Computing the three-point correlation function of galaxies in $\mathcal {O}(N^2)$ time

Abstract

We present an algorithm that computes the multipole coefficients of the galaxy threepoint correlation function (3PCF) without explicitly considering triplets of galaxies. Rather, centering on each galaxy in the survey, it expands the radially-binned density field in spherical harmonics and combines these to form the multipoles without ever requiring the relative angle between a pair about the central. This approach scales with number and number density in the same way as the two-point correlation function, allowing runtimes that are comparable, and 500 times faster than a naive triplet count. It is exact in angle and easily handles edge correction. We demonstrate the algorithm on the LasDamas SDSS-DR7 mock catalogs, computing an edge corrected 3PCF out to 90 Mpc/h in under an hour on modest computing resources. We expect this algorithm will render it possible to obtain the large-scale 3PCF for upcoming surveys such as Euclid, LSST, and DESI.

Streaming Velocities and the Baryon Acoustic Oscillation Scale

“…Figure 2 of Ref. [48]). Thus, the streaming velocity, v bc , rapidly changes at the acoustic scale, and advection can move tracers separated by roughly this scale between regions of different v bc .…”

“…The definitions of the bias coefficients are not standardized: while our b v is equivalent to that of [48], b r in [47] is related to both via b r = The mapping between x and q can be expanded to order δ lin , since we are only concerned with contributions to δ g up to O(δ 3 lin ). Lagrangian and Eulerian positions are related by x(q, η) = q + Ψ(q, η).…”

“…streaming velocity versus a region with zero streaming velocity. Studies based on perturbation theory have found that the BAO ruler shrinks (stretches) for b v > 0 (b v < 0) [46][47][48].…”

Estimating covariance matrices for two- and three-point correlation function moments in Arbitrary Survey Geometries

“…3.4. To form a sample covariance matrix, we first compute the 3PCF for each of 250 mocks using the algorithm of Slepian & Eisenstein (2015b), which has complexity O(N 2 ) for N galaxies, and {3, 4, 5, 6}); we here depict how these are chosen, as well as which term each configuration corresponds to. To compute the integrals we first draw three points in space, {i, j, k }, then add the l, m and n points successively to find a single contribution to each integral.…”

“…3.4, the difference between the matrices appears to be solely consistent with noise at this resolution level. Slepian & Eisenstein (2015b). The 3PCF is measured from ∼ 6 × 10 5 galaxies in each mock, using the Szapudi & Szalay (1998) estimator to correct for boundary effects, and the compression of Slepian & Eisenstein (2015a) to convert the 3PCF to a set of one-dimensional functions, averaging over radial bins which avoid the (non-Gaussian) squeezed limit.…”