class LensModel(object):
"""
class to handle an arbitrary list of lens models. This is the main lenstronomy LensModel API for all other modules.
"""
def __init__(self,
lens_model_list,
z_lens=None,
z_source=None,
lens_redshift_list=None,
cosmo=None,
multi_plane=False,
numerical_alpha_class=None,
observed_convention_index=None,
z_source_convention=None,
cosmo_interp=False,
z_interp_stop=None,
num_z_interp=100):
"""
:param lens_model_list: list of strings with lens model names
:param z_lens: redshift of the deflector (only considered when operating in single plane mode).
Is only needed for specific functions that require a cosmology.
:param z_source: redshift of the source: Needed in multi_plane option only,
not required for the core functionalities in the single plane mode.
:param lens_redshift_list: list of deflector redshift (corresponding to the lens model list),
only applicable in multi_plane mode.
:param cosmo: instance of the astropy cosmology class. If not specified, uses the default cosmology.
:param multi_plane: bool, if True, uses multi-plane mode. Default is False.
:param numerical_alpha_class: an instance of a custom class for use in NumericalAlpha() lens model
(see documentation in Profiles/numerical_alpha)
:param observed_convention_index: a list of indices, corresponding to the lens_model_list element with same
index, where the 'center_x' and 'center_y' kwargs correspond to observed (lensed) positions, not physical
positions. The code will compute the physical locations when performing computations
:param z_source_convention: float, redshift of a source to define the reduced deflection angles of the lens
models. If None, 'z_source' is used.
:param cosmo_interp: boolean (only employed in multi-plane mode), interpolates astropy.cosmology distances for
faster calls when accessing several lensing planes
:param z_interp_stop: (only in multi-plane with cosmo_interp=True); maximum redshift for distance interpolation
This number should be higher or equal the maximum of the source redshift and/or the z_source_convention
:param num_z_interp: (only in multi-plane with cosmo_interp=True); number of redshift bins for interpolating
distances
"""
self.lens_model_list = lens_model_list
self.z_lens = z_lens
self.z_source = z_source
self._z_source_convention = z_source_convention
self.redshift_list = lens_redshift_list
if cosmo is None:
from astropy.cosmology import default_cosmology
cosmo = default_cosmology.get()
self.cosmo = cosmo
self.multi_plane = multi_plane
if multi_plane is True:
if z_source is None:
raise ValueError(
'z_source needs to be set for multi-plane lens modelling.')
self.lens_model = MultiPlane(
z_source,
lens_model_list,
lens_redshift_list,
cosmo=cosmo,
numerical_alpha_class=numerical_alpha_class,
observed_convention_index=observed_convention_index,
z_source_convention=z_source_convention,
cosmo_interp=cosmo_interp,
z_interp_stop=z_interp_stop,
num_z_interp=num_z_interp)
else:
self.lens_model = SinglePlane(
lens_model_list,
numerical_alpha_class=numerical_alpha_class,
lens_redshift_list=lens_redshift_list,
z_source_convention=z_source_convention)
if z_lens is not None and z_source is not None:
self._lensCosmo = LensCosmo(z_lens, z_source, cosmo=cosmo)
def ray_shooting(self, x, y, kwargs, k=None):
"""
maps image to source position (inverse deflection)
:param x: x-position (preferentially arcsec)
:type x: numpy array
:param y: y-position (preferentially arcsec)
:type y: numpy array
:param kwargs: list of keyword arguments of lens model parameters matching the lens model classes
:param k: only evaluate the k-th lens model
:return: source plane positions corresponding to (x, y) in the image plane
"""
return self.lens_model.ray_shooting(x, y, kwargs, k=k)
def fermat_potential(self,
x_image,
y_image,
kwargs_lens,
x_source=None,
y_source=None):
"""
fermat potential (negative sign means earlier arrival time)
for Multi-plane lensing, it computes the effective Fermat potential (derived from the arrival time and
subtracted off the time-delay distance for the given cosmology). The units are given in arcsecond square.
