Implement Gamma prediction in our modeling

[eduardojsbarroso]

Description

For cluster predictions, the excess surface density ΔΣ is a weak-lensing quantity that depends on cosmology because it is measured using distances, while the tangential shear γₜ is cosmology-independent as it is measured in angles. The relevant equations are in arXiv:2107.10857.

Problem: we need to implement tangential shear (γₜ) support in Firecrown, TXPipe, and SACC. This issue focuses on the modeling / Firecrown + SACC side; the TXPipe data-vector change should be tracked in a separate issue (see related TXPipe issue: #12 ).

Current Firecrown prediction code for ΔΣ lives here:
firecrown/models/cluster/deltasigma.py and is used by the cluster likelihoods. We need the prediction routine to optionally produce γₜ (cosmology-independent) instead of ΔΣ (cosmology-dependent), and ensure SACC can store / represent the chosen data vector correctly.

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Steps to Reproduce / Inspect

1. Inspect current prediction implementation:
   https://github.com/LSSTDESC/firecrown/blob/master/firecrown/models/cluster/deltasigma.py
2. Inspect cluster likelihoods that consume those predictions:
   • firecrown/likelihood/binned_cluster_number_counts_deltasigma.py
   • firecrown/likelihood/binned_cluster_number_counts.py
3. Review the theory relating ΔΣ and γₜ (see section/equations in arXiv:2107.10857).

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What Needs to Be Done

• Add support in Firecrown to compute tangential shear γₜ as an option in the cluster prediction code:
  • Decide whether to create a new prediction object (e.g. cluster_shear or deltasigma_or_shear) or extend the existing ΔΣ object to optionally return γₜ.
  • Add a clear configuration flag (e.g. output: "deltasigma" | "shear") so users can choose at runtime.
  • Ensure the implementation computes γₜ correctly (angle-based, cosmology-independent data vector) when requested.
• Update SACC serialization:
  • Decide on SACC schema/metadata to represent whether a stored column is ΔΣ or γₜ (units / cosmology-dependence metadata).
  • Implement code to write/read γₜ vectors into/from SACC files consistently with existing conventions.
• Update Firecrown likelihoods / wrappers to accept either data vector type (ΔΣ or γₜ) and to use appropriate prediction outputs.
• Add tests:
  • Unit tests verifying ΔΣ vs γₜ outputs for a fixed input model (including consistency checks where applicable).
  • End-to-end test that Firecrown + SACC round-trip preserves the data vector type and values.
• Documentation:
  • Document the new option in Firecrown docs, including guidance on when to use ΔΣ vs γₜ.
  • Add notes to SACC documentation about the new representation (if schema changes).
• Coordinate with TXPipe team:
  • Reference the separate TXPipe issue (Tangential shear implementation in TXPipe data vector and N(z) source selection #12 ) that will update the data vector production to provide γₜ when appropriate.

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Extra Note (not the main focus)

The TXPipe data-vector change is a separate task (see #12 ). This Firecrown/SACC issue should not attempt to change TXPipe internals here, but must remain compatible with the TXPipe-side decision.

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Type of change

• New feature (adds option to output tangential shear γₜ)
• Refactoring (prediction object/design & SACC serialization)

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