def calc_pdf(luminosity,
z,
alpha,
a_m,
b_m,
absolute_magnitude_max=-16.0,
absolute_magnitude_min=-28.0):
luminosity_min = astro.luminosity_from_absolute_magnitude(
absolute_magnitude_max)
luminosity_max = astro.luminosity_from_absolute_magnitude(
absolute_magnitude_min)
lum_star = lum._calculate_luminosity_star(z, a_m, b_m)
c = np.fabs(special.gammaincc(alpha + 1, luminosity_min / lum_star))
d = np.fabs(special.gammaincc(alpha + 1, luminosity_max / lum_star))
return 1. / lum_star * np.power(
luminosity / lum_star,
alpha) * np.exp(-luminosity / lum_star -
scipy.special.gammaln(alpha + 1)) / (c - d)
def calc_pdf(luminosity,
z,
alpha,
a_m,
b_m,
absolute_magnitude_max=-16.0,
absolute_magnitude_min=-28.0):
luminosity_min = astro.luminosity_from_absolute_magnitude(
absolute_magnitude_max)
luminosity_max = astro.luminosity_from_absolute_magnitude(
absolute_magnitude_min)
lum_star = lum._calculate_luminosity_star(z, a_m, b_m)
c = special.upper_incomplete_gamma(alpha + 1,
luminosity_min / lum_star)
d = special.upper_incomplete_gamma(alpha + 1,
luminosity_max / lum_star)
return 1. / lum_star * np.power(luminosity / lum_star, alpha) * np.exp(
-luminosity / lum_star) / (c - d)
def herbel_pdf(redshift, alpha, a_phi, a_m, b_phi, b_m, cosmology,
luminosity_min):
"""Calculates the redshift pdf of the Schechter luminosity function
according to the model of Herbel et al. (2017) equation (3.6). That is,
changing the absolute magnitude M in equation (3.2) to luminosity L,
integrate over all possible L and multiplying by the comovin element
using a flat LamdaCDM model to get the corresponding pdf.
Parameters
----------
redshift : array_like
Input redshifts.
alpha : float or scalar
The alpha parameter in the Schechter luminosity function
a_phi : float or scalar
Parametrisation factor of the normalisation factor Phi_* as a function
of redshift according to Herbel et al. (2017) equation (3.4).
a_m : float or scalar
Parametrisation factor of the characteristic absolute magnitude M_* as
a function of redshift according to Herbel et al. (2017) equation (3.3)
b_phi : float or scalar
Parametrisation factor of the normalisation factor Phi_* as a function
of redshift according to Herbel et al. (2017) equation (3.4).
b_m : float or scalar
Parametrisation factor of the characteristic absolute magnitude M_* as
a function of redshift according to Herbel et al. (2017) equation
(3.3)
cosmology : instance
Instance of an Astropy Cosmology class.
luminosity_min : float or scalar
Cut-off luminosity value such that the Schechter luminosity function
diverges for L -> 0
Returns
-------
ndarray or float
Un-normalised probability density function as a function of redshift
according to Herbel et al. (2017)
"""
dv = cosmology.differential_comoving_volume(redshift).value
x = luminosity_min \
* 1./astro.luminosity_from_absolute_magnitude(a_m*redshift + b_m)
value_gamma = special.upper_incomplete_gamma(alpha + 1, x)
return dv * b_phi * np.exp(a_phi * redshift) * value_gamma
def herbel_redshift(alpha, a_phi, a_m, b_phi, b_m, cosmology, low=0.0,
high=2.0,
size=None, absolute_magnitude_max=-16.0, resolution=100):
r""" Redshift following the Schechter luminosity function marginalised over
luminosities following the Herbel et al. (2017) model.
Parameters
----------
alpha : float or scalar
The alpha parameter in the Schechter luminosity function
a_phi : float or scalar
Parametrisation factor of the normalisation factor Phi_* as a function
of redshift according to Herbel et al. (2017) equation (3.4).
a_m : float or scalar
Parametrisation factor of the characteristic absolute magnitude M_* as
a function of redshift according to Herbel et al. (2017) equation (3.3)
b_phi : float or scalar
Parametrisation factor of the normalisation factor Phi_* as a function
of redshift according to Herbel et al. (2017) equation (3.4).
b_m : float or scalar
Parametrisation factor of the characteristic absolute magnitude M_* as
a function of redshift according to Herbel et al. (2017) equation (3.3)
cosmology : instance
Instance of an Astropy Cosmology class.
low : float or array_like of floats, optional
The lower boundary of teh output interval. All values generated will be
greater than or equal to low.
It has to be larger than 0. The default value is 0.01.
high : float or array_like of floats, optional
Upper boundary of output interval. All values generated will be less
than high. The default value is 2.0
size : int or tuple of ints, optional
The number of redshifts to sample. If None one value is sampled.
absolute_magnitude_max : float or scalar, optional
Upper limit of the considered absolute magnitudes of the galaxies to
wanna sample.
resolution : int, optional
Characterises the resolution of the sampling. Default is 100.
Returns
-------
redshift_sample : ndarray or float
Drawn redshifts from the marginalised Schechter luminosity function.
Notes
-----
The Schechter luminosity function is given as
.. math::
\Phi(L, z) = \frac{\Phi_\star(z)}{L_\star(z)}
\left(\frac{L}{L_\star(z)}\right)^\alpha
/exp\left(-\frac{L}{L_\star(z)}\right) \;.