:param x_image: image position
:param y_image: image position
:param x_source: source position
:param y_source: source position
:param kwargs_lens: list of keyword arguments of lens model parameters matching the lens model classes
:return: fermat potential in arcsec**2 without geometry term (second part of Eqn 1 in Suyu et al. 2013) as a list
"""
if hasattr(self.lens_model, 'fermat_potential'):
return self.lens_model.fermat_potential(x_image, y_image,
kwargs_lens, x_source,
y_source)
elif hasattr(self.lens_model, 'arrival_time') and hasattr(
self, '_lensCosmo'):
dt = self.lens_model.arrival_time(x_image, y_image, kwargs_lens)
fermat_pot_eff = dt * const.c / self._lensCosmo.ddt / const.Mpc * const.day_s / const.arcsec**2
return fermat_pot_eff
else:
raise ValueError(
'In multi-plane lensing you need to provide a specific z_lens and z_source for which the '
'effective Fermat potential is evaluated')
def arrival_time(self, x_image, y_image, kwargs_lens, kappa_ext=0):
"""
:param x_image: image position
:param y_image: image position
:param kwargs_lens: lens model parameter keyword argument list
:param kappa_ext: external convergence contribution not accounted in the lens model that leads to the same
observables in position and relative fluxes but rescales the time delays
:return: arrival time of image positions in units of days
"""
if hasattr(self.lens_model, 'arrival_time'):
arrival_time = self.lens_model.arrival_time(
x_image, y_image, kwargs_lens)
else:
fermat_pot = self.lens_model.fermat_potential(
x_image, y_image, kwargs_lens)
if not hasattr(self, '_lensCosmo'):
raise ValueError(
"LensModel class was not initialized with lens and source redshifts!"
)
arrival_time = self._lensCosmo.time_delay_units(fermat_pot)
arrival_time *= (1 - kappa_ext)
return arrival_time
def potential(self, x, y, kwargs, k=None):
"""
lensing potential
:param x: x-position (preferentially arcsec)
:type x: numpy array
:param y: y-position (preferentially arcsec)
:type y: numpy array
:param kwargs: list of keyword arguments of lens model parameters matching the lens model classes
:param k: only evaluate the k-th lens model
:return: lensing potential in units of arcsec^2
"""
return self.lens_model.potential(x, y, kwargs, k=k)
def alpha(self, x, y, kwargs, k=None, diff=None):
"""
deflection angles
:param x: x-position (preferentially arcsec)
:type x: numpy array
:param y: y-position (preferentially arcsec)
:type y: numpy array
:param kwargs: list of keyword arguments of lens model parameters matching the lens model classes
:param k: only evaluate the k-th lens model
:param diff: None or float. If set, computes the deflection as a finite numerical differential of the lensing
potential. This differential is only applicable in the single lensing plane where the form of the lensing
potential is analytically known
:return: deflection angles in units of arcsec
"""
if diff is None:
return self.lens_model.alpha(x, y, kwargs, k=k)
elif self.multi_plane is False:
return self._deflection_differential(x, y, kwargs, k=k, diff=diff)
else:
raise ValueError(
'numerical differentiation of lensing potential is not available in the multi-plane '
'setting as analytical form of lensing potential is not available.'
)
def hessian(self, x, y, kwargs, k=None, diff=None, diff_method='square'):
"""
hessian matrix
:param x: x-position (preferentially arcsec)
:type x: numpy array
:param y: y-position (preferentially arcsec)
:type y: numpy array
:param kwargs: list of keyword arguments of lens model parameters matching the lens model classes
:param k: only evaluate the k-th lens model
:param diff: float, scale over which the finite numerical differential is computed. If None, then using the
exact (if available) differentials.
:param diff_method: string, 'square' or 'cross', indicating whether finite differentials are computed from a
cross or a square of points around (x, y)
:return: f_xx, f_xy, f_yx, f_yy components
"""
if diff is None:
return self.lens_model.hessian(x, y, kwargs, k=k)
elif diff_method == 'square':
return self._hessian_differential_square(x,
y,
kwargs,
k=k,
diff=diff)
elif diff_method == 'cross':
return self._hessian_differential_cross(x,
y,
kwargs,
k=k,
diff=diff)
else:
raise ValueError(
'diff_method %s not supported. Chose among "square" or "cross".'
