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GWDALI: A Fisher-matrix based software for gravitational wave parameter-estimation beyond Gaussian approximation

Josiel Mendonça Soares de Souza, Riccardo Sturani

TL;DR

The paper addresses the instability of traditional Fisher-matrix forecasts in GW parameter estimation, especially when the Fisher matrix is ill-conditioned. It introduces GWDALI, a Python tool implementing the Derivative Approximation for Likelihood (DALI) to extend beyond Gaussian assumptions by constructing higher-order derivative tensors (Flexion, Quarxion, P-, H-tensors) and forming Doublet- and Triplet-DALI likelihoods. The package can interface with LAL waveform models and Bilby for Monte Carlo sampling, enabling accurate uncertainty forecasts for arbitrary detector networks and parameters, and it demonstrates improved alignment with full likelihood results in luminosity-distance studies. This work enhances the reliability of GW parameter forecasts, supports informed detector-network design, and opens paths to cosmological applications and further derivative-order extensions.

Abstract

We introduce GWDALI, a new Fisher-matrix, python based software that computes likelihood gradients to forecast parameter-estimation precision of arbitrary network of terrestrial gravitational wave detectors observing compact binary coalescences. The main new feature with respect to analogous software is to assess parameter uncertainties beyond Fisher-matrix approximation, using the derivative approximation for Likelihood (DALI). The software makes optional use of the LSC algorithm library LAL and the stochastic sampling algorithm Bilby, which can be used to perform Monte-Carlo sampling of exact or approximate likelihood functions. As an example we show comparison of estimated precision measurement of selected astrophysical parameters for both the actual likelihood, and for a variety of its derivative approximations, which turn out particularly useful when the Fisher matrix is not invertible.

GWDALI: A Fisher-matrix based software for gravitational wave parameter-estimation beyond Gaussian approximation

TL;DR

The paper addresses the instability of traditional Fisher-matrix forecasts in GW parameter estimation, especially when the Fisher matrix is ill-conditioned. It introduces GWDALI, a Python tool implementing the Derivative Approximation for Likelihood (DALI) to extend beyond Gaussian assumptions by constructing higher-order derivative tensors (Flexion, Quarxion, P-, H-tensors) and forming Doublet- and Triplet-DALI likelihoods. The package can interface with LAL waveform models and Bilby for Monte Carlo sampling, enabling accurate uncertainty forecasts for arbitrary detector networks and parameters, and it demonstrates improved alignment with full likelihood results in luminosity-distance studies. This work enhances the reliability of GW parameter forecasts, supports informed detector-network design, and opens paths to cosmological applications and further derivative-order extensions.

Abstract

We introduce GWDALI, a new Fisher-matrix, python based software that computes likelihood gradients to forecast parameter-estimation precision of arbitrary network of terrestrial gravitational wave detectors observing compact binary coalescences. The main new feature with respect to analogous software is to assess parameter uncertainties beyond Fisher-matrix approximation, using the derivative approximation for Likelihood (DALI). The software makes optional use of the LSC algorithm library LAL and the stochastic sampling algorithm Bilby, which can be used to perform Monte-Carlo sampling of exact or approximate likelihood functions. As an example we show comparison of estimated precision measurement of selected astrophysical parameters for both the actual likelihood, and for a variety of its derivative approximations, which turn out particularly useful when the Fisher matrix is not invertible.
Paper Structure (10 sections, 22 equations, 11 figures, 1 table)

This paper contains 10 sections, 22 equations, 11 figures, 1 table.

Table of Contents

  1. Introduction
  2. Astrophysical parameter estimation for coalescing binaries and Fisher matrix
  3. Analytic Fisher Matrix Inversion
  4. DALI algorithm for Gravitational Waves
  5. Approximate likelihood: Doublet-DALI
  6. GWDALI python package
  7. Example of luminosity distance uncertainty
  8. Conclusions
  9. Derivative expansion beyond cubic order: P- and H-Tensors
  10. Example of parameter recovery

Figures (11)

  • Figure 1: ET etd and CE Srivastava:2022slt noise spectral densities.
  • Figure 2: Example of $\Delta d_L/d_L$ vs. $\iota$ for a 4-dimensional Fisher matrix in the parameters: $\{d_L, \iota, \psi, \phi\}$ from GWFISH code (blue line) Dupletsa:2022scg, full likelihood for the same parameters (black) and for the analytic curve given by $100/SNR$ (red). GW signals are generated by equal mass spin-less binaries with $M=3M_\odot$ with TaylorF2 approximant Sathyaprakash:2009xs, averaged over sky position and inclination angles for ET (left) and CE (right).
  • Figure 3: 1$\sigma$ relative uncertainty in $d_L$ fixing all other parameters (dashed red) or marginalizing over $\iota$ (solid black).
  • Figure 4: Example of Bayesian inference results obtained with one Einstein Telescope (triangle-shaped) detector, searching over parameters $\{d_L,\iota\}$, obtained with GWDALI.
  • Figure 5: Example of Bayesian inference results obtained with one Cosmic Explorer ($L$-shaped) detector, searching over parameters $\{d_L,\iota\}$, obtained with GWDALI.
  • ...and 6 more figures