Primordial Non-Gaussianity of Gravitational Waves in Ho\v{r}ava-Lifshitz Gravity

Abstract

In this paper, we study 3-point correlation function of primordial gravitational waves generated in the de Sitter background in the framework of the general covariant Ho\v{r}ava-Lifshitz gravity with an arbitrary coupling constant $\lambda$. We find that, at cubic order, the interaction Hamiltonian receives contributions from four terms built of the 3-dimensional Ricci tensor $R_{ij}$ of the leaves $t = $ constant. In particular, the 3D Ricci scalar $R$ yields the same $k$-dependence as that in general relativity, but with different magnitude due to coupling with the $U(1)$ field $A$ and a UV history. Interestingly, the two terms $R_{ij}R^{ij}$ and $\left(\nabla^{i}R^{jk}\right)\left(\nabla_{i}R_{jk}\right)$ exhibit peaks at the squeezed limit. We show that this is due to the effects of the polarization tensors. The signal generated by the fourth term, $R^i_j R^j_k R^k_i$, favors the equilateral shape when spins of the three tensor fields are the same, but peaks in between the equilateral and squeezed limits when spins are mixed. The consistency with the recently-released Planck observations on non-Gaussianity is also discussed and is found that $\left(H/M_*\right)^2\left(H/M_{pl}\right) \le 10^{-8}$, where $M_{*}$ denotes the suppression energy of high-order operators, $M_{pl}$ the Planck mass, and $H$ the energy of inflation.