  As experiment charts new territory at the electroweak scale, the enterprise
to characterise all possible theories becomes all the more necessary. In the
absence of new particles, this ambitious enterprise is attainable and has led
to the Higgs Effective Field Theory (HEFT) as the most general characterising
framework, containing the Standard Model Effective Field Theory (SMEFT) as a
subspace. The characterisation of this theory space led to the dichotomy SMEFT
vs. HEFT\SMEFT as the two possible realisations of symmetry breaking. The
criterion to distinguish these two possibilities is non-local in field space,
and phenomena which explore field space beyond the neighbourhood of the vacuum
manifold are in a singular position to tell them apart. Cosmology allows for
such phenomena, and this work focuses on HEFT\SMEFT, the less explored of the
two options, to find that first order phase transitions with detectable
gravitational wave remnants, domain wall formation and vacuum decay in the far,
far distant future can take place and single out HEFT\SMEFT. Results in
cosmology are put against LHC constraints, and the potential of future ground-
and space-based experiments to cover parameter space is discussed.
