Cosmological Perturbation Theory Using the FFTLog: Formalism and Connection to QFT Loop Integrals

TL;DR

This paper introduces a novel FFTLog-based method for calculating cosmological perturbation theory loops by approximating cosmologies as sums of power-law universes, simplifying complex integrals into reusable, analytic forms.

Contribution

The paper develops a new FFTLog-based approach that reduces loop calculations in cosmology to matrix operations, enabling efficient and reusable computations for various cosmological models.

Findings

Explicit formulas for one-loop and two-loop power spectra.

Analytic solutions for loop integrals using hypergeometric functions.

Demonstrated efficiency for higher-order correlation functions.

Abstract

We present a new method for calculating loops in cosmological perturbation theory. This method is based on approximating a $\Lambda$CDM-like cosmology as a finite sum of complex power-law universes. The decomposition is naturally achieved using an FFTLog algorithm. For power-law cosmologies, all loop integrals are formally equivalent to loop integrals of massless quantum field theory. These integrals have analytic solutions in terms of generalized hypergeometric functions. We provide explicit formulae for the one-loop and the two-loop power spectrum and the one-loop bispectrum. A chief advantage of our approach is that the difficult part of the calculation is cosmology independent, need be done only once, and can be recycled for any relevant predictions. Evaluation of standard loop diagrams then boils down to a simple matrix multiplication. We demonstrate the promise of this method for applications to higher multiplicity/loop correlation functions.