Polarized primordial gravitational waves in spatial covariant gravities

The spatial covariant gravities provide a natural way to including odd-order spatial derivative terms into the gravitational action, which breaks the parity symmetry at gravitational sector. A lot of parity-violating scalar-tensor theories can be mapped to the spatial covariant framework by imposing the unitary gauge. This provides us with a general framework for exploring the parity-violating effects in primordial gravitational waves (PGWs). The main purpose of this paper is to investigate the polarization of PGWs in the spatial covariant gravities and their possible observational effects. To this end, we first construct the approximate analytical solution to the mode function of the PGWs during the slow-roll inflation by using the uniform asymptotic approximation. With the approximate solution, we calculate explicitly the power spectrum and the corresponding circular polarization of the PGWs analytically. It is shown that the new contributions to power spectrum from spatial covariant gravities contain two parts, one from the parity-preserving terms and the other from the parity-violating terms. While the parity-preserving terms can only affect the overall amplitudes of PGWs, the parity-violating terms induce nonzero circular polarization of PGWs, i.e., the left-hand and right-hand polarization modes of GWs have different amplitudes. The observational implications of this nonzero circular polarization is also briefly discussed.