Relative Uncertainty

---
name: relative_uncertainty
description: Evaluates the precision of estimating globular cluster distance and mass parameters by quantifying how much gravitational wave modulation reduces parameter uncertainties compared to electromagnetic-only observations. Use when the user has predictions and gold and needs to compute relative_uncertainty.
metadata:
  skill_kind: metric
  source_arxiv: 2508.04021
  bibtex_key: wu2025probing
  confidence: medium
---

# relative_uncertainty

> Probing globular clusters using modulated gravitational waves from binary black holes — Wu et al. (2025) (arXiv:2508.04021, 2025)

## What this evaluates

Evaluates the precision of estimating globular cluster distance and mass parameters by quantifying how much gravitational wave modulation reduces parameter uncertainties compared to electromagnetic-only observations.

## Datasets

- **EM1** — total ?; splits: test (-1)
- **EM2** — total ?; splits: test (-1)

## Metrics

- `relative_uncertainty` **(primary)** — range: percent
  - Calculated as the ratio of the Fisher Information Matrix-derived standard deviation to the true parameter value, expressed as a percentage. Lower values indicate higher estimation precision.

## Input / output format

**Input**: Post-Newtonian gravitational wave waveforms modulated by binary black hole orbital motion within globular clusters, combined with existing electromagnetic prior measurements of cluster distance and mass.

**Output**: Estimated globular cluster distance and mass parameters with associated 1σ uncertainties derived from Fisher Information Matrix inversion.

## Scoring recipe

```python
def compute_relative_uncertainty(fisher_matrix, true_params):
    cov_matrix = np.linalg.inv(fisher_matrix)
    param_uncertainties = np.sqrt(np.diag(cov_matrix))
    rel_unc_dist = param_uncertainties[0] / true_params[0] * 100
    rel_unc_mass = param_uncertainties[1] / true_params[1] * 100
    return rel_unc_dist, rel_unc_mass
```

## Common pitfalls

- FIM analysis assumes Gaussian posteriors and unbiased estimators, which may fail for weak signals or highly non-linear parameter dependencies.
- GW amplitude scales inversely with distance, so extragalactic sources yield weaker signals and inherently larger uncertainties regardless of modulation.
- Modulation sidebands require sufficient orbital velocity and specific viewing angles to be resolvable; edge-on or face-on configurations may not produce detectable features.

## Evidence (verbatim from paper)

> Figure 3 shows how the relative uncertainties in GC parameters depend on the BBH orbital velocity. A clear negative correlation is observed: as the velocity increases, the relative uncertainties in both distance and mass decrease.

## Citation

```bibtex
@misc{wu2025probing,
  title={Probing globular clusters using modulated gravitational waves from binary black holes},
  author={Wu et al. (2025)},
  year={2025},
  note={arXiv:2508.04021}
}
```

- arXiv: 2508.04021