Basic Notation

Geometric model as described in the text

We assume a flat sky, so that the source is contained in a plane $S$ at distance $D_S$ from the observer, and the lens in a plane $L$ at distance $D_L$ from the observer. The optical axis is the line from the observer through the lens. The planes $S$ and $L$ are orthogonal on the optical axis, and have origin in the intersection therewith.

We consider a single source point at $\boldsymbol{\eta}$ in $S$ . The apparent position, as seen by the observer, is at $\nu$ and

\Delta\boldsymbol{\eta} = \boldsymbol{\nu} - \boldsymbol{\eta}

(1)

The apparent position in the lens plane $L$ is called

\boldsymbol{\xi} = \frac{D_L}{D_S} \boldsymbol{\nu}.

(2)

The deflection is most easily described in terms of angles, so we define $\beta$ and $\theta$ as the angles between the optical axis and respectively $\boldsymbol{\eta}$ and $\boldsymbol{\nu}$ . The deflection angle $\hat\alpha$ is the angle between the actual and apparent source in the source plane as seen from the apparent image in the lens plane.

With the flat sky approximation, the angles are related to lengths in the lens plane by a factor of $D_L$ , so that

\boldsymbol{\xi} = D_L\theta

(3)

Similarly, in the source plane^[1], we find

\boldsymbol{\eta} = D_S\beta,

(4)

but also

\boldsymbol{\Delta\eta}= D_{LS}\hat{\boldsymbol{\alpha}}.

(5)

Now, we can write the actual image as

\begin{align} \boldsymbol{\eta} = \frac{D_S}{D_L}\boldsymbol{\xi} - D_{LS}\boldsymbol{\hat\alpha} \end{align}

(6)

It is noteworthy that if we consider spherical geometries (a source sphere and a lens sphere), we may also write down the equation

\begin{align} \boldsymbol{\beta}=\boldsymbol{\theta}-\frac{\xi_0}{D_L}\boldsymbol{\alpha}. \end{align}

(7)

Considering small angles,

\sin\theta\approx\theta

(8)

it may readily be shown that Eq.~\eqref{raytracePhys} and Eq.~\eqref{raytraceAng} are the same. It is this latter equation that we shall take to be our constitutive relation. But before we get there, let us also introduce the standard way of normalizing.

Normalisation¶

It is customary to normalise using a constant factor $\xi_0$ . This gives the following entities, following Kormann 1994

\begin{align} \mathbf{x}= \frac{\boldsymbol{\xi}}{\xi_0}\quad,\quad\mathbf{y} = \frac{\boldsymbol{\eta}}{\eta_0} \quad\text{where } \nu_0 = \frac{D_S}{D_L}\xi_0\quad,\quad \boldsymbol{\alpha} = \frac{D_LD_{LS}}{D_S\xi_0}\hat{\boldsymbol{\alpha}} \end{align}

(9)

In addition to the previously mentioned (cosmological) distances $D_L$ , $D_S$ and $D_{LS}$ we must thus find a proper length scale $\xi_0$ from which we normalize everything else. In SEF, Kormann1994 and other standard sources one typically takes $\xi_0$ to be the so-called Einstein radius. This is the radius at which a spherically symmetric lens will produce a ring (so-called Einstein ring) whenever the source is directly behind the lens, along the optical axis.

In normalised coordinates the ray-trace equation reads

\begin{align} \mathbf{y}=\mathbf{x}-\boldsymbol{\alpha}, \end{align}

(10)

which also explains the particular definition of $\boldsymbol{\alpha}$ : it makes the normalized version of the ray-trace equation look very nice and tidy. The normalisation presented above is however somewhat different from the one we shall prefer in this work, where we shall prefer to work in angular coordinates, such as the ones given in Eq.~\eqref{raytraceAng}.In the sections to follow, such coordinates will therefore be our focus.

Einstein Radius

R_E

and

\Theta_E

Consider a source directly behind a spherically symmetric lens. Id est, consider a source at $y=0$ . In that case the normalized ray-trace equation yields

x=\alpha.

