Does exponential expansion ensue locally if in a certain hubble patch vacuum dominates?

As a concrete example, consider a universe scattered with low density extremely massive objects (e.g. black holes) so that there is 1 of these per many many Hubble patches. (schematically in drawing) In a random Hubble patch where there is no BH, vacuum energy clearly dominates. However, suppose that if we averaged over many many many Hubble patches the energy density in BHs would actually be greater. How does this universe evolve? Does the scale factor $$R(t)$$ follow the evolution of a matter dominated FRW universe? I suspect rather that the vast regions of space between the BHs will expand exponentially due to the cosmological constant and become completely decoupled from the BHs. How would one quantify conditions for this?

[The following is not vital to the question, just some extra thoughts concerning the example. Notice that the radius of the BH can be at most $$H^{-1}$$ (This H is the Hubble parameter when considering the scales over which matter domination is eveident) and so its top mass is $$1/2GH$$. Their energy density is thus at most $$\frac{1}{2GH} \frac{1}{N H^{-3}}$$, where N is the number of Hubble patches per 1 BH. To ensure that this is greater than the vacuum energy density $$\rho_\Lambda$$ one has a upper limit on $$N$$ $$N<\frac{1}{2GH^{-2} \rho_\Lambda},$$ although, for the purposes of an example, $$\rho_\Lambda$$ could be taken to as small as one likes. A very small value however would mean the expansion, although exponential $$R(t) \sim e^{H_\Lambda (t-t_0)}$$, would not really kick in until a time $$\sim H_\Lambda^{-1}= (8 \pi G \rho_\Lambda / 3)^{-1/2}$$. I'm wondering if this time is ever long enough to allow to 'causally latch on' the various BHs together so that one can then legitimately consider the system as a whole... ]

• This is an interesting question. I'm not sure that it has a definite answer unless you specify some boundary conditions. E.g., exponential contraction is also a solution to the Friedmann equations in the vacuum-dominated case, and other solutions are possible if there is no symmetry. Their energy density is thus at most 12GH1NH−3, where N is the number of Hubble patches per 1 BH. You can't really do this. The equation of state of dark energy is not the same as the equation of state of a gas of black holes. Black holes are basically a dust in this context, i.e., nonrelativistic matter. – Ben Crowell Mar 20 at 18:41
• I understand that the two have different equations of state and therefore the Hubble parameter most likely changes from patch to patch. I imagine that it is $H_\Lambda$, as I wrote, in the patches without BHs. In the patches containing a BH I'm a bit confused. Initially I thought there would be matter domination, but really the spread of density is incredibly inhomogeneous. I'm not sure even if one single BH would 'drive expansion'. Yet the fact remains that at some enormous scales the system is homogenous and matter dominated... – Rudyard Mar 20 at 19:23
• I guess I don't know, when considering a homogeneous universe made of 'dust' and therefore claiming the metric is FRW with $R(t) \propto t^{2/3}$, whether it matters if the dust particles are causally connected to one another. – Rudyard Mar 20 at 19:35
• Could you rewrite your “extra thoughts” part in terms of cosmological constant $\Lambda$ (the emphasis on constant) instead of Hubble parameter (which is defined only for particular class of solutions)? – A.V.S. Mar 21 at 8:15