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If we were to hypothesise that the Universe contained a significant mass of "dark matter" particles subject only to gravity, presumably general relativity would give us a good idea of how they would behave.

For example, what would happen in a region where they were the only mass?

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What you describe is the standard paradigm in cosmological physics, so it has been studied extensively. The basic consequence of dark matter not having significant nongravitational interactions is that it has no way to shed its orbital energy. Dark matter particles that cluster due to gravity cannot coalesce into disks or compact objects, because they retain the orbital velocities that they acquired upon first infall into a system. Thus, dark matter remains in the form of hot, diffuse "halos".

By the way, this remains true even in places where ordinary matter is also present, since the dark and ordinary matter do not interact with each other nongravitationally. The formation of disks and compact objects in the ordinary matter does not qualitatively alter the distribution of dark matter.

Also, dark matter is not unique in this behavior. Due to their tiny cross sections, stars also effectively only interact gravitationally. So the structures of dark matter halos are similar to those of globular star clusters and gas-poor elliptical galaxies, only differing due to their different initial conditions and the possibility of gravitational few-body interactions in star clusters.


More detail on the dark matter distribution

Since two-body interactions are negligible, dark matter systems do not thermalize, so their structures are tightly connected to the initial conditions. For example, material accreted earlier settles into lower orbits while material accreted later settles into higher orbits, which implies that the radial distribution of material of a halo is tightly connected to its assembly history. Let $M_\mathrm{acc}(\rho)$ be the mass accreted by the halo when the density of the universe was larger than $\rho$, keeping in mind that the density of the universe drops over time. Also, let $M_\mathrm{halo}(\rho)$ be the halo mass inside radii for which the average enclosed density exceeds $\rho$. Ludlow et al (2013) noted that $$M_\mathrm{halo}(800\rho)\simeq M_\mathrm{acc}(\rho),$$ that is, material accreted when the universe was denser than $\rho$ settles into regions denser than $\sim 800\rho$.

Despite this, the structures of dark matter halos tend to be pretty consistent.

  • The mass distribution is pretty close to spherical.
  • At each position, the distribution of particle velocities is pretty close to isotropic, although this becomes less true toward the edge of the system (where radial or tangential velocities might be favored, depending on the halo's current accretion rate).
  • As a function of radius, the mass distribution is reasonably well approximated by the NFW density profile. The density of a halo diverges toward the center of the system, although it's expected to converge to a finite value at radii smaller than what is resolved in simulations. It decreases at larger radii.
  • The gravitationally orbiting system ends at roughly the virial radius, where the enclosed density is $\sim 200$ times the average density of the universe. Outside this radius lie still accreting particles that have not completed full orbits yet.
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    $\begingroup$ What stops secondary shedding of this energy? E.g. when a dark matter particle has an n-body gravitational interaction with several normal matter particles, can it not transfer energy and angular momentum to the matter which can then be radiated away etc. by the matter as normal? $\endgroup$ Mar 5, 2023 at 0:57
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    $\begingroup$ @zibadawatimmy The collective impact of gravitational few-body encounters drops as the number of particles in the system grows (see the relaxation time scale). Few-body encounters between stars are negligible in galaxies (although relevant in star clusters); encounters between e.g. dark matter particles and gas molecules are even less important. $\endgroup$
    – Sten
    Mar 5, 2023 at 9:02
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    $\begingroup$ @zibadawatimmy Moreover, we have lots of reasons to believe that the dark matter particles are lighter than stars. This means the net energy exchange is from the stars to the dark matter, and not the reverse (see mass segregation and dynamical friction). But this effect isn't really significant in galaxies anyway. $\endgroup$
    – Sten
    Mar 5, 2023 at 9:02
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    $\begingroup$ @Sten thanks for the very helpful answer. Is there anything more concrete that can be said e.g. about the statistical distribution of particles. I realise that the maths involved might be rather advanced. $\endgroup$
    – mikado
    Mar 5, 2023 at 12:07
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    $\begingroup$ @Vikki It would, over extraordinarily long time scales. The rate of energy loss through gravitational radiation scales with the particle mass. Gravitational radiation from stars in galactic orbits is already negligible over the age of the universe; gravitational radiation from dark matter particles would be much less significant than that. $\endgroup$
    – Sten
    Mar 5, 2023 at 21:56

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