Supported mass profiles#
The equations for elliptical profiles provided in the following are valid for coordinates \((x,y)\) centered on the origin and with the major axis of the ellipsoid along the \(x\) axis which corresponds to a position angle \(\phi=-90^\circ\) following COOLEST conventions.
Total mass profiles#
Singular isothermal power-law#
Availability
Implemented in both coolest.template and coolest.api.
The singular power-law ellipsoid (SIE) lens potential is given by $$ \psi_{\rm SIE}(x,y) = \frac{\theta_{\rm E} \sqrt{q}~x}{\sqrt{1-q^2}} {\rm arctan} \left( \frac{\sqrt{1-q^2}~x}{\sqrt{q^2x^2+y^2}} \right) + \frac{\theta_{\rm E} \sqrt{q} ~y}{\sqrt{1-q^2}} {\rm arctanh} \left( \frac{\sqrt{1-q^2}~y}{\sqrt{q^2x^2+y^2}} \right) \ , $$ with \(\theta_{\rm E}\) being the Einstein radius (as the product average of the minor and major axis of the ellipse at mean enclose convergence is 1), and \(q\) being the axis ratio.
The SIE convergence has the following analytical formula $$ \kappa_{\rm SIE}(x,y) = \frac{\theta_{\rm E}}{2\sqrt{qx^2+y^2/q}} \ , $$
In COOLEST, the singular isothermal sphere (SIS) can be expressed as a SIE with fixed axis ratio \(q=1\) (the position angle value is irrelevant in this case). In this case, the potential and convergence are simply $$ \psi_{\rm SIS} (x,y)= \theta_E \sqrt{x^2+y^2} \ , \ \kappa_{\rm SIS}(x,y)=\frac{\theta_E }{2\sqrt{x^2+y^2}} \ . $$
Cored generic power-law#
Availability
Implemented in coolest.template.
The convergence of a SPEMD profile is the following : $$ \kappa_{\rm SPEMD}(x,y) = \frac{3-\gamma}{2} \left(\frac{b}{\sqrt{qx^2+y^2/q + s^2}} \right)^{\gamma -1} $$
with \(\gamma\) the logarithmic power-law slope (when \(\gamma=2\), the profile is isothermal), \(q\) the axis ratio, \(s\) the core-radius. For rather small scale radii, that is \(s<0.1~b\), \(\theta_E \simeq b - s^{3-\gamma} q^{(\gamma-1)/2}\) where \(\theta_E\) is as define in our conventions.
Generic power-law profile#
Implemented in coolest.template and coolest.api.
The PEMD profile is characterized by the same equation with \(s=0\); in this case, \(b\) is thus equal to the Einstein radius as define in our conventions.
Cored isothermal power-law#
Availability
Implemented in coolest.template.
The Non-singular Isothermal Ellipsoid (NIE) is the special case of a SPEMD, with isothermal slope \(\gamma=2\). The convergence is thus $$ \kappa_{\rm NIE}(x,y) = \frac12 \frac{b}{\sqrt{qx^2 + y^2/q + s^2}} \ , $$ where the Einstein radius is equal to \(b\) for no core (\(s=0\)). For a small cored radius \(s<0.1\), the approximation formula given for SPEMD should hold.
Dark matter profiles#
Baryonic matter profiles#
Availability
Implemented in coolest.template.
Following Dutton et al. 2011, the chameleon profile is defined as the difference between two NIE profiles with different core radii \(s_{\rm c}\) and \(s_{\rm t}\). The former defines an overall core radius, and latter defines a truncation radius (hence \(s_{\rm t} > s_{\rm c}\)). The two NIE components have the same normalization \(b\) to ensure the total mass is finite. The convergence of the chameleon profile is thus $$ \begin{align} \nonumber \kappa_{\rm chm}(x, y) &\equiv \kappa_{\rm NIE, c}(x, y) - \kappa_{\rm NIE, t}(x, y) \ \nonumber &= \frac{b}{2} \left[ \frac{1}{\sqrt{qx^2 + y^2/q + s_{\rm c}^2}} - \frac{1}{\sqrt{qx^2 + y^2/q + s_{\rm t}^2}} \right] \ . \end{align} $$ The lens potential and deflection angles are similarly defined, as the difference of the potentials and deflections of two NIEs.
Massive fields#
External shear#
Availability
Implemented in both coolest.template and coolest.api.
The lens potential due to shear only with respect to the origin and for an angle \(\phi\) measured counter-clockwise from the positive x-axis, is given by:
$$
\psi(\boldsymbol{r}) \equiv \psi(r,\phi) = \frac{r^2}{2} , \gamma_{\rm ext} , \cos\left[ 2 (\phi - \phi_{\rm ext}) \right],
$$
or equivalently:
$$
\psi(x,y) = \frac12 , \gamma_1 , (x^2 - y^2) + \gamma_2 , x , y,
$$
where in the last equation we set:
$$
\gamma_1 = \gamma_{\rm ext} , \cos (2 \phi_{\rm ext}) \quad \mathrm{and} \quad \gamma_2 = \gamma_{\rm ext} , \sin (2 \phi_{\rm ext}).
$$
The angle and magnitude of the shear are related to its \(\gamma_1\) and \(\gamma_2\) components through:
$$
\gamma_{\rm ext} = \sqrt{\gamma_1^2 + \gamma_2^2} \quad \mathrm{and} \quad \phi_{\rm ext} = \frac12 \tan^{-1} \left( \frac{\gamma_2}{\gamma_1} \right).
$$
Flexion shift#
Availability
Soon implemented in coolest.template.