FFT-PT: Reducing the two-loop large-scale structure power spectrum to low-dimensional radial integrals

TL;DR

FFT-PT extends a fast 1-loop framework to 2-loop corrections in Eulerian perturbation theory by recasting high-dimensional integrals as sequences of low-dimensional radial Hankel transforms, computed efficiently with 1D FFTs. The method leverages space-domain switching to avoid convolutions and integrates over orientations to isolate radial dependences, preserving generality with respect to the input linear power spectrum and BAO features. It provides exact reformulations for 2-loop terms without inverse Laplacians and extends to single and multiple inverse Laplacians with nested radial transforms, including clear pathways for halo bias and redshift-space distortions. The approach promises substantial speedups for Monte Carlo analyses and parameter inference, with practical extensions to LPT, LCDM variants, and higher-order statistics, albeit with challenges for multiple inverse Laplacians and potential infrared issues.

Abstract

Modeling the large-scale structure of the universe on nonlinear scales has the potential to substantially increase the science return of upcoming surveys by increasing the number of modes available for model comparisons. One way to achieve this is to model nonlinear scales perturbatively. Unfortunately, this involves high-dimensional loop integrals that are cumbersome to evaluate. Trying to simplify this, we show how two-loop (next-to-next-to-leading order) corrections to the density power spectrum can be reduced to low-dimensional, radial integrals. Many of those can be evaluated with a one-dimensional Fast Fourier Transform, which is significantly faster than the five-dimensional Monte-Carlo integrals that are needed otherwise. The general idea of this FFT-PT method is to switch between Fourier and position space to avoid convolutions and integrate over orientations, leaving only radial integrals. This reformulation is independent of the underlying shape of the initial linear density power spectrum and should easily accommodate features such as those from baryonic acoustic oscillations. We also discuss how to account for halo bias and redshift space distortions.

Figures (2)

• Figure 1: Diagrammatic representation of nontrivial 2-loop contributions to the dark matter power spectrum in standard Eulerian perturbation theory. Tadpole subdiagrams (red dashed) are evaluated at the same point, leading to zero-lag correlations $\xi(0)$. 'Connector' subdiagrams (blue) are evaluated at two different points, leading to correlations $\xi(r)$ at non-zero separation $r$.
• Figure 2: Numerical results for two-loop integrals $I_{15}$, $I_{24}$ and $I_{33}$ defined in Eqs. (\ref{['eq:I15def']}), (\ref{['eq:I24def']}) and (\ref{['eq:I33def']}) for the special case where $\boldsymbol\alpha=\boldsymbol\beta =0$ and the linear power spectrum has the simple form $P_\mathrm{lin}=k^2 \exp (-k^2)$. Using the radial integrals in Eqs. (\ref{['eq:I15WithXiN']}), (\ref{['eq:I24WithXiN']}) and (\ref{['eq:I33WithXiN']}) together with Eq. (\ref{['eq:QNIntegralExpanded']}) (red lines) is compared against the conventionally used direct Monte Carlo evaluation of two-loop integrals using the Cubacuba library (black points). As mentioned in the text, denominators in the $I_{15}$, $I_{24}$ and $I_{33}$ terms have been extended, $q^2 \rightarrow q^2 +\epsilon$ with $\epsilon=0.001$, in order order to avoid singular points.