Numerical relativity higher order gravitational waveforms of eccentric spinning nonprecessing binary black hole mergers Abhishek V. Joshi

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Numerical relativity higher order gravitational waveforms of eccentric, spinning,

nonprecessing binary black hole mergers

Abhishek V. Joshi ,1, 2, ∗Shawn G. Rosofsky,1, 2 Roland Haas ,1, 2 and E. A. Huerta 3, 4, 2

1NCSA, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA

2Department of Physics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA

3Data Science and Learning Division, Argonne National Laboratory, Lemont, Illinois 60439, USA

4Department of Computer Science, University of Chicago, Chicago, Illinois 60637, USA

We use the open source, community-driven, numerical relativity software, the Einstein Toolkit to

study the physics of eccentric, spinning, nonprecessing binary black hole mergers with mass-ratios

q={2,4,6}, individual dimensionless spin parameters χ1z=±0.6, χ2z=±0.3, that include higher

order gravitational wave modes `≤4, except for memory modes. Assuming stellar mass binary

black hole mergers that may be detectable by the advanced LIGO detectors, we ﬁnd that including

modes up to `= 4 increases the signal-to-noise of compact binaries between 3.5% to 35%, compared

to signals that only include the `=|m|= 2 mode. We use two waveform models, TEOBResumS

and SEOBNRE, which incorporate spin and eccentricity corrections in the waveform dynamics, to

quantify the orbital eccentricity of our numerical relativity catalog in a gauge-invariant manner

through ﬁtting factor calculations. Our ﬁndings indicate that the inclusion of higher order wave

modes has a measurable eﬀect in the recovery of moderately and highly eccentric black hole mergers,

and thus it is essential to develop waveform models and signal processing tools that accurately

describe the physics of these astrophysical sources.

I. INTRODUCTION

The modeling of eccentric compact binary mergers has

attracted signiﬁcant attention in recent years. The un-

derstanding of these astrophysical sources has gradually

increased through a variety of analytical and numerical

relativity studies that have shed new light into physics

of these systems, and the properties of the gravitational

wave signals that may be emitted by these sources [1–32].

Strides in the modeling and understanding of eccentric

compact binary mergers has been accompanied by pop-

ulation synthesis models [33–36] that have been signiﬁ-

cantly improved to be compatible with the observation

of stellar mass black holes in dense stellar environments,

such as globular clusters in our galaxy [37–39], and galac-

tic nuclei [40–42].

Impelled by these theoretical and observational ad-

vances, researchers have developed the required tools to

search for this astrophysical population in gravitational

wave data [43–48]. Some recent studies have attempted

to constrain the eccentricity of actual gravitational wave

sources [49]. A plethora of studies for the massive stellar

black hole merger named GW190521 [50] provide per-

suasive evidence for the existence of eccentric compact

binary mergers in dense stellar environments [28, 51, 52].

It is expected that several tens of eccentric compact bi-

nary mergers observed by advanced ground-based grav-

itational wave detectors will suﬃce to understand what

formation channels contribute or dominate the eccentric

merger rate [49].

In view of these developments, and the upcoming del-

uge of gravitational wave observations to be enabled by

∗avjoshi2@illinois.edu

advanced LIGO [53, 54] and its international counter-

parts VIRGO and KAGRA [47, 55, 56], it is timely and

relevant to continue developing adequate tools for the

identiﬁcation of gravitational wave signals that may be

produced by eccentric compact binary mergers.

The best tool at hand to gain insights about the

physics of eccentric binary black hole mergers is nu-

merical relativity, and thus we use the open source,

community-driven, numerical relativity software, the

Einstein Toolkit [57] to produce a suite of numerical rel-

ativity waveforms that describe eccentric, spinning, non-

precessing binary black hole mergers. Non-spinning, ec-

centric simulations were investigated in previous works

in [3, 46]. These waveforms include higher order modes

up to `≤4, except for memory modes. We use these

numerical relativity waveforms to carry out the following

studies:

•Gravitational wave detection We construct two

types of waveforms that include either quadrupole

modes, `=|m|= 2, or modes up to `≤4. We

assume stellar mass binary black holes that may

be observed by advanced LIGO-type detectors and

compute signal-to-noise ratio (SNR) calculations

for a variety of astrophysical scenarios, and explore

whether the inclusion of higher order wave modes

leads to measurable SNR increases.

•Gravitational wave modeling We use two

eﬀective-one-body (EOB) eccentric waveform mod-

els: TEOBResumS [58–65] and SEOBNRE [5, 66, 67] to

estimate the eccentricities of our numerical relativ-

ity waveforms. This exercise was useful to identify

areas of improvement for next generation waveform

models, and to get a better understanding of sig-

nals that may be discovered in upcoming gravita-

tional wave searches. Note that due to conven-

tions and diﬀerent deﬁnitions of eccentricity, the

arXiv:2210.01852v2 [gr-qc] 21 Mar 2023

inferred eccentricities cannot be directly compared

with each other. A detailed comparison between

the two waveform models is given in Knee et al.

[68].

•Parameter space degeneracy We quantiﬁed the

impact of including higher order modes in terms

of ﬁtting factor calculations that aim to pinpoint

an optimal quasicircular NRHybSur3dq8 waveform

signal [69] whose astrophysical parameters best re-

produce the complex morphology of moderately or

highly eccentric numerical relativity waveforms.

