy. 

The theoretical feasibility of bosonic stars has been extensively examined, particularly regarding their formation mechanisms, stability conditions, and dynamical behavior. While initially conceptually conceived as static and spherically symmetric objects, modern studies now routinely explore rotating configurations that manifest as axisymmetric, spinning solutions in both scalar and vector forms. Their versatility in emulating a range of astrophysical objects—including neutron stars, black holes, and intermediate-mass bodies—renders them a powerful tool in astrophysical modeling, allowing to explore the effect of purely gravitational entities. 

This paper explores the modeling of galactic DM halos using bosonic fields, extending the study initiated in [68] and addressing two open issues identified in that work. First, the previous analysis changed the properties of the vector field for each galaxy instead of treating it as a single DM candidate; and second, it introduced an additional dark component without a clear physical justification. As in [68] we employ bosonic vector fields coupled to gravity to represent the primary galactic halo, thus enhancing RC fits when combined with ordinary matter contributions. 

A significant modification in the present work, which addresses the first open issue, is that we now fix the vector boson mass to a specific scale relevant to the problem, allowing only for the field frequency to vary, thereby identifying constraints on this parameter. Regarding the second issue, in [68] the extra component was introduced through an ad-hoc mathematical adjustment, lacking physical motivation and coherence across the configurations. 

Here, we provide a physically meaningful explanation by employing a subhalo model consisting of a scalar bosonic field coupled to gravity to represent intermediate galactic structures. An illustration of our model is depicted in Fig. 1. 

It consists of the following components: the luminous matter contribution, represented by a yellow ellipsoid (the galaxy); a quasispherical main halo formed by rotating vector bosonic matter extending beyond the galaxy, shown in dark grey; and a set of spherical subhalos modeled as scalar boson star-like structures, depicted in light grey. As we show in this paper, this model significantly improves the RC fits presented in [68], in addition to provide a physically justified framework for them.

The structure of the paper is as follows: In Section II we introduce the theoretical framework for the bosonic fields we employ in our model, along with key clarifications regarding the rescalings and physical units used. Section III details how the individual contributions to the RC are obtained from both luminous and DM systems. In particular, this section discusses the model for DM subhalos under the assumption that they could be formed by a distribution of boson stars. Next, in Section IV we compare our model predictions with observational data, using the same sample of galaxies employed in [68]. This section also provides a quantitative assessment of our new model against the fits reported in [68]. A discussion of this study is presented in Section V along with our conclusions. Additional information on the equations of motion for the scalar and vector bosonic models is provided in Appendix A.

Are these models converging on the massless spin-2 graviton, with varying energies, as a dark matter particle that is really just a gravitational explanation for dark matter phenomena?

One wonders what a model with a single massive tensor dark matter particle with continuous range of masses from about 10^-26 eV to 10^-10 eV with masses distributed in something like a power law (or just an empirical estimate of the frequency of gravitational waves at each frequency which might have a lumpy and gappy distribution) fitted to the distribution of gravitational wave strengths observed by gravitational wave detectors would look like. This wide variation of graviton mass-energies is natural and expected in a graviton as dark matter candidate model.

A galaxy length wave-length would be something on the order of 10^-24 Hertz (which is basically undetectable with current gravitational wave detectors), while some rare phenomena could generate gravitational waves with much shorter wave lengths as indicated in the chart shown below from Wikipedia:

The sensitivity rang of existing gravitational wave observatories is shown in this chart from Wikipedia:

This would put the stochastic background gravitational wave wavelengths in the same vicinity as the scalar dark matter particles in the paper's model, while the vector dark matter particles would be far, far below the frequencies that existing gravitational wave observatories can detect in a far noisier background.

If the frequency range of gravitational waves has a very low floor value similar to that of the vector dark matter candidate in the paper, and there is a big gap between that floor value and the low frequency stochastic background, however, this could be a reasonable fit to the two type bosonic dark matter model in the paper.

But, Einstein's Field Equations are structured in a manner that does not depict gravitational waves and/or graviton which have mass-energy (but not rest mass) as possible sources on the right hand side in the stress-energy tensor, and also obscures their self-interactions which are hidden within the many non-linear differential equations on the left hand side. 

So, while the cumulative impact of graviton mass-energy and non-perturbative gravitational self-interactions is potentially significant in galaxy scale or larger systems, it is systemically ignored because it is very hard to extract from Einstein's Field Equations and is negligible in the strong field systems relative to first order general relative effects, like those seen in mergers of compact objects like stars and black holes, in inspiraling binary systems, where purely perturbative approximations like the Post-Newtonian approximations work well.

One way to overcome this would be to use a massive graviton model with a range of masses comparable to the range of graviton mass-energies, which have been better developed theoretically, instead, and to consider qualitatively and in order of magnitude quantities, how that model would behave differently if all of the gravitons were traveling at exactly the speed of light, rather than the slightly below the speed of light speed of a true massive graviton with a slight rest mass with that speed itself varying slightly based upon graviton mass.

Deur's gravitational work has used somewhat similar modeling, although using massless scalar gravitons, and ignoring the differences between scalar and tensor fields (which are quite plausibly comparatively small). And, I suspect that Deur's model would closely approximate this massive graviton model (and the real world data). But, I don't have the general relativity and mathematical expertise necessary to test this myself.