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. Therefore, even if the propagation speed of gravitational waves in a homogeneous and isotropic background is fine-tuned to that of light (at linear level), the model is subject to further constraints (at non-linear level) due to the presence of inhomogeneities. 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.

Figure 1

Schematic diagram of the transformations of the generalized Proca Lagrangian. An arrow indicates that after a disformal transformation extra terms will contribute to the Lagrangian to which the arrow is directed. A loop means that the Lagrangian is closed by itself under a disformal transformation. We have to include ${\mathcal{G}}_5$ and $U^2$ terms so that the generalized Proca Lagrangian is closed. The division of ${\mathcal{L}}_2$ is only for illustrative purposes and to visualize the contributions from ${\mathcal{L}}_4$ and ${\mathcal{L}}_6$ to parts of ${\mathcal{L}}_2$ through the transformation. See the main text for details.