(11)

Reinstating dimensionfull variables $\boldsymbol{\xi}=\xi_0\mathbf{x}$ we find $\boldsymbol{\xi}=\xi_0\boldsymbol{\alpha}$ . This radius is known as the Einstein radius.

R_E\equiv \xi_0\alpha.

(12)

Note that for a so-called SIS-lens and similarely for the Point-mass lens (PML), both of which will be discussed later, calculations yield

\boxed{R_E^{SIS}=R_E^{PML}=\xi_0.}

(13)

When we talk about the Einstein radius, we shall usually mean the one corresponding to the SIS /PML Einstein radius, without explicitely mentioning. The final relation above is therefore important to note. Also note that the Einstein radius is commonly referred to in angular variables (in the lens plane), in which one defines

\theta_E=\frac{R_E}{D_L}.

(14)

The thin-lens approximation and the lens-potential $\psi$ ¶

Considering a thin lens, it is customary to define the lens potential as the projection of the 3D gravitaitonal lens potential down on the lens plane. Such a simplification is typically warranted, due to $D_L \gt\gt \xi_0$ .

Our starting point here will be to define the lensing potential to so that its gradient is the reduced deflection angle $\boldsymbol{\alpha}$ the gradient of $\psi$ , i.e.

\boldsymbol{\alpha} = \nabla_{\mathbf{x}}\psi.

(15)

In angular coordintes, which we prefer here, the chain rule gives^[2],

\begin{align} \boldsymbol{\alpha} = \xi_0\cdot\nabla_{\xi}\psi = \frac{\xi_0}{D_L}\cdot\nabla_{\theta}\psi \end{align}

(16)

WThe normalised raytrace equation~\eqref{raytraceNorm} is thus rewritten to

\begin{align} \mathbf{y} = \mathbf{x} - \nabla_{\mathbf{x}}\psi \end{align}

(17)

Working on the sphere, id est; rewriting to our preferred angular coordinate system, the chain rule gives

\boldsymbol{\beta} = \theta - \frac{\xi_0}{D_L}\alpha = \theta - \frac{\xi_0}{D_L}\boldsymbol{\alpha} = \theta - \frac{\xi_0^2}{D_L^2}\nabla_{\theta}\psi.

(18)

This final equation shall be taken as motivation to redefine the standard lensing potential ever so slightly. Let $\psi$ be the usual lensing potential. Then we define

\boxed{\psi^{\mathrm{R}} = \frac{\xi_0^2}{D_L^2}\psi}

(19)

In this notation, we find the following pleasing expression for the angular version of the ray-trace equation:

\boxed{\boldsymbol{\beta} = \theta - \nabla_{\theta}\psi^R.}

(20)

This relation is implemented in the RaytraceModel::calculateEta() function in CosmoSim. It follows that

\boxed{\boldsymbol{\alpha}=\frac{1}{\theta_E}\nabla_\theta\psi^R}

(21)

Surface Mass Density¶

A final, very central concept in lensing, is the convergence $\kappa$ . This is the dimensionless, projected surface-mass density, which is related to $\psi$ through the Poisson equation. In our coordinates this gives

\boxed{\kappa(\boldsymbol{\theta})= \frac12\nabla_\theta\psi^R}

(22)

Writing it out more explicitely, we find

\kappa(\boldsymbol{\theta})= \frac12\left( \frac{\partial^2\psi^{\mathrm{R}}}{\partial\theta_1^2} + \frac{\partial^2\psi^{\mathrm{R}}}{\partial\theta_2^2} \right)

(23)

Notes¶

Footnotes¶

This is seen because $D_S\alpha$ and $D_{LS}\hat\alpha$ are the lengths of arcs between the actual and apparent source, and for small angles they are both approximately equal to the straight line $\Delta\boldsymbol{\eta}_S$ .
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We use here the chain rule with $\boldsymbol{\xi}= \xi_0\mathbf{x}$ and $\theta = \boldsymbol{\xi}/D_L$ .
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Normalisation¶

The thin-lens approximation and the lens-potential ψ\psiψ¶

Surface Mass Density¶

Notes¶

The thin-lens approximation and the lens-potential $\psi$ ¶