Why doesn't a dark matter halo just collapse to a disc?

Question

According to this and this answer, and as far as I understand these answers, dark matter halos cannot collapse to a black hole because, due to uncoupling from the EM field, they are unable to radiate their kinetic energy, and hence, getting closer to some gravitational center point also means that they get faster and so they resist further "collapse".

But what about the motion perpendicular to the galactic plane? I would naively expect the dark matter to gravitationally fall down on the galactic plane on both its sides, until it concentrates there. Depending on whatever the type of dark matter may be, this may cause other forces (e.g. weak interaction) to take over (possibly at nuclear distances) or it may oscillate until it becomes spherical again.

One way I imagine that this might happen is a small-scale alternating velocity variation in the dark matter field, so that dark matter is alternately falling to/moving away from the mid plane from location to location, and these regions simply pass by one another infinitely. If this alternating pattern in the velocity field exactly balances, the dark matter halo is able to maintain a spherical shape. But even the slightest imbalance might result in a global oscillatory motion between spherical and disc-like.

Have the available observations been investigated with respect to these possible oscillations of the dark matter halos? And isn't it likely that such oscillatory motion (if it existed) would eventually stop due to second order dissipation (dark-matter to ordinary matter to radiation).

Answer 1

The dark matter is (approximately) dissipationless. How can it collapse into a disk unless it has a means to rid itself of its kinetic energy? Any collapse would increase the kinetic energy at the expense of gravitational potential energy - without any dissipation then this process is then reversed.

The dark matter forms an approximately spheroidal distribution around a galaxy. The dark matter bodies will orbit in the galactic potential, just like other bodies that experience gravity - those orbits are non Keplerian but can be highly eccentric and would usually involve components that oscillate in both the radial and "vertical" directions along with any azimuthal component. If you randomly orient such orbits then you end up with a spherical distribution of density.

A disk is what you get when the orbiting bodies (could be particles, might be primordial black holes) are capable of losing their kinetic energy in some way, but without significant loss of angular momentum. Ordinary gas is capable of doing this because it radiates away energy when it is compressed and heated. Dark matter is (by definition) unable to do this. A better comparison for dark matter is the halo of very old stars that were thought to have formed very early in the history of our Galaxy, when the gas was distributed more spheroidally. These "halo stars" are now unable to lose their kinetic energy because stars basically behave like collisionless particles in the galaxy and so they maintain their spheroidal distribution.

In fact you can turn this answer entirely around. We know that dark matter is weakly dissipative and non-interacting because we observe (via its gravitational influence, e.g. an oblateness $q \simeq 0.8$, Piffl et al. 2014, but maybe even prolate with $q \sim 1.2$ in the inner 20 kpc, Dodd et al. 2021) that it forms a large, roughly spherical halo around the Galaxy and is not concentrated into a plane.

Edit: To address the possible long-term dissipation of energy.

In terms of whether the vertical component of dark matter orbital motion will ever be damped, then we are forced to make assumptions about what dark matter is and just how weakly interacting it is.

The inferred dark matter density at the solar galactocentric radius is about $0.013$ solar masses per cubic parsec and that of "normal matter" about $0.084$ solar masses per cubic parsec (McKee et al. 2015). If we assume the dark matter is in the form of WIMPS of mass 100 GeV the current experimental constraints appear to put an upper limit on the WIMP-nucleon cross-section of around $\sigma < 10^{-46}$ cm$^2$. The mean free path of a WIMP would be of order $(n \sigma)^{-1} = 10^{27}$ parsecs, where $n$ would be the average baryon number density of roughly $3 \times 10^6$ m$^{-3}$ (most of which is in stars).

The Galactic disc is of order 1000 pc thick, so a WIMP could oscillate back and forward through it about $10^{24}$ times before interacting. Given that the timescale of Galactic orbits are $\sim 10^8$ years, then it would take $10^{32}$ years to dissipate significant energy in this way.

Answer 2

Your idea of small scale alternating velocity variation is on the right track. The model that captures the properties of dark matter is a gas of particles with a temperature hot enough that the individual particles are moving a a few hundred km/s. In a gas, the motions of the individual particles are random, so a particle will generally have a substantial velocity out of the plane.