# Licensed under a 3-clause BSD style license - see LICENSE.rst """ Convenience functions for `astropy.cosmology`. """ from __future__ import (absolute_import, division, print_function, unicode_literals) import warnings import numpy as np from .core import default_cosmology as _default_cosmology from .core import CosmologyError from ..units import Quantity from ..utils import deprecated __all__ = ['H', 'angular_diameter_distance', 'arcsec_per_kpc_comoving', 'arcsec_per_kpc_proper', 'comoving_distance', 'critical_density', 'distmod', 'kpc_comoving_per_arcmin', 'kpc_proper_per_arcmin', 'lookback_time', 'luminosity_distance', 'scale_factor', 'z_at_value'] __doctest_requires__ = {'*': ['scipy.integrate']} def z_at_value(func, fval, zmin=0, zmax=1000, ztol=1e-5, maxfun=500): """ Find the redshift ``z`` at which ``func(z) = fval``. This finds the redshift at which one of the cosmology functions or methods (for example Planck13.distmod) is equal to a known value. .. warning:: Make sure you understand the behaviour of the function that you are trying to invert! Depending on the cosmology, there may not be a unique solution. For example, in the standard Lambda CDM cosmology, there are two redshifts which give an angular diameter distance of 1500 Mpc, z ~ 0.7 and z ~ 3.8. To force ``z_at_value`` to find the solution you are interested in, use the ``zmin`` and ``zmax`` keywords to limit the search range (see the example below). Parameters ---------- func : function or method A function that takes a redshift as input. fval : astropy.Quantity instance The value of ``func(z)``. zmin : float, optional The lower search limit for ``z`` (default 0). zmax : float, optional The upper search limit for ``z`` (default 1000). ztol : float, optional The relative error in ``z`` acceptable for convergence. maxfun : int, optional The maximum number of function evaluations allowed in the optimization routine (default 500). Returns ------- z : float The redshift ``z`` satisfying ``zmin < z < zmax`` and ``func(z) = fval`` within ``ztol``. Notes ----- This works for any arbitrary input cosmology, but is inefficient if you want to invert a large number of values for the same cosmology. In this case, it is faster to instead generate an array of values at many closely-spaced redshifts that cover the relevant redshift range, and then use interpolation to find the redshift at each value you're interested in. For example, to efficiently find the redshifts corresponding to 10^6 values of the distance modulus in a Planck13 cosmology, you could do the following: >>> import astropy.units as u >>> from astropy.cosmology import Planck13, z_at_value Generate 10^6 distance moduli between 23 and 43 for which we want to find the corresponding redshifts: >>> Dvals = (23 + np.random.rand(1e6) * 20) * u.mag Make a grid of distance moduli covering the redshift range we need using 50 equally log-spaced values between zmin and zmax. We use log spacing to adequately sample the steep part of the curve at low distance moduli: >>> zmin = z_at_value(Planck13.distmod, Dvals.min()) >>> zmax = z_at_value(Planck13.distmod, Dvals.max()) >>> zgrid = np.logspace(zmin, zmax) >>> Dgrid = Planck13.distmod(zgrid) Finally interpolate to find the redshift at each distance modulus: >>> zvals = np.interp(Dvals.value, zgrid, Dgrid.value) Examples -------- >>> import astropy.units as u >>> from astropy.cosmology import Planck13, z_at_value The age and lookback time are monotonic with redshift, and so a unique solution can be found: >>> z_at_value(Planck13.age, 2 * u.Gyr) 3.1981191749374629 The angular diameter is not monotonic however, and there are two redshifts that give a value of 1500 Mpc. Use the zmin and zmax keywords to find the one you're interested in: >>> z_at_value(Planck13.angular_diameter_distance, 1500 * u.Mpc, zmax=1.5) 0.68127769625288614 >>> z_at_value(Planck13.angular_diameter_distance, 1500 * u.Mpc, zmin=2.5) 3.7914918534022011 Also note that the luminosity distance and distance modulus (two other commonly inverted quantities) are monotonic in flat and open universes, but not in closed universes. """ from scipy.optimize import fminbound fval_zmin = func(zmin) fval_zmax = func(zmax) if np.sign(fval - fval_zmin) != np.sign(fval_zmax - fval): warnings.warn("""\ fval is not bracketed by func(zmin) and func(zmax). This means either there is no solution, or that there is more than one solution between zmin and zmax satisfying fval = func(z).""") if isinstance(fval_zmin, Quantity): unit = fval_zmin.unit val = fval.to(unit).value f = lambda z: abs(func(z).value - val) else: f = lambda z: abs(func(z) - fval) zbest, resval, ierr, ncall = fminbound(f, zmin, zmax, maxfun=maxfun, full_output=1) if ierr != 0: warnings.warn('Maximum number of function calls ({}) reached'.format( ncall)) if np.allclose(zbest, zmax): raise CosmologyError("Best guess z is very close the upper z limit.\n" "Try re-running with a different zmax.") elif np.allclose(zbest, zmin): raise CosmologyError("Best guess z is very close the lower z limit.