timmdeeg said:
I had the notion that the replacement of dark matter and dark energy by Deur's field self-interaction wouldn't change the development of the universe.
The replacement almost certainly changes the development of the universe (e.g. in the early universe, it leads to earlier galaxy formation).
Deur's replacement is not identical to dark energy which is a constant instead of an emergent function of the manner in which baryonic matter clumps itself in the course of this history of the universe.
timmdeeg said:
Deur ##3\ddot{a}/a=-4\pi{G}(\rho+3p)(1+\alpha)## (14) - whereby ##(1+\alpha)## accounts for the anisotropy of this universe.
L-CDM ##3\ddot{a}/a=-4\pi{G}(\rho+3p)+\Lambda##
So in the very far future the L-CDM universe reaches a state where it expands exponentially, driven by ##\Lambda## whereas Deur's universe seems to collapse in case ##(1+\alpha)>0##.
I have no doubt that the evolution of the universe is different from LambdaCDM.
But, I'm not sure how it works.
I don't have the GR and cosmology chops to rigorously evaluate the math.
But I do think that I have enough of an understanding to make some qualitative observations and to suggest some heuristic ways of thinking about the question and different components that would go into the overall analysis.
The Depletion Function Used In The Paper May Have A Limited Domain Of Applicability
One point I would make to urge caution, however, is that the the depletion function that Deur is using in the cited paper is an approximation approach used merely to simplify intractable exact calculations. The depletion function does not in and of itself represent a laws of nature, per se, and it is more of a statistical average effect than it is an analytical conclusion with basic assumptions involved the way that the Friedman equation is. It is closer to statistical mechanics than to the Friedman equation conceptually.
This approximation may have a domain of applicability that doesn't extend all of the way to the end of the universe tens of billions of years from now. Instead, this approximation's domain of applicability may differ from the FLW approximation typically used in cosmology. Deur's depletion function may be (and probably is) less general with a narrower domain of applicability. It might not be valid, for example, ten billion, or one hundred billion years in the future from now.
The Impact Of Conservation of Energy v. Dark Energy
The biggest fundamental reason to suspect a different late-time history of the universe between Deur's cosmology and LambdaCDM is that the growth in the amount and proportion of dark energy in a LambdaCDM universe violates conservation of mass-energy (you can make a convoluted argument that it is converting gravitational potential energy into dark energy but it doesn't really ring true).
In contrast, Deur's approach doesn't have a cosmological constant and conserves mass-energy not just locally but globally.
In Deur's analysis, phenomena attributed to dark energy arise from weaker gravitational fields between galaxies and between galaxy clusters because the gravitational pull ends up being focused on holding matter at the fringes of galaxies and galaxy clusters where gravitational fields are weak more tightly than in Newtonian gravity (which conventionally applied GR says is a near perfect approximation of GR in very weak fields).
This distinction should matter more and more as the proportion of dark energy in the total mass-energy budget of the universe gets larger in the far future late universe in LambdaCDM, and as the already diminished external fields of galaxies and galaxy clusters reach a point where making those fields even weaker doesn't change the strength of that as much in relative terms.
A Deur style cosmology analyzed in the context of a LambdaCDM model keeps the aggregate amount of inferred dark matter and inferred dark energy closer to each other, since they have a common source.
In contrast, in a LambdaCDM model, the model dependent by observationally based conclusion that the amount of dark matter in the mass-energy budget of the universe and the amount of dark energy in the mass-energy budget of the universe differ by only a factor of two is just a coincidence that is a function of when we happen to be observing it. In LambdaCDM, the early universe has much less dark energy than dark matter and the later universe has vastly more dark energy than dark matter, because the amount of dark matter is constant, while the amount of dark energy grows proportionately to the volume of space within the Big Bang's light cone.
In the present era and earlier parts of the universe, the observational impact of removing a pull between objects and imposing an external dark energy pull two is similar enough to be hard to distinguish within the range of uncertainty in cosmological constant measurements. These measurements are not very precise and has lots of methodological issues that are not fully resolved that are coming out in the Hubble tension debate, and the uncertainties in the underlying observations and in the analysis are also probably greatly understated.
But, at some point taking away a pull between objects becomes very different from adding an external pull from dark energy.
This suggests that the dark energy phenomena driven expansion in Deur's cosmology of the universe should lose steam in the far future, gradually and eventually, relative to what one would expect in a cosmological constant/dark energy driven cosmology.
The Impact Of Higher Order GR Lagrangian Terms
Another factor to consider about the evolution of the universe in late time Deur's analysis is that Deur's work to date which considers the present universe and the past, is looking at only one or two terms in the GR Lagrangian beyond the Newtonian first order term.
But, in the ultra weak fields of a very dispersed, very late time universe, one would expect that one or two orders of terms in the Largrangian beyond the first and/or second order corrections from Newtonian gravity that he is using for the self-interaction effects may be significant (at least for some geometries of matter distribution which would be non-spherically symmetric, which we would expect if Deur's universe is more anisotropic than LambdaCDM predicts, something which is a pretty natural things to suspect in a Deur-like analysis since it already favor, for example disk-like collections of matter over spherically symmetric ones).
Intuitively, this should not come up in the early universe, however.
This is first because the early universe is closer to being spherically symmetric before structure has really evolved.
Secondly, in the early universe, everything in the universe is closer together so the strength of the gravitational field between say any two average galaxies or proto-galaxies should be much greater than in the far future late universe. So, the Newtonian and earlier order terms in the GR Lagrangian should swamp the later order terms then, making the higher order terms negligible by comparison.
This said, I really have no idea what the magnitude and direction of the corrections that the higher order Lagrangian terms would make to Newtonian gravity plus Deur's first and/or second order self-interaction term.
Certainly, the relative magnitude of the later terms relative to the earlier terms should grow larger as the universe expands, but I couldn't quantify it. And, again, I don't know if adding terms weakens or strengthens the self-interaction effects, or if it, for example, makes different kinds of geometries than those that matter for first order self-interactions more relevant.
The only conclusion I can state with much confidence about this factor is that in the very late universe, relying on the first order self-interaction term alone in Deur's analysis should be less and less accurate the later in time you get.
Long Term Galaxy and Galaxy Cluster Evolution
Another issue which may or may not be a factor is the long term evolution of galaxies and galaxy clusters.
The default analysis would be that the amount of inferred dark matter in galaxies or galaxy clusters evaluated in a LambdaCDM model should be very stable over time and not really evolve.
If that analysis is right, then the dark matter phenomena of LambdaCDM should be pretty similar in the late universe to those of Deur's cosmology. But, if the default analysis is wrong, and galaxies and galaxy clusters become more tightly bound over time with more inferred dark matter effects and less of the system's gravitational field escaping the system over time, then this would be a boost to dark energy phenomena in the earlier part of the late universe.
On the other hand, if galaxies and galaxy clusters tend to disintegrate over time, or to become more spherically symmetric through their merger histories over time, then you'd expect both the dark matter phenomena effects and the dark energy phenomena effects quantified in the eyes of a LambdaCDM model to decline as the universe gets older.
I don't have good intuition regarding whether a static evolution, a disintegrating evolution, or a more tightly bound evolution of the average galaxy and galaxy cluster makes more sense over the next many billions of years from the point where we are now.
