ler.lens_galaxy_population.sampler_functions

Module for lens galaxy parameter sampling functions.

This module provides probability density functions (PDFs) and random variable samplers (RVS) for lens galaxy parameters including redshift, velocity dispersion, and axis ratio. It also includes rejection and importance sampling algorithms for sampling lens parameters weighted by gravitational lensing cross sections.

Key Components:

• Lens redshift samplers (SIS model from Haris et al. 2018)

• Velocity dispersion samplers (generalized gamma distribution)

• Axis ratio samplers (Rayleigh and Padilla-Strauss distributions)

• Rejection and importance sampling for cross-section weighted parameters

Copyright (C) 2026 Hemantakumar Phurailatpam. Distributed under MIT License.

Module Contents

Functions

lens_redshift_strongly_lensed_sis_analytical_pdf(zl, zs)	Compute lens redshift PDF for SIS model (Haris et al. 2018).
lens_redshift_strongly_lensed_sis_analytical_rvs(size, zs)	Sample lens redshifts for SIS model (Haris et al. 2018).
velocity_dispersion_ewoud_denisty_function(sigma, z[, ...])	Calculate the lens galaxy velocity dispersion function at redshift z (Oguri et al. (2018b) + Wempe et al. (2022)).
velocity_dispersion_bernardi_denisty_function(sigma, ...)	Calculate the local universe velocity dispersion function.
gengamma_function(sigma[, alpha, beta, phistar, sigmastar])	Compute unnormalized velocity dispersion function using generalized gamma.
gengamma_pdf(sigma[, sigma_min, sigma_max, alpha, ...])	Compute normalized velocity dispersion PDF using generalized gamma.
gengamma_rvs(size[, sigma_min, sigma_max, alpha, ...])	Sample velocity dispersions from generalized gamma distribution.
rayleigh_rvs(size, sigma[, q_min, q_max])	Sample axis ratios from velocity-dependent Rayleigh distribution.
rayleigh_pdf(q, sigma[, q_min, q_max])	Compute truncated Rayleigh PDF for axis ratio.
axis_ratio_padilla_strauss_rvs(size)	Sample axis ratios from Padilla & Strauss (2008) distribution.
axis_ratio_padilla_strauss_pdf(q)	Compute axis ratio PDF from Padilla & Strauss (2008).
rejection_sampler_full(size, zs_pdf, number_density, ...)	Joint rejection sample over (zs, zl, sigma, nuisance).
rejection_sampler_partial(size, zs, zl, ...[, ...])
importance_sampler_partial(size, zs, zl, ...[, ...])	Importance/resampling for lens nuisance parameters at fixed (zs, zl).
importance_sampler_full(size, zs_pdf, number_density, ...)	Full joint importance sample over (zs, zl, sigma, nuisance).
create_sampler(number_density, q_pdf, phi_pdf, ...[, ...])	Build a closure that runs one of the cross-section-weighted lens samplers.

Attributes

importance_sampler_partial_njit
importance_sampler_full_njit
rejection_sampler_partial_njit
rejection_sampler_full_njit

ler.lens_galaxy_population.sampler_functions.lens_redshift_strongly_lensed_sis_analytical_pdf(zl, zs, cosmo=LambdaCDM(H0=70, Om0=0.3, Ode0=0.7, Tcmb0=0.0, Neff=3.04, m_nu=None, Ob0=0.0))

Compute lens redshift PDF for SIS model (Haris et al. 2018).

Computes the probability density function for lens redshift between zl=0 and zl=zs using the analytical form from Haris et al. (2018) equation A7, based on the SIS (Singular Isothermal Sphere) lens model.

Parameters:

zlfloat

Redshift of the lens galaxy.

zsfloat

Redshift of the source.

cosmoastropy.cosmology

Cosmology object for distance calculations.

default: LambdaCDM(H0=70, Om0=0.3, Ode0=0.7, Tcmb0=0.0, Neff=3.04, m_nu=None, Ob0=0.0)

Returns:

pdffloat

Probability density at the given lens redshift.

