Quantifying the Redshift Space Distortion of the Bispectrum I: Primordial Non-Gaussianity

Abstract: The anisotropy of the redshift space bispectrum contains a wealth of cosmological information. This anisotropy depends on the orientation of three vectors ${\bf k_1,k_2,k_3}$ with respect to the line of sight. Here we have decomposed the redshift space bispectrum in spherical harmonics which completely quantify this anisotropy. To illustrate this we consider linear redshift space distortion of the bispectrum arising from primordial non-Gaussianity. In the plane parallel approximation only the first four even $\ell$ multipoles have non-zero values, and we present explicit analytical expressions for all the non-zero multipoles {\it i.e.} upto $\ell=6,m=4$. The ratio of the different multipole moments to the real space bispectrum depends only on $\beta_1$ the linear redshift distortion parameter and the shape of the triangle. Considering triangles of all possible shapes, we have studied how this ratio depends on the shape of the triangle for $\beta_1=1$. We have also studied the $\beta_1$ dependence for some of the extreme triangle shapes. If measured in future, these multipole moments hold the potential of constraining $\beta_1$. The results presented here are also important if one wishes to constrain $f_{\text{NL}}$ using redshift surveys.