Can a secluded self-interacting dark sector generate detectable gravitational waves?

In this work we study the possibility to detect the gravitational waves generated by a secluded self-interacting dark sector. ``Secluded'' means that the dark sector has almost no portal to the visible sector and thus its entropy is conserved by itself, and ``self-interacting'' means that dark matter in this model has a significant interaction to itself, making it consistent with the small-scale structure observations. A spontaneously broken $U(1)'$ is introduced for the interactions in the dark sector, and nearly massless dark radiation is also introduced to avoid the over-closure problem. Through a parameter space scan, we find that this model is highly constrained by the current observed effective number of neutrinos ($N_{\text{eff}}$) and the large-scale structure observable Lyman-$\alpha$. Combined together, these two constraints make such a secluded self-interacting dark sector almost impossible to generate gravitational waves accessible at future projects like SKA or LISA.