z_at_value

astropy.cosmology.z_at_value(func, fval, zmin=1e-08, zmax=1000, ztol=1e-08, 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 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.Quantity instance

The value of func(z).

zmin : float, 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, 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 24 and 43 for which we want to find the corresponding redshifts:

>>> Dvals = (24 + 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(np.log10(zmin), np.log10(zmax), 50)
>>> 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)
3.19812268...

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)
0.6812769577...
>>> z_at_value(Planck13.angular_diameter_distance, 1500 * u.Mpc, zmin=2.5)
3.7914913242...

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.