Lens Models - CosmoSim

Point Mass¶

Stars and other celestial objects may sometimes be modelled as so-called point-masses. The gravitational potential is in this case

\begin{aligned} \psi^R = \theta_E^2\ln \frac{\theta}{\theta_E} \end{aligned}

(1)

where $\theta_E$ is the previously defined angular Einstein radius, $\theta_E=R_E/D_L$ and where $\theta=\sqrt{\theta_1^2+\theta_2^2}.$

SIS (Singular isothermal sphere)¶

It is often necessary to model the lens as an extended object. The simplest such model, useful for instance in the modelling of a galaxy surrounded by dark amtter, is the so-called Singular isothermal sphere, or SIS-model for short. The following expression for $\psi^R$ is implemented in amplitudes.py

\psi^R_\mathrm{SIS}(x,y) = \theta_E\cdot\theta,

(2)

[CHECK: There was a negative sign here that I removed, HG] where notation is as for the point-mass case. In the code, we are working in angular variables. Hence what is called einsteinR in the code, is the angular Einstein radius $\theta_E$ . Note that the partial derivatives are readily calculated as

\frac{\partial\psi}{\partial \theta_1} = \theta_E\cdot\frac{\theta_1}{\sqrt{\theta_1^2+\theta_1^2}}=\theta_E\frac{\theta_1}{\theta}\quad\textrm{and}\quad \frac{\partial\psi}{\partial \theta_2} = \theta_E\cdot\frac{\theta_2}{\sqrt{x^2+y^2}}=\theta_E\frac{\theta_2}{\theta}

(3)

Thus the reduced deflection angle is in the SIS case given as

\boldsymbol{\alpha}=\frac{1}{\theta_E}\left(\theta_1,\theta_2\right).

(4)

See psiXvalue and psiYvalue in SIS.cpp.

Roulette amplitudes are calculated using the recursive formulæ.

SIE¶

The extension of the SIS to an ellipsoide is well described in KORMAANN 1994, and referred to as the Singular Isothermal Ellipsoid, or SIE-lens for short. It is a three-parameter family of lens models: The Einstein radius ( $\theta_E$ ), the excentricity factor $f$ and the orientation $\lambda$ of the lens relative to the polar axis. Given these parameters, the lens profile is given as:

\begin{aligned} \begin{split} \psi^R_\textrm{SIE}(\theta_E,f,\lambda;\theta,\phi) = \theta_E\sqrt{\frac{f}{1-f^2}}\theta\cdot &\Bigg([\sin(\phi-\lambda)]\cdot\sin^{-1}\left(\sqrt{1-f^2}\cdot \sin{(\phi-\lambda)}\right) \\\\& +[\cos(\phi-\lambda)]\cdot\sinh^{-1}\left(\frac{\sqrt{1-f^2}}{f}\cos(\phi-\lambda)\right)\Bigg). \end{split} \end{aligned}

(5)

where $(\theta,\phi)$ are (normalized) polar coordinates in the lens plane, i.e. $$

\boldsymbol{\theta}=\theta_E\mathbf{x}=\theta(\sin\phi,\cos\phi).

(6)

The deflection may now be calculated using that

\begin{aligned} \frac{\partial\psi}{\partial \theta_1} &= \theta_E\cdot\frac{\sqrt{f}}{\sqrt{1-f^2}}\cdot\big( \cos\lambda\cdot\sinh^{-1}(\frac{\sqrt{1-f^2}}{f}\frac{\theta_1^\prime}{\theta}) - \sin\lambda\cdot\sin^{-1}(\sqrt{1-f^2}\frac{\theta_2^\prime}{\theta}) \big) \\\\ \frac{\partial\psi}{\partial \theta_2} &= \theta_E\cdot\frac{\sqrt{f}}{\sqrt{1-f^2}}\cdot\big( \sin\lambda\cdot\sinh^{-1}(\frac{\sqrt{1-f^2}}{f}\frac{\theta_1^\prime}{\theta}) + \cos\lambda\cdot\sin^{-1}(\sqrt{1-f^2}\frac{\theta_2^\prime}{\theta}) \big) \end{aligned}

where

\begin{aligned} \theta_1^\prime &= \cos\lambda\cdot \theta_1 + \sin\lambda\cdot \theta_2 \ \theta_2^\prime &= -\sin\lambda\cdot \theta_1 + \cos\lambda\cdot \theta_2 \end{aligned}

and $\theta = \sqrt{\theta_1^2+\theta_2^2}$ as before. As for SIS, $\theta_E$ is called einsteinR in the code. See psiXvalue and psiYvalue in SIE.cpp.