UV‑Saturated Regular‑Core Black Holes: Ringdown, Eikonal/WKB, and Hierarchical Inference

Comprehensive, reproducible code to study how short‑distance, UV‑regular black‑hole cores modify quasinormal modes (QNMs) and related observables, and to infer bounds from gravitational‑wave ringdown and ancillary data. The code focuses on a one‑scale Hayward‑like mass profile and small‑core expansions, with both geodesic (eikonal) and wave‑based (WKB and time‑domain) validations, plus a lightweight hierarchical analysis for LVK O3b.

If you want a one‑shot demo, see the commands under “Run the Experiments”.

Summary

• Task: Constrain a UV‑regular core scale L (or fractional size ε = L/r_s) using BH ringdown (220) frequencies and damping times, cross‑checked by analytic limits (eikonal) and semi‑analytic wave computations (WKB), with time‑domain validation and a hierarchical posterior across multiple GW events.
• Method: Model a one‑scale interior via m(r) = M r^3/(r^3 + L^3), derive eikonal shifts δΩ/Ω, compute wave QNMs from WKB1/3 using barrier derivatives at the deformed peak, and validate with time‑domain Regge–Wheeler evolutions. Build per‑event residuals against GR fits (Berti–Nakano) and infer posteriors on ε and on absolute L0 via a simple likelihood with optional slope systematics and start‑time mixtures.
• Key findings (as produced by this repo’s scripts): fractional QNM shifts scale close to linear+cubic in ε at small cores; time‑domain and WKB trends agree at small L/rs; hierarchical posteriors are dominated by small shifts consistent with GR within current uncertainties, yielding informative upper bounds on ε and L0 at 95%.
• Code highlights: self‑contained Kerr (220) Leaver continued‑fraction solver, Schwarzschild TD evolution + Prony, symbolic eikonal/WKB scaffolds, and compact scripts to reproduce figures and CSV outputs under data/.

Note: For precise numerical values, run the scripts below; they generate the tables/figures in data/ and figures/ directly from this codebase and small CSV inputs.

Repository Structure

Only the analysis code and supporting utilities are summarized here. Simulation, paper build files, and AI‑agent notes are intentionally omitted.

• src/
  • td_rw.py: Time‑domain Regge–Wheeler evolution for Schwarzschild axial perturbations; includes evolve_td_RW(...), prony_single(...), and helpers.
  • teukolsky_cf.py: Kerr (l=2, m=2, n=0) QNM via Leaver’s angular+radial continued fractions; robust initial guess from Berti–Cardoso–Will fits.
  • leaver_schwarzschild.py: Convenience Schwarzschild (220) frequency using the Kerr CF in the a→0 limit.
  • eikonal_level.py and re-eikonal_level.py: Symbolic and numeric eikonal helpers for photon‑sphere r_c, frequency Ω_c, Lyapunov λ_c, and small‑ε fits.
  • uv_core_qnm_validation.py: End‑to‑end eikonal and WKB1 validation for a Hayward‑like core; produces plots and fit coefficients.
  • wkb_pade.py: WKB1/3 formulae and Pade(1,1) utilities; used in barrier‑based QNM estimates.
  • hayward_core.py: Core mass function and axial potential for Hayward‑type deformation.
  • hierarchical.py and hierarchical_cov.py: Per‑event residual construction, slope extraction from TD maps, hierarchical posteriors over ε and L0, leave‑one‑out diagnostics; hierarchical_cov.py adds simple start‑time mixtures and covariance options.
  • kerr_eikonal.py, kerr_surrogate.py: Eikonal reference for Kerr and small‑spin scaling surrogate for slope propagation.
  • entropy_calculation.py: EHT‑style shadow residuals and an entropy budget sanity check at the inferred bounds (toy calculation).
  • NCBH_numerical_constants.py: Numerics for a related non‑commutative BH profile and thermodynamic quantities (used in ancillary scans).
  • pending_modify_case_c.py: Scratchpad for alternative model cases and fits.
• scripts/
  • run_hier.py: Reproduce the hierarchical analysis end‑to‑end; writes data/hier/ CSVs and PDFs.
  • run_posteriors.py: Quick posterior prototype from a linearized TD surrogate; produces data/hayward_L_over_rs_posterior.csv and a PDF.
  • run_reomega.py: TD Hayward QNMs across L/rs and calibrated WKB‑3 sweep; produces comparison figures and CSVs in data/.
  • cf_audit.py: Sanity checks for Schwarzschild/Kerr CF frequencies; writes data/cf_audit/report.json.
  • run_diagnostics.py: Full‑covariance hierarchical posteriors and barrier diagnostics; writes under data/hier/ and data/diagnostics/.
  • plot_barrier_mprime.py, run_kerr_surrogate.py: Helper diagnostics and slope‑scaling plots.
• tests/
  • test_schwarzschild_qnm.py: Smoke tests comparing TD‑extracted (220) with a continued‑fraction solve and known “gold” values.
• data/
  • Small inputs and generated artifacts (CSVs, PDFs). Key inputs include data/hier/events_o3b_tablexiii.csv and TD/WKB sweeps generated by scripts.
• outputs/
  • Optional aggregated results.
• requirements.txt, Makefile
  • Minimal Python deps and convenience targets to run key experiments.

