ler.lens_galaxy_population.optical_depth

Module for optical depth and lens parameter distribution calculations.

This module provides the OpticalDepth class for computing strong lensing optical depth, cross-section, sampling velocity dispersion, axis ratio, and other lens galaxy population parameters. It supports multiple lens models including SIS, SIE, and EPL + external shear.

Key features:

• Optical depth computation for strong gravitational lensing

• Velocity dispersion sampling with multiple models

• Lens redshift distribution sampling

• Cross-section calculations for various lens models

Copyright (C) 2024 Hemantakumar Phurailatpam. Distributed under MIT License.

Module Contents

Classes

OpticalDepth

Class for computing optical depth and lens galaxy population parameters.

class ler.lens_galaxy_population.optical_depth.OpticalDepth(npool=4, z_min=0.0, z_max=10.0, cosmology=None, lens_type='epl_shear_galaxy', lens_functions=None, lens_functions_params=None, lens_param_samplers=None, lens_param_samplers_params=None, directory='./interpolator_json', create_new_interpolator=False, verbose=False)

Class for computing optical depth and lens galaxy population parameters.

This class calculates strong lensing optical depth, velocity dispersion, axis ratio, and other parameters for a lens galaxy population. It supports SIS, SIE, and EPL + external shear lens models with customizable samplers and interpolators for efficient computation.

Key Features:

• Multiple lens model support (SIS, SIE, EPL + shear)

• Configurable velocity dispersion distributions

• Cached interpolators for fast optical depth computation

• Flexible parameter sampling with user-defined priors

Parameters:

npoolint

Number of processors for multiprocessing.

default: 4

z_minfloat

Minimum redshift of the lens galaxy population.

default: 0.0

z_maxfloat

Maximum redshift of the lens galaxy population.

default: 10.0

cosmologyastropy.cosmology or None

Cosmology object for distance calculations.

default: LambdaCDM(H0=70, Om0=0.3, Ode0=0.7)

lens_typestr

Type of lens galaxy model.

Options:

• ‘epl_shear_galaxy’: Elliptical power-law with external shear

• ‘sie_galaxy’: Singular isothermal ellipsoid

• ‘sis_galaxy’: Singular isothermal sphere

default: ‘epl_shear_galaxy’

lens_functionsdict or None

Dictionary with lens-related functions.

default: None (uses defaults for lens_type)

lens_functions_paramsdict or None

Dictionary with parameters for lens-related functions.

default: None

lens_param_samplersdict or None

Dictionary of sampler functions for lens parameters.

default: None (uses defaults for lens_type)

lens_param_samplers_paramsdict or None

Dictionary with parameters for the samplers.

default: None

directorystr

Directory where interpolators are saved.

default: ‘./interpolator_json’

create_new_interpolatorbool or dict

Whether to create new interpolators.

default: False

verbosebool

If True, prints additional information.

default: False

Examples

Basic usage:

>>> from ler.lens_galaxy_population import OpticalDepth
>>> od = OpticalDepth()
>>> tau = od.optical_depth(zs=np.array([1.0, 2.0]))

Instance Methods

OpticalDepth has the following instance methods:

Instance Attributes

OpticalDepth has the following instance attributes:

Attribute	Type	Unit	Description
npool	int		Number of processors for multiprocessing
z_min	float		Minimum redshift
z_max	float		Maximum redshift
cosmo	astropy.cosmology		Cosmology object
lens_type	str		Type of lens galaxy model
directory	str		Directory for interpolator storage
optical_depth	FunctionConditioning		Optical depth calculator
velocity_dispersion	FunctionConditioning	km/s	Velocity dispersion sampler
axis_ratio	FunctionConditioning		Axis ratio sampler
axis_rotation_angle	FunctionConditioning	rad	Axis rotation angle sampler
lens_redshift	FunctionConditioning		Lens redshift sampler
external_shear	FunctionConditioning		External shear sampler
density_profile_slope	FunctionConditioning		Density profile slope sampler
cross_section	callable	rad²	Cross-section calculator
available_lens_samplers	dict		Available lens parameter samplers
available_lens_functions	dict		Available lens functions

property lens_type

Type of lens galaxy model.

