Madau-Dickinson

madau_dickinson ¶

Madau-Dickinson star-formation / merger-rate redshift model.

The rate shape follows the flexible "Madau-like" parametrization widely used in gravitational-wave population analysis (e.g. arXiv:2111.03634), which reduces to the Madau & Dickinson (2014, arXiv:1403.0007) cosmic star-formation history for the default parameters:

.. math::

\psi(z) = \mathcal{C}\,
    \frac{(1+z)^{\gamma}}{1 + \left(\frac{1+z}{1+z_p}\right)^{\gamma+\kappa}},
\qquad
\mathcal{C} = 1 + (1+z_p)^{-(\gamma+\kappa)},

so that psi(0) = 1 and the absolute source-frame merger-rate density is R(z) = R_0 * psi(z) for a local rate R_0. The redshift distribution of sources observed over a fixed detector-frame interval carries a single factor of 1 / (1 + z) (cosmological time dilation):

.. math::

p(z) \propto \frac{\psi(z)}{1+z}\,\frac{\mathrm{d}V_c}{\mathrm{d}z}.

Functions:¶

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`.