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I understand that dark matter does not collapse into dense objects like stars apparently because it is non-interacting or radiating and thus cannot lose energy as it collapses. However why then does it form galactic halos? Isn't that also an example of gravitational collapse?

Kyle Oman
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Astrobuf
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    Thanks for the nice question, it's well timed in that answering it has been useful in my efforts to study for my PhD qualifying/comprehensive/candidacy exam coming up next week :) – Kyle Oman Apr 08 '15 at 22:07

1 Answers1

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The answer comes from the virial theorem, which can be derived from the Jeans equations, which are the equivalent of the Euler equations of fluid dynamics for collisionless particles (i.e., dark matter). Incidentally, the virial theorem is also valid for an ideal fluid. For a derivation see Mo, van den Bosch & White 2010 (or I'm sure many other texts). The theorem is:

$$\frac{1}{2}\frac{{\rm d}^2I}{{\rm d}t^2} = 2K + W + \Sigma$$

$I$ is the moment of inertia, $K$ is the kinetic energy of the system, $\Sigma$ is the work done by any external pressure and $W$ is the gravitational energy of the system (if external masses can be ignored in the calculation of the potential).

If $\Sigma$ is negligible (as it is in the collapse of DM haloes), then a system which has $2K < -W$ will have a dynamical evolution that drives an increase in $I$, or in other words the system contracts. Collapse halts and a quasi-stable structure results when $2K\sim-W$.

To sum that up in somewhat less technical terms, the absence of dissipation (e.g. radiative cooling or collisions between particles) does not mean that collapse cannot occur. The dynamics of a collisionless system are described by the Jeans equations, and these equations allow for collapse until virialization occurs.

The difference with gas collapsing into a star is that radiation can carry away energy, so the system can dissipate $K$ and continue to collapse for longer. In the case of a star, collapse continues until pressure support is sufficient to halt it.

Kyle Oman
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    You should know - was that you on arXIv this morning? – ProfRob Apr 08 '15 at 22:06
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    @RobJeffries Yup, that was me :) It's a small world! My reaction was indeed "oh I should really know this!", though I did end up having to reach for a book. – Kyle Oman Apr 08 '15 at 22:08
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    Nice answer. Just wanted to add that there are additional relaxation processes that help a halo collapse. For instance, some particles are ejected from the system at the expense of reducing the energy of other particles. Also, there's Landau damping, where particles that overtake a density wave (from a disturbance of the system e.g. in case of mergers) with a speed comparable to that of the wave will have a net transfer of energy to the wave. – pela Apr 08 '15 at 22:46
  • "The dynamics of a collisionless system are described by the Jeans equations, and these equations allow for collapse until virialization occurs." Just to clarify your wording, are you saying here that when $K < -2W$, the system obeys the Jeans equations but does not obey the virial theorem, and that "virialization occurs" when $K = -2W$? Is this why your answer is compatible with John Baez's derivation here, in which he assumes a ball of ideal gas obeys the virial theorem and shows a decrease in volume would always mean a decrease in entropy? – Hypnosifl May 14 '15 at 19:54
  • @Hypnosifl I'm not sure what you're asking... I'm arguing that when $K<-2W$ the system obeys both the Jeans equations and the virial theorem, and indeed it is by the virial theorem that we can most easily see that the DM must collapse. I would agree that "virialization occurs" when $K=-2W$, but this is distinct from obeying the virial theorem, which is obeyed also when $K\neq -2W$, and drives the evolution of $I$ until virialization occurs. – Kyle Oman May 15 '15 at 08:46
  • Note I had my factor of 2 in the wrong place - have corrected this! – Kyle Oman May 15 '15 at 08:52
  • Got it, I wrongly assumed that "virialized" was synonymous with obeying the virial theorem. But hopefully I am correct in understanding that the reason Baez's derivation doesn't rule out dark matter collapse is because his analysis assumed the cloud of particles was virialized, whereas there were many regions of dark matter shortly after the Big Bang where $2K < - W$ so that these regions were non-virialized, and the Jeans equations would predict they would tend to collapse until $2K = W$ and they became virialized. – Hypnosifl May 15 '15 at 17:20
  • @Hypnosifl Yeah, at least that's my understanding. – Kyle Oman May 18 '15 at 08:26
  • Does this apply to massless particles? Does this claim the density of the cosmic microwave background photons is larger near galaxies? – PPenguin Apr 20 '17 at 17:38
  • @PPenguin no, and no. – Kyle Oman Apr 20 '17 at 18:03
  • Where does the virial argument break down? The stress-energy tensor for massless particles is not zero, so it still interacts via gravity. Why isn't this gravitational interaction enough if it is enough for dark matter? Why wouldn't the gas of photons collapse into a halo if dark matter can collapse into a halo with only gravitational interactions? – PPenguin Apr 20 '17 at 18:54
  • @PPenguin well, the formulation is non-relativistic, so I'd expect that to be a problem to start with. Then, I expect it's rather difficult to make a photon gas with $2K < -W$, I suspect you'd usually have $2K >> -W$, in which case the system would expand and the photons would fly apart. – Kyle Oman Apr 20 '17 at 23:57
  • What do you mean by "fly apart"? Fly apart into what? I thought you were describing a universe filled with a collisionless gas of particles interacting with gravity and having some small initial fluctuations in density. Wikipedia makes it sound like including relativistic particles the virial theorem is just modified with the ratio 2/ changing from 1 up to 2, with 2 being in the limit of ultra-relativistic (neutrinos) or massless particles. – PPenguin Apr 21 '17 at 03:05
  • moved to a separate question: https://physics.stackexchange.com/questions/327992/do-galaxies-have-a-halo-of-neutrinos-and-cosmic-microwave-background – PPenguin Apr 21 '17 at 03:17
  • Why is the gravitational energy negative? Is this showing a collapse? – Rick Feb 12 '21 at 21:25
  • @Rick you mean the -W? There's a usual convention that the gravitational potential is defined to approach 0 as distance tends to infinity, and it has to decrease at closer separations for the concept of energy to make sense. You could choose any constant at infinity since physics only cares about differences in potential (not its absolute normalization), but 0 is convenient in many situations, such as the discussion above. – Kyle Oman Feb 13 '21 at 11:11