Revisiting a parity violating gravity model without ghost instability:
  Local Lorentz covariance
  Recently, based on the theory of teleparallel gravity, a simple and ghost
free parity violating gravity model was proposed in [M. Li, H. Rao, and D.
Zhao, J. Cosmol. Astropart. Phys. 11 (2020) 023], where the Weitzenb\"{o}ck
connection was adopted for simplifying the calculations but breaks the local
Lorentz symmetry explicitly. In this paper, we restore the local Lorentz
symmetry of this model by giving up the Weitzenb\"{o}ck condition on the spin
connection. With full local Lorentz covariance, this model is not a pure tetrad
theory any more. We also apply the new version of this model to the universe
with general Friedmann-Robertson-Walker background. This further generalizes
the studies of [M. Li, H. Rao, and D. Zhao, J. Cosmol. Astropart. Phys. 11
(2020) 023] where only the spatially flat background was considered. Through
the investigations of this paper, we confirm the results obtained in [M. Li, H.
Rao, and D. Zhao, J. Cosmol. Astropart. Phys. 11 (2020) 023] and in addition
get some new results.
