More On Gravitational Self-Interactions

Deur's research program is arguably the most promising in quantum gravity today.

He demonstrates how modeling the self-interaction of the graviton in the case of a scalar graviton can reproduce all dark matter phenomena, predict previously unobserved correlations between the extent to which an elliptical galaxy and its apparent dark matter content, and explain at least some dark energy phenomena, without (in principle at least) introducing any new fundamental physical constants to general relativity, or any new particles beyond the Standard Model other than the graviton.

His model also plausibly suggests that the key insights secured from the self-interacting scalar graviton regime and by analogy to QCD might not be materially altered by generalizing these result to the full tensor graviton.

The elegance with which his research explains a huge range of phenomena that seem to call for beyond the Standard Model (and general relativity) physics, with minimal tweaks to existing knowledge, and integrates quantum gravity into a Standard Model-like framework, in the face of extreme computational barriers to addressing these questions in a brute force numerical manner, is remarkable.

We study two self-interacting scalar field theories in their strong regime. 

We numerically investigate them in the static limit using path integrals on a lattice. We first recall the formalism and then recover known static potentials to validate the method and verify that calculations are independent of the choice of the simulation's arbitrary parameters, such as the space discretization size. The calculations in the strong field regime yield linear potentials for both theories. 

We discuss how these theories can represent the Strong Interaction and General Relativity in their static and classical limits.  

In the case of Strong Interaction, the model suggests an origin for the emergence of the confinement scale from the approximately conformal Lagrangian. The model also underlines the role of quantum effects in the appearance of the long-range linear quark-quark potential. 

For General Relativity, 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 in their strong regime" (November 17, 2016) (Hat tip to Viljami from the comments).

From the body text:

The (gφ∂_µφ∂^µφ + g²φ²∂_µφ∂^µφ) theory calculations in strong field regime yield a static potential which varies approximately linearly with distance, see Figs. 5 and 6. This can be pictured as a collapse of the three dimensional system into one dimension. As discussed in Section V D and shown numerically, typical galaxy masses are enough to trigger the onset of the strong regime for GR.

Hence, for two massive bodies, such as two galaxies or two galaxy clusters, this would result in a string containing a large gravity field that links the two bodies –as suggested by the map of the large structures of the universe. That this yields quantitatively the observed dark mass of galaxy clusters and naturally explains the Bullet Cluster observation [18] was discussed in [19]. For a homogeneous disk, the potential becomes logarithmic. 

Furthermore, if the disk density falls exponentially with the radius, as it is the case for disk galaxies, it is trivial to show that a logarithmic potential yields flat rotation curves: a body subjected to such a potential and following a circular orbit (as stars do in disk galaxies to good approximation) follows the equilibrium equation:

v(r) = √ G'M(r), (14)


with v the tangential speed and M(r) the disk mass integrated up to r, the orbit radius. G' is an effective coupling constant of dimension GeV⁻¹ similar to the effective coupling σ (string tension) in QCD. Disk galaxies density profiles typically fall exponentially: ρ(r) = M₀e^−r/r0/(2πr₀² ), where M₀ is the total galactic mass and r_0/ is a characteristic length particular to a galaxy. Such a profile leads to, after integrating ρ up to r:

v(r) = √ G'M₀ (1 − (r/r₀ + 1)e^−r/r₀), (15)

At small r, the speed rises as v(r) approximately equal to √ G'M₀r/r₀ and flatten at large r: v approximately equal to √ G'M₀.

This is what is observed for disk galaxies and the present approach yields rotation curves agreeing quantitatively with observations [19]. For a uniform and homogeneous spherical distribution of matter, the system remains three-dimensional and the static potential stays proportional to 1/r. The dependence of the potential with the system’s symmetry suggested a search for a correlation between the shape of galaxies and their dark masses [19]. Evidence for such correlation has been found [20].

VI. SUMMARY AND CONCLUSION 

We have numerically studied non-linearities in scalar field theory.

Limiting ourselves to static systems allowed us to greatly simplify and speed-up the numerical calculations while providing a possible description of the strong regimes of the Strong Interaction and of General Relativity in the static case. The overall validity of the method is verified by recovering analytically known potentials. We further verified the validity of the simplifications in the case of General Relativity by recovering the post-Newtonian formalism. 

Lattice gauge calculations of QCD are well advanced. What justifies developing the present approximation is that it provides fast calculations that can be run on any personal computer, and it may help to isolate the important ingredients leading to confinement. This method is able to provide: 1) the expected field function, 2) a mechanism –straightforwardly applicable to QCD– for the emergence of a mass scale out of a conformal Lagrangian, 3) a running of the field effective mass in qualitative agreement with that seen in QCD, and 4) a potential agreeing with the phenomenological Cornell linear potential up to 0.8 fm, the relevant range for hadronic physics. Quantum effects, which cause couplings to run, are necessary for producing the linear potential and must be supplemented to the approach. A non-running coupling, even with a large value, would only yield a short range Yukawa potential. 

An important benefit of this method is that it may apply to theories, such as General Relativity, that are too CPU-demanding to be easily computed on a lattice. That the method recovers the essential features of QCD supports its application to GR. Since GR is a classical theory, no running coupling needs to be supplemented.

In QCD, the strong regime arises for distances greater than 2 × 10⁻¹⁶ m, while it should arise for gravity at galactic scales: Two massive bodies, such as galaxies, or at larger scale two galaxy clusters, would then be linked by a QCD-like string/flux tube. This would explain the universe’s large scale stringy structure observed by weak gravitational lensing. It also agrees quantitatively with cluster dynamics [19]. Furthermore, in the case of massive disks of exponentially falling density, such as disk galaxies, the logarithmic potential resulting from the strong field non-linearities trivially yields flat rotation curves. Those agree quantitatively with observations [19]. Finally, for a uniform and homogeneous spherical distribution of matter, the non-linearity effects should balance out [19]. Evidences for the consequent correlation expected between galactic ellipticity and galactic dark mass have been found [20].

Alexandre Deur has had some key insights into how to explore quantum gravity. 

The first has been to exploit parallels arising from the fact that both the strong force theory of  QCD and gravity involve self-interacting carrier bosons. 

The second has been to simplify the analysis of gravity by starting from a scalar graviton rather than a spin-2 tensor graviton. This produces identical results in a static case.  Even in the dynamic case, the deviations from the static case due to tensor components of linear momentum, angular momentum, pressure and electromagnetic flux are often negligible.

Also, the adjustments due to using a self-interacting tensor graviton (full GR) relative to a self-interacting scalar graviton (simplified GR) are unlikely to cancel out the differences between a self-interacting scalar graviton (simplified GR) and a non-self interacting scalar graviton (Newtonian gravity plus propagation at the speed of light and couplings to energy as well as mass).

His description of the impact in GR as pertaining to the "strong field regime" is somewhat misleading, because the effects in question, while involving large masses that generate gravitational fields much stronger than those generated by small masses, are generally only observable when the gravitational field is weak. In stronger gravitational fields (such as those in the vicinity of black holes, or even those involving solar system gravitational fields) the first order effects of gravitons pulling on other objects overwhelm any visible second order effects due to graviton self-interactions (particularly in the circularly symmetrical case).