Coping with Junk Radiation in Binary Black Hole Simulations

TL;DR

This paper investigates the impact of spurious junk radiation in binary black hole simulations, providing methods to estimate and correct its effects on initial parameters to improve gravitational wave template accuracy.

Contribution

It introduces fitting formulas based on single black hole simulations to estimate and adjust for junk radiation effects in binary black hole initial data.

Findings

Junk radiation effects are localized near black holes.

Fitting formulas can estimate changes in mass and spin due to junk radiation.

Adjusted initial parameters improve waveform accuracy for high SNR LIGO sources.

Abstract

Spurious junk radiation in the initial data for binary black hole numerical simulations has been an issue of concern. The radiation affects the masses and spins of the black holes, modifying their orbital dynamics and thus potentially compromising the accuracy of templates used in gravitational wave analysis. Our study finds that junk radiation effects are localized to the vicinity of the black holes. Using insights from single black hole simulations, we obtain fitting formulas to estimate the changes from junk radiation on the mass and spin magnitude of the black holes in binary systems. We demonstrate how these fitting formulas could be used to adjust the initial masses and spin magnitudes of the black holes, so the resulting binary has the desired parameters after the junk radiation has left the computational domain. A comparison of waveforms from raw simulations with those from simulations that have been adjusted for junk radiation demonstrate that junk radiation could have an appreciable effect on the templates for LIGO sources with SNRs above 30.

Tables

Table 1. Table 1: Fitting coefficients for δ ​ M i 𝛿 subscript 𝑀 𝑖 \delta M_{i} and δ ​ a 𝛿 𝑎 \delta a in Eq.  5 .

$c_{ij}$	$\delta M_i$	$\delta a$	$c_{ij}$	$\delta M_i$	$\delta a$
$c_{00}$	-0.09935	0.1294	$c_{40}$	-65.83	79.59
$c_{10}$	2.148	-3.169	$c_{31}$	-4.27	13.47
$c_{01}$	0.05612	-0.0977	$c_{22}$	-1.565	-5.56
$c_{20}$	-14.92	21.01	$c_{13}$	3.918	-13.17
$c_{11}$	-0.4674	1.118	$c_{04}$	-21.74	288.9
$c_{02}$	-0.6845	7.696	$c_{50}$	36.64	-40.72
$c_{30}$	46.3	-61.24	$c_{41}$	2.524	-7.763
$c_{21}$	2.386	-7.675	$c_{32}$	2.931	2.056
$c_{12}$	-0.2141	2.781	$c_{23}$	-0.7278	-35.24
$c_{03}$	5.791	-15.01	$c_{14}$	-5.623	12.7
			$c_{05}$	29.78	-413.8

Table 2. Table 2: Mass of the BH, its irreducible mass and spin magnitude at the beginning of the simulation and after the junk radiation has dissipated.

Type	$t=0$			$t>t_{\text{junk}}$
	$M_h/M$	$M_i/M$	$a$	$M_h/M$	$M_i/M$	$a$
raw	0.4994	0.4736	0.6013	0.5004	0.4744	0.6032
adj	0.4985	0.4729	0.6000	0.4994	0.4736	0.6018
raw	0.4972	0.4430	0.8090	0.5001	0.4471	0.8013
adj	0.4939	0.4387	0.8161	0.4971	0.4431	0.8079

Equations

$$\tilde{\Delta }\Phi +\frac{1}{8}{\Phi }^{-7}{\tilde{A}}_{ij}{\tilde{A}}^{ij}=0.$$

$${\tilde{A}}^{ij}$$

$$\Phi =1+\frac{m_1}{|r-r_1|}+\frac{m_2}{|r-r_2|}+u$$

$$F$$

$$M_i^{1,2}$$

$$a^{1,2}$$

Full text

Introduction: This letter presents a method to deal with the spurious junk radiation present in puncture-type initial data for binary black hole (BBH) simulations.

