Lens types
- class glow.lenses.Psi_SIS(p_phys={}, p_prec={})
Bases:
PsiAxisymLens object for the singular isothermal sphere (SIS).
Additional information: theory, default parameters.
- Parameters:
- p_physdict
Physical parameters, with keys:
psi0(float) – Normalization of the lens.
- Attributes:
- asymp_indexfloat
0.5
- asymp_amplitudefloat
\(\psi_0/\sqrt{2}\)
Methods
ddpsi_ddx(x)Second derivative of the lensing potential.
Initialize the default parameters.
dpsi_dx(x)First derivative of the lensing potential.
psi_x(x)Lensing potential.
- default_params()
Initialize the default parameters.
- Returns:
- p_physdict
Default physical parameters.
- p_precdict
Empty dictionary.
- psi_x(x)
Lensing potential.
\[\psi(x) = \psi_0 x\]
- dpsi_dx(x)
First derivative of the lensing potential.
\[\psi'(x) = \psi_0\]
- ddpsi_ddx(x)
Second derivative of the lensing potential.
\[\psi''(x) = 0\]
- class glow.lenses.Psi_CIS(p_phys={}, p_prec={})
Bases:
PsiAxisymLens object for the cored isothermal sphere (CIS).
Additional information: theory, default parameters.
- Parameters:
- p_physdict
Physical parameters, with keys:
psi0(float) – Normalization of the lens.rc(float) – Core radius.
- Attributes:
- asymp_indexfloat
0.5
- asymp_amplitudefloat
\(\psi_0/\sqrt{2}\)
Methods
Check that the core radius is not negative.
ddpsi_ddx(x)Second derivative of the lensing potential.
Initialize the default parameters.
dpsi_dx(x)First derivative of the lensing potential.
psi_x(x)Lensing potential.
- default_params()
Initialize the default parameters.
- Returns:
- p_physdict
Default physical parameters.
- p_precdict
Empty dictionary.
- check_input()
Check that the core radius is not negative.
- Raises:
- LensException
- psi_x(x)
Lensing potential.
\[\begin{split}\begin{align} r &\equiv \sqrt{x^2 + r_c^2}\\ \psi(x) &= \psi_0 r + \psi_0 r_c\log\left(\frac{2r_c}{r+r_c}\right) \end{align}\end{split}\]
- dpsi_dx(x)
First derivative of the lensing potential.
\[\begin{split}\begin{align} r &\equiv \sqrt{x^2 + r_c^2}\\ \psi'(x) &= \psi_0\frac{x}{r}\left(1 - \frac{r_c}{r + r_c}\right) \end{align}\end{split}\]
- ddpsi_ddx(x)
Second derivative of the lensing potential.
\[\begin{split}\begin{align} r &\equiv \sqrt{x^2 + r_c^2}\\ R &\equiv (x/r)^2\\ \psi''(x) &= \psi_0\frac{r_cR}{(r + r_c)^2} + \psi_0\left(1-\frac{r_c}{R + r_c}\right)\frac{1-R}{r} \end{align}\end{split}\]
- class glow.lenses.Psi_PointLens(p_phys={}, p_prec={})
Bases:
PsiAxisymLens object for the point lens.
Additional information: theory, default parameters.
- Parameters:
- p_physdict
Physical parameters, with keys:
psi0(float) – Normalization of the lens.
- p_precdict
Precision parameters, with keys:
xc(float) – Point mass regularization (Plummer sphere).
- Attributes:
- asymp_indexfloat
- asymp_amplitudefloat
\(\psi_0/2\)
Methods
ddpsi_ddx(x)Second derivative of the lensing potential.
Initialize the default parameters.
dpsi_dx(x)First derivative of the lensing potential.
psi_x(x)Lensing potential.
- default_params()
Initialize the default parameters.
- Returns:
- p_physdict
Default physical parameters.
- p_precdict
Default precision parameters.
- psi_x(x)
Lensing potential.
\[\psi(x) = \frac{1}{2}\psi_0\log(x^2 + x_c^2)\]
- dpsi_dx(x)
First derivative of the lensing potential.
\[\psi'(x) = \frac{\psi_0 x}{x^2 + x_c^2}\]
- ddpsi_ddx(x)
Second derivative of the lensing potential.
\[\psi''(x) = \psi_0\frac{x_c^2-x^2}{(x_c^2 + x^2)^2}\]
- class glow.lenses.Psi_Ball(p_phys={}, p_prec={})
Bases:
PsiAxisymLens object for the uniform density sphere.
