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.

Another Mirror Cosmology Paper

One of the cleaner solutions in physics to the idea of the Big Bang in cosmology is one in which the universe extends infinitely forward and backward in time from the Big Bang. 

This could also explain the matter-antimatter asymmetry of the Universe is that an antimatter dominated universe extends backward in time (in our coordinate system) from the Big Bang and in which the arrow of time runs in the opposite direction due to entropy, which is the boundary between our matter dominated universe and the antimatter dominate universe on the other side of the Big Bang, 

This paper explores such a cosmology model from a mathematical physics perspective, in a way which embraces not only mirror cosmologies of the kind that I have suggested above, but also "bounce" cosmologies that do not posit an antimatter dominated and time reversed universe before the Big Bang. 

There could be quantum entanglement connections between the two sides of the Big Bang, but as the body text of the paper explains (at page 19), "quantum-entanglement correlations cannot be used to send a message from one world to the other: the arguments are essentially the same as those against faster-than-light communication from quantum entanglement."

In this model, the paper suggests that "the maximal energy density and curvature values are very large but finite (the typical energy scale may be the so-called Planck scale given by a combination of sqrt ((h-bar x c^5)/G) ≈ 1.22 × 10^19 GeV ≈ 1.42 × 10^32 Kelvin)."

We review the suggestion that it is possible to eliminate the Big Bang curvature singularity of the Friedmann cosmological solution by considering a particular type of degenerate spacetime metric. Specifically, we take the 4-dimensional spacetime metric to have a spacelike 3-dimensional defect with a vanishing determinant of the metric. 

This new solution suggests the existence of another "side" of the Big Bang (perhaps a more appropriate description than "pre-Big-Bang" phase used in our original paper). The corresponding new solution for defect wormholes is also briefly discussed.

F.R. Klinkhamer, "Big Bang as spacetime defect" arXiv:2412.03538 (Submitted December 4, 2024, last revised August 21, 2025, published version with expanded references) (31 pages).

Predicted Value Of Neff Refined

A new calculation of the predicted value of the observable N(eff) (for the effective number of neutrinos) from cosmology data when there are exactly three neutrino flavors predicts that N(eff) = 3.0439 ± 0.0006. 

The commonly quoted value determined with a somewhat less precise model is 3.045 which is consistent with the new value at the two sigma level.

Astronomy observations are consistent with these predicted values. We know that there are at least three neutrino flavors, but astronomy observations rule out the possibility of four or more neutrino flavors at high significance. This measure of the number of neutrino flavors would include sterile neutrinos that oscillate with the three ordinary active neutrinos up to about 1-10 eV of mass (which is far more massive than any of the active neutrino masses). But a sterile neutrino that is much more massive than that would not impact the value of the N(eff) observable.

W and Z boson decays rule out a fourth generation active neutrino up to masses of about 45.6 GeV.

The Standard Model Constants Are Constant

The available evidence constrains changes in the strong force coupling constant over time (implicitly, through the quantum chromodynamics energy scale lambda QCD) to a value consistent with zero to high precision, from a robust set of four different sets of data going all of the way back to Big Bang Nucleosynthesis which happens about fifteen minutes after the Big Bang. 

The constraint from atomic clocks of 3.2 ± 3.5 parts per 10^17 per year, when the age of the universe is about 1.38 x 10^10 years, implying a maximum difference of 230 ± 250 parts per billion over the entire age of the universe. The constraint from a natural nuclear reactor on Earth that started to react 1.8 billion years ago is slightly more strict at less than 72 parts per billion over the entire age of the universe. A proposed extension of the paper would tighten that constraint by four orders of magnitude.

Of course, the strong force coupling constant, like all of the other experimentally determined Standard Model constants, run with energy scale, so when the universe was very hot, not so long after the Big Bang, its value at the prevailing temperature of the universe would have been smaller, because the strong force coupling is weaker at higher energies. The relationship between energy-scale and the strength of the strong force coupling constant is known exactly in the Standard Model and has been corroborated by particle accelerator experiment data.

This is also what efforts to determine the electromagnetic force coupling constant (i.e. the Fine Structure Constant) and electron and quark mass ratios, using astronomy to determine those ratios at high redshifts, has found.

