Baryons as a collisional fluid

The baryonic component, although subdominant in mass, plays a crucial role in observational cosmology, because it is what astronomers can see. To faithfully use baryons as tracers of the underlying dark matter mass distribution, we need to understand baryonic physics. For cosmologists, baryons are usually treated as a nuisance parameters, or a systematic effect that we need to understand and get around. For galaxy formation theorists, baryonic physics is the main topic: galaxy properties, as we will see in Section 11.5, are determined not only by the underlying cosmological evolution, but also, as importantly, by small-scale gas and radiative phenomena. Before describing the processes at play during galaxy formation, let us discuss the basic theoretical model for baryons, namely collisional gas dynamics.

The Euler—Poisson equations

Baryons are a highly collisional fluid, the exact opposite of dark matter. A collisional fluid can be described by the collisional Boltzmann equation, which can be written very similarly to the Vlasov-Poisson equation as

where the right-hand side is no longer zero, but is equal to the collision integral, denoted here as an operator on the distribution function f. The collision integral will scatter particles from one phase-space volume element to another phase-space volume element, at a rate that depends on the collision cross section of the underlying interaction processes (Coulomb interactions mostly). After enough time, the distribution function will evolve towards an equilibrium distribution function f0, for which the collision integral vanishes:

It can be shown that the only distribution that makes the collision integral vanish is the Maxwellian distribution function, for which

Interestingly enough, since at equilibrium the collision integral vanishes, f0 also satisfy the Vlasov equation, like dark matter. The main difference arises because for baryons, at equilibrium, we have the distribution function completely determined, while for dark matter, it is not. Using the particular form of the Maxwellian distribution function, one can then define various velocity moments

and derive from the Boltzmann equations the Euler equations, which are nothing other than simple conservation laws:

where the thermal pressure is P = (7 — 1) pe for a perfect rarefied gas (7 = 5/3). Note that the Euler equations are valid for a plasma in strict local thermodynamic equilibrium (LTE)—which never occurs strictly in practice. In other words, the distribution function is always slightly different from a pure Maxwellian. The Euler equations are therefore idealized, and this can lead to singularities in the computed solution with unphysical properties. One interesting way to quantify non-LTE effects is to explore possible distribution functions close to LTE, modelling the collision integral as a simple relaxation term such as

where т is the relaxation (or collision) timescale, assumed to be a constant here for simplicity. Using the famous Chapman-Enskog perturbative technique, we can insert this into the Boltzmann equation and find to leading order the perturbed distribution function as

Using this new distribution function in the Boltzmann equation leads to a new set of equation, usually referred to as the non-LTE fluid equations:

where the viscosity and the diffusion coefficients can be derived directly using the Chapman-Enskog approach as

This first-order modification of the Euler equations is of primary importance, because it highlights one of the most important properties of the equations that we want to solve: the Euler equations should be considered as the vanishing-viscosity limit of the collisional Boltzmann equation. Physical solutions are those that can be expressed as the limit of viscous solutions with p ^ 0. These are a more restrictive set of solutions, sometimes referred to as the weak solutions of the Euler equations. This fundamental property of the Euler equations will be used as a general guideline to design stable and accurate numerical schemes.

 
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