  Inflation driven by a generic self-gravitating medium is an interesting
alternative to study the impact of spontaneous spacetime symmetry breaking
during a quasi de-Sitter phase, in particular the 4-dimensional diffeomorphism
invariance of GR is spontaneously broken down to $ISO(3)$. The effective
description is based on four scalar fields that describe the excitations of a
supersolid. There are two phonon-like propagating scalar degrees of freedom
that mix non-trivially both at early and late times and, after exiting the
horizon, give rise to non-trivial correlations among the different scalar power
spectra. The non-linear structure of the theory allows a secondary
gravitational waves production during inflation, efficient enough to saturate
the present experimental bound and with a blue-tilted spectral index.