% diff_method)
def kappa(self, x, y, kwargs, k=None, diff=None, diff_method='square'):
"""
lensing convergence k = 1/2 laplacian(phi)
:param x: x-position (preferentially arcsec)
:type x: numpy array
:param y: y-position (preferentially arcsec)
:type y: numpy array
:param kwargs: list of keyword arguments of lens model parameters matching the lens model classes
:param k: only evaluate the k-th lens model
:param diff: float, scale over which the finite numerical differential is computed. If None, then using the
exact (if available) differentials.
:param diff_method: string, 'square' or 'cross', indicating whether finite differentials are computed from a
cross or a square of points around (x, y)
:return: lensing convergence
"""
f_xx, f_xy, f_yx, f_yy = self.hessian(x,
y,
kwargs,
k=k,
diff=diff,
diff_method=diff_method)
kappa = 1. / 2 * (f_xx + f_yy)
return kappa
def curl(self, x, y, kwargs, k=None, diff=None, diff_method='square'):
"""
curl computation F_xy - F_yx
:param x: x-position (preferentially arcsec)
:type x: numpy array
:param y: y-position (preferentially arcsec)
:type y: numpy array
:param kwargs: list of keyword arguments of lens model parameters matching the lens model classes
:param k: only evaluate the k-th lens model
:param diff: float, scale over which the finite numerical differential is computed. If None, then using the
exact (if available) differentials.
:param diff_method: string, 'square' or 'cross', indicating whether finite differentials are computed from a
cross or a square of points around (x, y)
:return: curl at position (x, y)
"""
f_xx, f_xy, f_yx, f_yy = self.hessian(x,
y,
kwargs,
k=k,
diff=diff,
diff_method=diff_method)
return f_xy - f_yx
def gamma(self, x, y, kwargs, k=None, diff=None, diff_method='square'):
"""
shear computation
g1 = 1/2(d^2phi/dx^2 - d^2phi/dy^2)
g2 = d^2phi/dxdy
:param x: x-position (preferentially arcsec)
:type x: numpy array
:param y: y-position (preferentially arcsec)
:type y: numpy array
:param kwargs: list of keyword arguments of lens model parameters matching the lens model classes
:param k: only evaluate the k-th lens model
:param diff: float, scale over which the finite numerical differential is computed. If None, then using the
exact (if available) differentials.
:param diff_method: string, 'square' or 'cross', indicating whether finite differentials are computed from a
cross or a square of points around (x, y)
:return: gamma1, gamma2
"""
f_xx, f_xy, f_yx, f_yy = self.hessian(x,
y,
kwargs,
k=k,
diff=diff,
diff_method=diff_method)
gamma1 = 1. / 2 * (f_xx - f_yy)
gamma2 = f_xy
return gamma1, gamma2
def magnification(self,
x,
y,
kwargs,
k=None,
diff=None,
diff_method='square'):
"""
magnification
mag = 1/det(A)
A = 1 - d^2phi/d_ij
:param x: x-position (preferentially arcsec)
:type x: numpy array
:param y: y-position (preferentially arcsec)
:type y: numpy array
:param kwargs: list of keyword arguments of lens model parameters matching the lens model classes
:param k: only evaluate the k-th lens model
:param diff: float, scale over which the finite numerical differential is computed. If None, then using the
exact (if available) differentials.
