Quantifying the Redshift Space Distortion of the Bispectrum II: Induced Non-Gaussianity at Second Order Perturbation

The anisotrpy of the redshift space bispectrum $B^s(mathbf{k_1},mathbf{k_2},mathbf{k_3})$, which contains a wealth of cosmological information, is completely quantified using multipole moments $bar{B}^m_{ell}(k_1,mu,t)$ where $k_1$, the length of the largest side, and $(mu,t)$ respectively quantify the size and shape of the triangle $(mathbf{k_1},mathbf{k_2},mathbf{k_3})$. We present analytical expressions for all the multipoles which are predicted to be non-zero ($ell le 8, m le 6$ ) at second order perturbation theory. The multipoles also depend on $beta_1,b_1$ and $gamma_2$, which quantify the linear redshift distortion parameter, linear bias and quadratic bias respectively. Considering triangles of all possible shapes, we analyse the shape dependence of all of the multipoles holding $k_1=0.2 , {rm Mpc}^{-1}, beta_1=1, b_1=1$ and $gamma_2=0$ fixed. The monopole $bar{B}^0_0$, which is positive everywhere, is minimum for equilateral triangles. $bar{B}_0^0$ increases towards linear triangles, and is maximum for linear triangles close to the squeezed limit. Both $bar{B}^0_{2}$ and $bar{B}^0_4$ are similar to $bar{B}^0_0$, however the quadrupole $bar{B}^0_2$ exceeds $bar{B}^0_0$ over a significant range of shapes. The other multipoles, many of which become negative, have magnitudes smaller than $bar{B}^0_0$. In most cases the maxima or minima, or both, occur very close to the squeezed limit. $mid bar{B}^m_{ell} mid $ is found to decrease rapidly if $ell$ or $m$ are increased. The shape dependence shown here is characteristic of non-linear gravitational clustering. Non-linear bias, if present, will lead to a different shape dependence.