May 13, 2021 - Lensing through deflections

Intro

After some hiatus we come back to the project with the hopes of lensing the CIB map "properly", by deflecting the sources and then getting the lensed CIB maps from these.

Jason says that the kappa maps are in

/scratch/r/rbond/jasonlee/cib_lensing2020/kappa_maps/total/

and his code is

/scratch/r/rbond/jasonlee/cib_lensing2020/gal_catalog/mapmaker/project_galaxies_by_redshift_0545.py

Conventions

We will be using the conventions from the paper "Lensed CMB simulation and parameter estimation" https://arxiv.org/pdf/astro-ph/0502469.pdf

There the lensed field [math]\displaystyle{ \tilde X(\hat n) }[/math] is equal to the unlensed field [math]\displaystyle{ X(\hat n') = X(\hat n + \nabla \psi) }[/math].

The relation connecting convergence and gravitational potential is

[math]\displaystyle{ \psi_{\ell m} = \frac{2\kappa_{\ell m}}{\ell(\ell+1)} }[/math]

but one must be careful to also know whether this is used to go from unlesed fields to the lensed ones, or vice versa. The implicit plus sign makes this consistent with

[math]\displaystyle{ \tilde X(\hat n) = X(\hat n + \nabla \psi) }[/math]

Simple analytic example

Let's play around with

[math]\displaystyle{ \kappa = A \cos \theta }[/math]

which contains only [math]\displaystyle{ \ell = 1 }[/math] multipole, so

[math]\displaystyle{ \psi = A \cos \theta }[/math]

The vector gradient is

[math]\displaystyle{ \nabla \psi = \left(\frac{\partial \psi}{\partial \theta}, \frac{1}{\sin \theta}\frac{\partial \psi}{\partial \varphi}\right) = \left(-A \sin \theta, 0\right) = A \sin \theta \left(\cos \pi, \sin \pi\right) }[/math]

which evaluates to