Gravitational-wave parameter inference with the Newman-Penrose scalar $ψ_4$

TL;DR

We introduce a framework for gravitational-wave parameter inference that operates directly on the Newman-Penrose scalar $\psi_4$, avoiding the problematic double integration required to obtain strain $h(t)$. By using second-order finite differences and a precise Fourier-domain mapping, we define $\Psi_4$-based templates and the corresponding noise PSD, enabling Bayesian inference and model comparison without integration artefacts. Our tests with head-on Proca-star mergers and real events GW190521 and S200114f show that integration artefacts can bias interpretation in some scenarios, while the $\psi_4$-based approach yields artefact-free, robust inferences and can reveal or constrain exotic compact-object scenarios. The method has broad applicability to non-quasi-circular mergers and offers a principled, gauge-invariant route to test beyond-Kerr/BBH physics, with implications for surrogate modeling and NR-data comparisons.

Abstract

Detection and parameter inference of gravitational-wave signals \ncor{from compact mergers} rely on the comparison of the incoming detector strain data $d(t)$ to waveform templates for the gravitational-wave strain $h(t)$ that ultimately rely on the resolution of Einstein's equations via numerical relativity simulations. These, however, commonly output a quantity known as the Newman-Penrose scalar $ψ_4(t)$ which, under the Bondi gauge, is related to the gravitational-wave strain by $ψ_4(t)=\mathrm{d}^2h(t) / \mathrm{d}t^2$. Therefore, obtaining strain templates involves an integration process that introduces artefacts that need to be treated in a rather manual way. By taking second-order finite differences on the detector data and inferring the corresponding background noise distribution, we develop a framework to perform gravitational-wave data analysis directly using $ψ_4(t)$ templates. We first demonstrate this formalism, and the impact of integration artefacts in strain templates, through the recovery of numerically simulated signals from head-on collisions of Proca stars injected in Advanced LIGO noise. Next, we re-analyse the event GW190521 under the hypothesis of a Proca-star merger, obtaining results equivalent to those in Ref.[1], where we used the classical strain framework. We find, however, that integration errors would strongly impact our analysis if GW190521 was four times louder. Finally, we show that our framework fixes significant biases in the interpretation of the high-mass GW trigger S200114f arising from the usage of strain templates. We remove the need to obtain strain waveforms from numerical relativity simulations, avoiding the associated systematic errors.

Figures (11)

• Figure 1: Schematic comparison of our proposed data analysis framework and the currently used one. To date, the $\psi_4$ magnitude outputted by numerical relativity simulations is converted to the strain $h$ outputted by gravitational-wave detectors via a double integration that is subject to systematic errors (red path). Instead, we transform both the simulation $\psi_4$ and the detector $h$ (and power-spectral-density $S_n$) into a third quantity that we label by $\Psi_4$, avoiding the integration process and the corresponding systematic errors (green paths).
• Figure 2: Demonstration of our transformation and whitening scheme on sine-Gaussian pulses. The left panel shows the analytical second derivative $\ddot{h}(t)$ of a sine-Gaussian strain time-series $h(t)$ and its second-order finite difference time-series $\delta^2 h(t)$. We obtain the latter both directly and from correcting $\ddot{h}(t)$ via Eq. \ref{['eq:psiPsi']}. The inset of the right panel shows the difference between the latter two time-series, whitened with a PSD $S_{n_{\Psi_4}}$, and the original strain $h(t)$ whitened by the corresponding $S_{n}$. These are of the order of 1 part in $10^{12}$. The main panel shows the (much larger) difference between the whitened strain and second derivative $\ddot{h}(t)$ whitened with $S_{n_{\Psi_4}}$.
• Figure 3: Whitening of strain and $\Psi_4$ detector data. Left: whitened strain and $\Psi_4$ gravitational-wave time-series from the Livingston detector around the time of GW190521. Right: difference between the absolute values of the corresponding Fourier-domain data. These are at the level of one part in $10^{12}$, so that both whitened detector data are equivalent for all practical purposes.
• Figure 4: Whitening of strain and $\Psi_4$ templates.Left: We show the raw time-domain data for the case of a) $\Psi_4$ directly coming from a numerical relativity simulation (through Eq. \ref{['eq:psiPsi']}) of a head-on Proca-star merger consistent with GW190521 (black), b) the strain obtained from $\psi_4$ through double integration (blue) and c) the $\Psi_4$ obtained from the latter through second-order finite differencing, denoted by $\delta^2 h(t)$. The strain in the left panel has been conveniently scaled to note the obvious morphological differences with respect to $\Psi_4$. Right: corresponding whitened time-series. The zoomed boxes show how the $h(t)$ and $\delta^2 h(t)$ are exactly identical while very small differences can be observed with respect to the original $\Psi_4$.
• Figure 5: Whitening of strain and $\Psi_4$ templates. Impact of aggressive choice of $\omega_0$. Same as in Fig. \ref{['fig:whitening_21gtemplate-1']} but for a waveform template consistent with S200114f O3IMBHpsi4_observations. In this case, the differences between the $\Psi_4$ directly extracted from the numerical simulation and the other two waveforms are significantly more noticeable.

Theorems & Definitions (6)

• Theorem A.1
• proof
• Theorem B.1
• proof
• Lemma B.2
• proof