🐍 Integrate Python Translations of LALSuite C Waveform Code

Summary

This PR integrates Python implementations of waveform generation routines translated from the corresponding C codes in LALSuite. The goal is to enable easier experimentation and validation within a Python-native workflow while preserving consistency with the original implementation.

Features / Changes

• Added Python implementations for the following spherical harmonic modes:
  • $(2, \pm 2), (2, \pm 1)$
  • $(3, \pm 3), (3, \pm 2), (3, \pm 1)$
  • $(4, \pm 4)$
• Implemented orbital evolution using SciPy’s solve_ivp with method='LSODA'
• Structured modules to mirror the original C code organization where possible

Numerical Details

• ODE integration handled via scipy.integrate.solve_ivp
• LSODA chosen for automatic stiffness detection and robustness across regimes
• Important difference: In LALSuite, absolute tolerance was chosen to be zero, here we have chosen atol=1e-25.

Validation

• Compared outputs against the original C implementation

• Mode mismatches found for this particular parameter set:

Limitations / TODO

• Remaining modes not yet implemented
• Performance not yet optimized (pure Python vs C)
• Further validation across wider parameter space needed

How to Test

• Run the example notebook: esigma_python.ipynb
• Verify waveform agreement / mismatch thresholds