Additional information: theory, default parameters.
- Parameters:
- p_physdict
Physical parameters, with keys:
psi0(float) – Normalization of the lens.b(float) – Radius of the sphere.
- Attributes:
- asymp_indexfloat
- asymp_amplitudefloat
\(\psi_0/2\)
Methods
Check that the radius is positive.
ddpsi_ddx(x)Second derivative of the lensing potential.
Second derivative of the lensing potential inside the sphere.
Initialize the default parameters.
dpsi_dx(x)First derivative of the lensing potential.
dpsi_small_x(x)First derivative of the lensing potential inside the sphere.
psi_small_x(x)Lensing potential inside the sphere.
psi_x(x)Lensing potential.
- default_params()
Initialize the default parameters.
- Returns:
- p_physdict
Default physical parameters.
- p_precdict
Empty dictionary.
- check_input()
Check that the radius is positive.
- Raises:
- LensException
- psi_small_x(x)
Lensing potential inside the sphere.
\[\psi(x) = \psi_0\log(b(1+X)) - \psi_0X(1+X^2/3)\ ,\quad x<b\]with \(X\equiv\sqrt{1-(x/b)^2}\).
- psi_x(x)
Lensing potential.
Piece together the lensing potential inside the sphere,
psi_small_x(), and outside,Psi_PointLens.psi_x().
- dpsi_small_x(x)
First derivative of the lensing potential inside the sphere.
\[\psi'(x) = \psi_0\frac{x}{b^2}\left(1+\frac{X^2}{1+X}\right)\ ,\quad x<b\]with \(X\equiv\sqrt{1-(x/b)^2}\).
- dpsi_dx(x)
First derivative of the lensing potential.
Piece together the derivative inside the sphere,
dpsi_small_x(), and outside,Psi_PointLens.dpsi_dx().
- ddpsi_small_x(x)
Second derivative of the lensing potential inside the sphere.
\[\psi''(x) = \psi_0\frac{2X^2 + 2X - 1}{b^2(1+X)}\ ,\quad x<b\]with \(X\equiv\sqrt{1-(x/b)^2}\).
- ddpsi_ddx(x)
Second derivative of the lensing potential.
Piece together the derivative inside the sphere,
ddpsi_small_x(), and outside,Psi_PointLens.ddpsi_ddx().
- class glow.lenses.Psi_NFW(p_phys={}, p_prec={})
Bases:
PsiAxisymLens object for the Navarro-Frenk-White (NFW) profile.
Additional information: theory, default parameters.
- Parameters:
- p_physdict
Physical parameters, with keys:
psi0(float) – Normalization of the lens.xs(float) – Rescaled NFW radius.
- p_precdict
Precision parameters, with keys:
eps_soft(float) – Softening factor.eps_NFW(float) – Switch on Taylor expansion when \(x/x_s < \epsilon_\text{NFW}\).
Methods
Auxiliary NFW function for radius larger than 1.
Auxiliary NFW function for radius smaller than 1.
F_nfw_v(x)Auxiliary NFW function for any value of the radius.
ddpsi_ddx(x)Second derivative of the lensing potential.
Second derivative of the lensing potential for large values of \(x/x_s\).
Expansion of
ddpsi_ddx_large_u()for \(x/x_s\ll 1\).Initialize the default parameters.
dpsi_dx(x)First derivative of the lensing potential.
First derivative of the lensing potential for large values of \(x/x_s\).
Expansion of
dpsi_dx_large_u()for \(x/x_s\ll 1\).psi_large_u(x)Lensing potential for large values of \(x/x_s\).
psi_small_u(x)Expansion of
psi_large_u()for \(x/x_s\ll 1\).psi_x(x)Lensing potential.
- default_params()
Initialize the default parameters.
- Returns:
- p_physdict
Default physical parameters.
- p_precdict
Default precision parameters.
- F_nfw_large_x(x)
Auxiliary NFW function for radius larger than 1.
\[\mathcal{F}(x) = \frac{1}{\sqrt{x^2-1}}\arctan\left(\sqrt{x^2-1}\right), \quad x>1\]
- F_nfw_small_x(x)
Auxiliary NFW function for radius smaller than 1.
\[\mathcal{F}(x) = \frac{1}{\sqrt{1-x^2}}\text{arctanh}\left(\sqrt{1-x^2}\right), \quad x<1\]
- F_nfw_v(x)
Auxiliary NFW function for any value of the radius.
