Deur's Work On Gravity

Background

MOND and other modified gravity theories 
that explain dark matter

A phenomenological toy-model theory developed in 1983 by M. Milgrom called MOND (for "Modified Newtonian Dynamics") has been used to successfully explain the weak field behavior of the gravitational force at all scales from Earth bound and solar system scales to the scale of galaxies and galaxy-satellite galaxy systems, although it tends to underestimate the non-Newtonian effects seen in weak fields in galaxy clusters.

Despite the fact that this theory is only a simple toy-model with a single universal acceleration parameter (a₀ = 1.2 × 10⁻¹⁰ ms⁻²), it can reproduce the dark matter phenomena in all systems of galaxy scale or smaller, and can reproduce some, but not the full extent of, dark matter phenomena in galactic cluster scale systems.

This toy model version of the theory isn't relativistic, however, so its doesn't capture many of the distinctive features of General Relativity, although generalizations of its in a relativistic sense (e.g. TeVeS), and a variety of alternative formulations that reproduce its core insight, have been attempted with mixed success. See also, e.g., MOG by John Moffat.

Deur's Work On Gravity

One of the most promising gravity based explanations of dark matter phenomena is Alexandre Deur's efforts to identify quantum gravity effects in a graviton based theory of quantum gravity. 

For mathematical simplicity, he models these effects in the static, scalar graviton case. It also exploits analogies to quantum chromodynamics (i.e. the Standard Model's theory of the strong force that binds quarks and gluons based upon their color charge). This model is useful because, like the self-interacting massless spin-1 gluons that are the carrier bosons of QCD (and unlike the massless spin-1 photons of quantum electrodynamics, which is the Standard Model theory of electromagnetism which don't interact with each other), the gravitons which are the carrier bosons of a hypothetical quantum gravity theory would have gravitational interactions with each other. Graviton based quantum gravity, like QCD, is a non-Abelian force mediated by a massless carrier boson.

Subsequently, he has replicated these rules using only classical general relativity equations without any quantum gravity component. While the idea of gravitational self-interaction is more intuitive in the context of a quantum gravity theory with the gravitational force mediated by gravitons, the effects described aren't reliant upon any quantum gravity specific properties or concepts.

Deur concludes, by analogy to QCD, that graviton-graviton interactions lead to effects that look like dark matter and dark energy, in the weak fields of very massive gravitational sources, to the extent that the systems are not spherically symmetric.

Deur claims that his theory is consistent with General Relativity, although this isn't entirely clear. In the limit of spherical systems and strong gravitational fields, it is simply General Relativity.

Deur's "day job" is as a QCD physicist at Jefferson Labs, so he brings to the field a different set of mathematical tools and insights than the typical quantum gravity researcher, who typically starts with a study of classical General Relativity and then expands from there.

Why Is Deur's Approach Promising?

Deur's papers are not widely cited, even though they have been published in peer reviewed journals, and his papers have not received in depth investigation from other scientists in the astronomy and gravity field.

But, they remain very promising as a possible solution to the dark matter and dark energy problems.

Observational Evidence Supports Deur's Model

* There is strong evidence, developed in the MOND context (and to a lesser extent in tests of other gravity modification approaches), that dark matter phenomena involved in galactic dynamics can be understood as a modification of conventional Newtonian approximations of gravity. This is in contrast to explanations of dark matter phenomena involving one or more dark matter particles beyond the Standard Model. Because the formula that Deur develops is observationally almost indistinguishable from MOND when applied to spiral galaxies, a circumstance where MOND works well, his theory benefits from this body of evidence. 

* Deur's approach also makes predictions similar to MOND in other contexts. For example, new 21 cm background radiation observations, that are contrary to the predictions of the Lambda-CDM Model, also support Deur's theory.

* Deur's solution elegantly solves the galactic cluster problem of MOND by resorting to the differences in shape of clusters and their subparts, and the geometry between bodies attracted to each other in galactic clusters, and the arrangements of matter found in galaxies. Thus, it cures one of the main short fallings of MOND.

* Deur's solution predicts and explains a previously unnoticed relationship between the apparent amount of dark matter in an elliptical galaxy and the extent to which the galaxy is not spherical, which other modified gravity and dark matter particle theories do not.

* Deur's solution predicts and explains a previously unnoticed relationship between the thickness of a disk galaxy and the apparent amount of dark matter in a disk galaxy.

* This quantum gravity theory overcomes problems arising from observational evidence from correlated visible light and gravitational wave observations of black holes merging with neutron stars that show that gravitational waves travel at a speed indistinguishable from the speed of light to high precision. This observation is inconsistent with massive gravitons in some modified gravity theories (e.g. many scalar-tensor or scalar-vector-tensor theories), because it utilizes only a single massless graviton.

* The Lambda-CDM Model does a great job of predicting the peaks in the cosmic background radiation of the universe, but does not do a good job of explaining dynamics of galaxies, or explaining why those dynamics are so tightly correlated with the distribution of baryonic matter in those systems. Simple cold dark matter models with a single "sterile" massive fermion do not accurately reproduce the inferred dark matter halos that are observed, nor do many more complicated dark matter particle theories. There are actually myriad discrepancies between observation and predicted behavior in the Lambda-CDM Model at the galaxy scale and some problems even at the galactic cluster scale. This is so even thought the Lambda-CDM Model is incomplete because it doesn't, by itself, explain how the cold dark matter in that matter came to have the very structured distribution that inferred dark matter distributions do observationally.

Deur's Model Is Attractive Theoretically

* Deur explains dark matter and dark energy phenomena as a natural outgrowth of quantum gravity, with no "moving parts" that can be adjusted to make it fit the data in advance.

