Distributions

distributions ¶

Core functions for distributions module.

Functions:¶

broken_power_law_log_normalization_constant ¶

broken_power_law_log_normalization_constant(x_min: float, x_max: float, transition: float, alpha_1: float, alpha_2: float) -> jax.Array

Log normalization constant of a broken power law distribution.

References: - GWTC-4.0: Population Properties of Merging Compact Binaries: https://arxiv.org/abs/2508.18083

Parameters:

Name	Type	Description	Default
`x_min`	`float`	The minimum value of the distribution.	required
`x_max`	`float`	The maximum value of the distribution.	required
`transition`	`float`	The transition point where the power law changes.	required
`alpha_1`	`float`	The power law index for x < transition.	required
`alpha_2`	`float`	The power law index for x >= transition.	required

Returns:

Type	Description
`Array`	The log normalization constant of the distribution.

broken_power_law_logpdf ¶

broken_power_law_logpdf(x: Array, x_min: float, x_max: float, transition: float, alpha_1: float, alpha_2: float) -> jax.Array

Log probability density function of a broken power law distribution.

References: - GWTC-4.0: Population Properties of Merging Compact Binaries: https://arxiv.org/abs/2508.18083

Parameters:

Name	Type	Description	Default
`x`	`Array`	The input value.	required
`x_min`	`float`	The minimum value of the distribution.	required
`x_max`	`float`	The maximum value of the distribution.	required
`transition`	`float`	The transition point where the power law changes.	required
`alpha_1`	`float`	The power law index for x < transition.	required
`alpha_2`	`float`	The power law index for x >= transition.	required

Returns:

Type	Description
`Array`	An array with the log probability density at x.

broken_power_law_unnormalized_logpdf ¶

broken_power_law_unnormalized_logpdf(x: Array, x_min: float, x_max: float, transition: float, alpha_1: float, alpha_2: float) -> jax.Array

Unnormalized log probability density function of a broken power law distribution.

References: - GWTC-4.0: Population Properties of Merging Compact Binaries: https://arxiv.org/abs/2508.18083

Parameters:

Name	Type	Description	Default
`x`	`Array`	The input value.	required
`x_min`	`float`	The minimum value of the distribution.	required
`x_max`	`float`	The maximum value of the distribution.	required
`transition`	`float`	The transition point where the power law changes.	required
`alpha_1`	`float`	The power law index for x < transition.	required
`alpha_2`	`float`	The power law index for x >= transition.	required

Returns:

Type	Description
`Array`	An array with the unnormalized log probability density at x.

madau_dickinson_rate ¶

madau_dickinson_rate(redshift: Array, gamma: float = MADAU_DICKINSON_GAMMA, kappa: float = MADAU_DICKINSON_KAPPA, z_peak: float = MADAU_DICKINSON_Z_PEAK) -> Array

Return the dimensionless Madau-like rate shape psi(z) normalised to psi(0) = 1.

Parameters:

Name	Type	Description	Default
`redshift`	`Array`	Redshift(s) at which to evaluate the rate shape.	required
`gamma`	`float`	Low-redshift power-law slope.	`MADAU_DICKINSON_GAMMA`
`kappa`	`float`	High-redshift fall-off; the denominator exponent is `gamma + kappa`.	`MADAU_DICKINSON_KAPPA`
`z_peak`	`float`	Redshift of the rate peak (the turnover scale `1 + z_peak`).	`MADAU_DICKINSON_Z_PEAK`

Returns:

Type	Description
`Array`	The rate shape `psi(z)`; multiply by a local rate `R_0` for the
`Array`	absolute source-frame merger-rate density.

madau_dickinson_redshift_pdf ¶

madau_dickinson_redshift_pdf(redshift: Array, *, z_max: float, z_min: float = 0.0, gamma: float = MADAU_DICKINSON_GAMMA, kappa: float = MADAU_DICKINSON_KAPPA, z_peak: float = MADAU_DICKINSON_Z_PEAK, hubble_constant: float = PLANCK18_H0_KM_S_MPC, omega_m: float = PLANCK18_OMEGA_M, n_grid: int = DEFAULT_LOOKUP_GRID_SIZE) -> Array

Return the normalised source redshift density on [z_min, z_max].

The density is p(z) proportional to psi(z) / (1 + z) * dVc/dz, normalised to integrate to one over [z_min, z_max] and zero outside it.

Parameters:

Name	Type	Description	Default
`redshift`	`Array`	Redshift(s) at which to evaluate the density.	required
`z_max`	`float`	Upper redshift bound of the population.	required
`z_min`	`float`	Lower redshift bound of the population.	`0.0`
`gamma`	`float`	Low-redshift power-law slope.	`MADAU_DICKINSON_GAMMA`
`kappa`	`float`	High-redshift fall-off exponent offset.	`MADAU_DICKINSON_KAPPA`
`z_peak`	`float`	Redshift of the rate peak.	`MADAU_DICKINSON_Z_PEAK`
`hubble_constant`	`float`	Hubble constant in km / s / Mpc.	`PLANCK18_H0_KM_S_MPC`
`omega_m`	`float`	Matter density.	`PLANCK18_OMEGA_M`
`n_grid`	`int`	Number of grid points used to normalise the density.	`DEFAULT_LOOKUP_GRID_SIZE`

Returns:

Type	Description
`Array`	Normalised probability density evaluated at `redshift`.

mass_ratio_pairing_cdf ¶

mass_ratio_pairing_cdf(mass_ratio: Array, primary_mass: Array, beta: float, secondary_minimum: float, minimum: float = 0.0) -> Array

Return the normalized conditional pairing CDF P(q' <= q | m1).

mass_ratio_pairing_pdf ¶

mass_ratio_pairing_pdf(mass_ratio: Array, primary_mass: Array, beta: float, secondary_minimum: float, minimum: float = 0.0) -> Array

Return the normalized conditional pairing density p(q | m1).

power_law_plus_peak_cdf ¶

power_law_plus_peak_cdf(mass: Array, alpha: float, minimum: float, maximum: float, lambda_peak: float, peak_mean: float, peak_sigma: float, peak_maximum: float) -> Array

Return the normalized Talbot-Thrane-style power-law-plus-peak CDF.

power_law_plus_peak_pdf ¶

power_law_plus_peak_pdf(mass: Array, alpha: float, minimum: float, maximum: float, lambda_peak: float, peak_mean: float, peak_sigma: float, peak_maximum: float) -> Array

Return the normalized Talbot-Thrane-style power-law-plus-peak density.

planck_tapering_function ¶

planck_tapering_function(x: Array, x_min: float, delta: float) -> Array

Planck tapering function to smoothly transition between 0 and 1.

References: - GWTC-4.0: Population Properties of Merging Compact Binaries: https://arxiv.org/abs/2508.18083

Parameters:

Name	Type	Description	Default
`x`	`Array`	The input value.	required
`x_min`	`float`	The minimum value of the transition region.	required
`delta`	`float`	The width of the transition region.	required

Returns:

Type	Description
`Array`	An array with the value of the tapering function at x.