Black Hole Swarm

[sevenperforce]

Some interesting research discussed by Phil Plait about black hole swarms in globular clusters:

In summary: we have observed many stellar-mass black holes and many supermassive black holes but no intermediate-mass black holes (IMBHs). It's odd because we would expect to find them somewhere. Astronomers were looking at globular clusters as potential candidates for containing IMBHs, but found that there is instead a swarm of stellar-mass black holes.

The Schwarzschild radius of the sun is about 3 km, which partially explains why these black holes don't coalesce into an IMBH. The collision cross-section is just extremely small, and in a cluster, the velocities will be extremely high, too high for the kind of gravitational frame-dragging that can slow black holes as they pass each other.

What would it look like if this swarm did finally collapse? Would it be a runaway chain reaction? What would the gravitational wave signal look like?

Another thought occurred to me. The Schwarzschild radius is directly proportional to mass, which is why the average density of a black hole decreases with size. A supermassive black hole actually has a relatively low average density; the ultramassive black hole in TON 618 has an average density of just 4.23 grams per cubic meter, half the density of hydrogen at STP.

I haven't done the maths on this yet, but what repeated mergers of globular clusters created a swarm of stellar-mass black holes so dense that it created a supermassive black hole without any actual collisions? What would that look like? Could that explain why we never see intermediate-mass black holes?

[K^2]

This is something you might want to reach out to a proper cosmologist or astrophysicist about. But I have two quick thoughts.

First of all, yes, merger of a cluster of stellar-mass black holes into a supermassive black hole is a runaway process. As you pointed out, Schwarzschild radius is proportional to the mass. That means the gravitational cross-section of a black hole goes as square of its mass.  And because the probability of collision is product of cross-sections, two 2M black holes are twice as likely to have a collision in a unit of time as four 1M black holes. There is a percentage of energy that's lost to gravity waves during merger, but it's not too high. So if it takes a billion years for your cluster to go from an average of 1 solar mass to 10 solar masses, it will only take 100My to go to an average of 100 solar masses, then 10My to go to 1,000 solar masses... And near the end, it's actually going to get even faster. The final black hole is likely to be comparable to size of the initial cluster, possibly a lot bigger. So you'll start running out of space, severely increasing the rate of merges.

What I can't tell you is what expected densities are for this runaway process to happen in time comparable to ages of galaxies. I suspect that critical density is very, very high, as stellar mass black holes are tiny. One thing that can speed it up drastically is if you start out with a swarm of stars with just a few black holes. Cross section between the star and stellar mass black hole is way higher than between two stellar mass black holes. But of course, now the black holes have to get way heavier before this is a runaway process again. Either way, all of these conditions sound like they'd be far more abundant in early universe. So one possibility for why we're not seeing this is because it just doesn't happen anymore. That's a very ad-hoc explanation, though.

Second, the signal. For the longest duration of this process, I don't expect anything special. You'll have a very occasional collision between two black holes, and we've already looked at these. And part of the problem is that while we have very clear expectations for what intermediate mass black hole merging with any other kind of black hole will look like, the signal isn't going to be very strong. This might be part of why we're not seeing a lot of that. If intermediate mass black holes are a brief note in a history of young galaxies, there might be nothing close enough for us to detect.

On the other hand, that final ring of black holes with thousands of solar masses merging into supermassive black hole should be distinctive, as you'll see a huge number of very heavy objects coming together almost at once. Trouble is, there might still not be any candidates close enough even at that strength, and also, I'm not sure we'll be able to recognize it. This is the sort of thing that will make any GR sim cry binary tears, and the only way we have right now to analyze signals is simulate event we think it might be, and compare signals.

But yeah, definitely run this by a cosmologist if you want concrete answers.

[sevenperforce]

7 minutes ago, K^2 said:

But yeah, definitely run this by a cosmologist if you want concrete answers.

You know how a kugelblitz would be formed by getting all the photons (which don't interact with each other) into a very small space at once?

That's sort of what I'm thinking. Ordinarily you get a black hole merger when two tightly-bound black holes are orbiting each other closely and frame-dragging begins to suck them in. They are already interacting. But what would happen if a bunch of black holes entered the same space at high rates of speed without being gravitationally bound, as in the center of two or more merging globular clusters? They really wouldn't be interacting with any of each other directly, just like the photons in a collapsing kugelblitz aren't interacting with each other. But yet there would nevertheless be a moment when the gravitational well collapsed everything into a black hole.