Returns:

lens_typestr

Lens type (‘epl_shear_galaxy’, ‘sie_galaxy’, or ‘sis_galaxy’).

property npool

Number of processors for multiprocessing.

Returns:

npoolint

Number of parallel processors.

property z_min

Minimum redshift of the lens galaxy population.

Returns:

z_minfloat

Minimum redshift.

property z_max

Maximum redshift of the lens galaxy population.

Returns:

z_maxfloat

Maximum redshift.

property cosmo

Cosmology object for distance calculations.

Returns:

cosmoastropy.cosmology

Cosmology object.

property directory

Directory for interpolator storage.

Returns:

directorystr

Path to interpolator JSON files.

property velocity_dispersion

Velocity dispersion sampler object.

Returns a FunctionConditioning object with methods:

• rvs(size, zl): Sample velocity dispersion values

• pdf(sigma, zl): Get probability density

• function(sigma, zl): Get number density function

Returns:

velocity_dispersionFunctionConditioning

Sampler object for velocity dispersion (km/s).

Examples

>>> from ler.lens_galaxy_population import OpticalDepth
>>> od = OpticalDepth()
>>> sigma = od.velocity_dispersion(size=100, zl=np.ones(100)*0.5)

property axis_ratio

Axis ratio sampler object.

Returns a FunctionConditioning object with methods:

• rvs(size, sigma): Sample axis ratio values

• pdf(q, sigma): Get probability density

• function(q, sigma): Get distribution function

Returns:

axis_ratioFunctionConditioning

Sampler object for axis ratio.

Examples

>>> from ler.lens_galaxy_population import OpticalDepth
>>> od = OpticalDepth()
>>> q = od.axis_ratio(size=100, sigma=np.ones(100)*200.)

property axis_rotation_angle

Axis rotation angle sampler object.

Returns a FunctionConditioning object with methods:

• rvs(size): Sample axis rotation angles

• pdf(phi): Get probability density

Returns:

axis_rotation_angleFunctionConditioning

Sampler object for axis rotation angle (rad).

Examples

>>> from ler.lens_galaxy_population import OpticalDepth
>>> od = OpticalDepth()
>>> phi = od.axis_rotation_angle(size=100)

property density_profile_slope

Density profile slope sampler object.

Returns a FunctionConditioning object with methods:

• rvs(size): Sample density profile slope values

• pdf(gamma): Get probability density

Returns:

density_profile_slopeFunctionConditioning

Sampler object for density profile slope.

Examples

>>> from ler.lens_galaxy_population import OpticalDepth
>>> od = OpticalDepth()
>>> gamma = od.density_profile_slope(size=100)

property external_shear

External shear sampler object.

Returns a FunctionConditioning object with methods:

• rvs(size): Sample shear components (gamma1, gamma2)

• pdf(gamma1, gamma2): Get probability density

Returns:

external_shearFunctionConditioning

Sampler object for external shear.

Examples

>>> from ler.lens_galaxy_population import OpticalDepth
>>> od = OpticalDepth()
>>> gamma1, gamma2 = od.external_shear(size=100)

property cross_section

Lensing cross-section calculator.

Returns a callable that computes lensing cross-section for individual

lensing events. Input parameters depend on lens type:

• EPL+shear: zs, zl, sigma, q, phi, gamma, gamma1, gamma2

• SIE: zs, zl, sigma, q

• SIS: zs, zl, sigma

Returns:

cross_sectioncallable

Cross-section function (rad² units).

Examples

>>> from ler.lens_galaxy_population import OpticalDepth
>>> od = OpticalDepth()
>>> cs = od.cross_section(zs=zs, zl=zl, sigma=sigma, ...)

property lens_redshift

Lens redshift sampler object.

Returns a FunctionConditioning object with methods:

• rvs(size, zs): Sample lens redshifts given source redshifts

• pdf(zl, zs): Get probability density

• function(zl, zs): Get effective lensing cross-section

Returns:

lens_redshiftFunctionConditioning

Sampler object for lens redshift.

