Qs re dark matter when all baryonic matter is in black holes

[Buzz Bloom]

Any collection of bound gravitation bodies which are far enough away from other bodies such that the gravitational influence of these other bodies is insignificant on the collection of bound bodies will ultimately all collapse into a single black hole. This is because all gravitational potential energy resulting from the separation of the bodies from each other is gradually radiated away as gravitational waves.

In my quote above from the thread

https://www.physicsforums.com/threads/ultimate-fate-of-the-universe.898640/​

I omitted consideration of dark matter because so very little is known about it. I am hoping someone here at the PF might be able to discuss the following questions.

QUESTIONS
Assume for the sake of simplicity in the context of this discussion, that some time in the very distant future all the matter in the universe which is not gravitationally part of the Milky Way has moved due the universe expansion to be too far away to gravitationally effect the Milky Way matter. For the reason discussed in the thread cited above (as well as the fact the baryonic matter interacts via photons in such a way that some of its gravitational potential energy is lost via photon radiation), all the Milky Way baryonic matter will after another very long period of time collapse into a single black hole.

Since the dark matter does not lose gravitational potential energy via photon radiation, I am guessing that after all the baryonic matter is in a single black hole, a lot of the dark matter will remain outside this vary large black hole.

1. Is the correct?
2. If so, what is the fate of this outside dark matter?
3. What happens to the mass of the dark matter which had entered the black hole during the evaporation process? Which of the following possibilities do you think is more plausible?

a. The Hawking radiation process turns this mass into photons.
b. The Hawking radiation process turns this mass into some currently unknown zero mass particles which are of a type that is a constituent of dark matter.
c. Something else happens.​

I am guessing that it is plausible that the dark matter may well will be captured by the black hole at a much slower rate than the black hole matter will escape due to Hawking radiation. If so, I think that this means that eventually the black hole will completely evaporate while there is still a lot of dark matter than never got into the black hole.

5. Is this plausibly correct?​

If so, I think this implies that at that time the universe contents will be just photons and dark matter?

6. is this correct?
7. If so, will the dark matter form another black hole?​

I will much appreciate any responses.

 

[PeterDonis]

I am hoping someone here at the PF might be able to discuss the following questions.

I'm not sure we know enough about dark matter to give answers with much confidence. But I'll try below, using the basic assumption that dark matter interacts gravitationally, but not in any other way, and that it's cold (i.e., not relativistic); i.e., using the basic model of dark matter that is part of our best current cosmological model of the universe.

I am guessing that after all the baryonic matter is in a single black hole, a lot of the dark matter will remain outside this vary large black hole.

I would say yes, because, as you say, baryonic matter can lose gravitational potential energy much faster than dark matter.

what is the fate of this outside dark matter?

It will still lose gravitational potential energy by emitting gravitational waves, so eventually it will all collapse into the black hole.

What happens to the mass of the dark matter which had entered the black hole during the evaporation process?

Hawking radiation, at least according to our best current model, does not change its particle composition based on the properties of whatever originally collapsed into the black hole. A hole formed from dark matter will not have Hawking radiation that is any different from that of a hole of the same mass formed from baryonic matter. At least, that's our best current model; but we don't really understand Hawking radiation all that well either.

Which of the following possibilities do you think is more plausible?

None of them, because Hawking radiation does not just produce photons anyway; photons are the most likely particle to be radiated but not the only one. The general answer is as given above.

I am guessing that it is plausible that the dark matter may well will be captured by the black hole at a much slower rate than the black hole matter will escape due to Hawking radiation.

I think it's the other way around. You might not realize how long it takes for a black hole with the mass of a galaxy to evaporate by Hawking radiation. A hole with the mass of our sun takes around $10^{67}$ years to evaporate, and the time goes like the cube of the mass, so a hole with the mass of the Milky Way, i.e., around $10^{11}$ suns, would take around $10^{100}$ years to evaporate. Even if dark matter can only lose gravitational potential energy via gravitational radiation, it still will lose it many, many, many orders of magnitude faster than that.

 

[Buzz Bloom]

I'm not sure we know enough about dark matter to give answers with much confidence.

Hi Peter:
I thank you very much for your response. It did clear up some confusion I had.

None of them, because Hawking radiation does not just produce photons anyway; photons are the most likely particle to be radiated but not the only one.

The reason I thought only photons would be created by Hawking radiation is what I have been reading in the 1975 Hawking paper, p.211. (Underlining is mine for easy reference.)