Examples

>>> from astropy.cosmology import LambdaCDM
>>> cosmo = LambdaCDM(H0=70, Om0=0.3, Ode0=0.7, Tcmb0=0.0, Neff=3.04, m_nu=None, Ob0=0.0)
>>> pdf = lens_redshift_strongly_lensed_sis_analytical_pdf(zl=0.5, zs=1.0, cosmo=cosmo)
>>> print(f"PDF at zl=0.5: {pdf:.4f}")
PDF at zl=0.5: 1.8750

ler.lens_galaxy_population.sampler_functions.lens_redshift_strongly_lensed_sis_analytical_rvs(size, zs, z_min=0.001, z_max=10.0, cosmo=LambdaCDM(H0=70, Om0=0.3, Ode0=0.7, Tcmb0=0.0, Neff=3.04, m_nu=None, Ob0=0.0))

Sample lens redshifts for SIS model (Haris et al. 2018).

Uses inverse transform sampling with the analytical CDF of the Haris et al. (2018) lens redshift distribution to efficiently generate samples.

Parameters:

sizeint

Number of samples to draw.

zsfloat

Redshift of the source.

z_minfloat

Minimum redshift for interpolation grid.

default: 0.001

z_maxfloat

Maximum redshift for interpolation grid.

default: 10.0

cosmoastropy.cosmology

Cosmology object for distance calculations.

default: LambdaCDM(H0=70, Om0=0.3, Ode0=0.7, Tcmb0=0.0, Neff=3.04, m_nu=None, Ob0=0.0)

Returns:

zlnumpy.ndarray

Array of sampled lens redshifts with shape (size,).

Examples

>>> from astropy.cosmology import LambdaCDM
>>> cosmo = LambdaCDM(H0=70, Om0=0.3, Ode0=0.7, Tcmb0=0.0, Neff=3.04, m_nu=None, Ob0=0.0)
>>> zl_samples = lens_redshift_strongly_lensed_sis_analytical_rvs(
...     size=1000,
...     zs=1.5,
...     z_min=0.001,
...     z_max=10.0,
...     cosmo=cosmo,
... )
>>> print(f"Mean lens redshift: {zl_samples.mean():.2f}")
>>> print(f"Redshift range: [{zl_samples.min():.3f}, {zl_samples.max():.3f}]")

ler.lens_galaxy_population.sampler_functions.velocity_dispersion_ewoud_denisty_function(sigma, z, alpha=0.94, beta=1.85, phistar=0.02099, sigmastar=113.78)

Calculate the lens galaxy velocity dispersion function at redshift z (Oguri et al. (2018b) + Wempe et al. (2022)).

Parameters:

sigmanumpy.ndarray

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

zfloat

Redshift of the lens galaxy.

alphafloat

Shape parameter of the velocity dispersion function.

default: 0.94

betafloat

Slope parameter of the velocity dispersion function.

default: 1.85

phistarfloat

Normalization of the velocity dispersion function (Mpc^-3).

default: 2.099e-2

sigmastarfloat

Characteristic velocity dispersion (km/s).

default: 113.78

Returns:

resultnumpy.ndarray

Velocity dispersion function values.

Negative values are clipped to 0.

Notes

The function name contains the misspelling denisty for backward compatibility with existing serialized configuration dictionaries.

ler.lens_galaxy_population.sampler_functions.velocity_dispersion_bernardi_denisty_function(sigma, alpha, beta, phistar, sigmastar)

Calculate the local universe velocity dispersion function.

This implements the velocity dispersion function from Bernardi et al. (2010).

\[\phi(\sigma) = \phi_* \left(\frac{\sigma}{\sigma_*}\right)^\alpha \exp\left[-\left(\frac{\sigma}{\sigma_*}\right)^\beta\right] \frac{\beta}{\Gamma(\alpha/\beta)\sigma}.\]

Parameters:

sigmanumpy.ndarray

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

alphafloat

Shape parameter (alpha/beta is used in the gamma function).

