Effect of Cosmic Neutrino Background on the Dark Matter Self-interaction via Neutrino force

Neutrino-pair exchange induces a neutrino force that can drive dark matter (DM) self-interactions and impact small-scale structure formation. In the presence of the cosmic neutrino background (C$ν$B), this force can be modified, with important consequences for DM phenomenology. We study the effect of the C$ν$B on neutrino forces, generated by the scalar and pseudoscalar interactions. We explore the significance of the background neutrino force on the scalar DM-neutrino portal model, including DM self-scattering and annihilation. Our results show that the interplay between attractive vacuum potential and repulsive background potential leads to a screening effect that varies across DM mass ($m_χ$) regimes, strongly affecting DM self-scattering in the DM mass $m_ν\lesssim m_χ\lesssim T_{C νB}$. Meanwhile, for DM annihilation, the screening completely vanishes the Sommerfeld Enhancement induced by the neutrino force. Overall, the C$ν$B substantially reshapes the viable coupling range for DM self-interactions while remaining compatible with current constraints, offering a pathway to small-scale structure problems.