Consider two tightly-bound stellar-mass black holes being pulled together by frame-dragging:

The black discs are the individual event horizons of the two black holes, and the gray circle represents their combined event horizon. I'm not really sure at which point the merger takes place.

Obviously, it hasn't yet happened over on the far left. What about step 2? The event horizons of the black holes have now entered their combined event horizon, but you do not yet have enough mass inside the event horizon for it to constitute a black hole. What about the next step? Now, you the centers of each black hole solidly within their combined event horizon but the two event horizons have not yet overlapped. Finally, you have contact, over on the right. Which one marks the actual point of merger?

In the swarm at the center of a black hole cluster, however, you no longer have tightly-bound black holes. You have a bunch of black holes zipping past each other at breakneck speeds. And since the Schwarzschild radius grows linearly with the number of black holes, the volume of the combined event horizons grows with the cube of the number of black holes:

This can only show in two dimensions, but as the number of black holes grows, the available volume for all the black holes to enter their combined Schwarzschild radius. 

The odds of individual collisions doesn't really go up, but the odds of a de facto black hole being created (because all the black holes are "inside" at once) goes up.

[K^2]

7 minutes ago, sevenperforce said:

The black discs are the individual event horizons of the two black holes, and the gray circle represents their combined event horizon. I'm not really sure at which point the merger takes place.

 

9 minutes ago, sevenperforce said:

In the swarm at the center of a black hole cluster, however, you no longer have tightly-bound black holes. You have a bunch of black holes zipping past each other at breakneck speeds. And since the Schwarzschild radius grows linearly with the number of black holes, the volume of the combined event horizons grows with the cube of the number of black holes:

You can only draw a combined event horizon for a swarm if the entire swarm is contained within that volume. So, like we can compute Schwarzschild radius for Earth, but since Earth isn't contained within that radius, Earth doesn't have an event horizon anywhere. Similarly, for a cluster of black holes that has enough objects to exhibit high symmetry, you only get a combined event horizon if all of the mass is within the combined Schwarzschild radius. This becomes more and more likely as the swarm grows, but I would still expect density near the center to be high enough for pair-wise merges to start way before you achieve that critical mass for the whole swarm to become a black hole all at once.

Then again, dynamics of a black holes in a near-critical swarm might be rather wild. It could very well be an aggregate sort of merger rather than pair-wise, starting at the core and growing out. Seems unlikely to me, but this is also why I'm saying you should run this past cosmologist, because this is definitely beyond my background.

[sevenperforce]

11 minutes ago, K^2 said:

Similarly, for a cluster of black holes that has enough objects to exhibit high symmetry, you only get a combined event horizon if all of the mass is within the combined Schwarzschild radius. This becomes more and more likely as the swarm grows, but I would still expect density near the center to be high enough for pair-wise merges to start way before you achieve that critical mass for the whole swarm to become a black hole all at once.

Then again, dynamics of a black holes in a near-critical swarm might be rather wild. It could very well be an aggregate sort of merger rather than pair-wise, starting at the core and growing out. Seems unlikely to me, but this is also why I'm saying you should run this past cosmologist, because this is definitely beyond my background.

I'll definitely try to find one, haha.

Thanks for the video!

My thought is that the velocities at the center of the swarm are going to be so wildly high that pairwise mergers at the center will not be any more likely as the cluster grows. Moreover, by the shell theorem, the behavior closer to the center will not necessarily be predictable at all -- I think you're correct that the orbital dynamics are wild and chaotic. When you have a bunch of bodies orbiting a single massive primary, the density increases as you get closer to the center, leading to collisions...but when you have no central massive primary, what happens?

Given the impact of the shell theorem on the whole thing...I am wondering if you could have an aggregate merger which starts at the edge and grows inward, rather than at the core and growing outward.

But yeah I need to dig up a proper cosmologist.

 

[K^2]

1 hour ago, sevenperforce said:

My thought is that the velocities at the center of the swarm are going to be so wildly high that pairwise mergers at the center will not be any more likely as the cluster grows.

