Multipole expansions

[mjyb16]

Is there a plan for adding multipole expansion parameters to lenses? I don't see any mentions of multipole capability at the moment in the codebase or the docs.

[ConnorStoneAstro]

Hmmm, could you describe more of what you are looking for here? Is there a reference where this is done? For a lot of lenses they are quite simple to evaluate and a multipole expansion would probably just increase the computational graph. An exception might be the NFW and TNFW lenses which are a bit of a pain right now. Or are you thinking of a new lens class that is just a multipole expansion up to a certain order and can take arbitrary values?

[AlexandreAdam]

Multipole should be an additional lens class. The external convergence is a very special case of a multipole extension, since it is the zeroth moment of the multipole expansion of the external convergence.

A good reference on the subject is Schneider & Bartelmann (1997), but one can find also a more modern treatment in Bernstein & Nakajima.

Note that Yashar's SDP.81 paper found that multiples were very important in the detection of substructure. He gives a simple definition of the multiple expansion of the convergence in polar coordinates

$$\kappa(r, \theta) = \left(\frac{r}{r_s}\right)^\alpha [A_m \cos (m \phi) + B_m \sin(m \phi)]$$

In his paper he describes that the logarithmic slope $\alpha$ is shared with his main lens parameters and he chooses $r_s$ based on the size of his lens (?).