Time delay does not work for "inverse" model in pycbc.population.population_models.p_tau

[divyajyoti09]

The inverse model for pycbc.population.population_models.p_tau does not work as intended because there are no minimum and maximum limits for the inverse function. This results in all probability being concentrated near zero, which effectively introduces no time delay in the merger_rate_density function (or other functions which use p_tau).

Following code tests the bug above:

from pycbc.population.population_models import merger_rate_density, sfr_madau_dickinson_2014

z_arr = np.arange(0.01, 10, 0.01)
md_arr = sfr_madau_dickinson_2014(z_arr)
plt.plot(z_arr, md_arr/md_arr.max(), 'r', label='No time delay')
for model, colr in zip(['inverse', 'gaussian'], ['yellow', 'green']):
    mrd_pycbc = merger_rate_density(sfr_func=sfr_madau_dickinson_2014, td_model=model, rho_local=105, npoints=100)
    mrd_pycbc_arr = mrd_pycbc(z_arr)
    plt.plot(z_arr, mrd_pycbc_arr/mrd_pycbc_arr.max(), '-.', c=colr, label=f'TD model = {model}')
plt.xlabel('z')
plt.ylabel('Normalised merger rate density')
plt.grid()
plt.legend()
plt.title('SFR Madau Dickinson')

The resulting plot is

In order to resolve this, a minimum and maximum limit on time delay function is needed. Something along of the lines of this should work.

[WuShichao]

See my comments in #5046