wcosmo.integrate.analytic_integral

wcosmo.integrate.analytic_integral(z, Om0, w0=-1, zpower=0, method='pade')

Compute an integral of the form \(\int_{\infty}^z \frac{(1+z)^k}{E(z)}\) using

\[f(z; \Omega_{m, 0}, w_0) = \int_{\infty}^{z} \frac{dz' (1 + z')^k}{E(z'; \Omega_{m, 0}, w_0)} = \frac{-2\Phi(z; \Omega_{m, 0}, w_0) (1+z)^k } {\sqrt{\Omega_{m, 0}(1 + z)}}.\]

The integral is approximated using the Pade approximation and is up to a factor the term in the braces in (1.1) of Adachi and Kasai.

Parameters:

z: array_like

Redshift

Om0: array_like

The matter density fraction

w0: array_like

The (constant) equation of state parameter for dark energy

Returns:

integral: array_like

The integral of \((1 + z)^k / E(z)\) from \(\infty\) to \(z\)

Examples

We can use this to calculate many cosmological distances. For example, the comoving distance \(d_{C}\) is given by

\[d_{C} = d_{H} \int_{0}^{z} \frac{dz'}{E(z'; \Omega_{m,0}, w_0)}\]

In this case, \(k=0\)

>>> from wcosmo.taylor import analytic_integral
>>> import wcosmo

>>> analytic_integral(z=2,Om0=0.3,zpower=0) * wcosmo.hubble_distance(H0=70)
5179.8621

We can check this against the comoving volume given by astropy and see that it gives the same answer.

>>> from astropy.cosmology import FlatLambdaCDM
>>> ap_cosmo = FlatLambdaCDM(H0=70,Om0=0.3)
>>> ap_cosmo.comoving_distance(2)
5179.8621 Mpc

This is how comoving_distance() is implemented in wcosmo, which gives the same result

>>> wcosmo.comoving_distance(2,H0=70,Om0=0.3)
5179.8621

As another example, the lookback time \(t_L\) is given by

\[t_{L} = t_{H}\int_{0}^{z} \frac{dz'}{(1+z)E(z')}\]

where \(t_L\) is the Hubble distance. In this case, \(k=-1\)

>>> analytic_integral(z=2,Om0=0.3,zpower=-1) * wcosmo.hubble_time(70) # Gyr
10.240357
We can again compare this to ``astropy`` and ``wcosmo``'s built-in commands

>>> ap_cosmo.lookback_time(2)
10.240357 Gyr
>>> wcosmo.lookback_time(2,H0=70,Om0=0.3)
10.240357

Finally, we can calculate the absorption distance, \(X\), commonly used in absorption line spectroscopy. This is not a physical distance, but is still a very useful quantity. See the book “Cosmological Absorption Line Spectroscopy” by Christopher W. Churchill for a full discussion.

\[X(z, \Omega_m) = \int_0^z \frac{dz' (1+z')^2}{ \sqrt{\Omega_m(1+z')^3 + (1 - \Omega_m)}}\]

Here, \(k=2\) and there are no factors of the Hubble distance.

>>> print(analytic_integral(z=2,Om0=0.3,zpower=2))
4.36995
>>> print(ap_cosmo.absorption_distance(2))
4.36995
>>> wcosmo.absorption_distance(2,Om0=0.3)
4.36995