When we try to reconstruct the surface mass density distribution of a lens, we can do the following:

Divide the picture into a grid, and for each "pixel" assign an effective distortion by finding $g$ (the distortion, or "reduced shear") that minimizes the average ellipticity of the galaxies in that pixel. This is based on the assumption that their actual orientations are independent and random.

$g$ is given by

$$ g = \frac{\gamma}{1-\kappa} $$ where $\gamma,\kappa$ are the shear and convergence respectively, and $\kappa := \Sigma / \Sigma_c$ gives the (dimensionless) mass density.

By measuring $g$ we shouldn't be able extract both $\gamma,\kappa$, and also $g$ is invariant to a transformation of multiplying both numerator and denominator by a common factor (known as the "mass-sheet degeneracy"). The $g$ factor arises from relating the true and observed ellipticity of an object in a standard way.

One way to extract the two parameters is by a second independent measurement, usually the magnification $\mu = ((1-\kappa)^2 + \gamma^2)^{-1}$.

But I've seen in Umetzu et al (2009) that the derivatives of $\gamma,\kappa$ can be related, yielding a PDE which can be solved to obtain $$\kappa (\theta) \sim \int d^2\theta' D(\theta - \theta ') \gamma(\theta')$$ where $D$ is some known function (kernel that arises from the PDE).

I find that pretty odd, since as geometric objects, $\gamma,\kappa$ are characterizations of the ellipticity and should be independent. How can one reconcile their dependence?

I was thinking that the lensing physics constrains them - the image that arises from lensing cannot create any arbitrary relation between the two because it is obtained in a particular physical way. This is supported by the fact that the deflection angle can be assigned a potential, namely some conservation gives the extra constraint.

But I don't know if this is either correct or satisfactory. Does anyone have a more convincing argument?


1 Answer 1


I have found the solution: this relation is still insufficient to remove the degeneracy. But if one can measure far enough from the cluster center, a reasonable assumption would be that the converges goes to zero. Hence the boundary conditions can be specified and the equation can be solved completely.


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