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:

  1. 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 $a^{-3}$.
  2. 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.
  3. The Friedmann equation gives that if $\rho$ is constant over time, then the pressure $p$ must also be constant and must equal $-\rho$.
  4. 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:

  1. 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?

  2. 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"?

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    $\begingroup$ relevant the comments and links therein n this question on the same subject physics.stackexchange.com/questions/750644/… $\endgroup$
    – anna v
    Commented Feb 23, 2023 at 5:22
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    $\begingroup$ Note, that this paper is first and foremost about observational evidence while most of the details for scenario with discrete sources of dark energy were worked out in a series of papers by Croker et al. Implications of Symmetry and Pressure in Friedmann Cosmology (part 1, part 2, part 3). $\endgroup$
    – A.V.S.
    Commented Feb 23, 2023 at 10:23
  • $\begingroup$ Also related: physics.stackexchange.com/questions/682414/… $\endgroup$
    – benrg
    Commented Feb 24, 2023 at 6:36

2 Answers 2


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 $k=0$:

  • 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)

  • $\begingroup$ In your point #1, do you mean that the scale factor is not defined within a single coordinate patch, or it isn't defined even globally? I thought that the scale factor (suitably defined) was physically measurable in principle. $\endgroup$
    – tparker
    Commented Feb 25, 2023 at 15:20
  • $\begingroup$ Yes, upon reading these accounts, my first impression was that this proposal was fundamentally misguided, for exactly the reasons that you give in your answer. But then (as I mentioned in another SE question) I started to question my initial assumption, because physicists of Bob Wald's caliber were taking the time to hold interviews with news outlets about this and only raise (what seemed to me to be) quite minor objections about the numbers not quite working out right at the quantitative level - which made me wonder if maybe the underlying idea did somehow make sense after all. $\endgroup$
    – tparker
    Commented Feb 25, 2023 at 15:26
  • $\begingroup$ I think I'm now satisfied that my initial impression was correct and this idea is fundamentally misguided, but I remained confused about the sociological aspect. It's very strange to me that fairly serious science journalists have quoted several GR experts giving either cautiously supportive reactions or skepticism on narrow technical grounds, but no GR experts saying "This entire proposal is fundamentally wrong," which you seem to present as being fairly obvious. $\endgroup$
    – tparker
    Commented Feb 25, 2023 at 15:32
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    $\begingroup$ @tparker (A)dS and Minkowski space specifically have no well defined scale factor because of their high degree of symmetry, but most FLRW spacetimes do. An expanding universe made of black holes has a somewhat fuzzy FLRW time and scale factor at scales large enough that FLRW is a good approximation, but individual black holes can't see that large-scale shape. Re your other comments, I have only the information that you have. I'm more convinced by my arguments than by the papers, and I don't know why few people make those arguments. (I'm not alone FWIW.) $\endgroup$
    – benrg
    Commented Feb 25, 2023 at 22:06
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    $\begingroup$ @tparker The answer in my previous comment has some good references (at least some published). A possibly analogous situation: everyone knows that the Casimir effect shows that vacuum fluctuations are real. Jaffe pointed out years ago that that isn't true. He says "point out" because anyone with the appropriate background could work out what he did if they thought to try. I don't think it's obvious, but I do think that once it's pointed out it should be possible to recognize it as correct without relying on any authority's say-so. Unless I'm wrong... $\endgroup$
    – benrg
    Commented Feb 25, 2023 at 22:10

I'd say most GR physicists are overconfident in their knowledge about FLRW metric. I was surprised as well, but digging a little bit into the literature that the paper presented in the introduction section, I noticed that it is a plain fact that we don't even have an exact solution to BH in FLRW background. Somebody confused the general ansatzs to exact solutions, while they are definitely not, and depending on the solution to those ansatzs coefficients, the BH mass could evolve.

So the possibility that the coupling solution as proposed in the paper, in simply GR and nothing exotic, does exist. But it is just a possibility, an open question. Who knows, many people showing a tough face to the public might be pulling an over-nighter to carry out their computations! And I would say that it is not that counter-intuitive to understand how BH works as a negative pressure object. To external observers, BH horizon could serve as a mirror reflecting the incoming modes, only reflecting onto a thermal background. In GR, when you are an outside observer, you will never see a particle `reach' the BH, the time consumed for that event to happen is indefinitely stretched as the particle travels closer and closer to the event horizon. One could buy it or not, such a phenomenon can be regarded effectively as a repulsive, or negative pressure physics process. And we are outside observers, at least with respect to those distant astrophysical BHs.

In summary, there is enough space in theory for this paper to work on. What is more concerning is the observation side. The BH observing experts around me do not have a very positive attitude to this paper as they believe the coupling evidence shown in this paper is just a selection effect. At higher redshift, further distance, the mass that you could measure for the BHs and the galaxies could be biased by whether you can see them, some galaxies or quasars might be too faint to see. This bias is different for galaxy and BH targets, and depending on how such bias difference evolves with redshift (scale factor), it might produce the illusion that BH mass is coupled to the scale factor.

So let us wait for more observation papers. This paper has definitely lit a fire.

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    $\begingroup$ While BH is local, it exists almost forever in semi-classical regime (only dissembled by Hawking radiation). So it is not local to the time dimension, thus the scale factor dimension. In cosmology t and scale factor could be used almost interchangeably. $\endgroup$
    – Angela
    Commented Nov 9, 2023 at 2:47

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