Self-interacting dark matter and cosmology of a light scalar mediator

We consider a fermionic dark matter candidate interacting via a scalar mediator coupled with the Standard Model through a Higgs portal. We consider general setting including both scalar and pseudoscalar interactions between the scalar and fermion, and illustrate the relevant features for dark matter abundance, direct search limits and collider constraints. The case where dark matter has a self-interaction strength $\left\langle \sigma_V \right\rangle/m_\psi \sim 0.1-1 \,\mathrm{cm}^2/\mathrm{g}$ is strongly constrained, in particular by the Big Bang Nucleosynthesis. We show that these constraints can be alleviated by introducing a new light sterile neutrino $N$. The allowed region for the extended model consists of a triangle at $10 (\sin \theta)^{-2/5} \,{\rm MeV} \lesssim m_N \lesssim 1\,{\rm GeV}$.