A simple parity violating model in the symmetric teleparallel gravity and its cosmological perturbations

Parity violating models based on the symmetric teleparallel gravity have been considered in the literature, but their applications in cosmology and especially the features in cosmological perturbations have not been fully explored. In this paper we consider such a model which modifies general relativity merely through a parity non-conserved coupling, ϕ(x)QQ˜, within the framework of the symmetric teleparallel gravity. We study in detail its cosmological applications, especially its linear perturbation theory. Besides the already known parity violation in the tensor perturbations, we find that the vector perturbations in this model are promoted to be dynamical degrees of freedom, and the left- and right-handed vector modes propagate with different velocities. More importantly, one of the vector modes is found to be a ghost at high momentum scales, this will give rise to the vacuum instability in the quantum theory of cosmological perturbations.