Well, this is the place where I actually have relevant experience. Subatomic particles are kind of the same way - we draw them as tiny little spheres, but it's all fields, so collisions are never really collisions - it's interactions. And because they're all just whizzing about at relativistic speeds, you don't care about specific details of a particular interaction. What you start talking about are cross-sections. It's the section area within which a particular interaction, such as capture, will happen with given probability. And yes, cross-section can depend on relative energies. So you start looking at it statistically. Given the average energy within a cluster and average mass, what is the probability per unit time that two black holes will come close enough to each other that gravitational wave this generates will steal enough kinetic energy to cause them to merge. And no matter what your cluster parameters may be, it's a finite number. So eventually, any cluster will merge - what we care about is how that "eventually" compares to age of the universe. But we're still talking billions of years here, potentially, so the odds don't have to be that high.

And the crucial part is that cross-section is increasing as square of the mass, because whatever the shape of that curve is going to be based on your threshold probability and energy, I know that it will scale with Schwarzschild radius. So if on average, a stellar mass black hole in your cluster will encounter another black hole in a billion years, then in two billion years, the entire cluster will be a single black hole.

So again, what this comes down to is, a) Is this the only mechanism, or can you really grow a cluster to the point where you can't treat stellar mass black holes as individual objects, and have to treat the whole thing as one GR mess, b) Which mechanism is dominant, and c) Is either of these relevant on the scale of life time of the universe for a typical galaxy. To answer these you need to actually build models and crunch numbers. Hopefully, somebody has done that already, and it's just something you can find out from somebody working in the field.

[YNM]

4 hours ago, K^2 said:

On the other hand, that final ring of black holes with thousands of solar masses merging into supermassive black hole should be distinctive, as you'll see a huge number of very heavy objects coming together almost at once. Trouble is, there might still not be any candidates close enough even at that strength, and also, I'm not sure we'll be able to recognize it. This is the sort of thing that will make any GR sim cry binary tears, and the only way we have right now to analyze signals is simulate event we think it might be, and compare signals.

I wonder if perhaps there's a timeframe where these kind of events happen the most often... Maybe in an elliptical galaxy ?

[sevenperforce]

On 2/12/2021 at 8:14 PM, K^2 said:

And the crucial part is that cross-section is increasing as square of the mass, because whatever the shape of that curve is going to be based on your threshold probability and energy, I know that it will scale with Schwarzschild radius.

The volume of the collective Schwarzschild radius scales with the cube of the total mass. If the cross-section only scales with the square of the total mass, then the collective gravity well is going to grow faster than the collision cross-section.

It’s almost like a mean free path analysis, isn’t it? You have a bunch of “particles” that basically only interact via elastic collisions, with a VERY small inelastic collision cross-section. Near-misses alter the paths, but since the motion is chaotic they don’t do anything other than reduce the overall kinetic energy in the swarm, bringing the BHs closer to fitting within their collective Schwarzschild radius.

Sgr A* is 4.154e6 solar masses so it has a Schwarzschild radius of 12.3 million km. Its event horizon encloses 7.8e21 cubic km. A single stellar-mass black hole has a Schwarzschild radius of 3 km and its event horizon encloses just 113 cubic km. Stuff in 4.154e6 of those and that’s still 99.999999999994% empty space. What does THAT collision cross-section look like?

[kerbiloid]

One could say that as any process dissipates energy, and even photon's can't leave the common horizon, all of them will quickly dissipate and turn into one honest black hole.

(If the BH theory is relevant at all).

[wumpus]

On 2/12/2021 at 5:39 PM, sevenperforce said:

You know how a kugelblitz would be formed by getting all the photons (which don't interact with each other) into a very small space at once?

[interesting theory far too big to quote again]

The odds of individual collisions doesn't really go up, but the odds of a de facto black hole being created (because all the black holes are "inside" at once) goes up.

Is it really an "combined event horizon" in the sense that a photon can't make its way through the "combined event horizon"?  The black holes themselves might be doomed, but you should be able to tell the difference from the outside between the separate black holes and the combined one (before the gravitational wave meter shows the event).  One of Scott Manley's black hole videos also shows that while the event horizon pretty much describes the edge between the universe and the black hole, the "region of certain doom" (for anything not a photon) is about twice the radius of the event horizon.

[kerbiloid]

As "no information can get out from behind the evnt horizon", or how is it said, it will be a bag of blackholes of Schroedinger, which either exist, or no.

Edited February 15, 2021 by kerbiloid