Quantifying the redshift space distortion of the bispectrum II: induced non-Gaussianity at second-order perturbation

ABSTRACT The anisotrpy of the redshift space bispectrum $B^s(\boldsymbol {k_1},\boldsymbol {k_2},\boldsymbol {k_3})$, which contains a wealth of cosmological information, is completely quantified using multipole moments $\bar{B}^m_{\ell }(k_1,\mu ,t)$, where k1, the length of the largest side, and (μ, t), respectively, quantify the size and the shape of the triangle $(\boldsymbol {k_1},\boldsymbol {k_2},\boldsymbol {k_3})$. We present analytical expressions for all the multipoles that are predicted to be non-zero (ℓ ≤ 8, m ≤ 6) at second-order perturbation theory. The multipoles also depend on β1, b1, and γ2, 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 γ2 = 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 ℓ 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.