Piece together
F_nfw_large_x()andF_nfw_xmall_x().\[\begin{split}\mathcal{F}(x) = \begin{cases} \mathcal{F}_\text{large}(x) & x > 1+\epsilon\\ \mathcal{F}_\text{small}(x) & x < 1-\epsilon\\ 1 + 2|x-1| & \text{otherwise} \end{cases}\end{split}\]where \(\epsilon\) is set by
p_phys['epsilon_soft'].
- psi_large_u(x)
Lensing potential for large values of \(x/x_s\).
This is the analytical expression of the lensing potential. The actual potential used in the code combines this one with an approximation for small radius in
psi_x().\[\psi(x) = \frac{1}{2}\psi_0\left(\log^2(u/2) + (u^2-1)\mathcal{F}^2(u)\right)\]with \(u\equiv x/x_s\).
- dpsi_dx_large_u(x)
First derivative of the lensing potential for large values of \(x/x_s\).
This is the analytical expression. The actual potential used in the code combines this one with an approximation for small radius in
dpsi_dx().\[\psi'(x) = \frac{\psi_0}{x_su}\left(\log(u/2) + \mathcal{F}(u)\right)\]with \(u\equiv x/x_s\).
- ddpsi_ddx_large_u(x)
Second derivative of the lensing potential for large values of \(x/x_s\).
This is the analytical expression. The actual potential used in the code combines this one with an approximation for small radius in
ddpsi_ddx().\[\psi''(x) = -\frac{\psi_0}{x_s^2u^2}\left(\log(u/2) + \frac{u^2 + \mathcal{F}(u)(1-2u^2)}{1-u^2}\right)\]with \(u\equiv x/x_s\).
- psi_small_u(x)
Expansion of
psi_large_u()for \(x/x_s\ll 1\).
- dpsi_dx_small_u(x)
Expansion of
dpsi_dx_large_u()for \(x/x_s\ll 1\).
- ddpsi_ddx_small_u(x)
Expansion of
ddpsi_ddx_large_u()for \(x/x_s\ll 1\).
- psi_x(x)
Lensing potential.
Piece together
psi_large_u()andpsi_small_u().\[\begin{split}\psi(x) = \begin{cases} \psi_\text{large}(x) & x > \epsilon_\text{NFW}\\ \psi_\text{small}(x) & x \leq \epsilon_\text{NFW} \end{cases}\end{split}\]where \(\epsilon_\text{NFW}\) is set by
p_phys['epsilon_NFW'].
- dpsi_dx(x)
First derivative of the lensing potential.
Piece together
dpsi_large_u()anddpsi_small_u().\[\begin{split}\psi'(x) = \begin{cases} \psi'_\text{large}(x) & x > \epsilon_\text{NFW}\\ \psi'_\text{small}(x) & x \leq \epsilon_\text{NFW} \end{cases}\end{split}\]where \(\epsilon_\text{NFW}\) is set by
p_phys['epsilon_NFW'].
- ddpsi_ddx(x)
Second derivative of the lensing potential.
Piece together
ddpsi_large_u()andddpsi_small_u().\[\begin{split}\psi''(x) = \begin{cases} \psi''_\text{large}(x) & x > \epsilon_\text{NFW}\\ \psi''_\text{small}(x) & x \leq \epsilon_\text{NFW} \end{cases}\end{split}\]where \(\epsilon_\text{NFW}\) is set by
p_phys['epsilon_NFW'].
- class glow.lenses.Psi_tSIS(p_phys={}, p_prec={})
Bases:
PsiAxisymLens object for the truncated singular isothermal sphere (tSIS).
Additional information: theory, default parameters.
- Parameters:
- p_physdict
Physical parameters, with keys:
psi0(float) – Normalization of the lens.xb(float) – radius.
Methods
Check that the radius is not negative.
ddpsi_ddx(x)Second derivative of the lensing potential.
Initialize the default parameters.
dpsi_dx(x)First derivative of the lensing potential.
psi_x(x)Lensing potential.
- default_params()
Initialize the default parameters.
- Returns:
- p_physdict
Default physical parameters.
- p_precdict
Empty dictionary.
- check_input()
Check that the radius is not negative.