\n" "Try re-running with a different zmin.") return zbest @deprecated(since='0.4', alternative='.kpc_comoving_per_arcmin') def kpc_comoving_per_arcmin(z, cosmo=None): """ Separation in transverse comoving kpc corresponding to an arcminute at redshift ``z``. Parameters ---------- z : array_like Input redshifts. Returns ------- d : `~astropy.units.Quantity` The distance in comoving kpc corresponding to an arcmin at each input redshift. """ if cosmo is None: cosmo = _default_cosmology.get() return cosmo.kpc_comoving_per_arcmin(z) @deprecated(since='0.4', alternative='.kpc_proper_per_arcmin') def kpc_proper_per_arcmin(z, cosmo=None): """ Separation in transverse proper kpc corresponding to an arcminute at redshift ``z``. Parameters ---------- z : array_like Input redshifts. Returns ------- d : `~astropy.units.Quantity` The distance in proper kpc corresponding to an arcmin at each input redshift. """ if cosmo is None: cosmo = _default_cosmology.get() return cosmo.kpc_proper_per_arcmin(z) @deprecated(since='0.4', alternative='.arcsec_per_kpc_comoving') def arcsec_per_kpc_comoving(z, cosmo=None): """ Angular separation in arcsec corresponding to a comoving kpc at redshift ``z``. Parameters ---------- z : array_like Input redshifts. Returns ------- theta : `~astropy.units.Quantity` The angular separation in arcsec corresponding to a comoving kpc at each input redshift. """ if cosmo is None: cosmo = _default_cosmology.get() return cosmo.arcsec_per_kpc_comoving(z) @deprecated(since='0.4', alternative='.arcsec_per_kpc_proper') def arcsec_per_kpc_proper(z, cosmo=None): """ Angular separation in arcsec corresponding to a proper kpc at redshift ``z``. Parameters ---------- z : array_like Input redshifts. Returns ------- theta : `~astropy.units.Quantity` The angular separation in arcsec corresponding to a proper kpc at each input redshift. """ if cosmo is None: cosmo = _default_cosmology.get() return cosmo.arcsec_per_kpc_proper(z) @deprecated(since='0.4', alternative='.distmod') def distmod(z, cosmo=None): """ Distance modulus at redshift ``z``. The distance modulus is defined as the (apparent magnitude - absolute magnitude) for an object at redshift ``z``. Parameters ---------- z : array_like Input redshifts. Returns ------- distmod : `~astropy.units.Quantity` Distance modulus at each input redshift. See Also -------- z_at_value : Find the redshift corresponding to a distance modulus. """ if cosmo is None: cosmo = _default_cosmology.get() return cosmo.distmod(z) @deprecated(since='0.4', alternative='.H') def H(z, cosmo=None): """ Hubble parameter (km/s/Mpc) at redshift ``z``. Parameters ---------- z : array_like Input redshifts. Returns ------- H : `~astropy.units.Quantity` Hubble parameter at each input redshift. """ if cosmo is None: cosmo = _default_cosmology.get() return cosmo.H(z) @deprecated(since='0.4', alternative='.scale_factor') def scale_factor(z, cosmo=None): """ Scale factor at redshift ``z``. The scale factor is defined as ``a = 1 / (1 + z)``. Parameters ---------- z : array_like Input redshifts. Returns ------- scalefac : ndarray, or float if input scalar Scale factor at each input redshift. """ if cosmo is None: cosmo = _default_cosmology.get() return cosmo.scale_factor(z) @deprecated(since='0.4', alternative='.critical_density') def critical_density(z, cosmo=None): """ Critical density in grams per cubic cm at redshift ``z``. Parameters ---------- z : array_like Input redshifts. Returns ------- critdens : `~astropy.units.Quantity` Critical density at each input redshift. """ if cosmo is None: cosmo = _default_cosmology.get() return cosmo.critical_density(z) @deprecated(since='0.4', alternative='.lookback_time') def lookback_time(z, cosmo=None): """ Lookback time in Gyr to redshift ``z``. The lookback time is the difference between the age of the Universe now and the age at redshift ``z``. Parameters ---------- z : array_like Input redshifts. Returns ------- t : `~astropy.units.Quantity` Lookback time at each input redshift. See Also -------- z_at_value : Find the redshift corresponding to a lookback time. """ if cosmo is None: cosmo = _default_cosmology.get() return cosmo.lookback_time(z) @deprecated(since='0.4', alternative='.comoving_distance') def comoving_distance(z, cosmo=None): """ Comoving distance in Mpc at redshift ``z``. The comoving distance along the line-of-sight between two objects remains constant with time for objects in the Hubble flow. Parameters ---------- z : array_like Input redshifts. Returns ------- codist : `~astropy.units.Quantity` Comoving distance at each input redshift. """ if cosmo is None: cosmo = _default_cosmology.get() return cosmo.comoving_distance(z) @deprecated(since='0.4', alternative='.angular_diameter_distance') def angular_diameter_distance(z, cosmo=None): """ Angular diameter distance in Mpc at a given redshift. This gives the proper (sometimes called 'physical') transverse distance corresponding to an angle of 1 radian for an object at redshift ``z``. Parameters ---------- z : array_like Input redshifts. Returns ------- angdist : `~astropy.units.Quantity` Angular diameter distance at each input redshift. """ if cosmo is None: cosmo = _default_cosmology.get() return cosmo.angular_diameter_distance(z) @deprecated(since='0.4', alternative='.luminosity_distance') def luminosity_distance(z, cosmo=None): """ Luminosity distance in Mpc at redshift ``z``. This is the distance to use when converting between the bolometric flux from an object at redshift ``z`` and its bolometric luminosity. Parameters ---------- z : array_like Input redshifts. Returns ------- lumdist : `~astropy.units.Quantity` Luminosity distance at each input redshift. See Also -------- z_at_value : Find the redshift corresponding to a luminosity distance. """ if cosmo is None: cosmo = _default_cosmology.get() return cosmo.luminosity_distance(z)