Similar results hold for the electromagnetic and linearised gravitational fields.
The fields produced on J- by positive frequency waves from J+ have the same
asymptotic form as (2.18) but with an extra blue shift factor in the amplitude.
This extra factor cancels out in the definition of the scalar product so that the
asymptotic forms of the coefficients e and fi are the same as in the Eqs. (2.19) and
(2.20). Thus one would expect the black hole also to radiate photons and gravitons
thermally.​

Since gravitons have not yet been verified to exist, I left them out. I also understand that the temperature of the Harking radiation is inversely proportional to the mass of the black hole. Therefore near the end of the evaporation the mass will become small enough so that massive particles would be created by the very high temperature of radiation. However, I am guessing that the final ratio of the number of particles with non-zero mass (including neutrinos) to the number of photons would be infinitesimal. Is this correct?

Regards,
Buzz

 

[PeterDonis]

The reason I thought only photons would be created by Hawking radiation is what I have been reading in the 1975 Hawking paper

Did you note the word "also" in what you quoted? He said the hole should also radiate photons (and gravitons). "Also" in addition to what?

In other words, Hawking had already shown that the hole should radiate something besides photons (IIRC it was scalar particles, particles with spin zero). So his paper does not show that only photons (if we leave out gravitons) should be radiated.

I also don't think gravitons should be left out, because gravitational waves have been verified to exist, and Hawking's paper was not making use of any particular particle-like aspects of the fields, so "gravitons" in his paper really means "gravitational waves".

 

[PeterDonis]

I am guessing that the final ratio of the number of particles with non-zero mass (including neutrinos) to the number of photons would be infinitesimal

I would expect the ratio of, say, electrons to photons to be very small. I'm not sure about neutrinos, because their masses are so much smaller than the mass of the electron; probably the ratio is still small (though much larger than the electron to photon ratio), but I would have to look at the math to be sure.

 

[timmdeeg]

Even if dark matter can only lose gravitational potential energy via gravitational radiation, it still will lose it many, many, many orders of magnitude faster than that.

Does this require necessarily that clouds of dark matter are rotating not symmetric with respect to their respective rotational axis? Or is there any other mechanism, such as contracting asymmetric (but I think they don't contract anyway)? I mean if such clouds would just move around randomly they wouldn't have a time dependent mass quadrupole moment, correct?

 

[PeterDonis]

if such clouds would just move around randomly they wouldn't have a time dependent mass quadrupole moment, correct?

Certainly they would. Not having a time-dependent mass quadrupole moment requires a very coordinated, symmetrical motion. It's certainly not something that's going to occur randomly, or even something that's going to occur non-randomly except in highly idealized models. In practical terms, any system of gravitating masses is going to emit some gravitational waves. Yes, in ordinary terms the emitted power is extremely small, way too small to matter; but it's still going to cause the systems to dissipate a lot, a lot sooner than $10^{100}$ years.

 

[timmdeeg]

In practical terms, any system of gravitating masses is going to emit some gravitational waves.

But this is not possible forever, right? Because the amount of potential energy which is convertible into radiation is finite. Would the particles of which dark matter consists reach asymptotically a final stage of "very coordinated, symmetrical motion"? If true, I can only think of comoving particles here, as they can't collapse to form a body.

 

[PeterDonis]

this is not possible forever, right?

Right; eventually the system will form a black hole and settle down to a stationary state in which no further GWs are emitted. That will happen in a time many orders of magnitude smaller than the time it takes for such a hole to evaporate by Hawking radiation.

Would the particles of which dark matter consists reach asymptotically a final stage of "very coordinated, symmetrical motion"?

Not before they form a black hole.

 

[timmdeeg]

Not before they form a black hole.

Thanks, but this is surprising. Or do you say the dark matter particles will vanish over time because they fall in an already existing black hole formed by baryonic matter. But even then I'd expect still stable orbits.
I mean due to the lack of electromagnetic forces of dark matter particles no "stickiness" should occur, their velocities shouldn't decrease therefor and thus not allow them to form compact objects..

 

[PeterDonis]

do you say the dark matter particles will vanish over time because they fall in an already existing black hole formed by baryonic matter

Not necessarily; they could, but they could also form black holes on their own.

even then I'd expect still stable orbits

I think at this point you need to stop waving your hands and do some math. I've given you the time it takes for black holes of the relevant mass to evaporate by Hawking radiation. Can you show a reasonable calculation that gives a longer time for dark matter to collapse into black holes?

due to the lack of electromagnetic forces of dark matter particles no "stickiness" should occur

EM forces are not the only forces. Dark matter is still "sticky" due to gravity and can still emit gravitational waves. You can't ignore that on the time scales we are considering. Stop waving your hands and do some math if you want to really consider the question.

