  We explore the capabilities of a future proton collider to probe the nature
of the electro-weak phase transition, following the hypothetical discovery of a
new scalar particle. We focus on the real singlet scalar field extension of the
Standard Model, representing the most minimal, and challenging to probe,
framework that can enable a strong first-order electro-weak phase transition.
By constructing detailed phenomenological methods for measuring the mass and
accessible couplings of the new scalar particle, we find that a 100 TeV proton
collider has the potential to explore the parameter space of the real singlet
model and provide meaningful constraints on the electro-weak phase transition.
We empirically find some necessary conditions for the realization of a strong
first order electro-weak phase transition and conjecture that additional
information, including through multi-scalar processes and gravitational wave
detectors, are likely needed to gauge the nature of the cosmological
electro-weak transition. This study represents the first crucial step towards
solving the inverse problem in the context of the electro-weak phase
transition.
