Gravitational Lensing Formalism - General Weak Lensing

General Weak Lensing

In weak lensing by large-scale structure, the thin-lens approximation may break down, and low-density extended structures may not be well approximated by multiple thin-lens planes. In this case, the deflection can be derived by instead assuming that the gravitational potential is slowly varying everywhere (for this reason, this approximation is not valid for strong lensing). This approach assumes the universe is well described by a Newtonian-perturbed FRW metric, but it makes no other assumptions about the distribution of the lensing mass.

As in the thin-lens case, the effect can be written as a mapping from the unlensed angular position to the lensed position . The Jacobian of the transform can be written as an integral over the gravitational potential along the line of sight

$\frac{\partial \beta_i}{\partial \theta_j} = \delta_{ij} + \int_0^{r_\infty} dr g(r) \frac{\partial^2 \Phi(\vec{x}(r))}{\partial x^i \partial x^j}$

where is the comoving distance, are the transverse distances, and

$g(r) = 2 r \int^{r_\infty}_r \left(1-\frac{r^\prime}{r}\right)W(r^\prime)$

is the lensing kernel, which defines the efficiency of lensing for a distribution of sources .

The Jacobian can be decomposed into convergence and shear terms just as with the thin-lens case, and in the limit of a lens that is both thin and weak, their physical interpretations are the same.

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