:param diff_method: string, 'square' or 'cross', indicating whether finite differentials are computed from a
cross or a square of points around (x, y)
:return: magnification
"""
f_xx, f_xy, f_yx, f_yy = self.hessian(x,
y,
kwargs,
k=k,
diff=diff,
diff_method=diff_method)
det_A = (1 - f_xx) * (1 - f_yy) - f_xy * f_yx
return 1. / det_A # attention, if dividing by zero
def flexion(self, x, y, kwargs, k=None, diff=0.000001, hessian_diff=True):
"""
third derivatives (flexion)
:param x: x-position (preferentially arcsec)
:type x: numpy array
:param y: y-position (preferentially arcsec)
:type y: numpy array
:param kwargs: list of keyword arguments of lens model parameters matching the lens model classes
:param k: int or None, if set, only evaluates the differential from one model component
:param diff: numerical differential length of Flexion
:param hessian_diff: boolean, if true also computes the numerical differential length of Hessian (optional)
:return: f_xxx, f_xxy, f_xyy, f_yyy
"""
if hessian_diff is not True:
hessian_diff = None
f_xx_dx, f_xy_dx, f_yx_dx, f_yy_dx = self.hessian(x + diff / 2,
y,
kwargs,
k=k,
diff=hessian_diff)
f_xx_dy, f_xy_dy, f_yx_dy, f_yy_dy = self.hessian(x,
y + diff / 2,
kwargs,
k=k,
diff=hessian_diff)
f_xx_dx_, f_xy_dx_, f_yx_dx_, f_yy_dx_ = self.hessian(
x - diff / 2, y, kwargs, k=k, diff=hessian_diff)
f_xx_dy_, f_xy_dy_, f_yx_dy_, f_yy_dy_ = self.hessian(
x, y - diff / 2, kwargs, k=k, diff=hessian_diff)
f_xxx = (f_xx_dx - f_xx_dx_) / diff
f_xxy = (f_xx_dy - f_xx_dy_) / diff
f_xyy = (f_xy_dy - f_xy_dy_) / diff
f_yyy = (f_yy_dy - f_yy_dy_) / diff
return f_xxx, f_xxy, f_xyy, f_yyy
def set_static(self, kwargs):
"""
set this instance to a static lens model. This can improve the speed in evaluating lensing quantities at
different positions but must not be used with different lens model parameters!
:param kwargs: lens model keyword argument list
:return: kwargs_updated (in case of image position convention in multiplane lensing this is changed)
"""
return self.lens_model.set_static(kwargs)
def set_dynamic(self):
"""
deletes cache for static setting and makes sure the observed convention in the position of lensing profiles in
the multi-plane setting is enabled. Dynamic is the default setting of this class enabling an accurate computation
of lensing quantities with different parameters in the lensing profiles.
:return: None
"""
self.lens_model.set_dynamic()
def _deflection_differential(self, x, y, kwargs, k=None, diff=0.00001):
"""
:param x: x-coordinate
:param y: y-coordinate
:param kwargs: keyword argument list
:param k: int or None, if set, only evaluates the differential from one model component
:param diff: finite differential length
:return: f_x, f_y
"""
phi_dx = self.lens_model.potential(x + diff / 2, y, kwargs=kwargs, k=k)
phi_dy = self.lens_model.potential(x, y + diff / 2, kwargs=kwargs, k=k)
phi_dx_ = self.lens_model.potential(x - diff / 2,
y,
kwargs=kwargs,
k=k)
phi_dy_ = self.lens_model.potential(x,
y - diff / 2,
kwargs=kwargs,
k=k)
f_x = (phi_dx - phi_dx_) / diff
f_y = (phi_dy - phi_dy_) / diff
return f_x, f_y
def _hessian_differential_cross(self, x, y, kwargs, k=None, diff=0.00001):
"""
computes the numerical differentials over a finite range for f_xx, f_yy, f_xy from f_x and f_y
The differentials are computed along the cross centered at (x, y).
:param x: x-coordinate
:param y: y-coordinate
:param kwargs: lens model keyword argument list
:param k: int, list of bools or None, indicating a subset of lens models to be evaluated
:param diff: float, scale of the finite differential (diff/2 in each direction used to compute the differential
:return: f_xx, f_xy, f_yx, f_yy
"""
alpha_ra_dx, alpha_dec_dx = self.alpha(x + diff / 2, y, kwargs, k=k)
alpha_ra_dy, alpha_dec_dy = self.alpha(x, y + diff / 2, kwargs, k=k)
alpha_ra_dx_, alpha_dec_dx_ = self.alpha(x - diff / 2, y, kwargs, k=k)
alpha_ra_dy_, alpha_dec_dy_ = self.alpha(x, y - diff / 2, kwargs, k=k)
dalpha_rara = (alpha_ra_dx - alpha_ra_dx_) / diff
dalpha_radec = (alpha_ra_dy - alpha_ra_dy_) / diff
dalpha_decra = (alpha_dec_dx - alpha_dec_dx_) / diff
dalpha_decdec = (alpha_dec_dy - alpha_dec_dy_) / diff
f_xx = dalpha_rara
f_yy = dalpha_decdec
f_xy = dalpha_radec
f_yx = dalpha_decra
return f_xx, f_xy, f_yx, f_yy
def _hessian_differential_square(self, x, y, kwargs, k=None, diff=0.00001):
"""
computes the numerical differentials over a finite range for f_xx, f_yy, f_xy from f_x and f_y
The differentials are computed on the square around (x, y). This minimizes curl.
