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Quantifying the Redshift Space Distortion of the Bispectrum II: Induced Non-Gaussianity at Second Order Perturbation
Arindam Mazumdar, Somnath Bharadwaj, Debanjan Sarkar
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The anisotrpy of the redshift space bispectrum Bˢ(𝐤₁,𝐤₂,𝐤₃), which contains a wealth of cosmological information, is completely quantified using multipole moments B̅ᵐ_ℓ(k₁,μ,t) where k₁, the length of the largest side, and (μ,t) respectively quantify the size and shape of the triangle (𝐤₁,𝐤₂,𝐤₃). We present analytical expressions for all the multipoles which 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 k₁=0.2 Mpc⁻¹, β₁=1, b₁=1 and γ₂=0 fixed. The monopole B̅⁰₀, which is positive everywhere, is minimum for equilateral triangles. B̅₀⁰ increases towards linear triangles, and is maximum for linear triangles close to the squeezed limit. Both B̅⁰₂ and B̅⁰₄ are similar to B̅⁰₀, however the quadrupole B̅⁰₂ exceeds B̅⁰₀ over a significant range of shapes. The other multipoles, many of which become negative, have magnitudes smaller than B̅⁰₀. In most cases the maxima or minima, or both, occur very close to the squeezed limit. |B̅ᵐ_ℓ | 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.
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