Fast Large Scale Structure Perturbation Theory using 1D FFTs
Marcel Schmittfull, Zvonimir Vlah, Patrick McDonald
TL;DR
This paper introduces a fast, exact method to compute nonlinear large-scale structure corrections in Eulerian Standard Perturbation Theory by recasting 3D convolution integrals as products in configuration space and evaluating them with 1D Hankel transforms via FFTs. By defining generalized linear correlations $\xi^l_n(r)$ and exploiting isotropy, the authors derive compact, fast expressions for the 2-2 and 1-3 loop contributions to the matter power spectrum, requiring only a small number of 1D transforms. The method achieves speedups of orders of magnitude over traditional quadrature, while maintaining high accuracy (better than $10^{-5}$ on relevant scales). It also provides a transparent, physically motivated framework that can be extended to higher loops, redshift-space distortions, or halos, offering practical benefits for parameter-space explorations and large-scale structure analyses. The work leverages configuration-space locality of the PT evolution to transform complex 3D integrals into manageable 1D Hankel transforms, with FFT-based implementations enabling broad applicability.
Abstract
The usual fluid equations describing the large-scale evolution of mass density in the universe can be written as local in the density, velocity divergence, and velocity potential fields. As a result, the perturbative expansion in small density fluctuations, usually written in terms of convolutions in Fourier space, can be written as a series of products of these fields evaluated at the same location in configuration space. Based on this, we establish a new method to numerically evaluate the 1-loop power spectrum (i.e., Fourier transform of the 2-point correlation function) with one-dimensional Fast Fourier Transforms. This is exact and a few orders of magnitude faster than previously used numerical approaches. Numerical results of the new method are in excellent agreement with the standard quadrature integration method. This fast model evaluation can in principle be extended to higher loop order where existing codes become painfully slow. Our approach follows by writing higher order corrections to the 2-point correlation function as, e.g., the correlation between two second-order fields or the correlation between a linear and a third-order field. These are then decomposed into products of correlations of linear fields and derivatives of linear fields. The method can also be viewed as evaluating three-dimensional Fourier space convolutions using products in configuration space, which may also be useful in other contexts where similar integrals appear.