Three-Point Statistics from a New Perspective

Abstract

Multipole expansion of spatial three-point statistics is introduced as a tool for investigating and displaying configuration dependence. The novel parametrization renders the relation between bi-spectrum and three-point correlation function especially transparent as a set of two-dimensional Hankel transforms. It is expected on theoretical grounds, that three-point statistics can be described accurately with only a few multipoles. In particular, we show that in the weakly non-linear regime, the multipoles of the reduced bispectrum, $Q_l$, are significant only up to quadrupole. Moreover, the non-linear bias in the weakly non-linear regime only affects the monopole order of these statistics. As a consequence, a simple, novel set of estimators can be constructed to constrain galaxy bias. In addition, the quadrupole to dipole ratio is independent of the bias, thus it becomes a novel diagnostic of the underlying theoretical assumptions: weakly non-linear gravity and perturbative local bias. To illustrate the use of our approach, we present predictions based on both power law, and CDM models. We show that the presently favoured SDSS-WMAP concordance model displays strong ``baryon bumps'' in the $Q_l$'s. Finally, we sketch out three practical techniques estimate these novel quantities: they amount to new, and for the first time edge corrected, estimators for the bispectrum.Comment: 5 pages 6 figures, ApL accepte

Introduction to Higher Order Spatial Statistics in Cosmology

“…As a consequence of spherical symmetry the three-point correlation function and the bispectrum, depend only on the shape of a triangle. [19] has observed that the three-point statistics can be expressed with two unit vectors, thus zero angular momentum bipolar expansion is suitable in 3 spatial dimensions under SO(3) symmetry. Zero angular momentum bipolar functions are proportional to the Legendre polynomials, thus in turn this becomes multi-pole expansion of the bispectrum.…”

“…E.g., the redshift space bispectrum is parametrized by the five parameters B(k 1,⊥ , k 2,⊥ , k 3,⊥ , k 1, , k 2, ), with ⊥ denoting transverse, and parallel quantities with respect to the line of sight in the distant observer approximation. Interestingly, the real space bispectrum can be estimated from taking k 1, ≃ k 2, ≃ 0 [19].…”

Fast Edge-corrected Measurement of the Two-Point Correlation Function and the Power Spectrum

“…This affects our method as it does the Feldman et al (1994) type direct methods. We conjecture that our method of inverting the correlation function can be generalized for redshift distortions by means of the formulae of Szapudi (2004b); this is left for future research.…”

“…Generalization of the proposed computa-1 2 tional and inversion techniques to gravitational lensing follows directly from the generalization of SpICE to CMB polarization (e.g., Chon et al 2004). Finally, generalization of equation (3) for a relation between the three-point function and the bispectrum (Szapudi 2004a), together with a fast algorithm to measure the three-point function, yields a new, edge-corrected method to measure the bispectrum. These generalizations will be presented elsewhere.…”

Measuring the Three‐Point Correlation Function of the Cosmic Microwave Background

“…Neither of these methods is able to scan through all possible configurations in megapixel maps with a reasonable amount of computing resources. Here we propose a new method, which uses both hierarchical pixelization and Fourier methods motivated by Szapudi (2004) and Szapudi et al (2005) to scan through all the triangles simultaneously. Note that Gaztañaga & Wagg (2003) come closest to our aims, but their simple two-step approach is not systematic enough to cover all possible triangles at a given resolution, and it is not fast enough for massive Monte Carlo simulations.…”