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I try to understand the difference between collisionless and collisional fluids in cosmology. My first question is the following.

In the context of FLRW cosmology, we suppose that the Universe can be described in terms of a mix of fluids with:

$T_{\mu\nu}=\left(\rho c^2 +P\right)u_{\mu}u_{\nu}+Pg_{\mu\nu}$

  • When we write that, do we suppose a collisionless or collisional nature of the fluids?
  • If this description corresponds to collisional fluids, why cosmological simulations are N-body simulations (collisionless) and are not simply based on hydrodynamics?
  • Can we solve equations of a collisionless system without using particles (and just cells with physical properties like in the collisional case)?
  • At very large scale (scale of homogeneity), when we are not interested in the formation of local structures (like galaxies and superclusters), does the collisionless/collisional description is important?
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    $\begingroup$ You can (and should!) assume both types of fluid models and calculate which one fits better. I am not aware that all cosmological calculations (numerical or not) are n-body simulations or that n-body simulations rule out modeling of collisions (they don't). Do collisions matter at very large scale? Depends on what you mean by "matter" and "very large scale", doesn't it? To first order the distribution of matter doesn't "matter" at all, as long as its homogeneous "enough". $\endgroup$ – CuriousOne May 10 '15 at 23:09
  • $\begingroup$ The density and temperatures in the interstellar medium are low and high enough that the plasma can be considered collisionless. In fact, the collision time scale in even the solar wind is roughly one Coulomb collision per day. Interstellar dust, however, may be neutral and could be influenced by collisions. Yet, I am not sure whether there is enough dust to significantly impact galaxies or clusters (an expert might help me here). $\endgroup$ – honeste_vivere May 11 '15 at 12:40
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  • Q: When we write that, do we suppose a collisionless or collisional nature of the fluids?
  • A: It's the energy-momentum tensor for a perfect fluid Chapter 2.26
  • Q: If this description corresponds to collisional fluids, why cosmological simulations are N-body simulations (collisionless) and are not simply based on hydrodynamics?
  • A: Cosmological simulations are not always N-body simulations. Physicists often assume dark matter is nearly collisionless. This assumption is justified by looking at e.g. the bullet cluster, where galaxies collide and dark matter does not interact with itself or baryonic matter: Calculations on dark matter self-interaction from the bullet cluster. Baryonic matter, on the other hand, may be assumed to be collisional, but often it can be neglected.
  • Q: Can we solve equations of a collisionless system without using particles (and just cells with physical properties like in the collisional case)?
  • A: Yes, that's possible. Collisionless Boltzmann equation describes collisionless matter (Section 30.5), and it can be discretized onto a mesh grid (into cells). One would also need to have the phase-space for the three velocity coordinates; these can also be discretized. Additionally, one would need to solve the relativistic Poisson equation which would describe the interaction with gravity.
  • Q: At very large scale (scale of homogeneity), when we are not interested in the formation of local structures (like galaxies and superclusters), does the collisionless/collisional description become important?
  • A: That depends on at least: a) whether the matter is highly collisional or not, and b) what kind of physics we are interested in. As an example, dark matter is usually assumed to be highly non-collisional, yet near black holes the chance for collisions is increased.

Additional notes:

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Your equation applies to a perfect fluid that has no viscosity. N-body simulations are for dark matter only and the dark matter is generally assumed to only interact gravitationally. There are hydrodynamical simulations that include baryons but those are more computationally intensive and so generally done for smaller simulations. If you are just interested in the average ("background") evolution on large scales the collisional nature is not important.

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