Laboratory and astrophysical tests of ''constant variation'' have so far concentrated on the dimensionless fine-structure constant α and on the electron or quark mass ratios Xe,q=me,q/ΛQCD, treating the QCD scale ΛQCD as unchangeable. 

Certain beyond Standard Model frameworks, most notably those with a dark matter or dark energy scalar field ϕ coupling with the gluon field, would make ΛQCD itself time dependent while leaving α and the electron mass untouched. Under the minimal assumption that this gluonic channel is the sole ϕ interaction, we recast state-of-the-art atomic clock comparisons into δΛQCD/ΛQCD=(3.2 ± 3.5) × 10^−17 yr^−1 limits, translate the isotope yields of the 1.8-Gyr-old Oklo natural reactor into a complementary geophysical limit of |δΛQCD/ΛQCD| < 2 × 10^−9 over that time span, corresponding to the linear drift limit |δΛQCD/ΛQCD| < 1 × 10^−18 yr^−1, and show that the proposed 8.4 eV 229Th nuclear clock would amplify a putative ΛQCD drift by four orders of magnitude compared with present atomic clocks. We also obtain constraints from quasar absorption spectra and Big Bang Nucleosynthesis data.

V. V. Mansour, A. J. Mansour, "Constraints on the Variation of the QCD Interaction Scale ΛQCD" arXiv:2508.07266 (August 10, 2025) (derivatives in the abstract above are shown with sigma notation rather than in the original superscript dot notation, because dot notation is hard to render in the blogger interface).

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.

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.

Conformal Cosmology

An interesting GR based response to dark matter and dark energy that makes one subtle adjustment to the standard analysis. I'll have to give it a more careful read before saying more about it.

This paper is published in a peer reviewed journal (although not a high profile one) and the author has seven prior peer reviewed journal publications since 2016, one with Pavel Kroupa, a leading astronomer in the MOND literature, as a co-author. So, this is not the work of a crackpot non-astronomer.

The cosmic time dilation observed in Type Ia supernova light curves suggests that the passage of cosmic time varies throughout the evolution of the Universe. This observation implies that the rate of proper time is not constant, as assumed in the standard FLRW metric, but instead is time-dependent. Consequently, the commonly used FLRW metric should be replaced by a more general framework, known as the Conformal Cosmology (CC) metric, to properly account for cosmic time dilation. 

The CC metric incorporates both spatial expansion and time dilation during cosmic evolution. As a result, it is necessary to distinguish between comoving and proper (physical) time, similar to the distinction made between comoving and proper distances. In addition to successfully explaining cosmic time dilation, the CC metric offers several further advantages: (1) it preserves Lorentz invariance, (2) it maintains the form of Maxwell's equations as in Minkowski space-time, (3) it eliminates the need for dark matter and dark energy in the Friedmann equations, and (4) it successfully predicts the expansion and morphology of spiral galaxies in agreement with observations.

Vaclav Vavrycuk, "Time dilation observed in Type Ia supernova light curves and its cosmological consequences" arXiv:2506.19099 (June 23, 2025) (published in 13 Galaxies 55 (2025)).

Kroupa has also published a new cosmology paper (although I'm not optimistic about this MOND plus sterile neutrinos in clusters approach).

The νHDM is the only cosmological model based on Milgromian Dynamics (MOND) with available structure formation simulations. While MOND accounts for galaxies, with a priori predictions for spirals and ellipticals, a light sterile neutrino of 11 eV can assist in recovering scaling relations on the galaxy-cluster scales. In order to perform MONDian cosmological simulations in this theoretical approach, initial conditions derived from a fit to the angular power spectrum of Cosmic Microwave Background (CMB) fluctuations are required. 

In this work, we employ CosmoSIS to perform a Bayesian study of the νHDM model. Using the best-fit values of the posterior, the CMB power spectrum is reevaluated. The excess of power in the transfer function implies a distinct evolution scenario, which can be used further as an input for a set of hydro-dynamical calculations. The resulting values H0 ≈ 56 km/s/Mpc and Ωm0≈0.5 are far from agreement with respect to the best fit ones in the canonical Cold Dark Matter model, but may be significant in MONDian cosmology. The assumed Planck CMB initial conditions are only valid for the ΛCDM cosmology. This work constitutes a first step in an iterative procedure needed to disentangle the model dependence of the derived initial density and velocity fields.

