The three‐point correlation function in cosmology

ABSTRACT With the advent of high‐quality surveys in cosmology, the full three‐point correlation function (3PCF) will be a valuable statistic for describing structure formation models. It contains information on cosmological parameters and detailed halo properties that cannot be extracted from the two‐point correlation function. We use the halo clustering model to calculate the three‐point correlation function analytically for general cosmological fields. We present detailed results for the configuration dependence of the three‐dimensional mass and galaxy distributions and the two‐dimensional cosmic shear field. We work in real space, where higher order correlation functions on small scales are easier to measure and interpret, but halo model calculations become rapidly intractable. Hence we develop techniques that allow for the accurate evaluation of all the contributing terms to real‐space correlations. We apply them to the 3PCF to show how its configuration and scale dependence change as one transits from the non‐linear to the quasilinear regime, and how this depends on the relative contributions from the one‐, two‐ and three‐halo terms. The 3PCF violates the hierarchical Ansatz in both its scale and configuration dependence. We study the behaviour of the coefficient Q in the expansion: ζ( r 12 , r 23 , r 31 ) = Q [ξ( r 12 ) ξ( r 23 ) +ξ( r 12 )ξ( r 31 ) +ξ( r 23 )ξ( r 31 )] from large, quasilinear scales down to approximately 20 kpc. We find that the non‐linear 3PCF is sensitive to the halo profile of massive haloes, especially its inner slope. We model the distribution of galaxies in haloes and show that the 3PCF of red galaxies has a weaker configuration and scale dependence than the mass, while for blue galaxies it is very sensitive to the parameters of the galaxy formation model. The 3PCF from weak lensing on the other hand shows different scalings arising from projection effects and a sensitivity to cosmological parameters. We discuss how our results can be applied to various analytical calculations: covariances of the two‐point correlation function, the pairwise peculiar velocity dispersion, higher order shear correlations, and to extend the halo model by including the effects of halo triaxiality and substructure on statistical measures.