What does it mean that "relativistic material becomes cosmologically coupled to the expansion rate" in the recent dark-energy black-hole paper? The recent paper "Observational Evidence for Cosmological Coupling of Black Holes and its Implications for an Astrophysical Source of Dark Energy" has made a splash in the popular press. From what I can tell, all of the paper's claims come from eq. (1), the proposal that a black hole's mass $M$ depends on the scale parameter $a$ as
$$M(a) \propto a^3. \tag{1}$$
The authors describe this proportionality by saying that "relativistic material can become cosmologically coupled to the expansion rate." If I understand correctly, their subsequent reasoning is as follows:

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*As the universe expands, the volume containing a given set of black holes expands as $a^3$, so the number density $n$ of black holes decreases as $a^{-3}$.

*Since the mass of each black hole grows as $a^3$, this growth exactly the cancels the decrease in the number density, and the mass-energy density $\rho = M n$ of black holes stays constant over time.

*The Friedmann equation gives that if $\rho$ is constant over time, then the pressure $p$ must also be constant and must equal $-\rho$.

*Any fluid with constant positive energy density $\rho$ and constant negative pressure $p = -\rho$ is equivalent to a positive cosmological constant. In particular, it corresponds to an exponentially increasing scale factor (in the absence of any other forms of matter).

Moreover, the paper claims to present observational evidence that over cosmological time scales, the mass of black holes does seem to be scaling as $M \propto a^3$. I'll take this observational evidence as a given; its validity is outside the scope of this question.
I understand the logic above except for the first step, eq. (1). In the standard classical cosmological model, the mass of an isolated black hole stays constant over time (neglecting Hawking radiation, which is a quantum effect). In such a universe, without an explicit cosmological constant, a universe consisting of a uniform fluid of far-separated black holes would eventually contract. But this paper claims that a universe whose black holes grow with the expansion of the universe would experience acceleration expansion. This seems extremely counterintuitive to me: how can attractive black holes that grow in mass over time effectively "push each other away" and cause the universe to expand - an effect that I would ordinarily describe as repulsive? Is there any physical intuition here beyond "GR is counterintuitive; get used to it"?
I know that question is somewhat vague, so here are two sharper versions:

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*I know that causation gets tricky to talk about when we have fully dynamical differential equations, but is there any sense in which it's more correct to say that either (a) the accelerated expansion of the universe causes black holes to grow more massive, or (b) black holes' growing more massive causes the universe's expansion to accelerate? Or is the better way to think about it that (c) the same local physics leads to both a time-varying black hole mass $M(t)$ and a time-varying scale parameter $a(t)$, and coincidentally $M(t)$ happens to be proportional to $a(t)^3$, but neither quantity is a priori a function of the other one? I understand what it means for fields to be coupled together within a given Lagrangian or Hamiltonian, but I have no idea what it means for matter to be "coupled to the acceleration rate". That seems like a category error, since the acceleration rate is not a dynamical field but a parameter within a given family of solutions to a differential equation. More specifically, what does it mean for a black hole, which is somewhat localized within stellar length scales (despite its ultimately teleological nature), to be "coupled" to a scale factor that describes behavior that is averaged over cosmological length scales?


*This analysis operates at cosmological length scales, where we can model the black holes as forming a continuous fluid of uniform density. But what does a single black hole satisfying eq. (1) look like at length scales smaller than the typical inter-black-hole spacing - where one black hole is much closer than any other ones, and the fluid approximation breaks down? Would such a black hole look attractive or repulsive at various distances? Is the answer just "Near the center, the black hole is attractive as usual, and the notion of 'distance' is so inherently ambiguous in GR that we can't meaningful talk about whether it's attractive or repulsive far away from the center"?
 A: Their equation 1 actually says $M\propto a^k$, and they claim $k\sim 3$ for a certain class of hypothesized objects, "Vacuum energy interior solutions with cosmological boundaries". I guess these are the GEODEs (GEneric Objects of Dark Energy) of Croker & Weiner (2019).
There are various ways to see that general relativity requires $k=0$:

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*The scale factor, in general, is not even well defined. On any patch of de Sitter space you can put FLRW coordinates with any scale factor satisfying $\ddot a/a = Λ/3 \triangleq H^2$, such as $a=e^{Ht}$, $a=e^{-Ht}$, or $a=\sinh H(t-t_0)$. If you add a GEODE to the background, using coordinates that asymptotically approach the FLRW coordinates, its mass varies as $e^{kHt}$, $e^{-kHt}$, etc. But these are all the same object, in the same background.


*Mass can't come from nowhere, because it is associated with a long-range gauge field—the same reason that charge must be conserved in electromagnetism. If a black hole gains mass, then that mass must have been conveyed to it somehow, with attendant ripples in the spacetime geometry. Nothing of the sort seems to be suggested in these papers. They seem to think that an object can gain mass by just "eating the scale factor" in some sense. That's not a thing in general relativity.
If they were speculating about alternate theories of gravity then I would say their theory is unlikely to be right. But they claim to be working within standard general relativity, so I have to say that they are just wrong.
As far as I can tell the mistake in the 2019 paper was assuming that the universe is FLRW with perturbations that are everywhere small in a certain sense, even in the neighborhood of black holes. This is inconsistent with the existence of black holes. They even mention that in the paper, and suggest the existence of black holes should be questioned, on the basis of their completely unjustified assumption.
It's possible that if curvature were really limited in that way, it would force the quintessence inside GEODEs to "leak out" in some way, lest it violate the curvature limit—i.e., their derivation from their assumptions may be correct. But curvature is not limited in that way. They just made that up.
Related earlier question: Is there a GR explanation for cosmological coupling causing mass increase as the universe expands? (see both answers)
