Construct simple halo biasing models which reproduce 2-pt statistics

[maho3]

Currently, there's an issue with the TruncatedPowerLaw model which results in emulated halo count fields with incorrect power on large scales. This problem is particularly significant in the low mass bins, wherein we see far too many halo counts in the voids.

This doesn't seem to be a problem with the fitting, since when we do self-consistent tests we achieve the fiducial parameters.

This problem is also echoed in the ROBIN tests, wherein forcing the bias model to match the 2-pt stats resulted in it breaking the 1-pt stats.

As a result, the problem likely has to do with the form of the bias model, and possibly the assumption of a Poisson distribution.

Some options:

• Bin the TruncatedPowerLaw fittings into rho-bins
• ADDGals - https://arxiv.org/abs/2105.12105
• Double power law (Talk to Stuart, who has looked into this a lot)

Other questions:

• Should we fit a general bias model for all cosmologies? Or continue to fit for specific cosmologies?

Definition of done

• Implementation of a simple, analytic bias model which produces unbiased 2-pt stats on aggregate (albeit with large scatter)

[maho3]

Notes of things to try from meeting with Stuart:, in no particular order:

• Beta is too low - Check variance of predicted vs true
• Beta prior 1-1.5
• Plot P(k) as a function of (beta)
• Fit one bias model over all cosmologies?
• Thicker mass bins?
• HMC over params, instead of minimize
• Generalized Poisson
• try doing this on the Quijote field rather than the low-res sim to see if that causes a problem

[maho3]

From Simon:

As you already know, we've been tinkering with the broken power law model in combination with Robin and you have the plots. One interesting thing might be that even after fixing nmean the optimizer would adjust the slope (alpha) to be flatter in order to get a better fit in the power spectrum. The two set of parameters that we get when optimizing without and with robin are:

• [0.3726995, 0.64506765, 2.06996142, 0.87577736] (Without Robin)
• [0.3726995, 0.31740525, 1.97158317, 0.84898678] (With Robin)

[maho3]

It seems that we've discovered why ltu-cmass bias fitting was producing too little power on large scales, while @AsianTaco 's bias models were producing too much.

The issue was that the Quijote density fields are measured at a half-voxel offset to [0,L]^3 volume, to have the first voxel centered at (0,0,0). That is, instead of creating a 128^3 meshgrid evenly spaced from [0,L], we should've spaced them from [-L/256, 255L/256].

Previously, we were fitting a bias model and then sampling the halo field like this:

And now, with the offset implemented, they look like this:

So, it is good that things are consistent among multiple attempts.

In any case, we still have an issue wherein a local-only bias model (no matter how flexible!) is insufficient to explain the 2-pt statistics. Our options to proceed with a simple bias model are:

• Create a simple bias model using successive shells of DM density as input
• Use an EFT approach like https://arxiv.org/abs/2403.03220 to aggregate field information

Since the latter is more common in the literature, and some BORG people are working on it, we will approach that one first...

[maho3]

Note this offset issue could explain the slight differences we see in the gravity solvers as in https://github.com/maho3/ltu-gobig/issues/15

[maho3]

Closed, shown to be difficult on small scales in : https://arxiv.org/abs/2405.00635

PRs #35 and #39 are intended to provide alternatives which are closely calibrated.