When Einstein’s theory of General Relativity is viewed as an initial-value problem, the initial data consist of the spatial metric $\gamma_{ij}$ and the extrinsic curvature $K_{ij}$ of the initial space-like hypersurface in the space-time foliation. In the pair $\{\gamma_{ij},K_{ij}\}$, not everything is freely specifiable. Four pieces are fixed by the Hamiltonian and momentum constraints. The York-Lichnerowicz approach Baumgarte and Shapiro (2010); Smarr (1979) provides a path to identify those four pieces via conformal transformations and tranverse-traceless decompositions. With this approach, after assuming conformal flatness ($\gamma_{ij}=\Phi^{4}\eta_{ij}$) and vanishing trace of the extrinsic curvature ($K^{i}_{i}=0$), the Hamiltonian constraint for vacuum space-times reads:

[TABLE]

Here, tildes denote tensors and operators in conformal space, and $\tilde{A}_{ij}$ is the conformal trace-free extrinsic curvature satisfying $\widetilde{\nabla}_{i}\tilde{A}^{ij}=0$, namely the momentum constraint.

To construct puncture-type initial data representing BBHs, one uses the Bowen-York  Bowen and York (1980) point-source solutions to $\widetilde{\nabla}_{i}\tilde{A}^{ij}=0$:

[TABLE]

with $l^{i}=x^{i}/r$ and $\epsilon^{ijk}$ the Levi-Civita symbol. Also, $P^{i}$ and $S^{i}$ are the linear and angular momentum of the point-source, respectively. Given (2) and (3), Eq. (1) is solved using the puncture approach introduced by Brandt and Brügmann Brandt and Bruegmann (1997). The essence of this approach is to factor out the black hole (BH) singularity, namely

[TABLE]

with $r_{1,2}$ the locations of the BHs and $u$ a regular function. The parameters $m_{1,2}$ are commonly referred as the bare or puncture masses. Ansorg Ansorg et al. (2004) developed an elegant solver for Eq. (1) based on spectral methods called 2Punctures. The solver is one of the most widely used codes by the numerical relativity (NR) community.

To construct astrophysically relevant BBH initial data, one thus needs to provide the values for the parameters $m_{1,2}$, $r_{1,2}$, $P^{i}_{1,2}$ and $S^{i}_{1,2}$. The most common approach for this is with the assistance of post-Newtonian (PN) approximations. PN equations of motion are used to evolve the BBH of interest from large separations until the separation at which the NR evolution will start. The parameters of the BBH at the end of the PN evolution are used as input parameters to solve Eq. (1). There is a subtlety here. The bare puncture masses $m_{1,2}$ are not the masses of the BHs. As first guess for these parameters, one uses the PN BH masses, but iterations are needed to adjust the puncture bare masses until the masses of the BHs match the desired PN masses.

There is an issue with the puncture BBH initial data as just described. For sufficiently large binary separations, one would expect the space-time in the neighborhood of each BH to be close to a boosted Kerr solution, or boosted Schwarzschild solution if the BH is not spinning. This is not the case for the puncture data with Bowen-York extrinsic curvatures, not even for a single BH. The reason for this are the conformal flatness assumption and the Bowen-York extrinsic curvatures. The space-time of a boosted or a Kerr BH is not conformally flat. Also, the Bowen-York extrinsic curvatures are not the extrinsic curvatures for a boosted or Kerr BH. As a consequence, the puncture Bowen-York initial data contain spurious or junk radiation.

In numerical evolutions, junk radiation manifests itself as a burst. Figure 1 depicts an example in terms of the Weyl scalar $\Psi_{4}$ as a function of time for the $l=2$, $m=2$ mode. There is an ongoing debate in the NR community about the extent to which the junk radiation introduces appreciable changes to the binary, specifically changes to the spins and masses of the BHs, and thus to the orbital dynamics of the binary Hannam et al. (2007); Lovelace (2009); Johnson-McDaniel et al. (2009); Chu et al. (2009); Kelly et al. (2010); Zhang and Szilágyi (2013); Slinker et al. (2018); Ruchlin (2015); Sperhake et al. (2011). Some of the studies attempt to tame the junk radiation by moving away from conformal flatness Lovelace (2009); Johnson-McDaniel et al. (2009), others introduce explicitly PN corrections Kelly et al. (2010). Our view here is to obtain first a detailed characterization of the effects from junk radiation on the holes and then introduce adjustments in the input parameters of the binary that anticipates the changes produced by the junk radiation. The expectation is that, after the junk radiation dissipates away, one is left with the BBH system one originally intends to have.