* Deur's theory provides a sound theoretical basis for an explanation of the dark matter phenomena with modifications of the Newtonian gravity approximation widely used in large scale astronomy contexts, that it utilizes, because it derives these modifications from first principles. It does so in a way that sidesteps the overwhelming calculation difficulties of doing the full fledged calculations of gravity with a spin-2 massless graviton that has been an insurmountable barrier to other quantum gravity theories, but without inducing significant systemic error in the systems to which the theory is applies (i.e. the differences between a spin-0 graviton theory and a spin-2 graviton theory in dark matter and dark energy contexts is slight except in gravitational systems that are far from equilibrium). It is not mere numerology or a purely phenomenological theory.

* While Deur's approach does not reproduce the conclusions of General Relativity as conventionally applied in the weak gravitational fields and spherically asymmetric systems where it dark matter and dark energy phenomena are observed, he does not make any assumptions about the properties of the graviton which are not utterly vanilla in the context of graviton based quantum gravity theories. None of the underlying assumptions from which this approach is derived contradict the underlying assumptions associated with General Relativity, except in ways generic to all quantum gravity theories (e.g. all quantum gravity theories with gravitons localize gravitational mass-energy, while classical General Relativity does not). He also reproduces the same results using classical, unmodified, General Relativity without a cosmological constant.

* Deur's approach builds on the common quantum gravity paradigm of gravity as QCD squared  (strictly speaking Yang-Mills squared, but QCD is an SU(3) Yang–Mills theory).

* Basically, if Deur's approach ends up being correct, then the way that gravitational field self-interactions are incorporated into General Relativity in the Einstein's equations as conventionally applied must be subtly flawed. 

* Deur's background as a professional QCD scientist pretty much assures that his non-abelian gauge field mathematics are sound. Independent efforts corroborate the validity of the main simplification he makes relative to quantum gravity with a spin-2 massless graviton, as do classical GR reproductions of the same effects.

* Deur's solution is pretty much the simplest possible resolution of the problems of quantum gravity, dark matter and dark energy, because (1) it does so with no new particles, (2) no new forces, and (3) one fewer fundamental physical constant than the existing core theory of the Standard Model and General Relativity (without dark matter), i.e. without the cosmological constant Λ.

* The ΛCDM Model, also known as the Lambda-CDM Model, also known as the Standard Model of Cosmology, requires that 97.8% of the mass-energy of the universe be made up of never observed dark matter and dark energy, while Deur's theory relies entirely on Standard Model fundamental particles.

* Many modified gravity theories assume new scalar and vector fields in addition to the tensor field of the graviton. Many dark matter particle theories require a new self-interaction force between dark matter particles or a new force governing interactions between dark matter and ordinary matter, or both. Deur's theory, in contrast, gives rise to no new forces or fields.

* This quantum gravity theory, in principle, replaces the three constants of general relativity plus MOND (Newton's constant G, the cosmological constant λ, and the MOND universal acceleration, a₀) and replaces them with a single fundamental constant, the gravitational coupling constant (whose value has already been measured moderately precisely). This coupling constant is Newton's constant G. Both the cosmological constant and the universal acceleration constant of MOND can be derived, in principle, from G in this theory (although he has not done this derivation himself). In contrast, MOND adds one physical constant to the existing core theory, and dark matter adds at least one dark matter particle mass (and more masses in the dark matter sector such as a mass and coupling constant for a dark boson that carries a self-interaction or ordinary matter-dark matter interaction or both, are present in many versions of dark matter theories), one dark matter abundance constant, and other properties related to the dark matter particle. Modified gravity theories other than MOND (such as Moffat's MOG theory) often have more new physical constants than MOND does.

* Deur's theory makes it possible to naturally harmonize gravity and the Standard Model with no particles beyond the Standard Model other than a massless graviton.

* Deur's theory explains the cosmic coincidence problem in a very natural way.

* Deur's theory solves the conservation of mass-energy problem with general relativity's cosmological constant solution to "dark energy." Conventional general relativity theory, in contrast, accepts that gravitational energy is only conserved locally and not globally. In Deur's theory, dark energy arises from self-interacting gravitons staying within the galaxy at rates higher than they would in the absence of self-interactions which causes mass of the edge of a galaxy to be pulled more tightly towards the galaxy. Because these gravitons leave the galaxy at a rate lower than they would in the absence of self-interactions, the gravitational pull between galaxies is weaker than it would be in the absence of gravitational self-interaction. Thus, dark energy is due to a weaker pull between galaxies than in the Newtonian or cosmological constant free general relativity model, rather than due to having something pervasive in space pulling apart distant objects.

* Deur's theory is not plagued with tachyons, causation violations, ghosts, unitarity violations and similar defects that are common in efforts to modify gravity.

A Quick Primer

One of the better and more intuitive introductions to his ideas is in this power point presentation. Some of the key points from that presentation are restated below:

Empirical parallels between Cosmology and Hadronic Physics

----------------------

Cosmology (1)

Galaxies (or clusters of galaxies) have a larger mass than the sum of their known constituents.

Alternatively Gravity is stronger than we think for these systems.

Hadronic Physics (1)

Hadronic physics 2 quarks ~10 MeV, Pion mass 140 MeV 3 quarks ~15 MeV, Nucleon: 938 MeV

For non-relativistic quarks, this extra mass comes from large binding energy.
----------------------

Cosmology (2)

Tully-Fisher relation: log(M)=γlog(v)+ε
(γ=3.9±0.2, ε ~1.5)
(M galaxy visible mass, v rotation speed)

Unexplained with dark matter. Assumed by MOND.

