FAST-PT: a novel algorithm to calculate convolution integrals in cosmological perturbation theory

TL;DR

FAST-PT introduces a fast algorithm for evaluating nonlinear mode-coupling integrals in cosmological perturbation theory by expressing angular dependence with Legendre polynomials and leveraging log-spaced power spectra. It recasts P22 and P13 integrals as a set of 1D convolutions that can be computed with FFTs, yielding O(N log N) performance and enabling seamless integration into MCMC pipelines. The paper demonstrates accurate 1-loop SPT results and efficient renormalization group flow calculations, with substantial speedups over traditional methods. The public Python implementation, documentation, and planned extensions to tensor quantities and higher-loop calculations make FAST-PT a broadly useful tool for fast perturbative predictions in cosmology.

Abstract

We present a novel algorithm, FAST-PT, for performing convolution or mode-coupling integrals that appear in nonlinear cosmological perturbation theory. The algorithm uses several properties of gravitational structure formation -- the locality of the dark matter equations and the scale invariance of the problem -- as well as Fast Fourier Transforms to describe the input power spectrum as a superposition of power laws. This yields extremely fast performance, enabling mode-coupling integral computations fast enough to embed in Monte Carlo Markov Chain parameter estimation. We describe the algorithm and demonstrate its application to calculating nonlinear corrections to the matter power spectrum, including one-loop standard perturbation theory and the renormalization group approach. We also describe our public code (in Python) to implement this algorithm, including the applications described here.

Figures (5)

• Figure 1: Power spectra in the log-periodic universe. Top panel shows the windowed linear power spectrum biased by $k^{-\nu}$ (we choose $\nu=-2$), with grey lines indicating the "satellite" power spectra, i.e. the contribution to the total power spectrum that arises due to the periodic assumption in a Fourier transform. The middle panel plots $\Delta^2(k)=k^3P(k)/(2 \pi^2)$, within the periodic universe. This is the quantity that sources the density variance $\sigma^2= \int d \ln k \Delta^2(k)$. The bottom panel plots the contribution to the displacement variance $\sigma_\xi= \int d \ln k \Delta^2(k)/k^2$.
• Figure 2: FAST-PT 1-loop power spectrum results versus those computed using a conventional fixed-grid method. The top panel shows FAST-PT results for $P_{22}(k) + P_{13}(k)$ (the dashed line is for negative values). The bottom panel plots the ratio between FAST-PT and the conventional method.
• Figure 3: Estimate of FAST-PT execution time to number of grid points scaling. The left panel plots the average one-loop evaluation time, after initialization of the FAST-PT class. The right panel plots the average time required for initialization of FAST-PT class for 1500 runs. For a sample of grid points, the error is computed by taking the standard deviation of 1500 runs.
• Figure 4: FAST-PT Renormalization group results for $k_\text{max}=\{5, 50 \} h \text{Mpc}^{-1}$. Left panel shows Renormalization group results and SPT results compared to the linear power spectrum (see legend in right panel). Right panel shows $n_\text{eff}= d \log P/d\log k$ for Renormalization group, SPT, and linear theory.
• Figure 5: Renormalization group results compared to standard 1-loop calculations and those taken from the Coyote Universe. Left panel plots power spectra. A plateau at high-$k$ develops due to boundary conditions. Right panel shows $n_\text{eff}(k)= d \log P/d\log k$.