# Comparing Numerical Relativity and Perturbation Theory Waveforms for a Non-Spinning Equal-Mass Binary

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## Abstract

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## 1. Introduction

## 2. NR and ppBHPT Data at $\mathbf{q}=\mathbf{1}$

`BHPTNRSur1dq1e4`[45] model (Direct simulation using our time-domain Teukolsky solver fails in the equal-mass limit due to limitations in trajectory generation for the secondary black hole that incorporates adiabatic radiative corrections).

`BHPTNRSur1dq1e4`is a reduced-order surrogate model trained on ppBHPT waveform data generated from a time-domain Teukolsky equation sourced by a test particle whose adiabatic inspiral is driven by energy fluxes [29,30,31,52]. This model is interfaced through the

`BHPTNRSurrogate`package [53], available in the

`black hole perturbation`

`Toolkit`[54]. While the model has been trained for mass ratios $2.5\le q\le {10}^{4}$, surrogate models have previously demonstrated good performance when extrapolated beyond their training range [5,6]. In our case, extrapolation to $q=1$ will bring in error, but these errors are expected to be small [6].

## 3. Comparing NR and ppBHPT at $\mathbf{q}=\mathbf{1}$

#### 3.1. Model Calibration Setup

#### 3.2. Setting a Common Mass Scale

#### 3.3. Effectiveness of the $\alpha $-$\beta $ Scaling at $q=1$

#### 3.4. Exploring the Time Dependence of $\alpha $-$\beta $ Parameters from Different Methods

#### 3.5. Understanding $\alpha $-$\beta $ Values at the Ringdown

## 4. Discussion and Conclusions

`BHPTNRSur1dq1e4`. We leave this for future work.

`BHPTNRSur1dq1e4`(a surrogate model) outside its training range. Yet direct simulation of a $q=1$ system with the ppBHPT framework was not possible as our time-domain Teukolsky solver [29,30,31,52] breaks down at the equal-mass limit (more specifically, the trajectory generation for the secondary black hole that incorporates adiabatic radiative corrections fails in the equal-mass limit). While surrogate models can be extrapolated a bit beyond their training range [5,6], this will bring in additional errors that might compromise studies that require very high accuracy.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**A non-spinning $q=1$ waveform for the $(2,2)$ mode from numerical relativity (black dashed line), ppBHPT (grey line), and the calibrated ppBHPT (red lines). The left panel shows the waveforms for the early inspiral, while the right panel focuses on the merger-ringdown stage. All waveforms have the same mass scale of $M={m}_{1}+{m}_{2}$.

**Figure 2.**The waveform’s $(2,2)$ mode from NR (black dashed line) and scaled ppBHPT (red solid line) calibrated over a short 5-cycle window of data: $t\in [-\mathrm{13,526}M,-\mathrm{10,983}M]$ and $t\in [-778.5M,-59.2M]$. This demonstrates that the $\alpha $ and $\beta $ parameters are nearly time-independent over in the early inspiral portion, an assumption that breaks down in the late inspiral. Further details can be found in Section 3.4.

**Figure 3.**Calibration parameters, $\alpha $ and $\beta $, for a $q=1$ BBH system obtained from different approaches outlined in Section 3.4; these are labeled {“5cycles”, “5000M”, “peaks”}. For comparison, we also show the scaling of $\frac{1}{1+1/q}=0.5$ required to change the ppBHPT mass scale to match the NR one. If $\alpha =\beta =0.5$, then the NR and ppBHPT waveform’s $(2,2)$ modes are identical. More details can be found in Section 3.4.

**Figure 4.**Time derivative of the scaling parameters ${\alpha}_{\mathrm{peak}}$ and ${\beta}_{\mathrm{peak}}$ obtained locally using the “peaks” method summarized in Section 3.4. While the derivatives are not zero, they remain small and increase in the late-inspiral stage. This explains why the $\alpha $-$\beta $ calibration technique, which assumes time-independent values for $\alpha $ and $\beta $, works particularly well in the inspiral. Further details can be found in Section 3.4.

**Figure 5.**The ringdown waveform’s $(2,2)$ mode from NR (black dashed line), ppBHPT waveform (blue solid line), ppBHPT waveform rescaled using only $\alpha $ and $\beta $ parameters determined from the inspiral data (red solid line), and ppBHPT waveform rescaled using $\alpha $ and $\beta $ parameters determined from the ringdown data (green solid line). While the $\alpha $ and $\beta $ parameters determined from the inspiral are unsuitable for the ringdown, a different set of parameter values—constant over the final $100M$ of the signal—can again deliver an accurate calibration. Further details can be found in Section 3.5.

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**MDPI and ACS Style**

Islam, T.; Field, S.E.; Khanna, G.
Comparing Numerical Relativity and Perturbation Theory Waveforms for a Non-Spinning Equal-Mass Binary. *Universe* **2024**, *10*, 25.
https://doi.org/10.3390/universe10010025

**AMA Style**

Islam T, Field SE, Khanna G.
Comparing Numerical Relativity and Perturbation Theory Waveforms for a Non-Spinning Equal-Mass Binary. *Universe*. 2024; 10(1):25.
https://doi.org/10.3390/universe10010025

**Chicago/Turabian Style**

Islam, Tousif, Scott E. Field, and Gaurav Khanna.
2024. "Comparing Numerical Relativity and Perturbation Theory Waveforms for a Non-Spinning Equal-Mass Binary" *Universe* 10, no. 1: 25.
https://doi.org/10.3390/universe10010025