# Source code for astropy.cosmology.funcs

```
# Licensed under a 3-clause BSD style license - see LICENSE.rst
"""
Convenience functions for `astropy.cosmology`.
"""
import warnings
import numpy as np
from astropy.units import Quantity
from astropy.utils.exceptions import AstropyUserWarning
from . import units as cu
from .core import CosmologyError
__all__ = ['z_at_value']
__doctest_requires__ = {'*': ['scipy']}
def _z_at_scalar_value(func, fval, zmin=1e-8, zmax=1000, ztol=1e-8, maxfun=500,
method='Brent', bracket=None, verbose=False):
"""
Find the redshift ``z`` at which ``func(z) = fval``.
See :func:`astropy.cosmology.funcs.z_at_value`.
"""
from scipy.optimize import minimize_scalar
opt = {'maxiter': maxfun}
# Assume custom methods support the same options as default; otherwise user
# will see warnings.
if str(method).lower() == 'bounded':
opt['xatol'] = ztol
if bracket is not None:
warnings.warn(f"Option 'bracket' is ignored by method {method}.")
bracket = None
else:
opt['xtol'] = ztol
# fval falling inside the interval of bracketing function values does not
# guarantee it has a unique solution, but for Standard Cosmological
# quantities normally should (being monotonic or having a single extremum).
# In these cases keep solver from returning solutions outside of bracket.
fval_zmin, fval_zmax = func(zmin), func(zmax)
nobracket = False
if np.sign(fval - fval_zmin) != np.sign(fval_zmax - fval):
if bracket is None:
nobracket = True
else:
fval_brac = func(np.asanyarray(bracket))
if np.sign(fval - fval_brac[0]) != np.sign(fval_brac[-1] - fval):
nobracket = True
else:
zmin, zmax = bracket[0], bracket[-1]
fval_zmin, fval_zmax = fval_brac[[0, -1]]
if nobracket:
warnings.warn(f"fval is not bracketed by func(zmin)={fval_zmin} and "
f"func(zmax)={fval_zmax}. This means either there is no "
"solution, or that there is more than one solution "
"between zmin and zmax satisfying fval = func(z).",
AstropyUserWarning)
if isinstance(fval_zmin, Quantity):
val = fval.to_value(fval_zmin.unit)
else:
val = fval
# 'Brent' and 'Golden' ignore `bounds`, force solution inside zlim
def f(z):
if z > zmax:
return 1.e300 * (1.0 + z - zmax)
elif z < zmin:
return 1.e300 * (1.0 + zmin - z)
elif isinstance(fval_zmin, Quantity):
return abs(func(z).value - val)
else:
return abs(func(z) - val)
res = minimize_scalar(f, method=method, bounds=(zmin, zmax),
bracket=bracket, options=opt)
# Scipy docs state that `OptimizeResult` always has 'status' and 'message'
# attributes, but only `_minimize_scalar_bounded()` seems to have really
# implemented them.
if not res.success:
warnings.warn(f"Solver returned {res.get('status')}: {res.get('message', 'Unsuccessful')}\n"
f"Precision {res.fun} reached after {res.nfev} function calls.",
AstropyUserWarning)
if verbose:
print(res)
if np.allclose(res.x, zmax):
raise CosmologyError(
f"Best guess z={res.x} is very close to the upper z limit {zmax}."
"\nTry re-running with a different zmax.")
elif np.allclose(res.x, zmin):
raise CosmologyError(
f"Best guess z={res.x} is very close to the lower z limit {zmin}."
"\nTry re-running with a different zmin.")
return res.x
[docs]def z_at_value(func, fval, zmin=1e-8, zmax=1000, ztol=1e-8, maxfun=500,
method='Brent', bracket=None, verbose=False):
"""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 behavior 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.units.Quantity`
The (scalar or array) value of ``func(z)`` to recover.
zmin : float or array-like['dimensionless'] or quantity-like, optional
The lower search limit for ``z``. Beware of divergences
in some cosmological functions, such as distance moduli,
at z=0 (default 1e-8).
zmax : float or array-like['dimensionless'] or quantity-like, optional
The upper search limit for ``z`` (default 1000).
ztol : float or array-like['dimensionless'], optional
The relative error in ``z`` acceptable for convergence.
maxfun : int or array-like, optional
The maximum number of function evaluations allowed in the
optimization routine (default 500).
method : str or callable, optional
Type of solver to pass to the minimizer. The built-in options provided
by :func:`~scipy.optimize.minimize_scalar` are 'Brent' (default),
'Golden' and 'Bounded' with names case insensitive - see documentation
there for details. It also accepts a custom solver by passing any
user-provided callable object that meets the requirements listed
therein under the Notes on "Custom minimizers" - or in more detail in
:doc:`scipy:tutorial/optimize` - although their use is currently
untested.
