LoganAMorrison
commented
5 years ago So it looks to me like this calculation simply involves looking at the elastic scattering processes of DM with the SM bath. The simplest method for computing the decoupling temperature looks to be determining the temperature at which:
sigma_el * n_rel / N ~ H
where N is the number of collisions required to keep the DM in thermal equilibrium, sigma_el is the elastic scattering cross-section of DM off dark matter batch particles and n_rel is the relativistic number density of the SM bath particles that the DM interacts with (which scales like T^3). N is given by:
N = (p / delta_p)^2 = m_dm / T
where delta_p is the momentum transfer/ collision and p is the momentum of DM (p = sqrt(2mT).) I'm guessing that we could just consider DM scattering off electrons and photons (assuming that m_dm / T ~ x_fo ~ 20 which would give T < m_dm / 20 ~ 10 MeV.) This calculation would just be a rough estimate (to do things correctly, one would need to solve an integrodifferential equation which would be way too much effort.)
The only thing that is slightly unclear is what to do if the DM freezes out pretty early, i.e. x <~ 1 in which the other particles (say muons) could potentially matter in determining the kinetic decoupling temperature or in cases where the scalar matters. For now, I think I'll ignore these cases and only consider scattering off photons and electrons.
@adam-coogan, what do you think? At any rate, I'm working on adding the elastic scattering cross-sections into Hazma.
Currently, it looks like we just set the kinetic decoupling temperature to 10^-4. We should compute a more rigorous value for the kinetic decoupling temperature.