Waveform Analysis: Our work is based on the Georgia Tech catalogue of BBH simulations Jani et al. (2016). The first step we took was to monitor in our simulations the behavior of the masses and spins of the BHs in a window between the initial time of the simulation and the end of the burst of junk radiation. Figure 2 shows the percent change $\delta M_{i}$ in the initial irreducible mass of the BHs as a function of time for three binary simulations from the catalogue: GT0406, GT0407 and GT0866. The irreducible mass $M_{i}$ is computed from the area $A$ of BH horizon: $M_{i}=\sqrt{A/4\,\pi}$. It is evident in Fig. 2 the increase in the masses of the BHs during the first $20\,M$ of the simulation. This trend was observed in all the simulations for which we tracked the masses of the holes. Similar jumps were also observed in the spins.

Learning from single black hole simulations: Next we investigated whether the jumps in mass and spin were due to local effects in the neighborhood of the BHs or if they involved correlations between the holes in the binaries. To answer this question, we looked at the effects of junk radiation on a single BH. We carried out simulations expanding the dimensionless spin parameter $a=S/M_{h}^{2}$ in the range $0\leq a\leq 0.8$ and the speed $v=P/M_{h}$ in the range $0\leq v\leq 0.3$, with $S$ the angular momentum, $P$ linear momentum, and $M_{h}$ the mass of the BH, where $M_{h}^{2}=M_{i}^{2}+S^{2}/(2M_{i})^{2}$. The spin of the BH was aligned with the $z$-direction and the momentum with the $y$-direction. To a good approximation, these configurations cover the initial setups of BHs in the non-precessing BBH simulations in our catalogue.

Not surprisingly, the single BH simulations also showed increases in mass and spin. Furthermore, those increases took place, as with the BBH simulations, during the first $20M$ of the simulation time. This suggests that the effects of junk radiation are localized near the hole and do not depend on the presence of the other hole.

Figure 3 shows with points the percentage change in the BH irreducible mass $\delta M_{i}$ and $\delta a$ for all the single BH simulations as a function of the dimensionless spin $a$ and speed $v$. The grey surfaces in Figure 3 are fits to the data where we use the following fitting function:

[TABLE]

The coefficients for the fit to $\delta M_{i}$ and $\delta a$ are given in Table 1.

Interestingly, from Fig. 3a, the effect of junk radiation on the mass correlates stronger with the initial spin than with the speed of the puncture. This is not the case with the effect on the spin of the BH. As it can be seen from Fig. 3b, the junk radiation reduces the spin for larger initial spin and increases the spin for larger speeds. As a consequence, there is a family of cases for which the effects cancel out. These are the cases when the surface in Fig. 3b intersects the $\delta a=0$ plane, and they are denoted with the black line. Another important finding was that the junk radiation only affected the magnitude of the spin but not its direction.

