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 <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:msub> <a:mi>ϕ</a:mi> <a:mn>1</a:mn> </a:msub> </a:math> and its excited partner <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:msub> <c:mi>ϕ</c:mi> <c:mn>2</c:mn> </c:msub> </c:math> 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 <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:msub> <e:mi>ϕ</e:mi> <e:mn>1</e:mn> </e:msub> </e:math> and <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"> <g:msub> <g:mi>ϕ</g:mi> <g:mn>2</g:mn> </g:msub> </g:math> 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 <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"> <i:msub> <i:mi>ϕ</i:mi> <i:mn>1</i:mn> </i:msub> </i:math> and <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"> <k:msub> <k:mi>ϕ</k:mi> <k:mn>2</k:mn> </k:msub> </k:math> , along with the heavy off shell mediator SM Higgs, leads to long-lived particle signatures of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"> <m:msub> <m:mi>ϕ</m:mi> <m:mn>2</m:mn> </m:msub> </m:math> 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.