For Oguri et al. (2018b): alpha=0.94

For Choi et al. (2008): alpha=2.32/2.67

betafloat

Slope parameter of the velocity dispersion function.

For Oguri et al. (2018b): beta=1.85

For Choi et al. (2008): beta=2.67

phistarfloat

Normalization of the velocity dispersion function (Mpc^-3).

For Oguri et al. (2018b): phistar=2.099e-2*(h/0.7)^3

For Choi et al. (2008): phistar=8.0e-3*h^3

sigmastarfloat

Characteristic velocity dispersion (km/s).

For Oguri et al. (2018b): sigmastar=113.78

For Choi et al. (2008): sigmastar=161.0

Returns:

philocnumpy.ndarray

Local velocity dispersion function values.

Notes

The function name contains the misspelling denisty for backward compatibility with existing serialized configuration dictionaries.

ler.lens_galaxy_population.sampler_functions.gengamma_function(sigma, alpha=0.94, beta=1.85, phistar=0.02099, sigmastar=113.78, **kwargs)

Compute unnormalized velocity dispersion function using generalized gamma.

Computes the galaxy velocity dispersion function (VDF) using the generalized gamma distribution formulation from Choi et al. (2007).

Parameters:

sigmafloat or numpy.ndarray

Velocity dispersion in km/s.

alphafloat

Power-law index governing the low-velocity slope.

default: 0.94

betafloat

Exponential parameter for high-velocity cutoff sharpness.

default: 1.85

phistarfloat

Normalization constant (comoving number density, Mpc^-3).

default: 8.0e-3

sigmastarfloat

Characteristic velocity scale in km/s.

default: 113.78

Returns:

densityfloat or numpy.ndarray

Unnormalized velocity dispersion function value.

Examples

>>> import numpy as np
>>> sigma = np.array([150.0, 200.0, 250.0])
>>> density = gengamma_function(sigma)
>>> print(f"Density at sigma=200 km/s: {density[1]:.6f}")

ler.lens_galaxy_population.sampler_functions.gengamma_pdf(sigma, sigma_min=100.0, sigma_max=400.0, alpha=0.94, beta=1.85, sigmastar=113.78)

Compute normalized velocity dispersion PDF using generalized gamma.

Computes the probability density function for velocity dispersion using the generalized gamma distribution, normalized over the specified range.

Parameters:

sigmafloat or numpy.ndarray

Velocity dispersion in km/s.

sigma_minfloat

Minimum velocity dispersion for normalization (km/s).

default: 100.0

sigma_maxfloat

Maximum velocity dispersion for normalization (km/s).

default: 400.0

alphafloat

Power-law index governing the low-velocity slope.

default: 0.94

betafloat

Exponential parameter for high-velocity cutoff sharpness.

default: 1.85

sigmastarfloat

Characteristic velocity scale in km/s.

default: 113.78

Returns:

pdffloat or numpy.ndarray

Normalized probability density at the given velocity dispersion.

Examples

>>> pdf = gengamma_pdf(
...     sigma=200.0,
...     sigma_min=100.0,
...     sigma_max=400.0,
... )
>>> print(f"PDF at sigma=200 km/s: {pdf:.6f}")

ler.lens_galaxy_population.sampler_functions.gengamma_rvs(size, sigma_min=100.0, sigma_max=400.0, alpha=0.94, beta=1.85, sigmastar=113.78)

Sample velocity dispersions from generalized gamma distribution.

Uses truncated generalized gamma sampling via rejection to generate velocity dispersion samples within the specified bounds.

Parameters:

sizeint

Number of samples to draw.

sigma_minfloat

Minimum velocity dispersion (km/s).

default: 100.0

sigma_maxfloat

Maximum velocity dispersion (km/s).

default: 400.0

alphafloat

Power-law index governing the low-velocity slope.

default: 0.94

betafloat

Exponential parameter for high-velocity cutoff sharpness.

default: 1.85

sigmastarfloat

Characteristic velocity scale in km/s.

default: 113.78

Returns:

sigmanumpy.ndarray

Sampled velocity dispersions in km/s with shape (size,).

