AI 中文总结
该研究将宇称破缺空间协变引力多项式构造扩展到五阶全导数阶次，通过多种方法化简得到59元基，区分不同单项式类型，针对张量微扰得出相关结论并推导两个关系，强制圆偏振光速相速度并允许相关动力学特性。
AI 中文摘要
我们将宇称破缺空间协变引力（SCG）的多项式构造扩展到全导数阶次\(d = 5\)，其中\(d = d_t + d_s\)计算时间和空间导数的总数。通过按\((d_t, d_s)\)组织单项式并利用分部积分、张量对称性、三维曲率恒等式、舒昂恒等式和凯莱 - 哈密顿关系进行化简，得到一个59元基。通过直接检查区分出不同类型的单项式，后两类需要专门的简并性分析。对于空间平坦宇宙学背景下的张量微扰，时间一阶显式部分的无加速度部分包含16个基元，其二阶作用仅取决于系数的四种组合，这些组合在动力学和梯度函数中产生与\(k/a\)和\((k/a)^3\)成比例的螺旋度奇数修正。我们推导了两个关系，强制两种圆偏振的光速相速度，同时仍允许依赖螺旋度的动力学归一化和阻尼。
英文摘要
We extend the polynomial construction of parity-violating spatially covariant gravity (SCG) to total derivative order $d=5$, where $d=d_{\mathrm{t}}+d_{\mathrm{s}}$ counts the total number of temporal and spatial derivatives. After organizing the monomials by $(d_{\mathrm{t}},d_{\mathrm{s}})$ 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 $\mathcal{L}_{\bm{u}}\ln N$ nor $\mathcal{L}_{\bm{u}}K_{ij}$ 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.
Commentsv2, Appendix B added, 21 pages, no figure
