APERTURE MULTIPOLE MOMENTS FROM WEAK GRAVITATIONAL LENSING

The projected mass of a gravitational lens inside an (circular) apertu re can be derived from the measured shear inside an annulus which is c aused by the tidal field of the deflecting mass distribution. Here we show that also the multipoles of the two-dimensional mass distribution can be derived from the shear in annuli. We derive several expression s for these mass multipole moments in terms of the shear, which allow large flexibility in the choice of a radial weight function. In contra st to determining multipole moments from weak-lensing mass reconstruct ions, this approach allows us to quantify the signal-to-noise (S/N) ra tio of the multipole moments directly from the observed galaxy ellipti cities, and thus to estimate the significance of the multipole detecti on. Radial weight functions tan therefore be chosen such as to optimiz e the significance of the detection given an assumed radial mass profi le. Application of our formulae to numerically simulated clusters demo nstrates that the quadrupole moment of realistic cluster models can be detected with high S/N ratio; in similar or equal to 85 per cent of t he simulated cluster fields S/N greater than or similar to 3. We also show that the shear inside a circular annulus determines multipole mom ents inside and outside the annulus. This is relevant for clusters who se central region is too bright to allow the observation of the shear of background galaxies, or which extend beyond the CCD. We also genera lize the aperture mass equation to the case of 'radial' weight functio ns that are constant on arbitrarily shaped curves that are not necessa rily self-similar.