Stability of dark matter admixed neutron stars

Dark matter is an essential ingredient in our understanding of the Universe. Although most studies have focused on investigating dark matter at large scales, recently there has been an exploration of the effect that dark matter can have on neutron stars, mostly referring to their instability. However, these studies have used a stability criterion that makes a number of, until now, unproven assumptions about the normal modes of the system. In this talk, I will prove the first few of these assumptions (finding a canonical energy and a Lagrangian) and I will discuss the remaining necessary steps to fully prove the validity of the stability criterion.