.. versionadded:: 4.3
bracket : sequence or object array[sequence], optional
For methods 'Brent' and 'Golden', ``bracket`` defines the bracketing
interval and can either have three items (z1, z2, z3) so that
z1 < z2 < z3 and ``func(z2) < func (z1), func(z3)`` or two items z1
and z3 which are assumed to be a starting interval for a downhill
bracket search. For non-monotonic functions such as angular diameter
distance this may be used to start the search on the desired side of
the maximum, but see Examples below for usage notes.
.. versionadded:: 4.3
verbose : bool, optional
Print diagnostic output from solver (default `False`).
.. versionadded:: 4.3
Returns
-------
z : `~astropy.units.Quantity` ['redshift']
The redshift ``z`` satisfying ``zmin < z < zmax`` and ``func(z) =
fval`` within ``ztol``. Has units of cosmological redshift.
Warns
-----
:class:`~astropy.utils.exceptions.AstropyUserWarning`
If ``fval`` is not bracketed by ``func(zmin)=fval(zmin)`` and
``func(zmax)=fval(zmax)``.
If the solver was not successful.
Raises
------
:class:`astropy.cosmology.CosmologyError`
If the result is very close to either ``zmin`` or ``zmax``.
ValueError
If ``bracket`` is not an array nor a 2 (or 3) element sequence.
TypeError
If ``bracket`` is not an object array. 2 (or 3) element sequences will
be turned into object arrays, so this error should only occur if a
non-object array is used for ``bracket``.
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 are 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 24 and 44 for which we
want to find the corresponding redshifts:
>>> Dvals = (24 + np.random.rand(1000000) * 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.geomspace(zmin, zmax, 50)
>>> Dgrid = Planck13.distmod(zgrid)
Finally interpolate to find the redshift at each distance modulus:
>>> zvals = np.interp(Dvals.value, Dgrid.value, zgrid)
Examples
--------
>>> import astropy.units as u
>>> from astropy.cosmology import Planck13, Planck18, 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) # doctest: +FLOAT_CMP
<Quantity 3.19812268 redshift>
The angular diameter is not monotonic however, and there are two
redshifts that give a value of 1500 Mpc. You can use the zmin and
zmax keywords to find the one you are interested in:
>>> z_at_value(Planck18.angular_diameter_distance,
... 1500 * u.Mpc, zmax=1.5) # doctest: +FLOAT_CMP
<Quantity 0.68044452 redshift>
>>> z_at_value(Planck18.angular_diameter_distance,
... 1500 * u.Mpc, zmin=2.5) # doctest: +FLOAT_CMP
<Quantity 3.7823268 redshift>
Alternatively the ``bracket`` option may be used to initialize the
function solver on a desired region, but one should be aware that this
does not guarantee it will remain close to this starting bracket.
For the example of angular diameter distance, which has a maximum near
a redshift of 1.6 in this cosmology, defining a bracket on either side
of this maximum will often return a solution on the same side:
>>> z_at_value(Planck18.angular_diameter_distance,
... 1500 * u.Mpc, bracket=(1.0, 1.2)) # doctest: +FLOAT_CMP +IGNORE_WARNINGS
<Quantity 0.68044452 redshift>
But this is not ascertained especially if the bracket is chosen too wide
and/or too close to the turning point:
>>> z_at_value(Planck18.angular_diameter_distance,
... 1500 * u.Mpc, bracket=(0.1, 1.5)) # doctest: +SKIP
<Quantity 3.7823268 redshift> # doctest: +SKIP
Likewise, even for the same minimizer and same starting conditions different
results can be found depending on architecture or library versions:
>>> z_at_value(Planck18.angular_diameter_distance,
... 1500 * u.Mpc, bracket=(2.0, 2.5)) # doctest: +SKIP
<Quantity 3.7823268 redshift> # doctest: +SKIP
>>> z_at_value(Planck18.angular_diameter_distance,
... 1500 * u.Mpc, bracket=(2.0, 2.5)) # doctest: +SKIP
<Quantity 0.68044452 redshift> # doctest: +SKIP
It is therefore generally safer to use the 3-parameter variant to ensure
the solution stays within the bracketing limits:
>>> z_at_value(Planck18.angular_diameter_distance, 1500 * u.Mpc,
... bracket=(0.1, 1.0, 1.5)) # doctest: +FLOAT_CMP
<Quantity 0.68044452 redshift>
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.