Examples

>>> from ler.lens_galaxy_population import OpticalDepth
>>> od = OpticalDepth()
>>> zl = od.lens_redshift(size=100, zs=np.ones(100)*2.0)

property density_profile_slope_sl

Density profile slope sampler object (strong lensing conditioned).

Returns a FunctionConditioning object with methods:

• rvs(size): Sample density profile slope values

• pdf(gamma): Get probability density

Returns:

density_profile_slope_slFunctionConditioning

Sampler object for density profile slope (strong lensing).

Examples

>>> from ler.lens_galaxy_population import OpticalDepth
>>> od = OpticalDepth()
>>> gamma = od.density_profile_slope_sl(size=100)

property external_shear_sl

External shear sampler object (strong lensing conditioned).

Returns a FunctionConditioning object with methods:

• rvs(size): Sample shear components (gamma1, gamma2)

• pdf(gamma1, gamma2): Get probability density

Returns:

external_shear_slFunctionConditioning

Sampler object for external shear (strong lensing).

Examples

>>> from ler.lens_galaxy_population import OpticalDepth
>>> od = OpticalDepth()
>>> gamma1, gamma2 = od.external_shear_sl(size=100)

property optical_depth

Strong lensing optical depth calculator.

Returns:

optical_depthFunctionConditioning

Function object with .function(zs) method that returns n optical depth for given source redshifts.

Examples

>>> from ler.lens_galaxy_population import OpticalDepth
>>> od = OpticalDepth()
>>> tau = od.optical_depth.function(np.array([1.0, 2.0]))

property available_lens_samplers

Dictionary of available lens parameter samplers and their default parameters.

Returns:

available_lens_samplersdict

Dictionary with sampler names and default parameters.

property available_lens_functions

Dictionary of available lens functions and their default parameters.

Returns:

available_lens_functionsdict

Dictionary with function names and default parameters.

comoving_distance

angular_diameter_distance

angular_diameter_distance_z1z2

differential_comoving_volume

lens_redshift_intrinsic

lens_redshift_strongly_lensed_numerical(size=1000, zs=None, get_attribute=False, **kwargs)

Sample lens redshifts conditioned on strong lensing (numerical method).

This method computes the lens redshift distribution by numerically integrating over the velocity dispersion distribution (galaxy density distribution wrt), cross-section and differential comoving volume.

Parameters:

sizeint

Number of samples to generate. n default: 1000

zsnumpy.ndarray

Source redshifts.

get_attributebool

If True, returns the sampler object instead of samples. n default: False

**kwargsdict

Additional parameters.

Returns:

zlnumpy.ndarray or FunctionConditioning

Lens redshift samples or sampler object.

Examples

>>> from ler.lens_galaxy_population import OpticalDepth
>>> od = OpticalDepth()
>>> zl = od.lens_redshift(size=100, zs=np.ones(100)*2.0)

lens_redshift_strongly_lensed_sis_haris(size, zs, get_attribute=False, **kwargs)

Sample SIS lens redshifts using Haris et al. (2018) distribution.

Parameters:

sizeint

Number of samples to generate.

zsnumpy.ndarray

Source redshifts.

get_attributebool

If True, returns the sampler object instead of samples.

default: False

**kwargsdict

Additional parameters.

Returns:

zlnumpy.ndarray or FunctionConditioning

Lens redshift samples or sampler object.

Examples

>>> from ler.lens_galaxy_population import OpticalDepth
>>> od = OpticalDepth(lens_type='sis_galaxy')
>>> zl = od.lens_redshift(size=100, zs=np.ones(100)*2.0)

velocity_dispersion_gengamma(size, get_attribute=False, **kwargs)

Sample velocity dispersion from generalized gamma distribution.

Parameters:

sizeint

Number of samples to generate.

get_attributebool

If True, returns the sampler object instead of samples.

default: False

**kwargsdict

Additional parameters.

