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

Abstract

The anisotrpy of the redshift space bispectrum $B^s({\mathbf{k}}_1,{\mathbf{k}}_2,{\mathbf{k}}_3)$, which contains a wealth of cosmological information, is completely quantified using multipole moments ${\bar{B}}_{\ell }^m(k_1,\mu ,t)$ 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 $k_1=0.2{\mathrm{Mpc}}^{-1},{\beta }_1=1,b_1=1$ 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 ${\bar{B}}_2^0$ and ${\bar{B}}_4^0$ are similar to ${\bar{B}}_0^0$, however the quadrupole ${\bar{B}}_2^0$ 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. $\mid {\bar{B}}_{\ell }^m\mid$ 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.

arXiv comment20 pages, 10 figures, Matches accepted version to MNRAS

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