Hadronic Physics (2)

Regge trajectories: log(M)=c log(J)+b
(c=0.5)
(M, hadron mass, J angular momentum)
----------------------
Cosmology (3)

Negative pressure pervades the universe and repels galaxies from each other. The attraction of galaxies is smaller than we think at very large distances.

Hadronic Physics (3)

Relatively weak effective force between hadrons (Yukawa potential) compared to QCD’s magnitude.

----------------------

Deur's Quantum Gravity LaGrangian and QCD Compared

The gravitational Lagrangian that Deur develops is as follows:

ℒGR=[∂ψ∂ψ]+√G[ψ∂ψ∂ψ]+G[ψ2∂ψ∂ψ]+ΣGn/2[ψn∂ψ∂ψ]+√ψμνTμν

This is derived by expanding the ℒGR in term of tensor gravity field ψμν by developing gμν around the Minkowsky metric: gμν~ημν+G^1/2ψμν+...

This is compared to the QCD Lagrangian:

ℒQCD=[∂ψ∂ψ]+√4πα_s[ψ²∂ψ]+ 4πα_s[ψ⁴]

The first terms of each are Newtonian gravity and perturbative QCD respectively (in the static case). The next two terms of the respective Lagrangians are field self-interaction terms.

How strong are the gravitational self-interaction terms? 

This is a function, roughly speaking, of system mass and system size:

Near a proton GM_p/r_p=4×10^-38 with M_p the proton mass and r_p its radius. ==>Self-interaction effects are negligible:

ℒGR=[∂ψ∂ψ]+√G[ψ∂ψ∂ψ]+G[ψ2∂ψ∂ψ]+...  the stricken terms are almost zero.

For a typical galaxy: Magnitude of the gravity field is proportionate to GM/size_system which is approximately equal to 10^-3.

Basically, the more thinly spread the mass is in space, the stronger the self-interaction terms are relative to the Newtonian term of the Lagrangian. For gravitons, the profoundly weak strength of the Newton's constant means that self-interaction terms are only significant at immense distances where the mass is spread thinly.

In QCD, by comparison, the profoundly greater strength of the QCD coupling constant allows the self-interaction terms to be significant even at tiny distances on the order of 10^-15 meters, despite the fact that the color charges are not spread thinly.

Matter Distribution Shapes and Self-Interaction Strength

Essentially, self-interacting gravitons, rather than going off randomly in all directions, tend to veer towards direct gravitational fields between clumps of mass, making those fields stronger, while weakening the fields in the direction of empty space.

Isolated Point Masses

For two significant point masses with nothing else nearby, self-interactions cause the system to reduce from a three dimensional one to a flux tube causing the force between them to remain nearly constant without regard to distance.

Disk-Like Masses

If the mass is confined to a disk, the self-interactions cause the system to reduce from a three dimensional one to a two dimensional one, causing the force to have a 1/r form that we see in the MONDian regime of spiral galaxies.

In the geometries where Deur's approach approximate's MOND, the following formula approximate's the self-interaction term:

F_G = G_NM/r² + c²(aπG_NM)^1/2/(2√2)r

where F_G is the effective gravitational force, G_N is Newton's constant, c is the speed of light, M is ordinary baryonic mass of the gravitational source, r is the distance between the source mass and the place that the gravitational force is measured, and a is a physical constant that is the counterpart of a₀  in MOND (that should in principle be possible to derive from Newton's constant) which is equal to 4*10⁻⁴⁴ m⁻³s².

Thus, the self-interaction term that it modifies is proportionate to (G_NM)^1/2/r. So, it is initially much smaller that the first order Newtonian gravity term, but it declines more slowly than the Newtonian term until it is predominant.

Spherically Symmetric Masses

If the mass is spherically symmetric, the self-interactions cancel out and the system remains three dimensional causing the force to have the 1/r² form that we associate with Newtonian gravity.

Why do galactic clusters have so much more apparent dark matter than spiral galaxies? 

Because geometrically, they are closer to the two point particle scenario, in which galaxies within the cluster are the point particles that exert a distance independent force upon each other (analogous to flux tubes in QCD), rather than being spherically symmetric or disk-like.

Why does the Bullet Cluster behave as it does? 

Since gas dominates the visible mass of a cluster, the observation that most of the total (dark) mass did not stay with the gas appears to rule out modifications of gravity as an alternative to dark matter. But, actually, this isn't the case in a self-interacting graviton scenario.

Because it has a gaseous component that is more or less spherically symmetric, that component has little apparent dark matter, while the galaxy components, which come close to the two point mass flux tube paradigm which is equivalent to a great amount of inferred dark matter. So, the gaseous portion and the core galaxy components are offset from each other. The apparent dark matter tracks the galaxy cores and not the interstellar gas medium between them.

Annotated Bibliography

The first article in the series by Deur on gravity is:

The non-abelian symmetry of a lagrangian invalidates the principle of superposition for the field described by this lagrangian. A consequence in QCD is that non-linear effects occur, resulting in the quark-quark linear potential that explains the quark confinement, the quarkonia spectra or the Regge trajectories. Following a parallel between QCD and gravitation, we suggest that these non-linear effects should create an additional logarithmic potential in the classical newtonian description of gravity. The modified potential may account for the rotation curve of galaxies and other problems, without requiring dark matter.