Parameter	Unit	Description
sigma_min	km/s	minimum velocity dispersion
sigma_max	km/s	maximum velocity dispersion
alpha	dimensionless	Power-law index governing the slope of the distribution at low velocities
beta	dimensionless	Exponential parameter determining the sharpness of the high-velocity cutoff
phistar	h^3 Mpc^-3	Normalization constant representing the comoving number density of galaxy
sigmastar	km/s	Characteristic velocity scale marking the “knee” transition in the VDF

Returns:

sigmanumpy.ndarray or FunctionConditioning

Velocity dispersion samples (km/s) or sampler object.

Examples

>>> from ler.lens_galaxy_population import OpticalDepth
>>> od = OpticalDepth(lens_param_samplers=dict(
...     velocity_dispersion="velocity_dispersion_gengamma"))
>>> sigma = od.velocity_dispersion(size=100)

velocity_dispersion_bernardi(size, get_attribute=False, **kwargs)

Sample velocity dispersion from Bernardi et al. (2010) distribution.

Uses inverse transform sampling on the velocity dispersion function.

Parameters:

sizeint

Number of samples to generate.

get_attributebool

If True, returns the sampler object instead of samples.

default: False

**kwargsdict

Additional parameters.

Parameter	Unit	Description
sigma_min	km/s	minimum velocity dispersion
sigma_max	km/s	maximum velocity dispersion
alpha	dimensionless	Power-law index governing the slope of the distribution at low velocities
beta	dimensionless	Exponential parameter determining the sharpness of the high-velocity cutoff
phistar	h^3 Mpc^-3	Normalization constant representing the comoving number density of galaxy
sigmastar	km/s	Characteristic velocity scale marking the “knee” transition in the VDF

Returns:

sigmanumpy.ndarray or FunctionConditioning

Velocity dispersion samples (km/s) or sampler object.

Examples

>>> from ler.lens_galaxy_population import OpticalDepth
>>> od = OpticalDepth(lens_param_samplers=dict(
...     velocity_dispersion="velocity_dispersion_bernardi"))
>>> sigma = od.velocity_dispersion(size=100)

velocity_dispersion_ewoud(size, zl, get_attribute=False, **kwargs)

Sample redshift-dependent velocity dispersion from Wempe et al. (2022).

Uses inverse transform sampling with redshift-dependent velocity dispersion function.

Parameters:

sizeint

Number of samples to generate.

zlnumpy.ndarray

Lens redshifts.

get_attributebool

If True, returns the sampler object instead of samples.

default: False

**kwargsdict

Additional parameters.

Parameter	Unit	Description
sigma_min	km/s	minimum velocity dispersion
sigma_max	km/s	maximum velocity dispersion
alpha	dimensionless	Power-law index governing the slope of the distribution at low velocities
beta	dimensionless	Exponential parameter determining the sharpness of the high-velocity cutoff
phistar	h^3 Mpc^-3	Normalization constant representing the comoving number density of galaxy
sigmastar	km/s	Characteristic velocity scale marking the “knee” transition in the VDF

Returns:

sigmanumpy.ndarray or FunctionConditioning

Velocity dispersion samples (km/s) or sampler object.

Examples

>>> from ler.lens_galaxy_population import OpticalDepth
>>> od = OpticalDepth()
>>> sigma = od.velocity_dispersion(size=100, zl=np.ones(100)*0.5)

axis_ratio_rayleigh(size, sigma, get_attribute=False, **kwargs)

Sample axis ratio from Rayleigh distribution conditioned on velocity dispersion.

Parameters:

sizeint

Number of samples to generate.

sigmanumpy.ndarray

Velocity dispersion of the lens galaxy (km/s).

get_attributebool

If True, returns the sampler object instead of samples.

default: False

**kwargsdict

Additional parameters (q_min, q_max).

Returns:

qnumpy.ndarray or FunctionConditioning

Axis ratio samples or sampler object.

Examples

>>> from ler.lens_galaxy_population import OpticalDepth
>>> od = OpticalDepth(lens_param_samplers=dict(axis_ratio="axis_ratio_rayleigh"))
>>> q = od.axis_ratio(size=100, sigma=np.ones(100)*200.)

axis_ratio_padilla_strauss(size=1000, get_attribute=False, **kwargs)

Sample axis ratio from Padilla & Strauss (2008) distribution.

