New feature to sample distance ratios in multi-lens-plane and multi-source-plane setting

[ajshajib]

I added a new feature to sample the distance ratios in the mult-lens-plane and multi-source-plane cases, with PR #318. To use this feature, kwargs_model needs to be set up as:

kwargs_model = {
    ...
    "multi_plane": True,
    "distance_ratio_sampling": True,
    "lens_redshift_list": ...,
    "z_source": ...,
    "source_redshift_list": ...
    "cosmo": ...
}

And, the kwargs_special needs to have "factor_a_{i}" and "factor_b_{i}" parameters defined in it, which are sampled (or can be fixed as desired like any other model parameter), for example:

special_params_init = {
    ...
    "factor_a_1": 1.,
    "factor_a_2": 1.,
    "factor_a_3": 1.,
    "factor_b_2": 1.,
    ...
}

In the multi-lens-plane setting with $P$ lens planes (and the last source plane being the $P+1$-th plane), the effective Fermat potential is defined as (eq. 9 of Shajib et al. 2020):

$$ \phi^{\rm eff} (\theta) \equiv \sum_{i=1}^{P} \frac{1+z_i}{1+z_{\rm d}} \frac{D_i D_{i+1} D_{\rm ds}}{D_{\rm d} D_{\rm s}D_{i\ i+1} } \left[ \frac{(\theta_{i} - \theta_{ i+1})^2}{2} - \zeta_{i,i+1} \psi_{i}(\theta_{i}) \right]. $$

$$ \zeta_{i, j} \equiv \frac{D_{ij}D_{\rm s}}{D_j D_{i{\rm s}}}, \quad i < j. $$

Then, the $a_i$ and $b_i$ distance ratios are defined as:

$$ a_i \equiv \frac{D_i D_{i+1} D_{1\ P+1}}{D_{\rm 1} D_{P+1}D_{i\ i+1} } = \frac{D_i D_{i+1}}{D_{i\ i+1} D_{\Delta t}^{\rm eff}}, $$

$$ b_i \equiv \frac{D_i D_{1\ P+1}}{D_1 D_{i\ P+1}} = \frac{D_i D_{P+1}}{D_{i\ P+1} D_{\Delta t}^{\rm eff}}. $$

Instead of sampling $a_i$ and $b_i$, lenstronomy samples "factor_a_{i}" and "factor_b_{i}" parameters that are defined using the fiducial cosmology cosmo provided in kwargs_model as follows:

    factor_a_i = a_i / a_i_fiducial
    factor_b_i = b_i / b_i_fiducial

"factor_b_1" and "factor_b_{P}" are not needed to be defined, as $b_1 = 1$ and $b_P = a_P$.