The authors asked themselves how their quantum gravity theory differs from General Relativity, and whether the successes of General Relativity in astrophysical settings would be fatal if their theory has strong differences, and that's the basis for this paper. The tl;dr is that their theory predicts different trajectories outside of large central masses, but that might not conflict with evidence from galactic-dynamics astronomy.
This is the second paper released in the past few days by the University College London Oppenheim group. It's a preliminary investigation of the longer length scale features of their classical stochastic theory. The central question is how its version of Schwarzschild-de Sitter (SdS) differs from standard General Relativity.
The first paper, and I think the more interesting one, is about the short length scale aspects of their asymptotically free theory, in which the gravitational interaction weakens as distances between interacting sources decreases. The asymptotic freedom means the theory is amenable to renormalization, unlike perturbative quantum gravity and a number of other approaches. That paper is at <https://arxiv.org/abs/2402.17844>. Note that they do not know how to make the gravitational part quantum mechanical without introducing problems (i.e., it is haunted by "bad" ghosts in the sense of <https://en.wikipedia.org/wiki/Ghost_(physics)>); their classical and stochastic gravitational sector is ghost-free (a point also made at the end of Appendix A in the large-scale paper), and it is reasonable for them to believe that could be good enough that it's worth continuing to investigate what the theory predicts and how its parameters are set.
The second paper was motivated by the first: "The theory was not developed to explain dark matter, but rather, to reconcile quantum theory with gravity. However, it was [noted] that diffusion in the metric could result in stronger gravitational fields when one might otherwise expect none to be present, and that this raised the possibility that gravitational diffusion may explain galactic rotation curves".
That MOND-like effects might arise in their approach to the problem of small-scale quantum gravity is at least interesting. It was not the starting point.
Moreover, they did not start with the idea of modifying General Relativity to get rid of the need for (some or all) cold dark matter. As they say: "While this study demonstrates that galactic rotation curves can undergo modification due to stochastic fluctuations, a phenomenon attributed to dark matter, it is important to acknowledge the existence of separate, independent evidence supporting ΛCDM. In particular, in the CMB power spectrum, in gravitational lensing, in the necessity of dark matter for structure formation, and in a varied collection of other methods used to estimate the mass in galaxies."
> Now explain the Bullet Cluster
This paper does not seek to do so. "To make it tractable analytically, we have restricted ourselves
to spherically symmetric and static spacetimes, with metrics of the form of Eqs. (17)." Eqn 17 describes out an adapted Schwarzschild-de Sitter spacetime and leans on an argument that Birkhoff's theorem applies (in particular that their model spacetime is stable against certain perturbations, notably those concentric upon the source mass). There is further detail in Appendix B.
Studying this restricted model, the de Sitter expansion of the spacetime and MOND-like anomalous Kepler orbits at some remove from the Schwarzschild central mass are in their theory driven by entropic forces generated by the fluctuations in the gravitational field of the central mass (and they do a good job in Appendix D explaining this).
In GR's Schwarzschild-de Sitter the free-fall trajectories of test particles around the central mass are totally determined by the mass; the gravitational field doesn't fluctuate. The (Boltzmann) gravitational entropy of the region outside the central mass is everywhere very high.
In GR-SdS we can consider adaptations where with M=const. we turn the pointlike central mass into a spherically symmetric shell, or a concentric set of such shells, or even a ball of fluid, or a ball of dust, or a ball of stars and other galactic matter. None of these symmetry-preserving adaptations changes the free-fall trajectories of test particles outside the outer surface, or the gravitational entropy at any outside point.
In the author's theory, the spacetime is stochastic. It fluctuates. Close to the central mass fluctuations are unnoticeably small; the gravitational entropy is very low. Far from the central mass the gravitational entropy is very high, and gravitational fluctuations are noticeable. A sort of thermodynamics leads to a diffusive flow outwards from the central mass, from the low entropy near there to the high entropy at increasing radial distance. This diffusion is carefully constructed so that the outwards flow is only really appreciable at large-scale distances. The effect is that large-radius orbits are statistically pulled inwards by something describable as stronger gravity at larger radiuses (see around Eqn (21)). This is an "entropic force", very roughly analogous to squashing a sponge ball in your hand then releasing the pressure and watching the sponge ball expand, where the material of the sponge represents the gravitational field.
Their stochastic fluctuations are still generated by the spherically-symmetric central mass. These fluctuations break the spherical symmetry of the outside metric. Consequently they have to do some work to make the outside metric look appropriately Schwarzschild-like in their "diffusion regime", and to keep that stable against the stochastic perturbations.
The authors contend that with reasonable choices of parameters, and restricted to static spherical symmetry of the central mass (and no additional dynamics), this effect comes close to duplicating MOND's low-acceleration regime.
They don't go into anything like a backreaction upon the Schwarzschild metric by large fluctuations.
(They do have an idea about how to get the de Sitter trajectories though, but that doesn't fit very naturally into this comment, which is already long.)
> Bullet cluster
The authors know full well that the metric for a gravitationally bound cluster of galaxies isn't well-represented by their choice of SdS-like metric. A galaxy cluster is too lumpy for the Schwarzschild part.
Two gravitationally bound galaxy clusters having passed through each other (trailing collided gas and dust, and tidally stripped stars and other matter) is even less like Schwarzschild. This is because SdS solutions of the Einstein Field Equations do not linearly superpose. So their metric is a poor description of any sort of "close call" interaction between galaxies or galaxy clusters, even if the individual components are "close enough" to Schwarzschild from the perspective of an observer sufficiently large (as in cosmologically large) distances. They do not (and within this initial paper should not really be expected to) offer a more suitable metric. I'm sure they'd love to look into things like that though.
The non-linear superposeability of useful solutions of General Relativity is a problem for asking how astrophysics differ in most theories that preserve the equivalence principle (this one does, it's a metric theory of gravitation). As the replacement for the Einstein Field Equations lose symmetries (sphericity, staticity) they tend to become analytically intractable and non-numerical approximations become unreliable.
The authors -- imho in a strikingly principled way -- call attention to various difficulties in using this work to describe astrophysical systems, particulary from the middle of the fifth page of the PDF.
They are not obviously worse off than the Verlinde programme of emergent-entropic gravity, where the gravitational field is generated by entropic forces rather than vice-versa.
