Background#

See also

Wang et al. [2023] for the companion journal article.

Given a catalogue of discrete particles (such as galaxies) with their spatial coordinates, Triumvirate computes estimators of the multipoles of the three-point correlation function, also known as the bispectrum in Fourier space, in the tri-polar spherical harmonic (TripoSH) decomposition proposed by Sugiyama et al. [2019]. The objective of Triumvirate is to provide efficient end-to-end measurement of clustering statistics which can be fed into downstream galaxy survey analyses to constrain and test cosmological models.

For the TripoSH decomposition mentioned above, Triumvirate offers the following functionalities:

  • triumvirate.threept measures three-point clustering statistics, namely multipoles of the bispectrum in Fourier space and of the three-point correlation function (3PCF) in configuration space, as in eqs. (41) & (48) of Sugiyama et al. [2019];

  • triumvirate.twopt measures two-point clustering statistics, namely multipoles of the power spectrum and two-point correlation function (2PCF), as in eqs. (40) & (49) of Sugiyama et al. [2018];

  • both modules mentioned above include the local plane-parallel estimator for a pair of survey-like data and random catalogues, and the global plane-parallel estimator for a simulation-like catalogue in a cuboid box, as in e.g. eq. (52) of Sugiyama et al. [2019];

  • both modules mentioned above can measure the configuration-space window function multipoles, which are used to convolve theoretical models derived in Fourier space through the Hankel transform, as in e.g. eqs. (58) & (63) of Sugiyama et al. [2019] and eq. (56) of Sugiyama et al. [2018].

For the global plane-parallel estimators, the simulation box is placed at the spatial infinity (or equivalently the observer is), so that the line of sight to each particle can be treated as the same and taken to be along the \(z\)-axis.

For the local plane-parallel estimators, the observer is placed at the origin in the survey coordinates, and the line of sight is chosen to point towards one of the particles in a doublet/triplet for two-/three-point clustering measurements.

The geometry of the survey leaves an imprint on the clustering statistics, where in Fourier space the effect is a convolution with the survey window function. This convolution mixes different multipoles of the underlying clustering statistics and the survey window, and the precise convolution formula (i.e. the number of multipoles to include in modelling) needed to achieve a given level of convergence depends on the precise survey geometry including any sample weights applied.

1(1,2,3,4)

Naonori S. Sugiyama, Shun Saito, Florian Beutler, and Hee-Jong Seo. A complete FFT-based decomposition formalism for the redshift-space bispectrum. Mon. Not. R. Astron. Soc., 484(1):364–384, 2019. doi:10.1093/mnras/sty3249.

2(1,2)

Naonori S. Sugiyama, Maresuke Shiraishi, and Teppei Okumura. Limits on statistical anisotropy from BOSS DR12 galaxies using bipolar spherical harmonics. Mon. Not. R. Astron. Soc., 473(2):2737–2752, 2018. doi:10.1093/mnras/stx2333.

3

Mike S. Wang, Florian Beutler, and Naonori S. Sugiyama. Triumvirate: A Python/C++ package for three-point clustering measurements. J. Open Source Softw., preprint (submitted): under review, 2023. arXiv:2304.03643, doi:10.48550/arXiv.2304.03643.