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

TL;DR

The paper introduces a fast, modular framework for cosmological perturbation theory loops by decomposing the linear power spectrum into a finite sum of complex power-laws using FFTLog, transforming loop integrals into massless QFT-like integrals with analytic gamma-function solutions. This decouples cosmology-dependent coefficients from the loop calculations, enabling precomputation of cosmology-independent kernels and rapid predictions via matrix multiplications for the one-loop power spectrum, the one-loop bispectrum, and (in principle) the two-loop power spectrum. The authors derive explicit analytic and series representations for the core building blocks ${\sf I}(\nu_1,\nu_2)$, ${\sf J}(\nu_1,\nu_2,\nu_3;x,y)$, and ${\sf K}(\nu_1,...,\nu_5)$, along with symmetry, inversion, and recursion relations to reduce computational cost. They demonstrate sub-percent agreement with standard results for the accessible cases and outline a feasible path to higher-order correlators, EFT refinements, and broader applications, including potential connections to quantum field theory techniques. The approach promises substantial speedups for parameter inference and provides a bridge between cosmology and QFT methods for loop integrals.

Abstract

We present a new method for calculating loops in cosmological perturbation theory. This method is based on approximating a $Λ$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.

Figures (12)

• Figure 1: Diagrammatic representation of two contributions to the one-loop power spectrum.
• Figure 2: Two contributions to the one-loop power spectrum calculated using direct numerical integration and eq. \ref{['eq:P22powers1']} and eq. \ref{['eq:P13powers1']} as described in the main text. Both plots are produced using $\nu=-0.3$, $N=150$, $k_{\rm min}=10^{-5}\,h{\rm Mpc}^{-1}$ and $k_{\rm max}=5\,h{\rm Mpc}^{-1}$. For these values of parameters the sum of two terms differs from the numerical one-loop power spectrum by less than $0.1\%$ at all scales.
• Figure 3: Two contributions to the one-loop power spectrum calculated using direct numerical integration and eq. \ref{['eq:P22powers1']} and eq. \ref{['eq:P13powers1']}. Both plots are produced using $\nu=-1.6$, $N=150$, $k_{\rm min}=3\cdot 10^{-4}\,h{\rm Mpc}^{-1}$ and $k_{\rm max}=180\,h{\rm Mpc}^{-1}$. For this value of bias both $P_{22}$ and $P_{13}$ are very different from their standard values.
• Figure 4: The full one-loop power spectrum calculated summing up contributions from Fig. \ref{['fig:p22p13wrong']}.
• Figure 5: Four different contributions to the one-loop power spectrum of biased tracers. All plots are produced using $\nu=-1.6$, $N=150$, $k_{\rm min}=10^{-5}\,h{\rm Mpc}^{-1}$ and $k_{\rm max}=5\,h{\rm Mpc}^{-1}$. For these values of parameters the difference with respect to the usual numerical calculation is less than $0.1\%$ at all scales.