Lensing shear

Weak lensing enables the direct study of mass in the universe. Lensing, weak or strong, provides a more direct probe of mass than other methods which rely on astrophysical assumptions (e.g. hydrostatic equilibrium in a galaxy cluster) or proxies (e.g. the galaxy distribution), and can potentially access a more redshift- independent sample of structures than can methods which depend on emitted light with its r−2 falloff. But strong lensing can be applied only to the centers of very dense mass concentrations. Weak lensing, in contrast, can be applied to the vast majority of the universe. It provides a direct probe of most areas of already-known mass concentrations, and a way to discover and study new mass concentrations which could potentially be dark. With sources covering a broad redshift range, it also has the potential to probe structure along the line of sight.

Weak gravitational lensing can be described as a linear transformation between unlensed coordinates (xu, yu; with the origin at the center of the distant light source) and the lensed coordinates in which we observe galax- ies (xl, yl; with the origin at the center of the observed image), 

Note that any symmetric 2×2 matrix can be written in the form 

Wrong? From: http://mwhite.berkeley.edu/Lensing/SantaFe04.pdf

How to understand the three parameters (r1, r2, k)?
   For the meaning of r and k, you can understand in this way. Keep the r1 in the equation, and if r1>0, the observed x will be enlarged by a factor of r1 with y shrunk by a factor of r1. Thus the observed shape of the source will looks like be elongated in x axis. For the other two parameters, you can think in the same way.

a positive (negative) γ1 results in an image being stretched along the x (y) axis direction 

   The convergence κ represents an isotropic magnification, and the shear γ represents a stretching in the direction φ. They are both related to physical properties of the lens as linear combinations of derivatives of the deflection angle. However, κ can be interpreted very simply as the projected mass density Σ divided by the critical density Σcrit, while γ has no such straight- forward interpretation. In fact, γ is nonlocal: its value at a given position on the sky depends on the mass distribution everywhere, not simply at that position. 

The surface brightness is unchanged after lensing!

this is a consequence of the fact that the lens equation is a gradient map. Magnification changes only angular sizes and shapes on the sky. Thus a constant surface brightness sheet stays a constant brightness sheet when lensed 

%There are only two free parameters!
%The r = sqrt(r1**2+r2**2) is called the cosmic shear. In

Weak lensing dominate the lensing field!
The region with strong lensing is quite limited, however, the weak lensed region is several factors larger and can help us to determine the total mass of galaxies.

What is reduced shear? why?
In some cases, the size of  galaxies is not available, so we cannot detect the size changes to determine the convergence parameter.
   The convergence κ de- scribes a change in apparent size for lensed objects: areas of the sky for which κ is positive have apparent changes in area (at fixed surface brightness) that make lensed images appear larger and brighter than if they were unlensed, and a modified galaxy density.
   In these cases, we use reduced shear, g1 = γ1/(1 − κ) and g2 = γ2/(1 − κ) to include the influence of k to the parameter of r. If sources with intrinsically circular isophotes (contours of equal brightness) could be identified, the observed sources (post-lensing) would have elliptical isophotes that we can characterize by their minor-to-major axis ratio b/a and the orientation of the major axis φ. For |g| < 1, these directly yield a value of the reduced shear

What is the challenge?
    It is hard to make the shear measurement even we have utilized statistic methods. I list some problems in the work.

   Variable PSFs: For a summary of some common methods of PSF estimation and interpolation, see Kitching et al. (2013)

   Realistic galaxy morphologies: ∼ 20% galaxies can be perfectly fit by simple galaxy models. nearly half can be fit by them, but with additional substructure clearly evident. ∼ 30% are true “irregulars” that cannot be fit by simple models at all.

   Combination of multiple exposures

 

  

Reference:

Gravitational lensing: an astrophysical tool 

THE THIRD GRAVITATIONAL LENSING ACCURACY TESTING (GREAT3) CHALLENGE HANDBOOK