# Licensed under a 3-clause BSD style license - see LICENSE.rst """ Convenience functions for `astropy.cosmology`. """ import warnings import numpy as np from astropy.cosmology import units as cu from astropy.cosmology.core import CosmologyError from astropy.units import Quantity from astropy.utils.exceptions import AstropyUserWarning __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, "xtol": ztol} # Assume custom methods support the same options as default; otherwise user # will see warnings. if callable(method): # can skip callables pass elif str(method).lower() == "bounded": opt["xatol"] = opt.pop("xtol") if bracket is not None: warnings.warn(f"Option 'bracket' is ignored by method {method}.") bracket = None # 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 # Construct bounds (Brent and Golden fail if bounds are not None) if callable(method) or str(method).lower() not in {"brent", "golden"}: bounds = (zmin, zmax) else: bounds = None # Objective function to minimize. # 'Brent' and 'Golden' ignore `bounds` but this keeps the domain witihin the bounds. def f(z): if z > zmax: return 1.0e300 * (1.0 + z - zmax) elif z < zmin: return 1.0e300 * (1.0 + zmin - z) elif isinstance(fval_zmin, Quantity): return abs(func(z).value - val) else: return abs(func(z) - val) # Perform the minimization res = minimize_scalar(f, method=method, bounds=bounds, 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')}:" f" {res.get('message', 'Unsuccessful')}\nPrecision {res.fun} reached after" f" {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 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 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 >>> z_at_value(Planck18.angular_diameter_distance, ... 1500 * u.Mpc, zmin=2.5) # doctest: +FLOAT_CMP 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, ... method="Brent", bracket=(1.0, 1.2)) # doctest: +FLOAT_CMP +IGNORE_WARNINGS 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 # 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 # doctest: +SKIP >>> z_at_value(Planck18.angular_diameter_distance, ... 1500 * u.Mpc, bracket=(2.0, 2.5)) # doctest: +SKIP # 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, method="Brent", ... bracket=(0.1, 1.0, 1.5)) # doctest: +FLOAT_CMP 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 ``fval`` can be any shape: >>> z_at_value(Planck13.age, [[2, 7], [1, 3]]*u.Gyr) # doctest: +FLOAT_CMP 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 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 """ # `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