Aperture multipole moments from weak gravitational lensing

The projected mass of a gravitational lens inside (circular) apertures can bederived from the measured shear inside an annulus which is caused by the tidalfield of the deflecting mass distribution. Here we show that also themultipoles of the two-dimensional mass distribution can be derived from theshear in annuli. We derive several expressions for these mass multipole momentsin terms of the shear, which allow large flexibility in the choice of a radialweight function. In contrast to determining multipole moments from weak-lensingmass reconstructions, this approach allows to quantify the signal-to-noiseratio of the multipole moments directly from the observed galaxy ellipticities,and thus to estimate the significance of the multipole detection. Radial weightfunctions can therefore be chosen such as to optimize the significance of thedetection given an assumed radial mass profile. Application of our formulae tonumerically simulated clusters demonstrates that the quadrupole moment ofrealistic cluster models can be detected with high signal-to-noise ratio S/N;in about 85 per cent of the simulated cluster fields S/N >~ 3. We also showthat the shear inside a circular annulus determines multipole moments insideand outside the annulus. This is relevant for clusters whose central region istoo bright to allow the observation of the shear of background galaxies, orwhich extend beyond the CCD. We also generalize the aperture mass equation tothe case of `radial' weight functions which are constant on arbitrarily-shapedcurves which are not necessarily self-similar.Comment: 14 pages including 3 figures; submitted to MNRAS; replaced to improve printing on non-A4 pape