Non-trivial gravitational waves and structure formation phenomenology
  from dark energy
  The detection of the GW170817/GRB170817A event improved the constraints on
the propagation speed of gravitational waves, thus placing possible variations
caused by dark energy under restraint. For models based on scalar fields
belonging to the family of Horndeski Lagrangians, non-minimal derivative
couplings are now severely constrained, entailing a substantially limited
phenomenology. In this work we want to stress that there is still a plethora of
dark energy models that get around this obstacle while still providing
interesting phenomenologies able to distinguish them from the standard
cosmology. We focus on a class involving vector fields as a proxy, but our
discussion is extensible to a broader class of models. In particular, we show
the possibility of having a non-minimal derivative coupling giving a
non-trivial effect on scalar modes without affecting gravitational waves and
the possibility of having a second tensor mode that can oscillate into
gravitational waves. We also present a novel class of configurations breaking
rotational invariance but with an energy-momentum tensor that is isotropic
on-shell. This peculiar feature makes the scalar and vector sectors of the
perturbations mix so that, even in a perfectly isotropic background cosmology,
preferred direction effects can appear in the perturbations. We also comment on
models that give rise to isotropic solutions when averaging over rapid
oscillations of the vector fields. The explored models are classified according
to distinctive field configurations that provide inequivalent realisations of
the Cosmological Principle.