:param x: x-coordinate
:param y: y-coordinate
:param kwargs: lens model keyword argument list
:param k: int, list of booleans or None, indicating a subset of lens models to be evaluated
:param diff: float, scale of the finite differential (diff/2 in each direction used to compute the differential
:return: f_xx, f_xy, f_yx, f_yy
"""
alpha_ra_pp, alpha_dec_pp = self.alpha(x + diff / 2,
y + diff / 2,
kwargs,
k=k)
alpha_ra_pn, alpha_dec_pn = self.alpha(x + diff / 2,
y - diff / 2,
kwargs,
k=k)
alpha_ra_np, alpha_dec_np = self.alpha(x - diff / 2,
y + diff / 2,
kwargs,
k=k)
alpha_ra_nn, alpha_dec_nn = self.alpha(x - diff / 2,
y - diff / 2,
kwargs,
k=k)
f_xx = (alpha_ra_pp - alpha_ra_np + alpha_ra_pn -
alpha_ra_nn) / diff / 2
f_xy = (alpha_ra_pp - alpha_ra_pn + alpha_ra_np -
alpha_ra_nn) / diff / 2
f_yx = (alpha_dec_pp - alpha_dec_np + alpha_dec_pn -
alpha_dec_nn) / diff / 2
f_yy = (alpha_dec_pp - alpha_dec_pn + alpha_dec_np -
alpha_dec_nn) / diff / 2
return f_xx, f_xy, f_yx, f_yy
class LensModel(object):
"""
class to handle an arbitrary list of lens models
"""
def __init__(self,
lens_model_list,
z_lens=None,
z_source=None,
lens_redshift_list=None,
cosmo=None,
multi_plane=False,
numerical_alpha_class=None):
"""
:param lens_model_list: list of strings with lens model names
:param z_lens: redshift of the deflector (only considered when operating in single plane mode).
Is only needed for specific functions that require a cosmology.
:param z_source: redshift of the source: Needed in multi_plane option only,
not required for the core functionalities in the single plane mode.
:param lens_redshift_list: list of deflector redshift (corresponding to the lens model list),
only applicable in multi_plane mode.
:param cosmo: instance of the astropy cosmology class. If not specified, uses the default cosmology.
:param multi_plane: bool, if True, uses multi-plane mode. Default is False.
:param numerical_alpha_class: an instance of a custom class for use in NumericalAlpha() lens model
(see documentation in Profiles/numerical_alpha)
"""
self.lens_model_list = lens_model_list
self.z_lens = z_lens
self.z_source = z_source
self.redshift_list = lens_redshift_list
self.cosmo = cosmo
self.multi_plane = multi_plane
if multi_plane is True:
self.lens_model = MultiPlane(
z_source,
lens_model_list,
lens_redshift_list,
cosmo=cosmo,
numerical_alpha_class=numerical_alpha_class)
else:
self.lens_model = SinglePlane(
lens_model_list, numerical_alpha_class=numerical_alpha_class)
if z_lens is not None and z_source is not None:
self._lensCosmo = LensCosmo(z_lens, z_source, cosmo=self.cosmo)
def ray_shooting(self, x, y, kwargs, k=None):
"""
maps image to source position (inverse deflection)
:param x: x-position (preferentially arcsec)
:type x: numpy array
:param y: y-position (preferentially arcsec)
:type y: numpy array
:param kwargs: list of keyword arguments of lens model parameters matching the lens model classes
:param k: only evaluate the k-th lens model
:return: source plane positions corresponding to (x, y) in the image plane
"""
return self.lens_model.ray_shooting(x, y, kwargs, k=k)
def fermat_potential(self, x_image, y_image, x_source, y_source,
kwargs_lens):
"""
fermat potential (negative sign means earlier arrival time)
:param x_image: image position
:param y_image: image position
:param x_source: source position
:param y_source: source position
:param kwargs_lens: list of keyword arguments of lens model parameters matching the lens model classes
:return: fermat potential in arcsec**2 without geometry term (second part of Eqn 1 in Suyu et al. 2013) as a list
"""
if hasattr(self.lens_model, 'fermat_potential'):
return self.lens_model.fermat_potential(x_image, y_image, x_source,
y_source, kwargs_lens)
else:
raise ValueError(
"Fermat potential is not defined in multi-plane lensing. Please use single plane lens models."
