An optimal <i>in situ</i> multipole algorithm for the isotropic three-point correlation function

ABSTRACT We present an optimized multipole algorithm for computing the three-point correlation function (3PCF), tailored for application to large-scale cosmological data sets. The algorithm builds on an in situ interpretation of correlation functions, wherein spatial displacements are implemented via translation window functions. In Fourier space, these translations correspond to plane waves, whose decomposition into spherical harmonics naturally leads to a multipole expansion framework for the 3PCF. To accelerate computation, we incorporate density field reconstruction within the framework of multiresolution analysis, enabling efficient summation using either grid-based or particle-based schemes. In addition to the shared computational cost of reconstructing the multipole-decomposed density fields – scaling as $\mathcal {O}(L^2_{\text{trun}} N_\mathrm{ g} \log N_\mathrm{ g})$ (where $N_\mathrm{ g}$ is the number of grids and $L_{\text{trun}}$ is the truncation order of the multipole expansion) – the final summation step achieves a complexity of $\mathcal {O}(D^6_{\text{sup}} N_\mathrm{ g})$ for the grid-based approach and $\mathcal {O}(D^3_{\text{sup}} N_\mathrm{ p})$ for the particle-based scheme (where $D_{\text{sup}}$ is the support of the basis function and $N_\mathrm{ p}$ is the number of particles). The proposed in situ multipole algorithm is fully GPU-accelerated and implemented in the open-source HERMES toolkit for cosmic statistics. This development enables fast, scalable higher order clustering analyses for large-volume data sets from current and upcoming cosmological surveys such as Euclid, the Dark Energy Spectroscopic Instrument (DESI), the Large Synoptic Survey Telescope (LSST), and the Chinese Space Station Telescope (CSST).