Accounting for the Known Unknowns: A Parametric Framework to Incorporate Systematic Waveform Errors in Gravitational-Wave Parameter Estimation

TL;DR

This paper tackles the challenge of systematic waveform-model errors in gravitational-wave parameter estimation by introducing a parametric WF-Error framework that models amplitude and phase uncertainties as frequency-dependent perturbations with abs-phase and rel-phase parametrizations. When priors on these perturbations are available, they are incorporated into Bayesian PE, and even modest 1–2% phase deviations can bias inferred source properties if unaccounted for; the WF-Error approach can correct biases and, in many cases, broaden posteriors to reflect true uncertainties. Through zero-noise injections, cubic-spline realizations, and missing-physics scenarios (e.g., precession), the framework demonstrates robust mitigation of biases and the ability to recover frequency-dependent error structures, while Fisher-matrix estimates prove unreliable outside the linear regime. The authors provide a Python plugin, pycbc_wferrors_plugin, enabling practitioners to embed WF-Error priors into the PyCBC pipeline and advocate waveform-developer-provided WF-Error envelopes to further quantify systematics. This approach offers a practical, data-driven path to account for waveform systematics in current and next-generation GW analyses, including potential missing-physics effects and calibration-wf degeneracies.

Abstract

The PE for GW merger events relies on a waveform model calibrated using numerical simulations. Within the Bayesian framework, this waveform model represents the GW signal produced during the merger and is crucial for estimating the likelihood function. However, these waveform models may possess systematic errors that can differ across the parameter space. Addressing these errors in the current data analysis pipeline is an active area of research. We introduce parametrizations for the uncertainties in the amplitude and phase of the reference waveform model. When the error budget in the amplitude and phase of the waveform model, as a function of frequency, is known, it can be used as a prior distribution in the Bayesian framework. We also show that conservative priors can be used to quantify uncertainties in waveform modeling without any knowledge of waveform uncertainty error budgets. Through zero-noise injections and PE recoveries, we demonstrate that even 1%-2% of errors in relative phase to the actual waveform model, for a GW150914-like signal and advanced LIGO detector sensitivity, can introduce biases in the recovered parameters. These biases can be corrected when we account for waveform uncertainties within the PE framework. By analyzing a series of simulated signals from mergers with precessing orbits and recovering them using a non-spinning waveform model, we demonstrate that we can reduce the ratio of systematic errors to statistical errors. This approach allows us to address scenarios where specific physical effects are missing in waveform modeling. The code that implements our parametrization for performing PE is available as a Python package "pycbc_wferrors_plugin", compatible with the PyCBC open source GW analysis library.

Figures (13)

• Figure 1: The Figure shows the calibration uncertainties as a function of frequency for the LIGO Hanford detectors around the time of the first gw detection: GW150914. The left panel shows the systematic error in the magnitude, while the right panel shows the systematic error in the phase angle (in radians). The shaded region represents the $\pm 1\sigma$ band around the median systematic error. The systematic error is described by the ratio of the true response function to the modeled response function of the detector. The LIGO and Virgo calibration uncertainty files for O1, O2, and O3 observation runs are available at ligoLIGOT2100313v3LIGO.
• Figure 2: We show the time domain waveform in a signal frame (before projecting it in the detector) for one of the polarization $(+)$ for a GW150914-type signal. Apart from the reference waveform model IMRPhenomPv2$h_{+, ref}^{SF}(t)$ (blue, denoted by '+' markers), we also show the modification to the reference signal by one of the parametrizations described in the text. Red and violet waveforms represent the abs-phase modification given by \ref{['eqn:wferror_modeling_abs']}. We use cubic splines for the parameters $(\delta\Tilde{A}, \delta\Tilde{\phi})$ with a realization from the normal distribution as shown in corresponding plot legends. Orange and green waveforms represent the modification with rel-phase parametrization given by equation \ref{['eqn:wferror_modeling_rel']}.
• Figure 3: We show the time domain waveform for $(+)$ polarization for the GW150914 type signal. The reference waveform model IMRPhenomPv2: $\Tilde{h}_{+, ref}^{SF}(t)$ is shown as dashed black curve. Other curves are modified waveform with abs-phase modification, described by equation \ref{['eqn:wferror_modeling_abs']}, with WF-Error parameters sampled from the distribution $\delta\Tilde{A}, \delta\Tilde{\phi} \sim \mathcal{N}(\mu=0, \sigma=0.05)$.
• Figure 4: On the x-axis, we show the mismatch between the reference waveform model $h_\mathrm{ref}$ and the modified waveform model $h_\mathrm{mod}$. The modified signal, $h_\mathrm{mod}$, is generated by either abs-phase or rel-phase WF-Error parametrization. The WF-Error parameters are taken from the normal distribution, and we generate cubic spline curves to modify the reference model. The rel-phase model is generally well suited to model comparatively more significant mismatches.
• Figure 5: Injection and recovery are shown for a non-spinning GW150914-like signal with source frame masses $(m^{src}_1, m^{src}_2)=(36,29)~M_{\odot}$, at a luminosity distance of 500 Mpc (left panel) and a distance of 100 Mpc (right panel). We also inject a modified signal with rel-phase modification with parameters $(\delta\Tilde{A}, \delta\Tilde{\phi})=(0.01,0.01)$. When we inject a modified signal and use the incorrect model to recover, we see a bias in the recovery of detector frame chirp mass $\mathcal{M}^\mathrm{z}$. We also show that using the correct parametrization (in this case, rel-phase) can correct the bias and get a broader marginalized posterior sample. The bias is more prominent in a louder signal (right panel) than in a comparatively weaker signal (left panel). We mark that combination of injection and recovery by an asterisk ($\star$) where we do not expect a bias. The vertical dashed line represents the injected value of $\mathcal{M}^\mathrm{z}$. For these pe runs, we fix the parameters RA, dec, and polarization to the injected values. The network SNR for reference injection is shown in each panel.