)
def arrival_time(self, x_image, y_image, kwargs_lens):
"""
:param x_image: image position
:param y_image: image position
:param kwargs_lens: lens model parameter keyword argument list
:return:
"""
if hasattr(self.lens_model, 'arrival_time'):
arrival_time = self.lens_model.arrival_time(
x_image, y_image, kwargs_lens)
else:
x_source, y_source = self.lens_model.ray_shooting(
x_image, y_image, kwargs_lens)
fermat_pot = self.lens_model.fermat_potential(
x_image, y_image, x_source, y_source, kwargs_lens)
if not hasattr(self, '_lensCosmo'):
raise ValueError(
"LensModel class was not initalized with lens and source redshifts!"
)
arrival_time = self._lensCosmo.time_delay_units(fermat_pot)
return arrival_time
def potential(self, x, y, kwargs, k=None):
"""
lensing potential
:param x: x-position (preferentially arcsec)
:type x: numpy array
:param y: y-position (preferentially arcsec)
:type y: numpy array
:param kwargs: list of keyword arguments of lens model parameters matching the lens model classes
:param k: only evaluate the k-th lens model
:return: lensing potential in units of arcsec^2
"""
return self.lens_model.potential(x, y, kwargs, k=k)
def alpha(self, x, y, kwargs, k=None):
"""
deflection angles
:param x: x-position (preferentially arcsec)
:type x: numpy array
:param y: y-position (preferentially arcsec)
:type y: numpy array
:param kwargs: list of keyword arguments of lens model parameters matching the lens model classes
:param k: only evaluate the k-th lens model
:return: deflection angles in units of arcsec
"""
return self.lens_model.alpha(x, y, kwargs, k=k)
def hessian(self, x, y, kwargs, k=None):
"""
hessian matrix
:param x: x-position (preferentially arcsec)
:type x: numpy array
:param y: y-position (preferentially arcsec)
:type y: numpy array
:param kwargs: list of keyword arguments of lens model parameters matching the lens model classes
:param k: only evaluate the k-th lens model
:return: f_xx, f_xy, f_yy components
"""
return self.lens_model.hessian(x, y, kwargs, k=k)
def kappa(self, x, y, kwargs, k=None):
"""
lensing convergence k = 1/2 laplacian(phi)
:param x: x-position (preferentially arcsec)
:type x: numpy array
:param y: y-position (preferentially arcsec)
:type y: numpy array
:param kwargs: list of keyword arguments of lens model parameters matching the lens model classes
:param k: only evaluate the k-th lens model
:return: lensing convergence
"""
f_xx, f_xy, f_yx, f_yy = self.hessian(x, y, kwargs, k=k)
kappa = 1. / 2 * (f_xx + f_yy)
return kappa
def gamma(self, x, y, kwargs, k=None):
"""
shear computation
g1 = 1/2(d^2phi/dx^2 - d^2phi/dy^2)
g2 = d^2phi/dxdy
:param x: x-position (preferentially arcsec)
:type x: numpy array
:param y: y-position (preferentially arcsec)
:type y: numpy array
:param kwargs: list of keyword arguments of lens model parameters matching the lens model classes
:param k: only evaluate the k-th lens model
:return: gamma1, gamma2
"""
f_xx, f_xy, f_yx, f_yy = self.hessian(x, y, kwargs, k=k)
gamma1 = 1. / 2 * (f_xx - f_yy)
gamma2 = f_xy
return gamma1, gamma2
def magnification(self, x, y, kwargs, k=None):
