We build on the notation defined in Lens Potential Principles of Gravitational Lensing
Basic Notation ¶ Working with angular co-ordinates, β \beta β θ \theta θ
The amplification tensor A A A
A ( θ ) = ∂ β ∂ θ = [ ∂ u ∂ θ 1 ∂ u ∂ θ 2 ∂ v ∂ θ 1 ∂ v ∂ θ 2 ] \mathcal{A}(\boldsymbol{\theta})
= \frac{\partial\boldsymbol{\beta}}{\partial\boldsymbol{\theta}}
=
\begin{bmatrix}
\frac{\partial u}{\partial \theta_1} &
\frac{\partial u}{\partial \theta_2} \\\\\\\\
\frac{\partial v}{\partial \theta_1} &
\frac{\partial v}{\partial \theta_2}
\end{bmatrix} A ( θ ) = ∂ θ ∂ β = ⎣ ⎡ ∂ θ 1 ∂ u ∂ θ 1 ∂ v ∂ θ 2 ∂ u ∂ θ 2 ∂ v ⎦ ⎤ where β = ( u , v ) \boldsymbol{\beta}=(u,v) β = ( u , v ) θ = ( θ 1 , y ) \boldsymbol{\theta}=(\theta_1,y) θ = ( θ 1 , y )
The raytrace equation is given as
β = θ − ∇ ψ \boldsymbol{\beta} = \boldsymbol{\theta} - \nabla\psi β = θ − ∇ ψ which gives
A ( θ ) = [ 1 − ψ θ 1 θ 1 − ψ θ 1 θ 2 − ψ θ 1 θ 2 1 − ψ θ 2 θ 2 ] \mathcal{A}(\boldsymbol{\theta})
=
\begin{bmatrix}
1 - \psi_{\theta_1\theta_1} & -\psi_{\theta_1\theta_2} \\\\\\\\
-\psi_{\theta_1\theta_2} & 1 - \psi_{\theta_2\theta_2}
\end{bmatrix} A ( θ ) = ⎣ ⎡ 1 − ψ θ 1 θ 1 − ψ θ 1 θ 2 − ψ θ 1 θ 2 1 − ψ θ 2 θ 2 ⎦ ⎤ We can equivalently write it in normalised screen space co-cordinates
A ( x ) = [ 1 − ψ x 1 x 1 − ψ x 1 y 2 − ψ x 1 y 2 1 − ψ y 2 y 2 ] \mathcal{A}(\mathbf{x})
=
\begin{bmatrix}
1 - \psi_{x_1x_1} & -\psi_{x_1y_2} \\\\\\\\
-\psi_{x_1y_2} & 1 - \psi_{y_2y_2}
\end{bmatrix} A ( x ) = ⎣ ⎡ 1 − ψ x 1 x 1 − ψ x 1 y 2 − ψ x 1 y 2 1 − ψ y 2 y 2 ⎦ ⎤ Decomposition ¶ Since A \mathcal{A} A
R − = [ 1 0 0 − 1 ] R / = [ 0 1 1 0 ] R_- =
\begin{bmatrix}
1 & 0 \\\\ 0 & -1
\end{bmatrix}
\quad
R_/ =
\begin{bmatrix}
0 & 1 \\\\ 1 & 0
\end{bmatrix} R − = ⎣ ⎡ 1 0 0 − 1 ⎦ ⎤ R / = ⎣ ⎡ 0 1 1 0 ⎦ ⎤ We get
A ( x ) = ( 1 − κ ) I − γ + R − − γ × R / \mathcal{A}(\mathbf{x})
=
(1-\kappa)I - \gamma_+R_- - \gamma_\times R_/ A ( x ) = ( 1 − κ ) I − γ + R − − γ × R / where this convergence (or mass distribution) is
κ ( θ ) = 1 2 ( ψ θ 1 θ 1 + ψ θ 2 θ 2 ) \kappa(\theta) =
\frac12( \psi_{\theta_1\theta_1} + \psi_{\theta_2\theta_2} ) κ ( θ ) = 2 1 ( ψ θ 1 θ 1 + ψ θ 2 θ 2 ) and
\begin{aligned}
\gamma_+(\theta) &=
\frac12( \psi_{\theta_1\theta_1} - \psi_{\theta_2\theta_2} ) \\\\
\gamma_\times(\theta) &= \psi_{\theta_1\theta_2}