Connecting with binary black holes simulations: Once we quantified the changes in mass and spin for the single BH simulations, the next step was to investigate whether these changes are the same as those observed in each of the holes in BBH simulations. We looked at 107 binary simulations in our catalogue: 67 precessing binaries, and 40 aligned spin, non-precessing binaries. Figures 4a and 4b show with points $\delta M_{i}$ and $\delta a$ for the BHs in the BBHs (red for aligned spins and green points for precessing binaries). Grey surfaces denote the single BH fit. Figures 4c and 4d show the corresponding residuals. The residuals in the masses are $O(10^{-2})\,\%$, and for $v\leq 0.24$ the spin residuals are $O(10^{-1})\,\%$. On the other hand, for $v\gtrsim 0.24$ the spin residuals are $O(1)\,\%$ and all negative. Since the residuals are data-fit, this implies that the single BH fit overestimates the effect of junk radiation for $v\gtrsim 0.24$. For reference, BHs with $v\approx 0.24$ involve binary systems with initial separations of $\approx 10\,M$ and gravitational wave frequency $\omega\,M\approx 0.048$. The levels of residuals for $\delta M_{i}$ and for $\delta a$ if $v\lesssim 0.24$ give us confidence that the fitting formula derived from single BH simulations provides good estimates applicable to binary simulations.

When to worry about junk radiation: Finally, we present a couple of examples of how the single BH junk radiation fits can be used in BBH simulations. From the waveforms in these simulations, we quantify whether one needs to worry about the effects of junk radiation in gravitational wave analysis for LIGO and LISA sources.

Assuming that one wants to simulate a binary with BHs having irreducible masses $M_{i}^{1,2}$, spins $a^{1,2}$ and speeds $v^{1,2}$, the task is to find irreducible masses $\bar{M}_{i}^{1,2}$ and spins $\bar{a}^{1,2}$ to use in the initial data such that the junk radiation modifies these values and yields the desired values $M_{i}^{1,2}$ and $a^{1,2}$. The adjusted values $\bar{M}_{i}^{1,2}$ and $\bar{a}^{1,2}$ can be found by solving the following equations:

[TABLE]

where the junk radiation changes in the formulas above are given as fractional changes not percentages. As mentioned before, for BHs moving with speeds $\gtrsim 0.24$, the formulas overestimate the correction on the spin magnitude. For those cases, we estimate that only a 10% correction should be applied.

Figure 5 depicts the $l=2$, $m=2$ mode of Weyl scalar $\Psi_{4}$ for two pairs of simulations. All four simulations consist of equal mass, aligned spin binaries. The top panel shows the cases with $a\approx 0.6$ and the bottom for $a\approx 0.8$. In blue is the waveform from the raw simulation and in orange the waveform in which the masses and spins have been adjusted according to the single BH fitting formulas. The mass of the BH, its irreducible mass and spin magnitude at the beginning of the simulation and after the junk radiation has dissipated are given in Table 2. Notice that with the junk adjustment $M_{i}/M(t>t_{\text{junk}};\text{adj})\approx M_{i}/M(t=0;\text{raw})$ and $a(t>t_{\text{junk}};\text{adj})\approx a(t=0;\text{raw})$, as needed.

The mismatches $\epsilon$ between the waveform from raw and adjusted simulations in Advanced LIGO are shown in Figure 6. For $a\approx 0.8$, $\epsilon\sim 10^{-3}$ in average, while for $a\approx 0.6$, $\epsilon\sim 10^{-5}$. To avoid astrophysical inference with a bias, one needs $\epsilon\,\rho^{2}\lesssim 1$ Lindblom et al. (2008). This implies that waveforms of sources with highly spinning BH measured with $S/N$ of $\rho\gtrsim 30$ will exhibit inference biases if they are not corrected for the junk radiation. For future detectors such as the Einstein Telescope and LISA, even low spin sources with $\rho\gtrsim 10^{2}$ would require junk radiation correction.

Conclusion: Using simulations of single punctures with different spins and linear momentum, we have investigated the effect of junk radiation on the mass and spin of the BH. We found that junk radiation does not affect the direction of the spin. With these results, we obtained fitting functions that can be used to predict changes in the masses and spin magnitudes of BHs in binary systems. We tested the effectiveness of the fitting functions and showed that for binary systems with highly spinning BHs, inference biases would be introduced by waveforms that not are corrected for junk radiation in sources with SNRs $\gtrsim 30$.

Acknowlegements: Work supported by NSF grants 1806580, 1809572, 1550461, and XSEDE allocation TG-PHY120016.