Investigating a strong first-order electroweak phase transition in the RxSM at future linear $e^+e^-$ colliders and LISA

The general real singlet extension of the Standard Model (SM), the RxSM, is one of the simplest theories Beyond-the-Standard Model (BSM) that can accommodate a strong first-order electroweak phase transition (SFOEWPT). We investigate the possible thermal histories of the scalar potential in the RxSM, and the regions of the model parameter space in which SFOEWPT can be realised. We then explore complementary avenues to probe such scenarios experimentally: either using searches for a stochastic background of gravitational waves (GWs), or using searches for di-Higgs production processes at future collider experiments, focusing on the case of a high-energy $e^+e^-$ collider. An important aspect of our work is that one-loop corrections to all relevant trilinear scalar couplings are consistently included both in the calculation of dynamics of the electroweak phase transition (EWPT) and in collider processes. We find entirely different phenomenological signatures for different parts of the RxSM parameter space giving rise to SFOEWPTs. On the one hand, if the SFOEWPT is driven by the singlet field, the 125 GeV Higgs boson is very SM-like and signs of BSM physics would be difficult to find at colliders, but strong GW signals could be produced. On the other hand, in scenarios where a SFOEWPT is driven by the doublet field, BSM deviations in properties of the detected Higgs boson, particularly in its trilinear self-coupling, typically lead to observable signals at colliders, while detectable GW signals are much more challenging to achieve. This work highlights the complementarity of collider experiments and cosmological observations to determine the dynamics of the EWPT and reconstruct the shape of the Higgs potential realised in Nature.