Jan 27, 2022 - Considering shear

During Jason's presentation at Penn it was commented that we should probably also include shear in the magnification coefficients, given that it is the same order as kappa^2 we find significant.

Based on information presented in the DES paper and Healpix convention we have

[math]\displaystyle{ \gamma_1 + i \gamma_2 = \gamma = \sum_{lm} \gamma_{lm} {}_2Y_{lm} = \sum_{lm} \left(- \sqrt{\frac{(l-1)(l+2)}{l(l+1)}}\right)\kappa_{lm}{}_2Y_{lm} }[/math] [math]\displaystyle{ \gamma_1 - i \gamma_2 = \gamma^* = \sum_{lm} \gamma_{lm}^* {}_2Y_{lm}^* = \sum_{lm} \left(- \sqrt{\frac{(l-1)(l+2)}{l(l+1)}}\right)\kappa_{lm}^*{}_2Y_{lm}^* = \sum_{lm} \left(- \sqrt{\frac{(l-1)(l+2)}{l(l+1)}}\right)\kappa_{lm}{}_{-2}Y_{lm} }[/math]

In the Healpix convention we have the values entering alm2map_spin (notice the minus sign at the end)

[math]\displaystyle{ {}_2a_{lm}^+ = -\left(\kappa_{lm}(-\sqrt{\dots}) + \kappa_{lm}(-\sqrt{\dots})\right)/2 = \kappa_{lm}\sqrt{\dots} = - \gamma_{lm} }[/math]

[math]\displaystyle{ {}_2a_{lm}^- = -\left(\kappa_{lm}(-\sqrt{\dots}) - \kappa_{lm}(-\sqrt{\dots})\right)/2i = 0 }[/math]

The two maps that are returned are then

[math]\displaystyle{ {}_2S^+ = (\gamma + \gamma^*)/2 = \gamma_1 }[/math]

[math]\displaystyle{ {}_2S^- = (\gamma - \gamma^*)/2i = \gamma_2 }[/math]