Rapid separable analysis of higher order correlators in large-scale structure

We present an efficient separable approach to the estimation and reconstruction of the bispectrum and the trispectrum from observational (or simulated) large-scale structure data. This is developed from general cosmic microwave background (poly) spectra methods that exploit the fact that the bispectrum and trispectrum in the literature can be represented by a separable mode expansion that converges rapidly (with n₌₀ₗ=O (30) terms). The underlying methodology can encompass a wide variety of modal types, including polynomials, trigonometric functions, wavelets, and bins. With an effective grid resolution l₌₀ₗ (number of particles/grid points N=l₌₀ₗ^3), we present a bispectrum estimator that requires only O (n₌₀ₗl₌₀ₗ^3) operations, along with a corresponding method for direct bispectrum reconstruction. This method is extended to the trispectrum revealing an estimator that requires only O (n₌₀ₗ^4/3l₌₀ₗ^3) operations. The complexity in calculating the trispectrum in this method is now involved in the original decomposition and orthogonalization process that need only be performed once for each model. However, for nondiagonal trispectra these processes present little extra difficulty and may be performed in O (l₌₀ₗ^4) operations. A discussion of how the methodology may be applied to the quadspectrum is also given. An efficient algorithm for the generation of arbitrary non-Gaussian initial conditions for use in N-body codes using this separable approach is described. This prescription allows for the production of non-Gaussian initial conditions for arbitrary bispectra and trispectra. A brief outline of the key issues involved in parameter estimation, particularly in the nonlinear regime, is also given.