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?

# [Physics] How to dark matter collapse without collisions or radiation

dark-mattergravitational-collapse

#### Related Solutions

*Note: I think my answer below may be incorrect, at least as far as dark matter is concerned. It seems Kyle Oman has studied the issue further since asking this question and given an answer to a different question about dark matter collapse here, if I'm understanding correctly his answer says that for an ideal fluid with kinetic energy $K$ and gravitational energy $W$, the Jeans equations say that it only "becomes virialized" and stops collapsing when $2K$ becomes approximately equal to $-W$, which meaning it is not virialized (though it does still obey the virial theorem, see Kyle's comment below) when $2K < -W$. And John Baez's derivation did assume the ball of ideal gas is virialized, so his demonstration that collapse of an ideal gas decreases the entropy presumably wouldn't apply to a non-virialized collection of dark matter with $2K < -W$, so I presume this means it could collapse without contradicting the second law, and without the dark matter particles needing to radiate or interact as I suggested in my original answer.*

If we want to examine gravitational collapse from a statistical mechanics point of view, we find that there's a tradeoff between the fact that a more spread-out collection of matter has more possible position states, whereas a more concentrated collection has more possible momentum states (because more of the system's potential energy has been converted to kinetic energy and thus the particles have higher average velocity/momentum). And in statistical mechanics, entropy is a function of the total number of states available, with higher entropy = more possible states. It turns out, though, that this tradeoff alone is not enough to explain why gravitational collapse can happen in some systems--the decrease in the number of possible position states when a cloud collapses is actually greater than the increase in the number of momentum states, as derived on this page from physicist John Baez, so if these were the only factors at play the entropy would be *lower* in the collapsed state than the diffuse state, and gravitational collapses would never occur. However, it turns out that if the collapsing matter can radiate energy away as it collapses, in *that* case the end state of "more concentrated, hotter matter distribution + outgoing radiation" can have a higher entropy than the initial state of "more spread out matter which hasn't yet radiated", and so this is the key to understanding why gravitational collapse respects the 2nd law of thermodynamics. As explained by Lubos Motl in this answer:

If you didn't allow the molecules to emit photons when they collide, they wouldn't ever shrink spontaneously by obeying the laws of gravity. The probability that a molecule slows down (or gets closer) under the gravitational influence of the other molecules would be equal to the probability that it speeds up (or gets further) - in average. If you introduce some objects and terms in the Hamiltonian that allow inelastic collisions, these inelastic collisions will selectively slow down the molecules that happened to be closer to each other, which is the mechanism that will be reducing the average distance between the molecules (the actual rate will depend on the gravitational attraction, too).

I wrote photons because, obviously, the probability of the emission of a photon is much higher for real-world gases because most of their interactions are electromagnetic interactions. Because a photon carries as much entropy as a graviton would, but you produce many more photons by random collisions, the entropy increase is stored in the photons. The entropy carried by gravitons is smaller by dozens of orders of magnitude.

And as explained in this answer by Ted Bunn, this is relevant to why dark matter would "clump" only very weakly (as seen in detailed physical simulations like the ones I have linked to)--dark matter particles would only experience irreversible interactions with other particles very rarely, from either occasional interactions involving the weak nuclear force (which would be infrequent, as with neutrinos which normally pass straight through the Earth, with only about 1 in 10^11 interacting with any of the particles that make up the Earth according to this page) or shedding gravitons:

But it's true that dark matter doesn't seem to have collapsed into very dense structures -- that is, things like stars and planets. Dark matter does cluster, collapsing gravitationally into clumps, but those clumps are much larger and more diffuse than the clumps of ordinary matter we're so familiar with. Why not?

The answer seems to be that dark matter has few ways to dissipate energy. Imagine that you have a diffuse cloud of stuff that starts to collapse under its own weight. If there's no way for it to dissipate its energy, it can't form a stable, dense structure. All the particles will fall in towards the center, but then they'll have so much kinetic energy that they'll pop right back out again. In order to collapse to a dense structure, things need the ability to "cool."

Ordinary atomic matter has various ways of dissipating energy and cooling, such as emitting radiation, which allow it to collapse and not rebound. As far as we can tell, dark matter is weakly interacting: it doesn't emit or absorb radiation, and collisions between dark matter particles are rare. Since it's hard for it to cool, it doesn't form these structures.

Detailed cosmological simulations like the "Illustris simulation" discussed in this article and this one indicate that there is some clustering with dark matter, but it doesn't form very condensed clumps on the scale of stars.

The problem with your hypothesis is that observation strongly suggest that dark matter behaves as a pressureless fluid: this is compatible with your hypothesis only if the interactions between dark matter particles are very much weaker than that between their standard partners. For long, this was only a hypothesis which was indirectly well confirmed by the success so-called $\Lambda$CDM cosmological model, which assumes a cold dark matter and a cosmological constant. So in addition to being weakly interacting, dark matter particles are slow in this model. By success, I refer to the inhomogeneities of the CMBR and of galaxy cluster distribution. But recently, [SR11] claim to have measured the equation of state for dark matter by combining weak gravitational lensing and rotation curve for clusters of galaxies. Their best fit is compatible with a pressureless dark matter.

So, of course, that does not rule out your hypothesis, but at least it shows that the symmetry between the standard particles and dark matter particles is extremely broken. At the very least, your dark matter world would be nothing like the standard one: with such weakly interacting particles, no way to have stars or any such assembly. So far the dark matter world is totally compatible with being a dust.

[SR11] Ana Laura Serra and Mariano Javier L. Dom ́ınguez Romero. Measuring the dark matter equation of state. Monthly Notices of the Royal Astronomical Society: Letters, 415(1):L74–L77, 2011. [Free access at arxiv]

## Best Answer

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.