A. Deur, “Non-Abelian Effects in Gravitation” (September 17, 2003) (not published).

The first of his papers published in a peer reviewed journal is:

Our present understanding of the universe requires the existence of dark matter and dark energy. We describe here a natural mechanism that could make exotic dark matter and possibly dark energy unnecessary. Graviton-graviton interactions increase the gravitational binding of matter. This increase, for large massive systems such as galaxies, may be large enough to make exotic dark matter superfluous. Within a weak field approximation we compute the effect on the rotation curves of galaxies and find the correct magnitude and distribution without need for arbitrary parameters or additional exotic particles. The Tully-Fisher relation also emerges naturally from this framework. The computations are further applied to galaxy clusters.

A. Deur, “Implications of Graviton-Graviton Interaction to Dark Matter” (May 6, 2009) (published at 676 Phys. Lett. B 21 (2009)). 

Deur also makes a theoretical prediction which neither dark matter nor MOND suggest, which is born out by observation. This prediction is that non-spherical elliptical galaxies have greater deviations from general relativity without dark matter than more spherically symmetric elliptical galaxies do. This is found in a 2014 paper:

We discuss the correlation between the dark matter content of elliptical galaxies and their ellipticities. We then explore a mechanism for which the correlation would emerge naturally. Such mechanism leads to identifying the dark matter particles to gravitons. A similar mechanism is known in Quantum Chromodynamics (QCD) and is essential to our understanding of the mass and structure of baryonic matter.

Alexandre Deur, “A correlation between the amount of dark matter in elliptical galaxies and their shape” (July 28, 2014).

Deur argues that most or all of observed dark energy phenomena results from gravitons being confined in galaxy and galactic cluster scale systems, which is what gives rise to the dark matter phenomena in this systems. The diversion of gravitons to more strongly bind matter in the galaxies leads to a small deficit of gravitons which escape the galaxy and cause galaxies and galactic clusters to bind to each other. It also neatly explains the "cosmic coincidence problem." He spells out this analysis in a 2018 pre-print (with an original pre-print date in 2017) which also examines cosmology implications of his approach more generally:

Numerical calculations have shown that the increase of binding energy in massive systems due to gravity's self-interaction can account for galaxy and cluster dynamics without dark matter. Such approach is consistent with General Relativity and the Standard Model of particle physics. The increased binding implies an effective weakening of gravity outside the bound system. In this article, this suppression is modeled in the Universe's evolution equations and its consequence for dark energy is explored. Observations are well reproduced without need for dark energy. The cosmic coincidence appears naturally and the problem of having a de Sitter Universe as the final state of the Universe is eliminated.

A. Deur, “A possible explanation for dark matter and dark energy consistent with the Standard Model of particle physics and General Relativity” (August 14, 2018) (Proceeding for a presentation given at Duke University, Apr. 2014. Based on A. D. PLB B676, 21 (2009); A.D, MNRAS, 438, 1535 (2014). The published version is https://link.springer.com/article/10.1140/epjc/s10052-019-7393-0). The body text of this paper explains at greater length that:

The framework used in Refs. [3, 4] is analogous to the well-studied phenomenology of Quantum Chromodynamics (QCD) in its strong regime. Both GR and QCD Lagrangians comprise field self-interaction terms. In QCD, their effect is important because of the large value of QCD’s coupling, typically αs ' 0.1 at the transition between QCD’s weak and strong regimes [8]. In GR, self-interaction becomes important for p GM/L large enough (G is Newton’s constant, M the mass of the system and p L its characteristic scale), typically for GM/L & 10−3 [4]. In QCD, a crucial consequence of self-interaction associated with a large αs is an increased binding of quarks, which leads to their confinement. Refs. [3, 4] show that GR’s self-interaction terms lead to a similar phenomenon for p GM/L large enough, which can explain observations suggestive of dark matter. Beside confinement, the other principal feature of QCD is a dearth of strong interaction outside of hadrons, the bound states of QCD. This is due to the confinement of the color field in hadrons. While the confined field produces a constant force between quarks that is more intense than the 1/r2 force expected from a theory without self-interaction, this concentration of the field inside the hadron means a depletion outside. If such phenomenon occurs for gravity because of trapping of the gravitational field in massive structures such as galaxies or clusters of galaxies, the suppression of gravity at large scale can be mistaken for a repulsive pressure, i.e. dark energy. Specifically, the Friedman equation for the Universe expansion is (assuming a matter-dominated flat Universe) H2 = 8πGρ/3, with H the Hubble parameter and ρ the density. If gravity is effectively suppressed at large scale as massive structures coalesce, the Gρ factor, effectively decreasing with time, would imply a larger than expected value of H at early times, as seen by the observations suggesting the existence of dark energy. Incidentally, beside dark matter and dark energy, QCD phenomenology also suggests a solution to the problem of the extremely large value of Λ predicted by Quantum Field Theory [9].

Reference [9] in that paper is worth noting because is provides a simple and analogous solution to the gross disparity between the quantum mechanical expectation for the cosmological constant and its actual, tiny value. It is as follows:

Casher and Susskind [Casher A, Susskind L (1974) Phys Rev 9:436–460] have noted that in the light-front description, spontaneous chiral symmetry breaking is a property of hadronic wave functions and not of the vacuum. Here we show from several physical perspectives that, because of color confinement, quark and gluon condensates in quantum chromodynamics (QCD) are associated with the internal dynamics of hadrons. We discuss condensates using condensed matter analogues, the Anti de Sitter/conformal field theory correspondence, and the Bethe–Salpeter–Dyson–Schwinger approach for bound states. Our analysis is in agreement with the Casher and Susskind model and the explicit demonstration of “in-hadron” condensates by Roberts and coworkers [Maris P, Roberts CD, Tandy PC (1998) Phys Lett B 420:267–273], using the Bethe–Salpeter–Dyson–Schwinger formalism for QCD-bound states. These results imply that QCD condensates give zero contribution to the cosmological constant, because all of the gravitational effects of the in-hadron condensates are already included in the normal contribution from hadron masses.

