Unlocking the Higgs Potential: from Colliders to the Cosmos

The upcoming decades in particle physics will offer an unprecedented amount of data, opening new avenues to deepen our understanding of the fundamental laws of nature. On one hand, the High-Luminosity Large Hadron Collider (HL-LHC) will significantly enhance our experimental reach at the energy frontier. On the other hand, the Laser Interferometer Space Antenna (LISA) will inaugurate the era of the early Universe gravitational wave astronomy. The data they will collect may shed light on some of the most profound open questions in physics. At the centre of many unresolved questions in the Standard Model (SM), which include the origin of electroweak symmetry breaking, the matter-antimatter asymmetry, and the nature of dark matter, lies the scalar potential. In particular, the trilinear Higgs self-coupling offers a unique window to determine the shape of this potential. While collider experiments probe it as realised today, cosmological observations can provide insights into its evolution in the early Universe. Together, they offer complementary perspectives on one of the most fundamental ingredients of particle physics.This thesis investigates the phenomenological implications of deviations in the Higgs trilinear self-coupling within well-motivated Beyond the Standard Model (BSM) scenarios featuring extended scalar sectors, with a particular focus on the Two Higgs Doublet Model (2HDM). We perform a detailed study of Higgs pair production at the HL-LHC, the process most directly sensitive to trilinear scalar couplings, examining the effects of additional scalar states both through direct resonant production channels and through radiative corrections to the trilinear Higgs coupling. Our results show that interference effects between resonant and non-resonant contributions, affected by loop-induced modifications to scalar self-interactions, can significantly alter both the total production cross section and the invariant mass distribution, while remaining consistent with all current experimental and theoretical constraints. To account for these effects, we develop and apply dedicated computational frameworks that enable precision BSM analyses incorporating these significant loop effects.Turning to the early Universe, we examine the thermal evolution predicted by BSM scenarios and identify conditions required for a strong first-order electroweak phase transition, which is a necessary ingredient for electroweak baryogenesis. We analyse the characteristic mass hierarchies that favour such transitions and identify the most important collider signatures capable of probing the relevant parameter space. At the same time, we explore the complementary reach of cosmological observables, focusing on stochastic gravitational wave (GW) backgrounds that may be sourced by such strong transitions. We find that space-based GW astronomy could become a complementary tool for exploring fundamental questions of particle physics.