Stability Analysis of the Possible Consistent Model of Parity Violations in the Symmetric Teleparallel Gravity: Generalized Background Solutions

In this paper, we consider a symmetric teleparallel gravity model that extends the general relativity equivalent model by several parity violating interactions between the gravitational field and a scalar field. We derive three different families of background solutions in flat FRW universe, with three classes of different connections. Through investigations on the linear cosmological perturbations, we show that one of the vector modes of this model will evolve into a ghost field at high energy, and the ghost instability can be cancelled only under specific combinations of the coefficients. On two of three families of backgrounds, such combination remains the same as the one we have investigated in our previous work; while on the other family of background, one additional condition should be taken into consider.