Stanley J. Brodsky and Robert Shrock, "Condensates in quantum chromodynamics and the cosmological constant" 108(1) PNAS 45 (January 4, 2011).

Analyses of internal galaxy and cluster dynamics typically employ Newton's law of gravity, which neglects the field self-interaction effects of General Relativity. This may be why dark matter seems necessary. 

The Universe evolution, on the other hand, is treated with the full theory, General Relativity. However, the approximations of isotropy and homogeneity, normally used to derive and solve the Universe evolution equations, effectively suppress General Relativity's field self-interaction effects and this may introduce the need for dark energy. 

Calculations have shown that field self-interaction increases the binding of matter inside massive systems, which may account for galaxy and cluster dynamics without invoking dark matter. 

In turn, energy conservation dictates that the increased binding must be balanced by an effectively decreased gravitational interaction outside the massive system. In this article, such suppression is estimated and its consequence for the Universe's evolution is discussed. Observations are reproduced without need for dark energy.

A. Deur, "An explanation for dark matter and dark energy consistent with the Standard Model of particle physics and General Relativity" arXiv:1709.02481  (September 7, 2017 published October 22, 2019) (published in Eur. Phys. J.C.) doi10.1140/epjc/s10052-019-7393-0 

The finding in the MINRAS paper for elliptical galaxies is further explored in the following (117 page) pre-print from 2020 and a follow up MINRAS paper in 2023.

Observations indicate that the baryonic matter of galaxies is surrounded by vast dark matter halos, which nature remains unknown. This document details the analysis of the results published in MNRAS 438, 2, 1535 (2014) reporting an empirical correlation between the ellipticity of elliptical galaxies and their dark matter content. Large and homogeneous samples of elliptical galaxies for which their dark matter content is inferred were selected using different methods. Possible methodological biases in the dark mass extraction are alleviated by the multiple methods employed. Effects from galaxy peculiarities are minimized by a homogeneity requirement and further suppressed statistically. After forming homogeneous samples (rejection of galaxies with signs of interaction or dependence on their environment, of peculiar elliptical galaxies and of S0-type galaxies) a clear correlation emerges. Such a correlation is either spurious --in which case it signals an ubiquitous systematic bias in elliptical galaxy observations or their analysis-- or genuine --in which case it implies in particular that at equal luminosity, flattened medium-size elliptical galaxies are on average five times heavier than rounder ones, and that the non-baryonic matter content of medium-size round galaxies is small. It would also provides a new testing ground for models of dark matter and galaxy formation.

A. Deur, "A correlation between the dark content of elliptical galaxies and their ellipticity" (October 13, 2020).

The use of a scalar graviton approximation used by Deur is justified in a published 2017 paper:

We study two self-interacting scalar field theories in their high-temperature limit using path integrals on a lattice. We first discuss the formalism and recover known potentials to validate the method. We then discuss how these theories can model, in the high-temperature limit, the strong interaction and General Relativity. For the strong interaction, the model recovers the known phenomenology of the nearly static regime of heavy quarkonia. The model also exposes a possible origin for the emergence of the confinement scale from the approximately conformal Lagrangian. Aside from such possible insights, the main purpose of addressing the strong interaction here --given that more sophisticated approaches already exist-- is mostly to further verify the pertinence of the model in the more complex case of General Relativity for which non-perturbative methods are not as developed. The results have important implications on the nature of Dark Matter. In particular, non-perturbative effects naturally provide flat rotation curves for disk galaxies, without need for non-baryonic matter, and explain as well other observations involving Dark Matter such as cluster dynamics or the dark mass of elliptical galaxies.

A. Deur, “Self-interacting scalar fields at high temperature” (June 15, 2017) (published at Eur. Phys. J. C77 (2017) no.6, 412).

A paper in the same journal by independent authors confirms that scalar approximations can reproduce experimental tests:

We construct a general stratified scalar theory of gravitation from a field equation that accounts for the self-interaction of the field and a particle Lagrangian, and calculate its post-Newtonian parameters. Using this general framework, we analyze several specific scalar theories of gravitation and check their predictions for the solar system post-Newtonian effects.

Diogo P. L. Bragança, José P. S. Lemos “Stratified scalar field theories of gravitation with self-energy term and effective particle Lagrangian” (June 29, 2018) (open access) (pre-print here).

A 2019 paper with two collaborators restates the basis premise of Deur's previous work.

The discrepancy between the visible mass in galaxies or galaxy clusters, and that inferred from their dynamics is well known. The prevailing solution to this problem is dark matter. Here we show that a different approach, one that conforms to both the current Standard Model of Particle Physics and General Relativity, explains the recently observed tight correlation between the galactic baryonic mass and its observed acceleration. Using direct calculations based on General Relativity's Lagrangian, and parameter-free galactic models, we show that the nonlinear effects of General Relativity make baryonic matter alone sufficient to explain this observation.

A. Deur, Corey Sargent,  Balša Terzić, "Significance of Gravitational Nonlinearities on the Dynamics of Disk Galaxies" (August 31, 2019, last revised January 11, 2020) (pre-print).  Latest update May 18, 2020. https://arxiv.org/abs/1909.00095v3

Another of Deur's papers approaches the question of dark matter phenomena in disk galaxies using more a more conventional gravitational field approach rather than an quantum gravity Lagrangian simplified to a scalar graviton approximation, with the same basic analysis focusing on self-interactions of the gravitational field, and derives a predicted relationship between disk thickness and inferred dark matter amount which he then verifies with empirical data.