Parameters:

sizeint

Number of samples to generate.

default: 1000

get_attributebool

If True, returns the sampler object instead of samples.

default: False

**kwargsdict

Additional parameters (q_min, q_max).

Returns:

qnumpy.ndarray or FunctionConditioning

Axis ratio samples or sampler object.

Examples

>>> from ler.lens_galaxy_population import OpticalDepth
>>> od = OpticalDepth(lens_param_samplers=dict(axis_ratio="axis_ratio_padilla_strauss"))
>>> q = od.axis_ratio(size=100)

axis_rotation_angle_uniform(size, get_attribute=False, **kwargs)

Sample axis rotation angle from uniform distribution.

Parameters:

sizeint

Number of samples to generate.

get_attributebool

If True, returns the sampler object instead of samples.

default: False

**kwargsdict

Additional parameters (phi_min, phi_max).

Returns:

phinumpy.ndarray or FunctionConditioning

Axis rotation angle samples (rad) or sampler object.

Examples

>>> from ler.lens_galaxy_population import OpticalDepth
>>> od = OpticalDepth()
>>> phi = od.axis_rotation_angle(size=100)

axis_ratio_uniform(size, get_attribute=False, **kwargs)

Sample axis ratio from uniform distribution.

Parameters:

sizeint

Number of samples to generate.

get_attributebool

If True, returns the sampler object instead of samples.

default: False

**kwargsdict

Additional parameters (q_min, q_max).

Returns:

qnumpy.ndarray or FunctionConditioning

Axis ratio samples or sampler object.

Examples

>>> from ler.lens_galaxy_population import OpticalDepth
>>> od = OpticalDepth(lens_param_samplers=dict(axis_ratio="axis_ratio_uniform"))
>>> q = od.axis_ratio(size=100)

external_shear_normal(size, get_attribute=False, **kwargs)

Sample external shear parameters from 2D normal distribution.

Parameters:

sizeint

Number of samples to generate.

get_attributebool

If True, returns the sampler object instead of samples.

default: False

**kwargsdict

Additional parameters (mean, std).

Returns:

shearnumpy.ndarray or FunctionConditioning

Array of shape (2, size) with gamma1, gamma2 or sampler object.

Examples

>>> from ler.lens_galaxy_population import OpticalDepth
>>> od = OpticalDepth()
>>> gamma1, gamma2 = od.external_shear(size=100)

density_profile_slope_normal(size, get_attribute=False, **kwargs)

Sample density profile slope from normal distribution.

Parameters:

sizeint

Number of samples to generate.

get_attributebool

If True, returns the sampler object instead of samples.

default: False

**kwargsdict

Additional parameters (mean, std).

Returns:

gammanumpy.ndarray or FunctionConditioning

Density profile slope samples or sampler object.

Examples

>>> from ler.lens_galaxy_population import OpticalDepth
>>> od = OpticalDepth()
>>> gamma = od.density_profile_slope(size=100)

optical_depth_numerical(zs, get_attribute=False, **kwargs)

Helper to compute optical depth numerically by integrating lens redshift.

Parameters:

zsnumpy.ndarray

Source redshifts.

get_attributebool

If True, returns the function object instead of values. n default: False

**kwargsdict

Additional parameters.

Returns:

taunumpy.ndarray or FunctionConditioning

Optical depth values or function object.

compute_einstein_radii(sigma, zl, zs)

Function to compute the Einstein radii of the lens galaxies

Parameters:

sigmafloat

velocity dispersion of the lens galaxy

zlfloat

lens redshifts

zsfloat

source redshifts

Returns:

theta_Efloat

Einstein radii of the lens galaxies in radians. Multiply by

Examples

>>> from ler.lens_galaxy_population import LensGalaxyParameterDistribution
>>> lens = LensGalaxyParameterDistribution()
>>> sigma = 200.0
>>> zl = 0.5
>>> zs = 1.0
>>> lens.compute_einstein_radii(sigma, zl, zs)

optical_depth_sis_analytic(zs, get_attribute=False, **kwargs)

Function to compute the strong lensing optical depth (SIS). LambdaCDM(H0=70, Om0=0.3, Ode0=0.7) was used to derive the following equation. This is the analytic version of optical depth from z=0 to z=zs.

