Bug fix: Adding min and max tau values for inverse td_model in p_tau function

[divyajyoti09]

This change introduces a minimum and maximum tau for the inverse model in pycbc.population.population_models.p_tau() and fixes this issue: #5043

Standard information about the request

This is a: bug fix

This change affects: PyCBC population models module

This change changes: scientific output

This change has been tested at the level of creating merger_rate_density plots.

Motivation

This change is made because the current state of the inverse model in the p_tau function is practically non-functional as it doesn't include any time delay. This has been demonstrated in the issue here: #5043

Links to any issues or associated PRs

#5043

Testing performed

This code has been tested to generate plots for merger_rate_density function

• The author of this pull request confirms they will adhere to the code of conduct

[WuShichao]

Thank you for this PR, I will review it this week.

[WuShichao]

Can you show the new curve for the inverse time delay model? I noticed in the issue #5043 that you normalized those curves. Actually, SFR itself has a different unit (solar-mass Mpc^-3 yr^-1 ) than the curve convoluted by the time delay model. Have you tried the power_law delay model? The inverse is just a special case of it. Previously, I knew the shape of the inverse model should be the same as SFR.

[divyajyoti09]

@WuShichao The new plot looks like this

I agree that the units are different for SFR and merger rate but I normalized them to see whether the time delay was introducing a shift in the peak (which it wasn't because all the probabilities were peaking at 0 time delay). After introducing the td_min limit, the curve is shifting as expected.

Also, although I agree that power_law is very close to the inverse model, I still think this bug in the inverse model should be fixed since it is being used as default in various functions.