We present a method to investigate the effect of relativistic corrections arising from large masses to the rotation curves of disk galaxies. The method employs a mean-field approximation and gravitational lensing. Applying it to a basic model of disk galaxy, we find that these corrections become important and magnified at large distances. The magnitude of the effect is sufficient to explain the galactic missing mass problem without requiring a significant amount of dark matter. A prediction of the model is that there should be a strong correlation between the inferred galactic dark mass and the galactic disk thickness. We use two independent sets of data to verify this.

Alexandre Deur, "Relativistic corrections to the rotation curves of disk galaxies" (April 10, 2020) (lated updated February 8, 2021 in version accepted for publication in Eur. Phys. Jour. C).

This paper also makes clear that the primary effect is actually a classical gravitational field self-interaction effect, rather than a genuinely quantum gravitational effect. As the introduction in the body text explains:

The total mass of a nearby disk galaxy is typically obtained from measuring its rotation curve and deducing from it the mass using Newton’s dynamics. The rationale for this non-relativistic treatment is the small velocity of stars: v/c << 1 sufficiently far from the central galactic black hole. However, the assumption that relativistic corrections are negligible may be questioned on several grounds. Inspecting the post-Newtonian [1] Lagrangian, e.g. for two masses M1 and M2 separated by r, shows non-Newtonian potential terms of the type G^2M1M2(M1+M2)/2r^2 (G is the gravitational constant) that are independent of v, thus not suppressed at small v, and can be non-negligible for large enough M1 and M2. These terms express the non-linear nature of General Relativity (GR), which arises from its field self-interaction: the gravitational field has an energy and hence gravitates too.

Field self-interactions are well-known in particle physics: Quantum Chromodynamics (QCD), the gauge theory of the strong force between quarks, features color-charged fields that self-interact. In fact, GR and QCD have similar Lagrangians, including self-interacting terms, as can be seen when the Einstein-Hilbert Lagrangian of GR is expanded in a polynomial form [2, 3]. Field self-interaction in QCD, which causes quark confinement, exists even for static sources, as shown by the existence of numerous heavy quark bound states (in which v ≈ 0 for quarks) [4] and by classic numerical lattice calculations for v = 0 quarks [5]. This, as well as the correspondence between the respective terms of the GR and QCD Lagrangians, shows that for bodies massive enough a relativistic treatment is required regardless of their velocity. Finally, the measured speeds at the rotation curve plateaus are of several hundreds of km/s, e.g. 300 km/s (or v/c = 0.1%) for NGC 2841. They are similar to that of stars orbiting the central black hole of our galaxy and clearly display the relativistic dynamics expected in the strong regime of GR [6]. 

These arguments suggest that one should investigate the importance of relativistic dynamics in galaxies and how it affects the missing mass problem. From experience with QCD, a non-perturbative approach is required to fully account for field self-interaction, making post-Newtonian formalism inadequate. In Refs. [2, 3], a nonperturbative numerical lattice method was used. Here, we propose to approach the problem with a mean-field technique combined with gravitational lensing. There are several advantages of the approach compared to the lattice method used in [2, 3]: (1) it is an entirely independent method, thereby providing a thorough check of the lattice result; (2) it is not restricted to the static limit of the lattice method and can be applied to systems with complex geometries; (3) it is significantly less CPU-intensive than a lattice calculation, and hence much faster; (4) it clarifies that the effect calculated in Refs. [2, 3] is classical. The lattice approach – an inherently quantum field theory (QFT) technique – used in Refs. [2, 3] may misleadingly suggest that a quantum phenomenon is involved. In fact, the classical nature of the effect is consistent with these lattice calculations being performed in the high-temperature limit in which quantum effects disappear, as discussed in Ref. [3]; (5) the lensing formalism is more familiar to astrophysicists and cosmologists, in contrast to lattice techniques with its QFT underpinning and terminology.

The references in the 2020 paper are as follows, with references to Deur's own prior works in bold.

[1] A. Einstein, L. Infeld and B. Hoffmann, Annals Math. 39, 65 (1938)
[2] A. Deur, Phys. Lett. B 676, 21 (2009) [3] A. Deur A, Eur. Phys. J. C 77, 412 (2017) [4] M. Tanabashi et al. [Particle Data Group], Phys. Rev. D 98, 030001 (2018)
[5] G. S. Bali, Phys. Rept. 343, 1 (2001)
[6] A. Hees et al., Phys. Rev. Lett. 118, 211101 (2017)
[7] L. D. Landau, E. M. Lifshitz, The Classical Theory of Fields (Oxford: Butterworth-Heinemann, 2000)
[8] S. J. Brodsky, G. F. de Teramond, H. G. Dosch and J. Erlich, Phys. Rept. 584, 1 (2015)
[9] B. S. DeWitt, Phys. Rev. 162, 1195 (1967); Phys. Rev. 7 162, 1239 (1967); L. F. Abbott, Acta Phys. Polon. B 13, 33 (1982)
[10] W. J. G. de Blok, Advances in Astronomy, 2010, 789293 (2010)
[11] M. Milgrom, Astrophys. J. 270, 365 (1983)
[12] E. B. Amores, A. C. Robin and C. Reyle A&A 602, A67 (2017)
[13] A. Deur, S. J. Brodsky, G. F. de Teramond, Prog. Part. Nucl. Phys. 90, 1 (2016)
[14] Y. Sofue, PASJ 68, 2 (2016)
[15] T. P. K. Martinsson et al. Astron. Astrophys. 557, A131 (2013)
[16] A. Deur, Mon. Not. Roy. Astron. Soc. 438, 2, 1535 (2014)
[17] A. Deur, C. Sargent and B. Terzic, arXiv:1909.00095 [astro-ph.GA]
[18] A. Deur, Eur. Phys. J. C 79, 883 (2019) 

