On cosmological perturbations of quasidilaton

A theory of the quasidilaton is an extension of massive gravity by a scalar field, nonlinearly realizing a certain new global symmetry of the Lagrangian. It has been shown that unlike pure massive gravity, this theory does admit homogeneous and isotropic spatially flat solutions. Among the latter, the self-accelerated solutions attract a special attention. Previous studies of perturbations, performed in the decoupling limit, revealed one healthy scalar mode, while the second relevant scalar was not captured in that limit. Here we study full cosmological perturbations above the simplest self-accelerated background. We show that the fluctuations of a mixed state of the quasidilaton and the helicity-0 graviton necessarily have a negative kinetic term at short distances, making this background unphysical. In addition, these cosmologies exhibit an order-1 sensitivity to higher dimensional terms suppressed by an energy scale that is parametrically higher than the strong coupling scale of the quasidilaton effective theory: such terms include Galileons, Goldstone-like self-interactions and derivatives of the quasidilaton coupled to curvature, none of which introduce extra Ostrogradsky states. As one consequence, cosmology at the Hubble distances for this particular class of solutions depends on an unknown extension of the quasidilaton below its strong coupling distance scale. We note that non-FRW solutions that are similar to those of the pure massive gravity should not necessarily suffer from these pathologies.