Dark matter as a collisionless fluid

The Vlasov—Poisson equations

In the current cosmological paradigm, mass in the universe is mostly made of “dark matter”, a mysterious undetected particle for which we have only indirect evidence thanks to its gravitational effects. Dark matter is the dominant component inside galactic haloes and galaxy clusters (roughly 85% of the total matter in the universe). Since it is so difficult to detect, either using laboratory experiments or through possible astronomical radiation sources, it is usually assumed that dark matter is a collisionless fluid. In practical terms, this means that the probability for a dark matter particle to interact with another dark matter particle or with baryonic matter is virtually zero; we can therefore neglect the collision integral in the Boltzmann equation, to obtain the Vlasov equation that describes the evolution of the distribution function f (x, v,t) of dark matter in phase space. The Vlasov equation reads

where F = —Уф is the gravitational acceleration and ф is the gravitational potential. The gravitational potential depends on the distribution function through the Poisson equation

where m is the dark matter particle mass. For cold dark matter (CDM) models, the initial “temperature” of the dark matter fluid is so small that the distribution function is a delta function in velocity space. For warm dark matter models, the spread in

506 Computational cosmology

velocities of the distribution function is more significant and the actual temperature plays a more direct role in the dynamics.

Interestingly enough, direct solution of the Vlasov-Poisson system of equations in phase space became possible only recently thanks to the increasing computational power of supercomputers (see e.g. Yoshikawa et al., 2013; Colombi et al., 2013), but at the prize of modelling the dynamics on a very low-resolution grid (typically 646 elements). Exploiting the fact that for CDM the distribution function is a delta function in velocity space, Hahn et al. (2013) have tried to model its dynamics following the trajectory of an infinitely thin sheet in phase space. These first attempts are still work in progress, but they are very promising for the future.

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