A&A 432, 783-795 (2005)

DOI: 10.1051/0004-6361:20041923

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

**P. Schneider and P. Watts**

Institut f. Astrophysik u. Extr. Forschung, Universität Bonn, Auf dem Hügel 71, 53121 Bonn, Germany

e-mail: [peter;pwatts]@astro.uni-bonn.de

(Received 30 August 2004 / Accepted 21 October 2004)

** Abstract **

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, , and a
galaxy-galaxy-shear correlation, . 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 ) and the average tangential shear around
foreground galaxy pairs (for ). 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.

**Key words:**gravitational lensing

**--**cosmology: large-scale structure of the Universe

**--**galaxies: evolution

**--**galaxies: statistics

**©**

*ESO 2005*