Probing the effects of Lorentz-symmetry violating Chern-Simons and Ricci-Cotton terms in higher derivative gravity

The combined effects of the Lorentz-symmetry violating Chern-Simons and Ricci-Cotton actions are investigated for the Einstein-Hilbert gravity in the second-order formalism modified by higher derivative terms, and their consequences on the spectrum of excitations are analyzed. We follow the lines of previous works and build up an orthonormal basis of projector-like operators for the degrees of freedom, rather than for the spin modes of the fields. With this new basis, the attainment of the propagators is remarkably simplified and the identification of the physical and unphysical modes becomes more immediate. Our conclusion is that the only tachyon- and ghost-free model is the Einstein-Hilbert action added up by the Chern-Simons term with a timelike vector of the type $v^{\mu }=(\mu ,\overrightarrow{0})$. Spectral consistency imposes that the Ricci-Cotton term must be switched off. We then infer that gravity with Lorentz-symmetry violation imposes a drastically different constraint on the background if compared to ordinary gauge theories whenever conditions for the suppression of tachyons and ghosts are imposed.