"""
magnification
mag = 1/det(A)
A = 1 - d^2phi/d_ij
:param x: x-position (preferentially arcsec)
:type x: numpy array
:param y: y-position (preferentially arcsec)
:type y: numpy array
:param kwargs: list of keyword arguments of lens model parameters matching the lens model classes
:param k: only evaluate the k-th lens model
:return: magnification
"""
f_xx, f_xy, f_yx, f_yy = self.hessian(x, y, kwargs, k=k)
det_A = (1 - f_xx) * (1 - f_yy) - f_xy * f_yx
return 1. / det_A # attention, if dividing by zero
def flexion(self, x, y, kwargs, diff=0.000001):
"""
third derivatives (flexion)
:param x: x-position (preferentially arcsec)
:type x: numpy array
:param y: y-position (preferentially arcsec)
:type y: numpy array
:param kwargs: list of keyword arguments of lens model parameters matching the lens model classes
:param diff: numerical differential length of Hessian
:return: f_xxx, f_xxy, f_xyy, f_yyy
"""
f_xx, f_xy, f_yx, f_yy = self.hessian(x, y, kwargs)
f_xx_dx, f_xy_dx, f_yx_dx, f_yy_dx = self.hessian(x + diff, y, kwargs)
f_xx_dy, f_xy_dy, f_yx_dy, f_yy_dy = self.hessian(x, y + diff, kwargs)
f_xxx = (f_xx_dx - f_xx) / diff
f_xxy = (f_xx_dy - f_xx) / diff
f_xyy = (f_xy_dy - f_xy) / diff
f_yyy = (f_yy_dy - f_yy) / diff
return f_xxx, f_xxy, f_xyy, f_yyy
class LensModel(object):
"""
class to handle an arbitrary list of lens models
"""
def __init__(self, lens_model_list, z_source=None, redshift_list=None, cosmo=None, multi_plane=False):
"""
:param lens_model_list: list of strings with lens model names
:param foreground_shear: bool, when True, models a foreground non-linear shear distortion
"""
self.lens_model_list = lens_model_list
self.z_source = z_source
self.redshift_list = redshift_list
self.cosmo = cosmo
self.multi_plane = multi_plane
if multi_plane is True:
self.lens_model = MultiLens(z_source, lens_model_list, redshift_list, cosmo=cosmo)
else:
self.lens_model = SinglePlane(lens_model_list)
def ray_shooting(self, x, y, kwargs, k=None):
"""
maps image to source position (inverse deflection)
:param x: x-position (preferentially arcsec)
:type x: numpy array
:param y: y-position (preferentially arcsec)
:type y: numpy array
:param kwargs: list of keyword arguments of lens model parameters matching the lens model classes
:param k: only evaluate the k-th lens model
:return: source plane positions corresponding to (x, y) in the image plane
"""
return self.lens_model.ray_shooting(x, y, kwargs, k=k)
def fermat_potential(self, x_image, y_image, x_source, y_source, kwargs_lens):
"""
fermat potential (negative sign means earlier arrival time)
:param x_image: image position
:param y_image: image position
:param x_source: source position
:param y_source: source position
:param kwargs_lens: list of keyword arguments of lens model parameters matching the lens model classes
:return: fermat potential in arcsec**2 without geometry term (second part of Eqn 1 in Suyu et al. 2013) as a list
"""
if self.multi_plane:
raise ValueError("Fermat potential is not defined in multi-plane lensing. Please use single plane lens models.")
