Weak lensing goes bananas: what flexion really measures

Abstract

In weak gravitational lensing, the image distortion caused by shear measures the projected tidal gravitational field of the deflecting mass distribution. To lowest order, the shear is proportional to the mean image ellipticity. If the image sizes are not small compared to the scale over which the shear varies, higher-order distortions occur, called flexion. For ordinary weak lensing, the observable quantity is not the shear, but the reduced shear, owing to the mass-sheet degeneracy. Likewise, the flexion itself is unobservable. Instead, higher-order image distortions measure the reduced flexion, i.e., derivatives of the reduced shear. We derive the corresponding lens equation in terms of the reduced flexion and calculate the resulting relation between brightness moments of source and image. Assuming an isotropic distribution of source orientations, estimates for the reduced shear and flexion are obtained and then tested with simulations. In particular, the presence of flexion affects the determination of the reduced shear. The results of these simulations yield the amount of bias of the estimators as a function of the shear and flexion. We point out and quantify a fundamental limitation of the flexion formalism in terms of the product of reduced flexion and source size. If this product increases above the derived threshold, multiple images of the source are formed locally, and the formalism breaks down. Finally, we show how a general (reduced) flexion field can be decomposed into its four components. Two of them are due to a shear field, carrying an E-and B-mode in general. The other two components do not correspond to a shear field, and they can also be split up into corresponding E-and B-modes.

Observational probes of cosmic acceleration

“…Leonard et al 2007;Velander et al 2011;Leonard et al 2011), and there is a growing literature on the theory of flexion measurement that parallels the formalism required for shear measurement (e.g. Massey et al 2007c;Schneider and Er 2008;Rowe et al 2012). However, because of the extra derivative it is sensitive mainly to structure at the very smallest scales, so it is primarily a tool for cluster lensing rather than cosmological applications on larger scales.…”

Improving lensing cluster mass estimate with flexion

“…It is worth noting that, as for the shear, the flexion is actually not directly observable. Indeed, what is measured from the data is (Schneider & Er 2008) …”

Estimate of dark halo ellipticity by lensing flexion

“…The full formalism described here can be found in Bacon et al (2006); Schneider & Er (2008). Weak lensing shear and flexion are conveniently described using a complex notation.…”

“…Different techniques were developed to measure flexion (Irwin & Shmakova 2006;Okura et al 2007;Massey et al 2007;Schneider & Er 2008). It was noted that flexion can contribute to studies of the dark matter halos in galaxies and clusters, especially for the detection of mass substructure (Bacon et al 2010;Er et al 2010).…”