Reference [16] in the 2020 paper is:

The QCD running coupling ${\alpha }_s(Q^2)$ sets the strength of the interactions of quarks and gluons as a function of the momentum transfer Q. The $Q^2$ dependence of the coupling is required to describe hadronic interactions at both large and short distances. In this article we adopt the light-front holographic approach to strongly-coupled QCD, a formalism which incorporates confinement, predicts the spectroscopy of hadrons composed of light quarks, and describes the low-$Q^2$ analytic behavior of the strong coupling ${\alpha }_s(Q^2)$. The high-$Q^2$ dependence of the coupling ${\alpha }_s(Q^2)$ is specified by perturbative QCD and its renormalization group equation. The matching of the high and low $Q^2$ regimes of ${\alpha }_s(Q^2)$ then determines the scale $Q_0$ which sets the interface between perturbative and nonperturbative hadron dynamics. The value of $Q_0$ can be used to set the factorization scale for DGLAP evolution of hadronic structure functions and the ERBL evolution of distribution amplitudes. We discuss the scheme-dependence of the value of $Q_0$ and the infrared fixed-point of the QCD coupling. Our analysis is carried out for the $\overline{\mathit{MS}}$, $g_1$, MOM and V renormalization schemes. Our results show that the discrepancies on the value of ${\alpha }_s$ at large distance seen in the literature can be explained by different choices of renormalization schemes. We also provide the formulae to compute ${\alpha }_s(Q^2)$ over the entire range of space-like momentum transfer for the different renormalization schemes discussed in this article.

Alexandre Deur, Stanley J. Brodsky and Guy F. de Termond, "On the interface between perturbative and nonperturbative QCD" 90 Prog. Part. Nucl. Phys. 1 (2016).

We check whether General Relativity's field self-interaction alleviates the need for dark matter to explain the universe's large structure formation. We found that self-interaction accelerates sufficiently the growth of structures so that they can reach their presently observed density. No free parameters, dark components or modifications of the known laws of nature were required. This result adds to the other natural explanations provided by the same approach to the, inter alia, flat rotation curves of galaxies, supernovae observations suggestive of dark energy, and dynamics of galaxy clusters, thereby reinforcing its credibility as an alternative to the dark universe model.

Alexandre Deur, "Effect of gravitational field self-interaction on large structure formation" arXiv: 2018:04649 (July 9, 2021) (Accepted for publication in Phys. Lett. B) DOI: 10.1016/j.physletb.2021.136510

Deur's approach to gravity, emphasizing gravitational field self-interactions in weak fields, that are generally neglected on the assumption that they are negligible in aggregate effect, is used to explain the Cosmic Microwave Background power spectrum which is the crowing achievement of the LambdaCDM model.

Field self-interactions are at the origin of the non-linearities inherent to General Relativity. We study their effects on the Cosmic Microwave Background anisotropies. We find that they can reduce or alleviate the need for dark matter and dark energy in the description of the Cosmic Microwave Background power spectrum.

A. Deur, "Effect of the field self-interaction of General Relativity on the Cosmic Microwave Background Anisotropies" arXiv:2203.02350 (March 4, 2022).

We investigate a correlation between the dark matter content of elliptical galaxies and their ellipticity that was initially reported in 2014. We use new determinations of dark matter and ellipticities that are posterior to that time. Our data set consists of 237 elliptical galaxies passing a strict set of criteria. We find a relation between the mass-to-light ratio and ellipticity that is well fit by M/L = (14.1 ± 5.4)ϵ, which agrees with the result reported in 2014.

David Winters, Alexandre Deur, Xiaochao Zheng, "New Analysis of Dark Matter in Elliptical Galaxies" arXiv:2207.02945 (July 6, 2022) (published at 518 (2) MNRAS 2845-2852 (2023)).

One of the most important problems vexing the ΛCDM cosmological model is the Hubble tension. It arises from the fact that measurements of the present value of the Hubble parameter performed with low-redshift quantities, e.g., the Type IA supernova, tend to yield larger values than measurements from quantities originating at high-redshift, e.g., fits of cosmic microwave background radiation. It is becoming likely that the discrepancy, currently standing at 5σ, is not due to systematic errors in the measurements. 

Here we explore whether the self-interaction of gravitational fields in General Relativity, which are traditionally neglected when studying the evolution of the universe, can explain the tension. We find that with field self-interaction accounted for, both low- and high-redshift data are simultaneously well-fitted, thereby showing that gravitational self-interaction could explain the Hubble tension. Crucially, this is achieved without introducing additional parameters.

Corey Sargent, Alexandre Deur, Balsa Terzic, "Hubble Tension and Gravitational Self-Interaction" arXiv:2301.10861 (January 25, 2023).

The short answer is probably no. Specifically, this paper considers a recent body of work which suggests that general relativity requires neither the support of dark matter halos, nor unconventional baryonic profiles, nor any infrared modification, to be consistent after all with the anomalously rapid orbits observed in many galactic discs. In particular, the gravitoelectric flux is alleged to collapse nonlinearly into regions of enhanced force, in an analogue of the colour-confining chromoelectric flux tube model which has yet to be captured by conventional post-Newtonian methods. However, we show that the scalar gravity model underpinning this proposal is wholly inconsistent with the nonlinear Einstein equations, which themselves appear to prohibit the linear confinement-type potentials which could indicate a disordered gravitational phase. Our findings challenge the fidelity of the previous Euclidean lattice analyses. We confirm by direct calculation using a number of perturbation schemes and gauges that the next-to-leading order gravitoelectric correction to the rotation curve of a reasonable baryonic profile would be imperceptible. The `gravitoelectric flux collapse' programme was also supported by using intragalactic lensing near a specific galactic baryon profile as a field strength heuristic. We recalculate this lensing effect, and conclude that it has been overstated by three orders of magnitude. As a by-product, our analysis suggests fresh approaches to (i) the fluid ball conjecture and (ii) gravitational energy localisation, both to be pursued in future work. In summary, whilst it may be interesting to consider the possibility of confinement-type effects in gravity, we may at least conclude here that confinement-type effects cannot play any significant part in explaining flat or rising galactic rotation curves without dark matter halos.