 

[timmdeeg]

I think at this point you need to stop waving your hands and do some math. I've given you the time it takes for black holes of the relevant mass to evaporate by Hawking radiation. Can you show a reasonable calculation that gives a longer time for dark matter to collapse into black holes?

EM forces are not the only forces. Dark matter is still "sticky" due to gravity and can still emit gravitational waves.

Oh, I understand, I should have seen that myself . Sorry for taking your time unnecessarily and thanks for clarifying.

 

[mfb]

Most of the stars (and gas) of the Milky Way will be ejected, only a tiny part will fall in - angular conservation shows this quickly.

For a binary system with circular orbits, the orbital decay due to gravitational waves is given by this formula, plugging in 1 billion solar masses for the central black hole, 1 billion solar masses for dark matter (guessed number for the non-homogeneous component) and 20000 light years, we get an orbital decay of about 1 fm/s. Fast enough, give or take 20 orders of magnitude.

If we take a single particle with a mass of 100 GeV, orbital decay drops to 10^-79 m/s, or ~3*10⁹¹ years to fall in. That is the lifetime of a black hole with 130 million solar masses. Will our central black hole, currently at 4 million solar masses, get that large? I don't know.

 

[timmdeeg]

For a binary system with circular orbits, the orbital decay due to gravitational waves is given by this formula, plugging in 1 billion solar masses for the central black hole, 1 billion solar masses for dark matter (guessed number for the non-homogeneous component) and 20000 light years, we get an orbital decay of about 1 fm/s. Fast enough, give or take 20 orders of magnitude.

Your examples make me curios. What is the decay time for a pair of binary dark matter particles? I will try to calculate that hopefully at the weekend.

 

[timmdeeg]

If we take a single particle with a mass of 100 GeV, orbital decay drops to 10^-79 m/s, or ~3*10⁹¹ years to fall in. That is the lifetime of a black hole with 130 million solar masses. Will our central black hole, currently at 4 million solar masses, get that large? I don't know.

If I didn't make a mistake I have received these results (of course disregarding the accelerated expansion of the universe) for the orbital decay time of two masses separated 100 Million LJ initially (roughly the diameter of the milky way):

Two black matter particles, 100 GEV each: 10²²³ years

A black matter particle and a black hole with 180 billion solar masses (assuming the milky way has formed a black hole already): 10⁹⁸ years

 

[mfb]

100 million light years is 1000 times the diameter of the Milky Way, and of the scale of superclusters.

The first scenario won't happen: At 10^-65 eV, the particles are bound too weakly to be actually bound: even very weak interactions with the CMB will make them move essentially randomly, which means they will probably move away from each other until expansion separates them forever. The accelerated expansion, assuming it stays like this, will also keep the CMB temperature finite.

180 billion solar masses looks way too high, as most stars will get ejected.

 

[timmdeeg]

100 million light years is 1000 times the diameter of the Milky Way, and of the scale of superclusters.

Sorry, and thanks for correcting. I have written erroneously 100 Million but have calculated using 100000 light years.

The first scenario won't happen

Yes, not in this universe. Therefor I mentioned "disregarding the accelerated expansion".

180 billion solar masses looks way too high, as most stars will get ejected.

Do you consider the merger of Milky Way and Andromeda?
I was assuming the theoretical case that most of the stars in a galaxy like ours are forming a black hole, due to the emission of gravitational waves.

 

[mfb]

"Thermal" emission of stars will happen on timescales shorter than gravitational waves.
Wikipedia references one arXiv article I can access and one book I don't have, claiming that 90%-99% of all stellar remnants will be ejected. That is not identical to 90%-99% of the mass, because lighter stellar remnants are more likely to get ejected. The rest falls into the central black hole.

The arXiv article also discusses some points brought up here, like suppression of gravitational waves if the mass distribution is homogeneous.

 

[timmdeeg]

"Thermal" emission of stars will happen on timescales shorter than gravitational waves.
Wikipedia references one arXiv article I can access and one book I don't have, claiming that 90%-99% of all stellar remnants will be ejected.

I see, good to know, thanks. Very interesting article.