Vector disformal transformation of generalized Proca theory
  Motivated by the GW170817/GRB170817A constraint on the deviation of the speed
of gravitational waves from that of photons, we study disformal transformations
of the metric in the context of the generalized Proca theory. The constraint
restricts the form of the gravity Lagrangian, the way the electromagnetism
couples to the gravity sector on cosmological backgrounds, or in general a
combination of both. Since different ways of coupling matter to gravity are
typically related to each other by disformal transformations, it is important
to understand how the structure of the generalized Proca Lagrangian changes
under disformal transformations. For disformal transformations with constant
coefficients we provide the complete transformation rule of the Lagrangian. We
find that additional terms, which were considered as beyond generalized Proca
in the literature, are generated by the transformations. Once these additional
terms are included, on the other hand, the structure of the gravity Lagrangian
is preserved under the transformations. We then derive the transformation rules
for the sound speeds of the scalar, vector and tensor perturbations on a
homogeneous and isotropic background. We explicitly show that they transform
following the natural expectation of metric transformations, that is, according
to the transformation of the background lightcone structure. We end by arguing
that inhomogeneities due to structures in the universe, e.g. dark matter halos,
generically changes the speed of gravitational waves from its cosmological
value. We give a rough estimate of the effect of inhomogeneities and find that
the fine-tuning should not depend on the background or that the fine-tuned
theory has to be further fine-tuned to pass the tight constraint.