\end{aligned}
We can then write
A ( θ ) = [ 1 − κ − γ + − γ × − γ × 1 − κ + γ + ] \mathcal{A}(\boldsymbol{\theta})
=
\begin{bmatrix}
1 - \kappa - \gamma_+ & - \gamma_\times \\\\\\\\
- \gamma_\times & 1 - \kappa + \gamma_+
\end{bmatrix} A ( θ ) = ⎣ ⎡ 1 − κ − γ + − γ × − γ × 1 − κ + γ + ⎦ ⎤ It is then straight forward to prove that the eigenvalues of
A ( θ ) \mathcal{A}(\boldsymbol{\theta}) A ( θ )
λ ± = ( 1 − κ ) ± γ + 2 + γ × 2 \lambda_\pm =
(1 - \kappa) \pm \sqrt{\gamma_+^2 + \gamma_\times^2} λ ± = ( 1 − κ ) ± γ + 2 + γ × 2 and we can verify that the inverse is given as
A − 1 ( θ ) = 1 ( 1 − κ ) 2 − γ + 2 − γ × 2 [ 1 − κ + γ + γ × γ × 1 − κ − γ + ] \mathcal{A}^{-1}(\boldsymbol{\theta})
=
\frac1{(1-\kappa)^2 - \gamma_+^2 - \gamma_\times^2}
\begin{bmatrix}
1 - \kappa + \gamma_+ & \gamma_\times \\\\\\\\
\gamma_\times & 1 - \kappa - \gamma_+
\end{bmatrix} A − 1 ( θ ) = ( 1 − κ ) 2 − γ + 2 − γ × 2 1 ⎣ ⎡ 1 − κ + γ + γ × γ × 1 − κ − γ + ⎦ ⎤ Magnification ¶ The magnification of an image is given as
μ ( θ ) = det A − 1 ( θ ) = 1 det A = λ − λ + = 1 ( 1 − κ ) 2 − γ + 2 − γ × 2 \mu(\boldsymbol{\theta}) = \det \mathcal{A}^{-1}(\boldsymbol{\theta})
= \frac1{\det\mathcal{A}} = \lambda_-\lambda_+ =
\frac1{(1-\kappa)^2 - \gamma_+^2 - \gamma_\times^2} μ ( θ ) = det A − 1 ( θ ) = det A 1 = λ − λ + = ( 1 − κ ) 2 − γ + 2 − γ × 2 1 TODO
What is the significance of this?
TODO
Is this consistent with the inverse magnification given in some sources?
The inverse magnification is
μ − 1 ( θ ) = β θ ⋅ d β d θ \mu^{-1}(\theta) = \frac{\beta}{\theta}\cdot\frac{d\beta}{d\theta} μ − 1 ( θ ) = θ β ⋅ d θ d β Critical Curves and Caustics ¶ The critical curve is defined as the set of points where the map from source to image plane
breaks down, i.e. when the magnification tends to infinity.
Thus the critical curve is the solution of the equation det A = 0 \det\mathcal{A}=0 det A = 0
0 = λ − λ + = ( 1 − κ ) 2 − γ + 2 − γ × 2 0 = \lambda_-\lambda_+ = (1-\kappa)^2 - \gamma_+^2 - \gamma_\times^2 0 = λ − λ + = ( 1 − κ ) 2 − γ + 2 − γ × 2 TODO Define caustic
Congdon, A. B., & Keeton, C. R. (2018). Principles of Gravitational Lensing: Light Deflection as a Probe of Astrophysics and Cosmology . Springer International Publishing. 10.1007/978-3-030-02122-1