Constraint-Calibrated Reliability Gates for Weyl Diagnostics in Numerical Relativity Cosmology

Technical Summary: Constraint-Calibrated Reliability Gates for Weyl Diagnostics in Numerical Relativity Cosmology

Problem Statement
In numerical relativity cosmology, the local gravitational structure of inhomogeneous universes is often characterized using the electric ($E_{ab}$) and magnetic ($B_{ab}$) parts of the Weyl tensor. These components allow for the classification of spacetime regions into regimes such as tidal, gravitomagnetic, radiative, or "silent" (where $B_{ab} \approx 0$). However, numerical data never satisfies the Einstein constraint equations (Hamiltonian and momentum constraints) exactly. Small constraint residuals can contaminate the computed Weyl scalars. Specifically, near a "silent" regime, a small numerical error can generate a spurious magnetic signal, leading to a false classification of radiative or gravitomagnetic structure. Current practices monitor constraint residuals globally to ensure solution quality but lack a local, quantitative link between the magnitude of a residual and the trustworthiness of a specific mechanism classification. There is no established method to determine how small a residual must be to guarantee a specific accuracy in Weyl diagnostics.

Methodology
The paper proposes treating the Hamiltonian ($I_H$) and momentum ($I_M$) constraint residuals not merely as passive monitors, but as sector-specific "reliability gates." The core methodology involves:

1. Diagnostic Vector Definition: A scalar vector $X$ is defined containing physical invariants (matter inhomogeneity, shear, electric/magnetic Weyl energies) and the two reliability axes ($I_H, I_M$).
2. Controlled Synthetic Tests: The authors construct matched electric and magnetic synthetic solutions on an expanding background. They inject constraint-violating perturbations of varying amplitude ($\eta$) into the metric (for the electric sector) and extrinsic curvature (for the magnetic sector).
3. Real-Data Validation: A perturbed Friedmann initial-data slice (silent class) is analyzed using the AUREL package. Constraint-violating perturbations are injected, and the resulting errors in Weyl scalars are measured relative to the clean, unperturbed evaluation.
4. Calibration Procedure:
   • The relationship between the constraint residual ($I$) and the induced relative error ($\varepsilon$) in the dominant Weyl axis is measured.
   • This relationship is fitted to a power law: $\varepsilon = C I^p$.
   • For a desired target accuracy $\varepsilon_\star$, the required tolerance $I_\star$ is calculated by inverting the fit: $I_\star = (\varepsilon_\star / C)^{1/p}$.
5. Dimensionless Analysis: To test for unit dependence, the residuals are re-expressed in a standard dimensionless fractional form ($\hat{I}_H$) by normalizing against the root-mean-square of the constraint terms.

Key Contributions and Results

• Sector-Specific Pairing: The study empirically establishes a specific pairing between constraint residuals and Weyl diagnostics:
  • The Hamiltonian residual ($I_H$) determines the reliability of the Electric Weyl diagnostic ($I_E$).
  • The Momentum residual ($I_M$) determines the reliability of the Magnetic Weyl diagnostic ($I_B$).
• Non-Portability of Fixed Tolerances: The paper demonstrates that a single, fixed tolerance (e.g., $10^{-3}$) is inadequate. The quantitative relationship between residual size and classification error varies significantly between synthetic fields and real cosmological data.
  • On synthetic fields, a fixed tolerance yields errors of order $10^{-4}\%$.
  • On real data, the same tolerance yields errors of approximately $0.27\%$.
  • This discrepancy persists even when using dimensionless fractional residuals, indicating the gap is not an artifact of units but a solution-dependent sensitivity.
• Power-Law Behavior: The residual-to-error coupling follows a near-quadratic power law ($p \approx 2$) across both synthetic and real data. However, the prefactor ($C$) differs by orders of magnitude (several hundred times) between synthetic and real slices.
  • The difference in prefactors arises from two factors: a genuine difference in absolute sensitivity (factor of ~5) and the small baseline amplitude of the electric Weyl field in the real silent slice, which amplifies relative errors (factor of ~20).
• Resolution Dependence: The calibration is resolution-dependent. The residual-to-error map shifts with grid resolution ($N$), necessitating that calibration be performed at the specific resolution of the data being classified.

Significance and Claims
The paper claims to provide a "reliability layer" for existing invariant characterisation frameworks (such as those in [10, 17, 18]). It does not introduce new Weyl implementations or new gravitational regimes. Instead, it offers a procedure to attach a calibrated statement of epistemic reliability to every mechanism label in a Weyl atlas.

The central claim is that no universal tolerance exists for Weyl diagnostics. The reliability of a classification (e.g., distinguishing a silent slice from a radiative one) must be determined by measuring the slice-level sensitivity of the specific solution and resolution. By replacing hand-set thresholds with measured, model-dependent tolerances, the framework ensures that classifications near critical boundaries (like the silent regime) are not mistaken for genuine physical structure due to numerical noise.

Limitations Noted by the Authors

• The injected perturbations are smooth, low-wavenumber random fields, which may not perfectly replicate the structured, formulation-dependent constraint violations found in live evolutions (e.g., in Z4 or conformal systems).
• The dimensionless robustness check was performed only for the electric sector; the magnetic sector check requires a non-silent solution with a non-zero momentum constraint scale, which was not included in this study.
• The calibration is specific to the discretization scheme, gauge, and domain, and is not portable across different numerical setups without re-calibration.