These three complementary studies underscore the im-

portance of improving our understanding of compact bi-

nary mergers in dense stellar environments. It is not

enough to hope for the best and expect that burst or ma-

chine learning searches identify complex signals in gravi-

tational wave data [46, 48]. It is also necessary to develop

a comprehensive toolkit that encompasses numerical rel-

ativity waveforms, semi-analytical or machine learning

based models, and signal processing tools to detect and

then infer the astrophysical properties of eccentric com-

pact binary mergers. Not doing so would be a disservice

to the proven detection capabilities of advanced gravita-

tional wave detectors, and would limit the science reach

of gravitational wave astrophysics. To contribute to this

important endeavor, we release our catalog of numerical

relativity waveforms along with this article.

This article is organized as follows. We describe our

approach to create a catalog of eccentric numerical rel-

ativity waveforms in Sec. II. Sec. IV presents our wave-

form catalog, and a systematic study on the importance

of including higher order wave modes in terms of SNR

calculations. In Sec. V we study whether surrogate mod-

els based on quasicircular, spinning, nonprecessing binary

black hole numerical relativity waveforms can capture the

physics of spinning, nonprecessing eccentric mergers. We

summarize our ﬁndings and future directions of work in

Sec. VI.

II. NUMERICAL SETUP AND SIMULATION

DETAILS

We used the Einstein Toolkit to generate a catalog

of numerical relativity waveforms. Initial data for the bi-

naries was computed using the TwoPunctures code. The

evolution was done with the CTGamma code implement-

ing the 3+1 BSSN formulation. The outer boundary of

the simulation domain was placed far enough (2500M) to

avoid any contamination of the signal until 200M after

the merger. Each simulation was run at three resolutions

to check for convergence (see appendix A): N= 36,40,44

where Nis the resolution across the ﬁnest grid radius.

The highest resolution simulations were used for all anal-

yses. Further details of the simulation setup are given in

[3]. Waveforms extracted at future null inﬁnity were com-

puted for 1 < l ≤4 and 1 ≤ |m| ≤ lmodes using the

POWER code [70] by extrapolating the observed signals

from 7 detectors located 100–700M. m= 0 modes were

not used since these modes (so-called memory modes)

are many orders of magnitude smaller than the dominant

modes of the waveform making a reliable estimation diﬃ-

cult due to numerical resolution (for more details see Sec.

6.2 in [71]). A plot of all the h+simulation waveforms

is shown in Fig. 1. Note that the simulations are also

dimensionalized in units of M.

Table I describes the properties of our waveform cata-

log, including the mass-ratio, individual spins and orbital

eccentricity of each binary (measured from both wave-

form templates). The library consists of 27 simulations

across 3 mass ratios, q={2,4,6}, and a combination of

nonprecessing individual spins, namely ±0.6 and ±0.3,

for the primary (heavier) and secondary (lighter) binary

components, respectively.

III. ECCENTRICITY MEASUREMENTS

Orbital eccentricity in a Keplerian interpretation can

only be deﬁned for a BBH system during the early in-

spiral, where the orbits of the binaries are nearly closed

(the adiabatic approximation). This deﬁnition breaks

down close to the merger, which is when our simulations

begin. Thus, the deﬁnitions of eccentricity used to gen-

erate the initial conditions are ill-deﬁned, even though

they produce eccentric simulations.

Using evolution information of the binary, such as the

separation between the components, throughout the sim-

ulation to obtain a measure of orbital eccentricity is not

useful, as such a concept is gauge-dependent by assum-

ing a coordinate system. To obtain a useful measure of

eccentricity, we calibrate our numerical simulations to

the spin-aligned eccentric EOB models TEOBResumS and

SEOBNRE. For both of these models, a reference eccentric-

ity e0and reference GW frequency fref are used as in-

puts to generate adiabatic initial conditions of the binary

from which the waveform is computed. As investigated

in Knee et al. [68], each waveform model’s deﬁnition of e0

may vary, due to diﬀerent conventions of fref and initial

condition constructions.

The method is similar to that used in [18, 66]. The key

idea consists of using `=|m|= 2 waveforms to compute

the ﬁtting factor between a given numerical relativity

waveform, and an array of templates. In this work, we

have assigned fref = 10Hz, which is at the lower end of

the detectability range for LIGO. To estimate the eccen-

tricity of our numerical relativity waveforms, we need to

compute a few objects. The ﬁrst of them is the inner

product between one of our numerical relativity wave-

forms, hNR

22 , and a waveform template, htemplate

22 , given

by:

hhNR

22 |htemplate

22 i=RZt2

hNR

22 h* template

22 .(1)

Where Rrepresents the real component. Note that the

FIG. 1. Numerical relativity waveform catalog Each column is associated with a given mass-ratio q={2,4,6}. From

top to bottom, simulations are ordered in (χ1z, χ2z). The eccentricity e0inferred from TEOBResumS is given in the label. Each

panel presents two types of waveforms: a `=|m|= 2 signal (orange), and one that includes higher order modes (blue). We

have selected the inclination of the binary that maximizes the contribution higher order modes.

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Numericalrelativityhigherordergravitationalwaveformsofeccentric,spinning,nonprecessingbinaryblackholemergersAbhishekV.Joshi,1,2,ShawnG.Rosofsky,1,2RolandHaas,1,2andE.A.Huerta3,4,21NCSA,UniversityofIllinoisatUrbana-Champaign,Urbana,Illinois61801,USA2DepartmentofPhysics,UniversityofIllinoisatUrbana-C...

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Numerical relativity higher order gravitational waveforms of eccentric spinning nonprecessing binary black hole mergers Abhishek V. Joshi

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