- Raises:
- LensException
- psi_x(x)
Lensing potential.
\[\begin{split}\begin{align} u &\equiv x/x_b\\ \psi(x) &= \psi_0 x\left(\text{erfc}(u) + \frac{1-\text{e}^{-u^2}}{u\sqrt{\pi}} +\frac{1}{2u\sqrt{\pi}}\left[2\log(u) + E_1(u^2) + \gamma_E\right]\right) \end{align}\end{split}\]
- dpsi_dx(x)
First derivative of the lensing potential.
\[\begin{split}\begin{align} u &\equiv x/x_b\\ \psi'(x) &= \psi_0\left(\text{erfc}(u) + \frac{1-\text{e}^{-u^2}}{u\sqrt{\pi}}\right) \end{align}\end{split}\]
- ddpsi_ddx(x)
Second derivative of the lensing potential.
\[\begin{split}\begin{align} u &\equiv x/x_b\\ \psi''(x) &= -\frac{\psi_0}{x_b\sqrt{\pi}}\frac{1-\text{e}^{-u^2}}{u^2} \end{align}\end{split}\]
- class glow.lenses.Psi_eSIS(p_phys={}, p_prec={})
Bases:
PsiGeneralLens object for the elliptical SIS (eSIS).
Additional information: theory, default parameters.
- Parameters:
- p_physdict
Physical parameters, with keys:
psi0(float) – Normalization of the lens.q(float) – Ellipticity parameter.alpha(float) – Orientation angle.xc1(float) – Location in the x1 axis.xc2(float) – Location in the x2 axis.
- Attributes:
- asymp_indexfloat
0.5
- asymp_amplitudefloat
\(\psi_0/\sqrt{2}\)
Methods
ddpsi_vec(x1, x2)Hessian of the lensing potential.
ddpsi_vec_a0(x1, x2)Second derivatives of the lensing potential (zero angle).
Initialize the default parameters.
dpsi_vec(x1, x2)Gradient of the lensing potential.
dpsi_vec_a0(x1, x2)First derivatives of the lensing potential (zero angle).
psi(x1, x2)Lensing potential.
psi_a0(x1, x2)Lensing potential (zero angle).
rotate_grad(f1, f2)Rotation of a gradient.
rotate_hessian(h11, h12, h22)Rotation of a Hessian.
rotate_vector(x1, x2)Rotation of a vector.
- default_params()
Initialize the default parameters.
- Returns:
- p_physdict
Default physical parameters.
- p_precdict
Empty dictionary.
- rotate_vector(x1, x2)
Rotation of a vector.
Linear transformation defined as
\[\begin{split}x'_1 &= \cos\alpha\,x_1 - \sin\alpha\,x_2\\ x'_2 &= \sin\alpha\,x_1 + \cos\alpha\,x_2\end{split}\]- Parameters:
- x1, x2float or array
Input vector.
- Returns:
- xx1float or array
\(x'_1\)
- xx2float or array
\(x'_2\)
- rotate_grad(f1, f2)
Rotation of a gradient.
Linear transformation defined as
\[\begin{split}f'_1 &= \cos\alpha\,f_1 + \sin\alpha\,f_2\\ f'_2 &= -\sin\alpha\,f_1 + \cos\alpha\,f_2\end{split}\]- Parameters:
- f1, f2float or array
Input vector.
- Returns:
- ff1float or array
\(f'_1\)
- ff2float or array
\(f'_2\)
- rotate_hessian(h11, h12, h22)
Rotation of a Hessian.
Linear transformation defined as
\[\begin{split}A &\equiv 2\cos\alpha\sin\alpha\,h_{12}\\ B &\equiv \sin\alpha (h_{11} - h_{22})\\ h'_{11} &= h_{11} + A - \sin\alpha\,B\\ h'_{12} &= h_{12}(1-2\sin^2\alpha) - \cos\alpha\,B\\ h'_{22} &= h_{22} - A + \sin\alpha\,B\end{split}\]- Parameters:
- h11, h12, h22float or array
Input vector.
- Returns:
- hh11float or array
\(h'_{11}\)
- hh12float or array
\(h'_{12}\)
- hh22float or array
\(h'_{22}\)
- psi_a0(x1, x2)
Lensing potential (zero angle).
\[\psi_{\alpha=0}=\sqrt{x_1^2 + x_2^2/q^2}\]
- dpsi_vec_a0(x1, x2)
First derivatives of the lensing potential (zero angle).
\[\begin{split}\begin{align} \partial_{1}\psi_{\alpha=0} &= \psi_0 x_1/X\\ \partial_{2}\psi_{\alpha=0} &= \psi_0 x_2/X/q^2 \end{align}\end{split}\]with \(X \equiv \sqrt{x_1^2 + x^2_2/q^2}\).
