The fluid treatment of the neutrino component would only be adequate if the neutrinos were self-collisional, which

Question:

The fluid treatment of the neutrino component would only be adequate if the neutrinos were self-collisional, which they are not. The phenomenon of Landau damping alerts us to the need for a kinetic approach. We develop this in stages.

Following the discussion, we introduce the neutrino distribution function η_ν(t , xⁱ , p_j), where xⁱ is the (contravariant) comoving (spatial) coordinate, and p_j is the (covariant) conjugate 3-momentum.The function η_ν satisfies the collisionless Boltzmann equation [Eq. (3.65)]:

We work with this equation to linear order.

(a) Show that the neutrino equation of motion in phase space can be written as

and explain why it is necessary to express the right-hand sides in terms of t , xⁱ, and p_i.
(b) Interpret the momentum equation in terms of the expansion of the universe and gravitational lensing.

(c) Introduce locally orthonormal coordinates in the rest frame of the FOs, and define p^α̂ = {ε, p^1̂, p^2̂, p^3̂}. Carefully interpret the density, velocity, and pressure in this frame, remembering that it is only necessary to work to linear order in Φ.

(d) Multiply the Boltzmann equation successively by 1, ε, and p^î , and integrate over the momentum space volume element dp^1̂dp^2̂dp^3̂ to show that

where δ̃_nν is the fractional fluctuation in neutrino number density, and S̃_ν is the associated number flux. In deriving these equations, it is necessary to impose the same closure relation as was used to derive the fluid equations (28.57). These kinetic equations have the same form as the fluid equations, although the coefficients are different and would change again if we changed the closure relation. This demonstrates that fluid equations can only be approximate, even when derived using the Boltzmann equation.

(e) One standard way to handle the neutrino perturbations accurately is to expand the distribution function in spherical harmonics. Outline how you would carry this out in practice, and how you would then use the more accurate neutrino distribution to improve the evolution equation for the dark-matter, baryon, and photon components.

Equation.