All the arguments except ``func``, ``method`` and ``verbose`` accept array
inputs. This does NOT use interpolation tables or any method to speed up
evaluations, rather providing a convenient means to broadcast arguments
over an element-wise scalar evaluation.
The most common use case for non-scalar input is to evaluate 'func' for an
array of ``fval``:
>>> z_at_value(Planck13.age, [2, 7] * u.Gyr) # doctest: +FLOAT_CMP
<Quantity [3.19812061, 0.75620443] redshift>
``fval`` can be any shape:
>>> z_at_value(Planck13.age, [[2, 7], [1, 3]]*u.Gyr) # doctest: +FLOAT_CMP
<Quantity [[3.19812061, 0.75620443],
[5.67661227, 2.19131955]] redshift>
Other arguments can be arrays. For non-monotic functions -- for example,
the angular diameter distance -- this can be useful to find all solutions.
>>> z_at_value(Planck13.angular_diameter_distance, 1500 * u.Mpc,
... zmin=[0, 2.5], zmax=[2, 4]) # doctest: +FLOAT_CMP
<Quantity [0.68127747, 3.79149062] redshift>
The ``bracket`` argument can likewise be be an array. However, since
bracket must already be a sequence (or None), it MUST be given as an
object `numpy.ndarray`. Importantly, the depth of the array must be such
that each bracket subsequence is an object. Errors or unexpected results
will happen otherwise. A convenient means to ensure the right depth is by
including a length-0 tuple as a bracket and then truncating the object
array to remove the placeholder. This can be seen in the following
example:
>>> bracket=np.array([(1.0, 1.2),(2.0, 2.5), ()], dtype=object)[:-1]
>>> z_at_value(Planck18.angular_diameter_distance, 1500 * u.Mpc,
... bracket=bracket) # doctest: +SKIP
<Quantity [0.68044452, 3.7823268] redshift>
"""
# `fval` can be a Quantity, which isn't (yet) compatible w/ `numpy.nditer`
# so we strip it of units for broadcasting and restore the units when
# passing the elements to `_z_at_scalar_value`.
fval = np.asanyarray(fval)
unit = getattr(fval, 'unit', 1) # can be unitless
zmin = Quantity(zmin, cu.redshift).value # must be unitless
zmax = Quantity(zmax, cu.redshift).value
# bracket must be an object array (assumed to be correct) or a 'scalar'
# bracket: 2 or 3 elt sequence
if not isinstance(bracket, np.ndarray): # 'scalar' bracket
if bracket is not None and len(bracket) not in (2, 3):
raise ValueError("`bracket` is not an array "
"nor a 2 (or 3) element sequence.")
else: # munge bracket into a 1-elt object array
bracket = np.array([bracket, ()], dtype=object)[:1].squeeze()
if bracket.dtype != np.object_:
raise TypeError(f"`bracket` has dtype {bracket.dtype}, not 'O'")
# make multi-dimensional iterator for all but `method`, `verbose`
with np.nditer(
[fval, zmin, zmax, ztol, maxfun, bracket, None],
flags=['refs_ok'],
op_flags=[*[['readonly']] * 6, # ← inputs output ↓
['writeonly', 'allocate', 'no_subtype']],
op_dtypes=(*(None,)*6, fval.dtype),
casting="no",
) as it:
for fv, zmn, zmx, zt, mfe, bkt, zs in it: # ← eltwise unpack & eval ↓
zs[...] = _z_at_scalar_value(func, fv * unit, zmin=zmn, zmax=zmx,
ztol=zt, maxfun=mfe, bracket=bkt.item(),
# not broadcasted
method=method, verbose=verbose)
# since bracket is an object array, the output will be too, so it is
# cast to the same type as the function value.
result = it.operands[-1] # zs
return result << cu.redshift
```