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.
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.
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.
Despite the fact that he claims that his theory is consistent with General Relativity, this is not accurate, even though in the limit of spherical systems and strong gravitational fields, it closely approximates General Relativity's predictions.
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.
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, in the circumstances 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 shows 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).
* 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. This also explains why quantum gravity researchers trying to build a quantum gravity theory that exactly reproduces Einstein's equations have failed. They have tried to reproduce a slightly erroneous equation and the theoretical difficulties with doing this become more apparent in the quantum gravity context.
* 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.
* 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 (other than the graviton found in all quantum gravity theories), (2) no new forces, and (3) one fewer fundamental physical constants than the existing core theory of the Standard Model and General Relativity (without dark matter).
* 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 and massless gravitons.
* 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 basically Newton's constant G, although possibly in different units. 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 even more new physical constants than MOND does.
* Deur's theory harmonizes gravity and the Standard Model with no particles beyond the Standard Model other than the massless graviton. The deep theoretical inconsistencies of the two models that make up core theory are eliminated (almost). Deur's formulation of the theory as a quantum field theory simplifies its integration as a quantum gravity theory with the Standard Model, which is also a quantum field theory.
* 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.
Empirical parallels between Cosmology and Hadronic Physics
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.
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
(M, hadron mass, J angular momentum)
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.
This is compared to the QCD Lagrangian:
This is a function, roughly speaking, of system mass and system size:
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.
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:
FG = GNM/r2 + c2(aπGNM)1/2/(2√2)r
where FG is the effective gravitational force, GN 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 modifies is proportionate to (GNM)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.
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.
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.
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.
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.
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 . 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 . 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 .
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.
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.
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.
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.
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.
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.
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  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)  and by classic numerical lattice calculations for v = 0 quarks . 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 .
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. ; (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.
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The QCD running coupling sets the strength of the interactions of quarks and gluons as a function of the momentum transfer Q. The 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- analytic behavior of the strong coupling . The high- dependence of the coupling is specified by perturbative QCD and its renormalization group equation. The matching of the high and low regimes of then determines the scale which sets the interface between perturbative and nonperturbative hadron dynamics. The value of 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 and the infrared fixed-point of the QCD coupling. Our analysis is carried out for the , , MOM and V renormalization schemes. Our results show that the discrepancies on the value of at large distance seen in the literature can be explained by different choices of renormalization schemes. We also provide the formulae to compute over the entire range of space-like momentum transfer for the different renormalization schemes discussed in this article.
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.
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.
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.