Examples

>>> sigma_samples = gengamma_rvs(
...     size=1000,
...     sigma_min=100.0,
...     sigma_max=400.0,
... )
>>> print(f"Mean sigma: {sigma_samples.mean():.2f} km/s")
>>> print(f"Range: [{sigma_samples.min():.1f}, {sigma_samples.max():.1f}] km/s")

ler.lens_galaxy_population.sampler_functions.rayleigh_rvs(size, sigma, q_min=0.2, q_max=1.0)

Sample axis ratios from velocity-dependent Rayleigh distribution.

Generates axis ratio samples using the Rayleigh distribution with scale parameter dependent on velocity dispersion, as described in Wierda et al. (2021) Appendix C.

Parameters:

sizeint

Number of samples to draw.

sigmanumpy.ndarray

Velocity dispersions in km/s with shape (size,).

q_minfloat

Minimum allowed axis ratio.

default: 0.2

q_maxfloat

Maximum allowed axis ratio.

default: 1.0

Returns:

qnumpy.ndarray

Sampled axis ratios with shape (size,).

Examples

>>> import numpy as np
>>> sigma = np.random.uniform(100, 300, 1000)
>>> q_samples = rayleigh_rvs(size=1000, sigma=sigma, q_min=0.2, q_max=1.0)
>>> print(f"Mean axis ratio: {q_samples.mean():.2f}")
>>> print(f"Range: [{q_samples.min():.2f}, {q_samples.max():.2f}]")

ler.lens_galaxy_population.sampler_functions.rayleigh_pdf(q, sigma, q_min=0.2, q_max=1.0)

Compute truncated Rayleigh PDF for axis ratio.

Computes the probability density function for axis ratio using the truncated Rayleigh distribution with velocity-dependent scale parameter (Wierda et al. 2021 equation C16).

Parameters:

qnumpy.ndarray

Axis ratios at which to evaluate PDF.

sigmanumpy.ndarray

Velocity dispersions in km/s (same shape as q).

q_minfloat

Minimum axis ratio for truncation.

default: 0.2

q_maxfloat

Maximum axis ratio for truncation.

default: 1.0

Returns:

pdfnumpy.ndarray

Probability density values with same shape as q.

Examples

>>> import numpy as np
>>> q = np.array([0.5, 0.7, 0.9])
>>> sigma = np.array([150.0, 200.0, 250.0])
>>> pdf = rayleigh_pdf(q, sigma)
>>> print(f"PDF values: {pdf}")

ler.lens_galaxy_population.sampler_functions.axis_ratio_padilla_strauss_rvs(size)

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

Uses inverse transform sampling with the empirical PDF from Padilla & Strauss (2008) for early-type galaxy axis ratios.

Parameters:

sizeint

Number of samples to draw.

Returns:

qnumpy.ndarray

Sampled axis ratios with shape (size,).

Examples

>>> q_samples = axis_ratio_padilla_strauss_rvs(size=1000)
>>> print(f"Mean axis ratio: {q_samples.mean():.2f}")
>>> print(f"Range: [{q_samples.min():.2f}, {q_samples.max():.2f}]")

ler.lens_galaxy_population.sampler_functions.axis_ratio_padilla_strauss_pdf(q)

Compute axis ratio PDF from Padilla & Strauss (2008).

Evaluates the probability density function for axis ratio using cubic spline interpolation of the Padilla & Strauss (2008) data.

Parameters:

qnumpy.ndarray

Axis ratios at which to evaluate PDF.

Returns:

pdfnumpy.ndarray

Probability density values with same shape as q.

