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FAST-PT II: an algorithm to calculate convolution integrals of general tensor quantities in cosmological perturbation theory

Xiao Fang, Jonathan A. Blazek, Joseph E. McEwen, Christopher M. Hirata

TL;DR

This work extends the FAST-PT framework to 1-loop tensor convolution integrals that depend on the observer’s line of sight, enabling rapid, $O(N\log N)$-scaling computations for orientation-dependent cosmological statistics. It achieves this by expanding tensor kernels into Legendre polynomials and translating angular structure into spherical harmonics with Wigner symbols, ultimately reducing 3D mode-coupling integrals to a suite of 1D Hankel transforms computable via FFTs. The authors present a complete algorithm, address potential divergences through biasing and regularization, and validate the method with four applications—quadratic intrinsic alignments, Ostriker-Vishniac, kinetic CMB polarization, and redshift-space distortions—showing high-precision agreement with conventional methods and runtimes around 0.1 s for hundreds of k-values. The public code at https://github.com/JoeMcEwen/FAST-PT makes tensor FAST-PT readily usable for orientation-dependent perturbation theory in cosmology, enabling fast, accurate predictions for a broader class of observables.

Abstract

Cosmological perturbation theory is a powerful tool to predict the statistics of large-scale structure in the weakly non-linear regime, but even at 1-loop order it results in computationally expensive mode-coupling integrals. Here we present a fast algorithm for computing 1-loop power spectra of quantities that depend on the observer's orientation, thereby generalizing the FAST-PT framework (McEwen et al., 2016) that was originally developed for scalars such as the matter density. This algorithm works for an arbitrary input power spectrum and substantially reduces the time required for numerical evaluation. We apply the algorithm to four examples: intrinsic alignments of galaxies in the tidal torque model; the Ostriker-Vishniac effect; the secondary CMB polarization due to baryon flows; and the 1-loop matter power spectrum in redshift space. Code implementing this algorithm and these applications is publicly available at https://github.com/JoeMcEwen/FAST-PT .

FAST-PT II: an algorithm to calculate convolution integrals of general tensor quantities in cosmological perturbation theory

TL;DR

This work extends the FAST-PT framework to 1-loop tensor convolution integrals that depend on the observer’s line of sight, enabling rapid, -scaling computations for orientation-dependent cosmological statistics. It achieves this by expanding tensor kernels into Legendre polynomials and translating angular structure into spherical harmonics with Wigner symbols, ultimately reducing 3D mode-coupling integrals to a suite of 1D Hankel transforms computable via FFTs. The authors present a complete algorithm, address potential divergences through biasing and regularization, and validate the method with four applications—quadratic intrinsic alignments, Ostriker-Vishniac, kinetic CMB polarization, and redshift-space distortions—showing high-precision agreement with conventional methods and runtimes around 0.1 s for hundreds of k-values. The public code at https://github.com/JoeMcEwen/FAST-PT makes tensor FAST-PT readily usable for orientation-dependent perturbation theory in cosmology, enabling fast, accurate predictions for a broader class of observables.

Abstract

Cosmological perturbation theory is a powerful tool to predict the statistics of large-scale structure in the weakly non-linear regime, but even at 1-loop order it results in computationally expensive mode-coupling integrals. Here we present a fast algorithm for computing 1-loop power spectra of quantities that depend on the observer's orientation, thereby generalizing the FAST-PT framework (McEwen et al., 2016) that was originally developed for scalars such as the matter density. This algorithm works for an arbitrary input power spectrum and substantially reduces the time required for numerical evaluation. We apply the algorithm to four examples: intrinsic alignments of galaxies in the tidal torque model; the Ostriker-Vishniac effect; the secondary CMB polarization due to baryon flows; and the 1-loop matter power spectrum in redshift space. Code implementing this algorithm and these applications is publicly available at https://github.com/JoeMcEwen/FAST-PT .

Paper Structure

This paper contains 32 sections, 74 equations, 5 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Method
  3. Transformation To 1D Integrals
  4. Algorithm
  5. Implementation For $\mathcal{J}_{ J_1J_2J_k}^{\alpha\beta}(k)$ Integral
  6. Summary of the Algorithm
  7. Removing Possible Divergences
  8. Divergence From Kernel Expansions
  9. Divergence From Periodic Power Spectrum and Choice of Bias Indices
  10. Applications
  11. Quadratic Intrinsic Alignments Model
  12. Theory
  13. Conversion to FAST-PT Format
  14. Ostriker-Vishniac Effect
  15. Theory
  16. ...and 17 more sections

Figures (5)

  • Figure 1: The convergence region of the bias indices $\nu_1,\nu_2$ is indicated by the shaded region.
  • Figure 2: The FAST-PT result for the intrinsic alignment integrals $P_{\rm IA,quad}^{(EE,BB)}(k)$ in Eq. (\ref{['eq:Pk-IA']}) (upper panel) and the fractional difference compared to the conventional method (lower panel).
  • Figure 3: The FAST-PT result for the Ostriker-Vishniac effect integral $S(k)$ in Eq. (\ref{['eq:Sk-OV']}) (upper panel) and the fractional difference compared to the conventional method (lower panel).
  • Figure 4: The FAST-PT results for the kinectic CMB polarization integrals $P^{(m)}(k)$ in Eq. (\ref{['eq:Pmk-kpol']}) (upper panels) and the fractional difference compared to the conventional method (lower panels).
  • Figure 5: The FAST-PT result for the redshift space distortion nonlinear corrections $A(k,\mu_n)+B(k,\mu_n)$ in the TNS model, Eq. (\ref{['eq:TNS']}) (upper panels) and the fractional difference compared to the conventional method result (lower panels).