Implementing number density as an observable

[epaillas]

• The observed galaxy number density measured from spectroscopic surveys places a strong constrain on the range of viable HOD models for a given summary statistic.

• We should be able to produce a dense sample of HOD catalogues for each of the AbacusSummit cosmologies, saving the (unfiltered) number densities to disk. We can then train a simple FCN to learn to predict this number density based on the HOD + cosmological parameters.

• During the MCMC, we can calculate the usual \chi^2 for the multipoles, but now we additionally add a \chi^2 that compares the observed v/s predicted number density. This can be a truncated Gaussian distribution as in Eq. (18) from http://arxiv.org/abs/2110.11412, which penalizes number densities that are lower than the target, but for higher number densities we invoke an incompleteness factor f_ic that in practice downsamples the HOD catalogues to the correct number density (this step already takes place when we create the HODs to train the summary statistics, so it's just a matter of reflecting this in the likelihood).

I'll go ahead and measure the HOD number densities that we can use to train the FCN, and then we can decide on what's the best way to incorporate this into the pipeline.

[SBouchard01]

Updating the discussion on this point, as we talked about it a few weeks ago:
What's the better approach for the number density constraint wrt the model training ? If we let the number density free in the model, we might have a prediction that does not match the observed number density unless we combine all models to the number density one. Alternatively, we can impose a training on mocks downsampled at a given number density (as done in acm.hod.BoxHOD with the target_density constraint, but at the cost of some HOD parameter space not being explored - though in theory this information will be encoded in the model weights.

Example of the HOD parameter space with a density constraint on the computed mocks