Here the luminosity is defined as
.. math::
L = 10^{-0.4M} \;,
with absolute magnitude :math:`M`. Furthermore, Herbel et al. (2017)
introduced
.. math::
\Phi_\star(z) = b_\phi \exp(a_\phi z) \;,
M_\star(z) = a_M z + b_M \;.
Now we have to rescale the Schechter function by the comoving element and
get
.. math::
\phi(L,z) = \frac{d_H d_M^2}{E(z)} \Phi(L,z)
Examples
--------
>>> from skypy.galaxy.redshift import herbel_redshift
>>> from astropy.cosmology import FlatLambdaCDM
Sample 100 redshift values from the Schechter luminosity function with
a_m = -0.9408582, b_m = -20.40492365, a_phi = -0.10268436,
b_phi = 0.00370253, alpha = -1.3.
>>> cosmology = FlatLambdaCDM(H0=70, Om0=0.3, Tcmb0=2.725)
>>> redshift = herbel_redshift(size=1000, low=0.01, alpha=-1.3,
... a_phi=-0.10268436,a_m=-0.9408582, b_phi=0.00370253,
... b_m=-20.40492365, cosmology=cosmology,
... absolute_magnitude_max=-16.)
References
----------
.. [1] Herbel J., Kacprzak T., Amara A. et al., 2017, Journal of Cosmology
and Astroparticle Physics, Issue 08, article id. 035 (2017)
"""
luminosity_min = astro.luminosity_from_absolute_magnitude(
absolute_magnitude_max)
redshift = np.linspace(low, high, resolution)
pdf = herbel_pdf(redshift, alpha, a_phi, a_m, b_phi, b_m, cosmology,
luminosity_min)
cdf = scipy.integrate.cumtrapz(pdf, redshift, initial=0)
cdf = cdf / cdf[-1]
u = np.random.uniform(size=size)
redshift_sample = np.interp(u, cdf, redshift)
return redshift_sample
def herbel_pdf(redshift, alpha, a_phi, b_phi, a_m, b_m, cosmology,
luminosity_min):
r"""Calculates the redshift pdf of the Schechter luminosity function
according to the model of Herbel et al. [1]_ equation (3.6).
That is, changing the absolute magnitude M in equation (3.2) to luminosity
L, integrate over all possible L and multiplying by the comoving element
using a flat :math:`\Lambda \mathrm{CDM}` model to get the corresponding
pdf.
Parameters
----------
redshift : array_like
Input redshifts.
alpha : float or scalar
The alpha parameter in the Schechter luminosity function
a_phi, b_phi : float or scalar
Parametrisation factors of the normalisation factor Phi_* as a function
of redshift according to Herbel et al. [1]_ equation (3.4).
a_m, b_m : float or scalar
Parametrisation factors of the characteristic absolute magnitude M_* as
a function of redshift according to Herbel et al. [1]_ equation (3.3).
cosmology : instance
Instance of an Astropy Cosmology class.
luminosity_min : float or scalar
Cut-off luminosity value such that the Schechter luminosity function
diverges for L -> 0
Returns
-------
pdf : ndarray or float
Un-normalised probability density function as a function of redshift
according to Herbel et al. [1]_.
Notes
-----
This module calculates the function
.. math::
\mathrm{pdf}(z) = \Phi_\star(z) \cdot \frac{d_H d_M^2}{E(z)} \cdot
\Gamma\left(\alpha + 1, \frac{L_\mathrm{min}}{L_\star(z)}\right)\:,
with :math:`\Phi_\star(z) = b_\phi \exp(a_\phi z)` and the second term the
comoving element.
References
----------
.. [1] Herbel J., Kacprzak T., Amara A. et al., 2017, Journal of Cosmology
and Astroparticle Physics, Issue 08, article id. 035 (2017)
Examples
--------
>>> from skypy.galaxy.redshift import herbel_pdf
>>> import skypy.utils.astronomy as astro
>>> from astropy.cosmology import FlatLambdaCDM
>>> import numpy as np
Calculate the pdf for 100 redshift values between 0 and 2 with
a_m = -0.9408582, b_m = -20.40492365, a_phi = -0.10268436,
b_phi = 0.00370253, alpha = -1.3 for a flat cosmology.
>>> cosmology = FlatLambdaCDM(H0=70, Om0=0.3, Tcmb0=2.725)
>>> redshift = np.linspace(0, 2, 100)
>>> luminosity_min = astro.luminosity_from_absolute_magnitude(-16.0)
>>> redshift = herbel_pdf(redshift=redshift, alpha=-1.3,
... a_phi=-0.10268436,a_m=-0.9408582, b_phi=0.00370253,
... b_m=-20.40492365, cosmology=cosmology,
... luminosity_min=luminosity_min)
"""
dv = cosmology.differential_comoving_volume(redshift).value
x = luminosity_min \
* 1./astro.luminosity_from_absolute_magnitude(a_m*redshift + b_m)
lg = sc.gammaln(alpha + 1)
gx = np.fabs(special.gammaincc(alpha + 1, x))
pdf = dv * b_phi * np.exp(a_phi * redshift + lg) * gx
return pdf
def _calculate_luminosity_star(redshift, a_m, b_m):
absolute_magnitude_star = a_m * redshift + b_m
return astro.luminosity_from_absolute_magnitude(absolute_magnitude_star)
def test_convert_abs_mag_to_lum():
assert round(astro.luminosity_from_absolute_magnitude(-22), 1) \
== 630957344.5
