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 ϕ₁ and its excited partner ϕ₂ 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 ϕ₁ and ϕ₂ 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 ϕ₁ and ϕ₂, along with the heavy off shell mediator SM Higgs, leads to long-lived particle signatures of ϕ₂ 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.