- ddpsi_vec_a0(x1, x2)
Second derivatives of the lensing potential (zero angle).
\[\begin{split}\begin{align} \partial_{11}\psi_{\alpha=0} &= \psi_0(1-r_1^2)/X\\ \partial_{12}\psi_{\alpha=0} &= -\psi_0r_1r_2/X\\ \partial_{22}\psi_{\alpha=0} &= \psi_0(1/q^2-r_2^2)/X \end{align}\end{split}\]with \(X \equiv \sqrt{x_1^2 + x^2_2/q^2}\), \(r_{1}\equiv x_{1}/X\) and \(r_{2}\equiv x_{2}/X/q^2\).
- psi(x1, x2)
Lensing potential.
Defining the transformed coordinates \(\pmb{\tilde{x}}\equiv R(\alpha)(\pmb{x}-\pmb{x_c})\) where \(R(\alpha)\) is the rotation defined in
rotate_vector(), we can write the lensing potential frompsi_a0()as\[\psi(x_1, x_2) = \psi_{\alpha=0}(\tilde{x}_1, \tilde{x}_2)\]
- dpsi_vec(x1, x2)
Gradient of the lensing potential.
Defining the transformed coordinates \(\pmb{\tilde{x}}\equiv R(\alpha)(\pmb{x}-\pmb{x_c})\) where \(R(\alpha)\) is the rotation defined in
rotate_vector(), we can write the first derivatives of the potential fromdpsi_a0()as\[\begin{split}\begin{align} \pmb{g} &\equiv \pmb{\nabla}\psi(\pmb{x})\\ \pmb{\tilde{g}} &\equiv \pmb{\nabla}\psi_{\alpha=0}(\tilde{\pmb{x}})\\ \pmb{g} &= R_g(\alpha)\pmb{\tilde{g}} \end{align}\end{split}\]where the linear transformation \(R_g\) is defined in
rotate_grad().
- ddpsi_vec(x1, x2)
Hessian of the lensing potential.
Defining the transformed coordinates \(\pmb{\tilde{x}}\equiv R(\alpha)(\pmb{x}-\pmb{x_c})\) where \(R(\alpha)\) is the rotation defined in
rotate_vector(), we can write the second derivatives of the potential fromddpsi_a0()as\[\begin{split}\begin{align} \pmb{h} &\equiv (\partial_{11}\psi,\,\partial_{12}\psi,\,\partial_{22}\psi)(\pmb{x})\\ \pmb{\tilde{h}} &\equiv (\partial_{11}\psi_{\alpha=0},\, \partial_{12}\psi_{\alpha=0},\,\partial_{22}\psi_{\alpha=0})(\pmb{\tilde{x}})\\ \pmb{h} &= R_h(\alpha)\pmb{\tilde{h}} \end{align}\end{split}\]where the linear transformation \(R_h\) is defined in
rotate_hessian().
- class glow.lenses.Psi_Ext(p_phys={}, p_prec={})
Bases:
PsiGeneralLens object for an external field (convergence & shear).
Additional information: theory, default parameters.
- Parameters:
- p_physdict, optional
Physical parameters, with keys:
kappa(float) – Convergence.gamma1(float) – Shear along x1.gamma2(float) – Shear along x2.
- Attributes:
- kappafloat
Convergence of the lens.
- gamma1float
Shear along x1 of the lens.
- gamma2float
Shear along x2 of the lens.
Methods
ddpsi_vec(x1, x2)Second derivatives of the lensing potential.
Initialize the default parameters.
dpsi_vec(x1, x2)First derivatives of the lensing potential.
psi(x1, x2)Lensing potential.
- default_params()
Initialize the default parameters.
- Returns:
- p_physdict
Default physical parameters.
- p_precdict
Empty dictionary.
- psi(x1, x2)
Lensing potential.
\[\begin{align} \psi(x_1, x_2) &= \kappa (x_1^2+x_2^2)/2 + \gamma_1(x_1^2-x_2^2)/2 + \gamma_2x_1x_2 \end{align}\]
- dpsi_vec(x1, x2)
First derivatives of the lensing potential.
\[\begin{split}\begin{align} \partial_{1}\psi &= \kappa x_1 + \gamma_1 x_1 + \gamma_2 x_2\\ \partial_{2}\psi &= \kappa x_2 - \gamma_1 x_2 + \gamma_2 x_1 \end{align}\end{split}\]
- ddpsi_vec(x1, x2)
Second derivatives of the lensing potential.
\[\begin{split}\begin{align} \partial_{11}\psi &= \kappa + \gamma_1\\ \partial_{12}\psi &= \gamma_2\\ \partial_{22}\psi &= \kappa - \gamma_1 \end{align}\end{split}\]