Examples

>>> import numpy as np
>>> q = np.array([0.3, 0.5, 0.7, 0.9])
>>> pdf = axis_ratio_padilla_strauss_pdf(q)
>>> print(f"PDF at q=0.5: {pdf[1]:.4f}")

ler.lens_galaxy_population.sampler_functions.rejection_sampler_full(size, zs_pdf, number_density, q_pdf, phi_pdf, gamma_pdf, shear1_pdf, shear2_pdf, cross_section, dVdz, ranges, threshold_factor=0.0001, n_prop=1000, lens_type='epl_shear_galaxy', npool=4)

Joint rejection sample over (zs, zl, sigma, nuisance).

Intrinsic target (before cross section) uses p(zs) * dV/dz_l * n(sigma,zl) * ...; z_l is proposed uniform in [z_min, z_s) — no separate p(zl|zs) factor (see importance_sampler_full docstring).

ler.lens_galaxy_population.sampler_functions.rejection_sampler_partial(size, zs, zl, number_density, q_pdf, phi_pdf, gamma_pdf, shear1_pdf, shear2_pdf, cross_section, ranges, threshold_factor=0.0001, n_prop=1000, lens_type='epl_shear_galaxy', npool=4)

ler.lens_galaxy_population.sampler_functions.importance_sampler_partial(size, zs, zl, number_density, q_pdf, phi_pdf, gamma_pdf, shear1_pdf, shear2_pdf, cross_section, ranges, threshold_factor=0.0001, n_prop=100, lens_type='epl_shear_galaxy')

Importance/resampling for lens nuisance parameters at fixed (zs, zl).

Redshifts are fixed (typically drawn from zs_sl and lens_redshift_sl). Conditional weights target p(sigma,lambda|zs,zl,SL) up to a constant: proportional to n(sigma,zl), axis-ratio / shear / slope PDFs, and the strong-lensing cross section. Terms that depend only on (zs,zl) --- e.g. P(zs|SL), P(zl|zs,SL), dV/dz_l --- are omitted (constant for fixed redshifts and identical scale across sigma proposals).

ler.lens_galaxy_population.sampler_functions.importance_sampler_full(size, zs_pdf, number_density, q_pdf, phi_pdf, gamma_pdf, shear1_pdf, shear2_pdf, cross_section, dVdz, ranges, threshold_factor=0.0001, n_prop=50, lens_type='epl_shear_galaxy')

Full joint importance sample over (zs, zl, sigma, nuisance).

Target for lens+source intrinsic factors (before cross section) is taken as

p(zs) * (dV_c/dz_l) * n(sigma, zl) * (axis ratio, shear, slope PDFs),

i.e. no separate p(zl|zs) factor: z_l is proposed uniformly in [z_min, z_s) and the comoving volume and Schechter-style n(sigma,zl) carry the lens-redshift abundance. This matches P(SL|...) p(zs) n(sigma,zl|...) dV/dz_l ... up to overall normalisation.

ler.lens_galaxy_population.sampler_functions.importance_sampler_partial_njit

ler.lens_galaxy_population.sampler_functions.importance_sampler_full_njit

ler.lens_galaxy_population.sampler_functions.rejection_sampler_partial_njit

ler.lens_galaxy_population.sampler_functions.rejection_sampler_full_njit

ler.lens_galaxy_population.sampler_functions.create_sampler(number_density, q_pdf, phi_pdf, gamma_pdf, shear1_pdf, shear2_pdf, cross_section, dVdz, zs_pdf=None, use_njit_sampler=True, sampler_type='importance_sampler_partial', threshold_factor=0.0001, n_prop=50, lens_type='epl_shear_galaxy', npool=4, **range_kwargs)

Build a closure that runs one of the cross-section-weighted lens samplers.

use_njit_sampler selects the implementation, not only a print message:

• True (default): Numba njit kernels. cross_section and other callables must be Numba-compatible (e.g. @njit cross sections).

• False: interpreted Python samplers, which accept ordinary callables (e.g. numerical / SciPy-backed cross sections).