Probing Purely Inelastic Scalar Dark Matter Across Colliders and Gravitational Wave Observatories

We propose and study a purely inelastic scalar dark matter model, where two real scalars-dark matter $\phi_1$ and its excited partner $\phi_2$ interact with the Standard Model via a Higgs portal. After mass diagonalization, only inelastic couplings remain, allowing the model to evade stringent bounds from direct detection. We show that thermal (co-)annihilation between $\phi_1$ and $\phi_2$ naturally yields the observed dark matter relic abundance. The same interaction structure can induce a strongly first-order phase transition in the early universe, generating detectable gravitational waves in upcoming experiments. Meanwhile, the slight mass splitting between $\phi_1$ and $\phi_2$, along with the heavy off-shell mediator SM Higgs, leads to long-lived particle signatures of $\phi_2$ at the HL-LHC via the displaced muon-jets technique. We pinpoint a feasible parameter space where the correct relic abundance, observable gravitational waves, and collider signals can all be achieved concurrently, presenting a valuable chance to validate this scenario through a comprehensive examination encompassing cosmological, astrophysical, and collider investigations.