tokens: gravitational lensing distortion matrix ( 1 ) across the image, on the employed weight function W , and also the employed ellipticity definition. In the case of a flat weight, W ()const., and the centre of light as image centre, equations ( 9 ) and ( 13 ) yield unweighted moments. The observed ellipticity ( 10 ) and the intrinsic ellipticity ( 14 ) are then related by An advantage of based on unweighted moments is that for g 1, one may use the observed ellipticity as an unbiased estimator, for the reduced shear g , i.e. , where angular brackets denote averages over the galaxy intrinsic ellipticity distribution (which is assumed rotation invariant), regardless of the details of the intrinsic ellipticity distribution (Seitz Schneider 1997 ). Unweighted moments and ellipticities derived from them are susceptible to image noise in the outer image regions [due to the factor i j in the integral in equation ( 9 )]. Moreover, unweighted moments and ellipticities are strongly affected by errors in the separation of the galaxy light from the light of neighbouring galaxies. Ellipticity estimation from real galaxy images thus often employs down-weighting of the outer regions of the image. The drawback of such radially weighted moments is that they complicate the relation between observed and intrinsic ellipticities. For a general choice of ellipticity definition and weight function W (), and sufficiently small reduced shear g , the observed ellipticity as a function of reduced shear g can be approximated by a linear response in g : where the intrinsic ellipticity , and the shear polarizability If P can be estimated with sufficient accuracy (e.g. from the observed galaxy image itself) and is invertible, one may obtain an unbiased shear estimator by applying P 1 as a shear polarizability correction factor: If the average shear polarizability P for the observed galaxy population is known, another unbiased shear estimator can be constructed that employs P 1 as polarizability correction: The above image ellipticity-based shear estimators can be treated in a uniform manner, 7 with appropriate definitions for the observed ellipticity and intrinsic ellipticity . For example, equation ( 17 ) suggests with and from unweighted moments. For e computed from unweighted moments, one obtains (e.g. Bernstein Jarvis 2002 ) 8 A simplified version of the estimator by Kaiser, Squires Broadhurst ( 1995 , KSB) yields with and from moments with a Gaussian radial weight function, and estimated from the individual observed galaxy images. As for the shear, the ellipticities and can be decomposed into tangential and cross components with respect to a given angular direction or comoving transverse direction : with , , and . 2.4 Density and intrinsic shape correlations The intrinsic shapes of galaxies may have a preferred orientation towards nearby other galaxies and matter overdensities. The preferred orientation of galaxies towards matter overdensities may be quantified by the
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