Galaxy-galaxy-galaxy lensing: Third-order correlations between the galaxy and mass distributions in the Universe

Galaxy- galaxy lensing (GGL) measures the 2-point cross-correlation between galaxies and mass in the Universe. In this work we seek to generalise this effect by considering the third-order correlations between galaxies and mass: galaxy-galaxy-galaxy lensing. Third-order correlations in the cosmic shear field have recently been reported in the VIRMOS-DESCART and CTIO surveys. Such data should also be ideal for measuring galaxy-galaxy- galaxy lensing. Indeed, the effects of these higher-order correlations may have already been detected in recent studies of galaxy-galaxy lensing. Higher-order cross-correlation functions contain invaluable information about the relationship between galaxies and their mass environments that GGL studies alone cannot detect. In this paper we lay out the basic relations for third-order cross correlations and their projections and introduce a new set of scale dependent third-order bias parameters. We define three new observables: two galaxy-shear-shear correlation functions, G(+/-), and a galaxy-galaxy- shear correlation, G. We relate these to the various projected cross-bispectra and give practical estimators for their measurement. We note that the observational signature of these correlators is simply the excess shear-shear correlation measured about foreground galaxies ( for G(+/-)) and the average tangential shear around foreground galaxy pairs ( for G). These quantities are no more than second order in the shear and so should be more easily measurable than the shear 3-point correlation. Finally we derive expressions for the third order aperture mass statistics in terms of both the cross-bispectra and the real-space correlation functions. Such statistics provide a very localized measurement of the bispectra, thus encapsulating essentially all of the available third-order information, while remaining easily obtainable from observations of 3-point cross-correlation functions. In addition we find that utilising aperture statistics has the further benefit that they measure only the connected part of the third order correlation.