Nick Samaras, Sebastian Grandis, Pavel Kroupa, "On the initial conditions of the νHDM cosmological model" arXiv:2506.19196 (June 23, 2025) (accepted for publication in MNRAS).

Quantum Gravity Can't Violate CPT

The reasoning in this article is sound and means that quantum gravity should preserve Charge-Parity-Time (CPT) symmetry, not just in an emergent low energy approximation, but at all energy scales.

CPT symmetry is at the heart of the Standard Model of particle physics and experimentally very well tested, but expected to be broken in some approaches to quantum gravity. It thus becomes pertinent to explore which of the two alternatives is realized: (i) CPT symmetry is emergent, so that it is restored in the low-energy theory, even if it is broken beyond the Planck scale, (ii) CPT symmetry cannot be emergent and must be fundamental, so that any approach to quantum gravity, in which CPT is broken, is ruled out. 

We explore this by calculating the Renormalization Group flow of CPT violating interactions under the impact of quantum fluctuations of the metric. We find that CPT symmetry cannot be emergent and conclude that quantum-gravity approaches must avoid the breaking of CPT symmetry. 

As a specific example, we discover that in asymptotically safe quantum gravity CPT symmetry remains intact, if it is imposed as a fundamental symmetry, but it is badly broken at low energies if a tiny amount of CPT violation is present in the transplanckian regime.

Astrid Eichhorn, Marc Schiffer, "No dynamical CPT symmetry restoration in quantum gravity" arXiv:2506.12001 (June 13, 2025).

The Missing Baryons Are Probably In Deep Space

The missing baryon problem is that we can't find where all of the ordinary matter that should exist is located. New studies such as this one strongly suggest that it is mostly spread diffusely between galaxies.

Fast radio bursts (FRBs) are emerging as powerful cosmological probes for constraining the baryon fraction in the intergalactic medium (IGM), offering a promising avenue to address the missing baryon problem. In this paper, we analyze constraints on the IGM baryon fraction (fIGM) using 92 localized FRBs, incorporating a corrected probability distribution function for the IGM dispersion measure within three different cosmological models. We find that variations in the underlying cosmological model have a negligible impact on the inferred values of fIGM. While the NE2001 Galactic electron density model yields slightly higher fIGM values compared to the YMW16 model, the results are consistent within the 1σ confidence level. Additionally, there is no statistically significant evidence for redshift evolution in fIGM. Our analysis constrains fIGM to the range 0.8∼0.9, providing strong support for the idea in which the majority of the missing baryons reside in the diffuse IGM.

Yang Liu, Yuchen Zhang, Hongwei Yu, Puxun Wu, "Constraining the Baryon Fraction in the Intergalactic Medium with 92 localized Fast Radio Bursts" arXiv:2506.03536 (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.

Inflationary Cosmology

I'm skeptical of all of the theories described in the abstract below, but I lack the expertise to say with confidence that none of them are correct. 

I am skeptical because, in short, I think that a more accurate description of gravity which gives rise to apparent dark matter and dark energy phenomena, and a mirror universe model in which an anti-matter universe very similar to our own flows backwards in time from the Big Bang, is likely to explain the observations that inflationary cosmology seeks to explain without requiring an exceedingly brief moment of cosmological inflation very shortly after the Big Bang. 

There are peer reviewed published articles that make claims along these lines, but I haven't devoted the time necessary to gain a firm grasp of this literature.

We give a brief review of the basic principles of inflationary theory and discuss the present status of the simplest inflationary models that can describe Planck/BICEP/Keck observational data by choice of a single model parameter. In particular, we discuss the Starobinsky model, Higgs inflation, and α-attractors, including the recently developed α-attractor models with SL(2,ℤ) invariant potentials. We also describe inflationary models providing a good fit to the recent ACT data, as well as the polynomial chaotic inflation models with three parameters, which can account for any values of the three main CMB-related inflationary parameters A(s), n(s) and r.

Renata Kallosh, Andrei Linde, "On the Present Status of Inflationary Cosmology" arXiv:2505.13646 (May 19, 2025).

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.