W. E. V. Barker, M. P. Hobson and A. N. Lasenby, "Does gravitational confinement sustain flat galactic rotation curves without dark matter?" arXiv:2303.11094 (March 20, 2023).

We comment on the methods and the conclusion of Ref. [1], "Does gravitational confinement sustain flat galactic rotation curves without dark matter?" The article employs two methods to investigate whether non-perturbative corrections from General Relativity are important for galactic rotation curves, and concludes that they are not. This contradicts a series of articles [2-4] that had determined that such corrections are large. We comment here that Ref. [1] use approximations known to exclude the specific mechanism studied in [2-4] and therefore is not testing the finding of Refs. [2-4].

Alexandre Deur, "Comment on "Does gravitational confinement sustain flat galactic rotation curves without dark matter?'' arXiv:2306.00992 (May 13, 2023).

Context. Deur (2014) and Winters et al. (2023) proposed an empirical relation between the dark to total mass ratio and ellipticity in elliptical galaxies from their observed total dynamical mass-to-light ratio data M/L = (14.1 +/- 5.4){\epsilon}. In other words, the larger is the content of dark matter in the galaxy, the more the stellar component would be flattened. Such observational claim, if true, appears to be in stark contrast with the common intuition of the formation of galaxies inside dark halos with reasonably spherical symmetry. 

Aims. Comparing the processes of dissipationless galaxy formation in different theories of gravity, and emergence of the galaxy scaling relations therein is an important frame where, in principle one could discriminate them. 

Methods. By means of collisionless N-body simulations in modified Newtonian dynamics (MOND) and Newtonian gravity with and without active dark matter halos, with both spherical and clumpy initial structure, I study the trends of intrinsic and projected ellipticities, Sérsic index and anisotropy with the total dynamical to stellar mass ratio. 

Results. It is shown that, the end products of both cold spherical collapses and mergers of smaller clumps depart more and more from the spherical symmetry for increasing values of the total dynamical mass to stellar mass, at least in a range of halo masses. The equivalent Newtonian systems of the end products of MOND collapses show a similar behaviour. The M/L relation obtained from the numerical experiments in both gravities is however rather different from that reported by Deur and coauthors.

Pierfrancesco Di Cintio "Dissipationless collapse and the dynamical mass-ellipticity relation of elliptical galaxies in Newtonian gravity and MOND" arXiv:2310.12114 (October 18, 2023).

See also:

We investigate the possible existence of graviballs, a system of bound gravitons, and show that two gravitons can be bound together by their gravitational interaction. This idea connects to black hole formation by a high-energy 2→N scattering and to the gravitational geon studied by Brill and Hartle. Our calculations rely on the formalism and techniques of quantum field theory, specifically on low-energy quantum gravity. By solving numerically the relativistic equations of motion, we have access to the space-time dynamics of the (2-gravitons) graviball formation. We argue that the graviball is a viable dark matter candidate and we compute the associated gravitational lensing.

B. Guiot, A. Borqus, A. Deur, K. Werner, "Graviballs and Dark Matter" (June 3, 2020 revised September 3, 2020). A follow up paper from February 9, 2022 is here.

Deur's work on the apparent dark matter fraction in elliptical galaxies and its relation to galaxy morphology is cited at:

Valentina Cesare, "Dark Coincidences: Small-Scale Solutions with Refracted Gravity and MOND" arXiv:2301.07115 (January 17, 2023) (at page 23). The abstract of that paper is as follows:

General relativity and its Newtonian weak field limit are not sufficient to explain the observed phenomenology in the Universe, from the formation of large-scale structures to the dynamics of galaxies, with the only presence of baryonic matter. The most investigated cosmological model, the ΛCDM, accounts for the majority of observations by introducing two dark components, dark energy and dark matter, which represent ∼95% of the mass-energy budget of the Universe. Nevertheless, the ΛCDM model faces important challenges on the scale of galaxies. For example, some very tight relations between the properties of dark and baryonic matters in disk galaxies, such as the baryonic Tully-Fisher relation (BTFR), the mass discrepancy-acceleration relation (MDAR), and the radial acceleration relation (RAR), which see the emergence of the acceleration scale a0≃1.2×10−10 m s−2, cannot be intuitively explained by the CDM paradigm, where cosmic structures form through a stochastic merging process. An even more outstanding coincidence is due to the fact that the acceleration scale a0, emerging from galaxy dynamics, also seems to be related to the cosmological constant Λ. Another challenge is provided by dwarf galaxies, which are darker than what is expected in their innermost regions. These pieces of evidence can be more naturally explained, or sometimes even predicted, by modified theories of gravity, that do not introduce any dark fluid. I illustrate possible solutions to these problems with the modified theory of gravity MOND, which departs from Newtonian gravity for accelerations smaller than a0, and with Refracted Gravity, a novel classical theory of gravity introduced in 2016, where the modification of the law of gravity is instead regulated by a density scale.

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