A Field-Theoretic Approach to Fermionic Cold Dark Matter

Abstract

Inspired by the novel theory developed by Friedrich and Prokopec based on scalar dark matter, we derive the corresponding theory for Dirac fermions. Even though much interest is recently shown in ultra-light scalar dark matter, very little is known about fermionic dark matter. While exhibiting similarities, the fermionic dark matter theory is much richer in possibilities, such as particle-antiparticle and helicity mixing. The theory finds applications in large-scale structure formation, and also allows for a description of dark matter from sub-dominant neutrinos. By starting from a fundamental Lagrangian theory, we take a heavy-particle limit and show that the resulting equations reduce to the classical dark matter description, while allowing for systematic treatment of gradient as well as relativistic corrections. We derive the integro-differential equations for fermionic phase-space densities, resulting in a generalized field-theoretic version of a Vlasov-Poisson equation.