Installation

Use Python 3.10+.

python3 -m venv .venv && source .venv/bin/activate
pip install -r requirements.txt

Optional: for notebooks, pip install jupyter.

Run the Experiments

The following commands reproduce the main figures and CSVs. All outputs go under data/ (and figures/ where applicable).

• Core audit and quick posterior prototype:
  • python3 scripts/cf_audit.py → data/cf_audit/report.json with CF frequencies.
  • python3 scripts/run_posteriors.py → TD surrogate, posterior over L/r_s, figures in data/.
• Time‑domain vs WKB comparison across L/rs:
  • python3 scripts/run_reomega.py → data/hayward_td_qnms_dense.csv, data/hayward_wkb3_qnms_dense.csv, and comparison figures.
• Hierarchical ringdown inference (LVK O3b‑style table):
  • python3 scripts/run_hier.py → data/hier/per_event_deltas.csv, data/hier/summary.json, and posterior PDFs.
  • python3 scripts/run_diagnostics.py → covariance/mixture variants and barrier diagnostics in data/hier/ and data/diagnostics/.
• Symbolic eikonal + WKB1 validation and fits:
  • python3 src/uv_core_qnm_validation.py → figures under figures/ and figures/fit_coefficients.json.

Makefile shortcuts bundle several of the above:

make data     # runs key scripts to regenerate CSVs/plots under data/
make hier     # only hierarchical analysis
make post     # posterior prototype and diagnostics

Tip: Some TD and CF sweeps are modestly intensive. Start with the defaults in each script; they are sized to finish quickly on a laptop.

How the Pieces Fit Together

• Theory side: src/eikonal_level.py, src/uv_core_qnm_validation.py, and src/wkb_pade.py derive/validate how small cores perturb photon‑sphere quantities and the barrier, capturing the leading ε scaling (often linear+cubic fits).
• Wave side: src/td_rw.py provides direct TD evolutions and QNM extraction; src/teukolsky_cf.py/src/leaver_schwarzschild.py provide independent frequency checks using continued fractions.
• Inference side: src/hierarchical*.py turn observed (f, τ) into fractional residuals relative to GR fits and combine them across events to posteriors on ε or L0 with simple systematics.

Reproducing Paper‑Style Results

The repo recreates, from scratch, the core figures/tables needed for a “ringdown bound on regular cores” analysis:

• TD and WKB‑3 trends: data/fig_Reomega_vs_L.pdf, data/fig_Imomega_vs_L.pdf, residuals in data/fig_residuals_vs_L.pdf.
• Hierarchical posteriors: data/hier/posterior_eps_frac.pdf and data/hier/posterior_L0_abs.pdf plus LOO diagnostics.
• Eikonal/WKB fits: figures/eikonal_caseA_deltaOmega.pdf, figures/wkb1_deltaReomega.pdf, etc., with fit coefficients in figures/fit_coefficients.json.

Exact numerical values may differ slightly across environments due to precision and solver settings; each script exposes tolerances and depths you can adjust.

Testing

Tests live under tests/. After installing pytest, run:

pytest -q

Note: The CF API in tests/test_schwarzschild_qnm.py assumes a convenience function that may differ from the current leaver_schwarzschild.py surface. The TD gold‑value test is the primary smoke check.

Data and Reproducibility

• Inputs are small and versioned under data/ (e.g., data/hier/events_o3b_tablexiii.csv). Scripts generate all other CSVs/figures.
• Paths are relative; no absolute paths are required. Precision knobs (e.g., CF depth, TD window) are encoded in each script. Randomness is not used.

Citation

If you use this repository in academic work, please cite the associated manuscript

@misc{Dinu_Ringdown_2025,
  author = {Dinu, Marius-Constantin},
  title = {{Ringdown Bounds on UV-Regularized Black-Hole Cores}},
  url = {https://github.com/ExtensityAI/gr_qm},
  month = {10},
  year = {2025}
}

and, where appropriate, the original QNM/CF/WKB references used in the code (Leaver 1985; Berti–Cardoso–Will 2009; Schutz–Will 1985; Iyer–Will 1987).