Tensor Integrals in the Large-Scale Structure

TL;DR

This paper introduces a novel analytical method for evaluating tensor integrals in large-scale structure, leveraging FFTLog decomposition and spherical harmonics to simplify complex loop integrals in cosmology.

Contribution

The paper develops an analytical framework that reduces tensor loop integrals in large-scale structure to solvable one-dimensional radial integrals using FFTLog and spherical harmonics.

Findings

Derived analytic expressions for one-loop power spectrum, bispectrum, and trispectrum.

Reduced complex tensor integrals to radial integrals solvable analytically.

Applicable to arbitrary multipole moments in scaling universe models.

Abstract

We present a new method for evaluating tensor integrals in the large-scale structure. Decomposing a $\Lambda$CDM-like universe into a finite sum of scaling universes using the FFTLog, we can recast loop integrals for biased tracers in the large-scale structure as certain tensor integrals in quantum field theory. While rotational symmetry is spontaneously broken by the fixed reference frame in which biased tracers are observed, the tensor structures can still be organized to respect the underlying symmetry. Projecting the loop integrands for scaling universes onto spherical harmonics, the problem effectively reduces to the evaluation of one-dimensional radial integrals, which can be solved analytically. Using this method, we derive analytic expressions for the one-loop power spectrum, bispectrum, and trispectrum for arbitrary multipole moments in the basis of scaling universes.