  Supersolid inflation is a class of inflationary theories that simultaneously
breaks time and space reparameterization invariance during inflation, with
distinctive features for the dynamics of cosmological fluctuations. We
investigate concrete realizations of such a scenario, including non-minimal
couplings between gravity and the fields driving inflation. We focus in
particular on the dynamics of primordial gravitational waves and discuss how
their properties depend on the pattern of symmetry breaking that we consider.
Tensor modes can have a blue spectrum, and for the first time we build models
in which the squeezed limit of primordial tensor bispectra can be
parametrically enhanced with respect to standard single-field scenarios. At
leading order in a perturbative expansion, the tensor-to-scalar ratio depends
only on the parameter controlling the breaking of space-reparameterization. It
is independent from the quantities controlling the breaking of
time-reparameterization, and this represents a difference with respect to
standard single-field inflationary models.
