wcosmo.taylor.indefinite_integral_pade#
- wcosmo.taylor.indefinite_integral_pade(z, Om0, w0=-1, zpower=0)[source]#
Compute the Pade approximation to \((1+z)^k / E(z)\) as described in arXiv:1111.6396. We extend it to include a variable dark energy equation of state, other integrands via powers of \(1 + z\) and include more terms in the expansion
\[I(z; \Omega_{m, 0}, w_0) = \frac{-2 (1 + z)^{\frac{1}{2}-k}} {\Omega_{m, 0}^{\frac{1}{2}}} \Phi(z; \Omega_{m, 0}, w_0).\]\[\Phi(z; \Omega_{m, 0}, w_0) = \frac{\sum_i^n \alpha_i x^i}{1 + \sum_{j=1}^m \beta_j x^j}.\]Here the expansion is in terms of
\[x = \left(\frac{1 - \Omega_{m, 0}}{\Omega_{m, 0}}\right) (1 + z)^{3 w_0}.\]In practice we use \(m=n=7\) whereas Adachi and Kasai use \(m=n=3\).
- 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:
- I: array_like
The indefinite integral of \((1+z)^k / E(z)\)