Parameters:

zsnumpy.ndarray

Source redshifts.

get_attributebool

If True, returns the function object instead of values. n default: False

**kwargsdict

Additional parameters.

Returns:

taunumpy.ndarray or FunctionConditioning

Optical depth values or function object.

cross_section_sis(zs=None, zl=None, sigma=None, get_attribute=False, **kwargs)

Function to compute the SIS cross-section

Parameters:

sigmafloat

velocity dispersion of the lens galaxy

zlfloat

redshift of the lens galaxy

zsfloat

redshift of the source galaxy

Returns:

cross_sectionfloat

SIS cross-section

Examples

>>> from ler.lens_galaxy_population import OpticalDepth
>>> od = OpticalDepth()
>>> print(self.cross_section_sis(sigma=200., zl=0.5, zs=1.0))

cross_section_sie_feixu(zs=None, zl=None, sigma=None, q=None, get_attribute=False, **kwargs)

Function to compute the SIE cross-section from Fei Xu et al. (2021)

Parameters:

sigmafloat

velocity dispersion of the lens galaxy

zlfloat

redshift of the lens galaxy

zsfloat

redshift of the source galaxy

Returns:

cross_sectionfloat

SIE cross-section

Examples

>>> from ler.lens_galaxy_population import OpticalDepth
>>> od = OpticalDepth()
>>> print(self.cross_section_sie_feixu(sigma=200., zl=0.5, zs=1.0, q=1.0))

cross_section_epl_shear_numerical(zs, zl, sigma, q, phi, gamma, gamma1, gamma2)

Function to compute the strong lensing cross-section numerically for EPL + external shear lenses.

Parameters:

zsnumpy.ndarray

redshift of the source galaxies

zlnumpy.ndarray

redshift of the lens galaxies

sigmanumpy.ndarray

velocity dispersion of the lens galaxies

qnumpy.ndarray

axis ratios of the lens galaxies

phinumpy.ndarray

axis rotation angles of the lens galaxies in radians

gammanumpy.ndarray

external shear magnitudes of the lens galaxies

gamma1numpy.ndarray

external shear component 1 of the lens galaxies

gamma2numpy.ndarray

external shear component 2 of the lens galaxies

Returns:

cross_sectionnumpy.ndarray

strong lensing cross-section of the lens galaxies in square radians

cross_section_epl_shear_numerical_mp(theta_E, gamma, gamma1, gamma2, q=None, phi=None, e1=None, e2=None, verbose=False, **kwargs)

Function to compute the strong lensing cross-section numerically for EPL + external shear lenses.

Parameters:

theta_Enumpy.ndarray

Einstein radii of the lens galaxies in radians

gammanumpy.ndarray

external shear magnitudes of the lens galaxies

gamma1numpy.ndarray

external shear component 1 of the lens galaxies

gamma2numpy.ndarray

external shear component 2 of the lens galaxies

qnumpy.ndarray

axis ratios of the lens galaxies

phinumpy.ndarray

axis rotation angles of the lens galaxies in radians

e1numpy.ndarray

ellipticity component 1 of the lens galaxies

e2numpy.ndarray

ellipticity component 2 of the lens galaxies

Returns:

cross_sectionnumpy.ndarray

strong lensing cross-section of the lens galaxies in square radians

create_parameter_grid(size_list=[25, 25, 45, 15, 15])

Create a parameter grid for lens galaxies.

Parameters:

size_listlist

List of sizes for each parameter grid.

Returns:

zlnumpy.ndarray

Lens redshifts.

sigmanumpy.ndarray

Velocity dispersions.

qnumpy.ndarray

Axis ratios.

cross_section_epl_shear_interpolation_init(file_path, size_list)

cross_section_epl_shear_interpolation(zs, zl, sigma, q, phi, gamma, gamma1, gamma2, get_attribute=False, size_list=[25, 25, 45, 15, 15], **kwargs)

Function to compute the cross-section correction factor