Computing the Small-Scale Galaxy Power Spectrum and Bispectrum in Configuration-Space

TL;DR

This paper introduces a new efficient estimator for small-scale galaxy power spectra and bispectra in configuration space, avoiding Fourier transforms and correcting for survey geometry, enabling high-$k$ analysis and joint cosmological parameter constraints.

Contribution

The paper presents a novel class of estimators for small-scale spectra that are more efficient and accurate than traditional Fourier-based methods, especially at high wavenumbers.

Findings

Estimator complexity scales as $\\mathcal{O}(NnR_0^3)$ for power spectrum.

Estimator complexity scales as $\\mathcal{O}(Nn^2R_0^6)$ for bispectrum.

The method accurately measures high-$k$ power spectra and covariance from BOSS DR12 simulations.

Abstract

We present a new class of estimators for computing small-scale power spectra and bispectra in configuration-space via weighted pair- and triple-counts, with no explicit use of Fourier transforms. Particle counts are truncated at $R_0\sim 100h^{-1}\,\mathrm{Mpc}$ via a continuous window function, which has negligible effect on the measured power spectrum multipoles at small scales. This gives a power spectrum algorithm with complexity $\mathcal{O}(NnR_0^3)$ (or $\mathcal{O}(Nn^2R_0^6)$ for the bispectrum), measuring $N$ galaxies with number density $n$. Our estimators are corrected for the survey geometry and have neither self-count contributions nor discretization artifacts, making them ideal for high-$k$ analysis. Unlike conventional Fourier transform based approaches, our algorithm becomes more efficient on small scales (since a smaller $R_0$ may be used), thus we may efficiently estimate spectra across $k$-space by coupling this method with standard techniques. We demonstrate the utility of the publicly available power spectrum algorithm by applying it to BOSS DR12 simulations to compute the high-$k$ power spectrum and its covariance. In addition, we derive a theoretical rescaled-Gaussian covariance matrix, which incorporates the survey geometry and is found to be in good agreement with that from mocks. Computing configuration- and Fourier-space statistics in the same manner allows us to consider joint analyses, which can place stronger bounds on cosmological parameters; to this end we also discuss the cross-covariance between the two-point correlation function and the small-scale power spectrum.