2026/07/22 by Jin-Jian Song, Xian Gao
#gr-qc
We extend the polynomial construction of parity-violating spatially covariant gravity (SCG) to total derivative order d=5, where d=dt+ds counts the total number of temporal and spatial derivatives. After organizing the monomials by (dt,ds) and reducing them using integrations by parts, tensor symmetries, three-dimensional curvature identities, the Schouten identity, and the Cayley-Hamilton relation, we obtain a 59-element basis: 2, 40, and 17 monomials in the sectors (0,5), (2,3), and (4,1), respectively. A direct inspection separates 35 monomials containing neither the lapse velocity L_\bmuln N nor L_\bmuKij from 11 lapse-velocity monomials and 13 monomials containing higher normal derivatives of the spatial metric. The latter two sectors require a dedicated degeneracy analysis. For tensor perturbations about a spatially flat cosmological background, the acceleration-free part of the manifestly first-order-in-time sector contains 16 basis elements, whose quadratic action depends on only four combinations of coefficients. These combinations generate helicity-odd corrections proportional to k/a and (k/a)3 in the kinetic and gradient functions. We derive two relations that enforce luminal phase velocity for both circular polarizations while still allowing helicity-dependent kinetic normalization and damping.