  In this thesis, we start by reviewing all constraints applying on
self-interacting DM with a light mediator considering well-known Higgs portal
and Kinetic Mixing portal models as concrete examples. Then, we elaborate on
the Sommerfeld effect, and we study its many implications on self-interacting
DM models with light mediator. Next, we investigate how to account for the DM
relic abundance in such portal models which present a generic structure with
three populations (Standard Model, DM and mediator particles) connected to each
other through three interactions resulting from two coupling strengths. It
appears that this can be done through five different dynamical ways which give
rise to nine regimes among which four are new. Those new production regimes are
relevant for a situation where the hidden sector particles are not thermally
connected to the visible vector particles. We further study the possibility of
a thermally disconnected Hidden Sector and establish the general allowed
parameter space for DM thermal candidates in the hidden-to-visible temperature
ratio versus DM mass plane. We show that, in this framework, simple portal
models such as the two we consider can alleviate tensions at small scale while
offering suitable DM candidates. Other minimal self-interacting DM with light
mediator frameworks which consistently fulfil all constraints are also proposed
and explored.
