Constraints on self-interaction cross-sections of dark matter in universal bound states from direct detection

\(\varLambda \)-Cold Dark Matter (\(\varLambda \)CDM) has been successful at explaining the large-scale structures in the universe but faces severe issues on smaller scales when compared to observations. Introducing self-interactions between dark matter particles claims to provide a solution to the small-scale issues in the \(\varLambda \)CDM simulations while being consistent with the observations at large scales. The existence of the energy region in which these self-interactions between dark matter particles come close to saturating the S-wave unitarity bound can result in the formation of dark matter bound states called darkonium. In this scenario, all the low energy scattering properties are determined by a single parameter, the inverse scattering length \(\gamma \). In this work, we set bounds on \(\gamma \) by studying the impact of darkonium on the observations at direct detection experiments using data from CRESST-III and XENON1T. The exclusion limits on \(\gamma \) are then subsequently converted to exclusion limits on the self-interaction cross-section and compared with the constraints from astrophysics and N-body simulations.