Equations used for the lookback time

An equation for the calculation of the lookback time is given by Carroll, Press & Turner [1]:

$$t_0-t_1 = H_0^{-1} \int_{0}^{z_1} (1+z)^{-1} [(1+z)^2\, (1+\Omega_M\, z)-z\, (2+z)\, \Omega_\Lambda]^{-1/2}\, dz$$ (1)

This equation, however, isn't used for the calculation but a modified version of it which makes use of the w parameter:

$$t_0-t_1 = H_0^{-1} \int_{0}^{z_1} (1+z)^{-1}\, [(1+z)^2\, (\O... ...M (1+z)+\Omega_\Lambda\, (1+z)^{1+3\, w}+\Omega_K]^{-1/2}\, dz$$ (2)

If w = -1 this equation reduces to eq.  1. In the function, $\Omega_K$ is calculated from $\Omega_M + \Omega_\Lambda + \Omega_K = 1$.