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


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?

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.

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