else:
return self.lens_model.fermat_potential(x_image, y_image, x_source, y_source, kwargs_lens)
def arrival_time(self, x_image, y_image, kwargs_lens):
"""
:param x_image:
:param y_image:
:param kwargs_lens:
:return:
"""
if self.multi_plane:
return self.lens_model.arrival_time(x_image, y_image, kwargs_lens)
else:
raise ValueError(
"arrival_time routine not defined for single plane lensing. Please use Fermat potential instead")
def mass(self, x, y, epsilon_crit, kwargs):
"""
:param x: position
:param y: position
:param epsilon_crit: critical mass density of a lens
:param kwargs: list of keyword arguments of lens model parameters matching the lens model classes
:return: projected mass density in units of input epsilon_crit
"""
kappa = self.kappa(x, y, kwargs)
mass = epsilon_crit * kappa
return mass
def potential(self, x, y, kwargs, k=None):
"""
lensing potential
:param x: x-position (preferentially arcsec)
:type x: numpy array
:param y: y-position (preferentially arcsec)
:type y: numpy array
:param kwargs: list of keyword arguments of lens model parameters matching the lens model classes
:param k: only evaluate the k-th lens model
:return: lensing potential in units of arcsec^2
"""
return self.lens_model.potential(x, y, kwargs, k=k)
def alpha(self, x, y, kwargs, k=None):
"""
deflection angles
:param x: x-position (preferentially arcsec)
:type x: numpy array
:param y: y-position (preferentially arcsec)
:type y: numpy array
:param kwargs: list of keyword arguments of lens model parameters matching the lens model classes
:param k: only evaluate the k-th lens model
:return: deflection angles in units of arcsec
"""
return self.lens_model.alpha(x, y, kwargs, k=k)
def kappa(self, x, y, kwargs, k=None):
"""
lensing convergence k = 1/2 laplacian(phi)
:param x: x-position (preferentially arcsec)
:type x: numpy array
:param y: y-position (preferentially arcsec)
:type y: numpy array
:param kwargs: list of keyword arguments of lens model parameters matching the lens model classes
:param k: only evaluate the k-th lens model
:return: lensing convergence
"""
f_xx, f_xy, f_yy = self.hessian(x, y, kwargs, k=k)
kappa = 1./2 * (f_xx + f_yy) # attention on units
return kappa
def gamma(self, x, y, kwargs, k=None):
"""
shear computation
g1 = 1/2(d^2phi/dx^2 - d^2phi/dy^2)
g2 = d^2phi/dxdy
:param x: x-position (preferentially arcsec)
:type x: numpy array
:param y: y-position (preferentially arcsec)
:type y: numpy array
:param kwargs: list of keyword arguments of lens model parameters matching the lens model classes
:param k: only evaluate the k-th lens model
:return: gamma1, gamma2
"""
f_xx, f_xy, f_yy = self.hessian(x, y, kwargs, k=k)
gamma1 = 1./2 * (f_xx - f_yy) # attention on units
gamma2 = f_xy # attention on units
return gamma1, gamma2
def magnification(self, x, y, kwargs, k=None):
"""
magnification
mag = 1/det(A)
A = 1 - d^2phi/d_ij
:param x: x-position (preferentially arcsec)
:type x: numpy array
:param y: y-position (preferentially arcsec)
:type y: numpy array
:param kwargs: list of keyword arguments of lens model parameters matching the lens model classes
:param k: only evaluate the k-th lens model
:return: magnification
"""
f_xx, f_xy, f_yy = self.hessian(x, y, kwargs, k=k)
det_A = (1 - f_xx) * (1 - f_yy) - f_xy*f_xy
return 1./det_A # attention, if dividing by zero
def hessian(self, x, y, kwargs, k=None):
"""
hessian matrix
:param x: x-position (preferentially arcsec)
:type x: numpy array
:param y: y-position (preferentially arcsec)
:type y: numpy array
:param kwargs: list of keyword arguments of lens model parameters matching the lens model classes
:param k: only evaluate the k-th lens model
:return: f_xx, f_xy, f_yy components
"""
return self.lens_model.hessian(x, y, kwargs, k=k)
def mass_3d(self, r, kwargs, bool_list=None):
"""
computes the mass within a 3d sphere of radius r
:param r: radius (in angular units)
:param kwargs: list of keyword arguments of lens model parameters matching the lens model classes
:param bool_list: list of bools that are part of the output
:return: mass (in angular units, modulo epsilon_crit)
"""
if self.multi_plane is True:
raise ValueError("mass_3d is not supported for multi-lane lensing. Please use single plane instead.")
else:
return self.lens_model.mass_3d(r, kwargs, bool_list=bool_list)
def mass_2d(self, r, kwargs, bool_list=None):
"""
computes the mass enclosed a projected (2d) radius r
:param r: radius (in angular units)
:param kwargs: list of keyword arguments of lens model parameters matching the lens model classes
:param bool_list: list of bools that are part of the output
:return: projected mass (in angular units, modulo epsilon_crit)
"""
if self.multi_plane is True:
raise ValueError("mass_2d is not supported for multi-lane lensing. Please use single plane instead.")
else:
return self.lens_model.mass_2d(r, kwargs, bool_list=bool_list)