# High accuracy PN and NR comparisons involving higher modes for eccentric
BBHs and a dominant mode eccentric IMR model

Abhishek Chattaraj<EMAIL_ADDRESS>Department of Physics, Indian
Institute of Technology Madras, Chennai 600036, India  Centre for Strings,
Gravitation and Cosmology, Department of Physics, Indian Institute of
Technology Madras, Chennai 600036, India  Tamal RoyChowdhury
<EMAIL_ADDRESS>Department of Physics, Indian Institute of Technology
Madras, Chennai 600036, India  Centre for Strings, Gravitation and Cosmology,
Department of Physics, Indian Institute of Technology Madras, Chennai 600036,
India  Divyajyoti<EMAIL_ADDRESS>Department of Physics, Indian
Institute of Technology Madras, Chennai 600036, India  Centre for Strings,
Gravitation and Cosmology, Department of Physics, Indian Institute of
Technology Madras, Chennai 600036, India  Chandra Kant Mishra<EMAIL_ADDRESS>Department of Physics, Indian Institute of Technology Madras, Chennai 600036,
India  Centre for Strings, Gravitation and Cosmology, Department of Physics,
Indian Institute of Technology Madras, Chennai 600036, India  Anshu Gupta
<EMAIL_ADDRESS>Inter-University Centre for Astronomy and Astrophysics, Post
Bag 4, Ganeshkhind, Pune 411 007, India

###### Abstract

Spherical harmonic modes of gravitational waveforms for inspiraling compact
binaries in eccentric orbits from post-Newtonian (PN) theory accurate to third
post-Newtonian order, and those extracted from Numerical Relativity (NR)
simulations for binary black holes (BBHs) are compared. We combine results
from the two approaches (PN and NR) to construct time-domain hybrid waveforms
that describe the complete evolution of BBH mergers through inspiral(I),
merger(M) and the ringdown(R) stages. These hybrids are then used in
constructing a fully analytical dominant mode ($\ell$=2, $|m|$=2) eccentric
IMR model. Overlaps with quasi-circular IMR waveform models including the
effect of higher modes, maximized over a time- and phase-shift, hint at the
importance (mismatches $>1\%$) of including eccentricity in gravitational
waveforms when analysing BBHs lighter than $\sim 80M_{\odot}$, irrespective of
the binary’s eccentricity (as it enters the LIGO bands), or mass-ratio.
Combined impact of eccentricity and higher modes seems to become more apparent
through smaller overlaps with increasing inclination angles and mass ratios.
Finally, we show that the state-of-the-art quasi-circular models including the
effect of higher modes will not be adequate in extracting source properties
for signals with initial eccentricities $e_{0}$ $\gtrsim 0.1$.

##  I Introduction and summary

Since the _first_ detection of gravitational waves (GWs) from the merger of
two black holes Abbott  _et al._ (2016a), the LIGO-Virgo-KAGRA (LVK)
collaboration has reported over 90 compact binary mergers Abbott  _et al._
(2019a, 2021a, 2021b, 2021c). These include two confirmed mergers of neutron
stars Abbott  _et al._ (2017a, 2020a) and two neutron star-black hole mergers
Abbott  _et al._ (2021d) apart from close to 85 binary black hole (BBH)
mergers (see Ref. LSC  for a complete catalog). Besides providing numerous new
insights into the compact binary physics and astrophysics, these observations
have proved to be unique probes into binary’s ultra-relativistic dynamics
Abbott  _et al._ (2016b, 2019b, 2019c, 2021e) and have improved our
understanding of the underlying astrophysical population of these objects
Abbott  _et al._ (2019d, 2021f, 2021g). However, questions linked with compact
binary formation channels largely remain unanswered Abbott  _et al._ (2016c,
2020b) (see also Ref. Abbott  _et al._ (2021f) and references therein). A
definitive answer to these although can come from the measurements of the
orbital eccentricity Lower  _et al._ (2018).

Current template-based search methods make use of quasi-circular templates
owing to the expected circularisation of compact binary orbits due to
radiation reaction forces Peters (1964). However, binaries formed through the
dynamical interactions in dense stellar environments or through Kozai-Lidov
processes Kozai (1962); Lidov (1962) (if part of stable triples), are likely
to be observed in ground based detectors such as advanced LIGO Aasi  _et al._
(2015) and advanced Virgo Acernese (2015) with residual eccentricities
$e_{0}\gtrsim 0.1$ Abbott  _et al._ (2019e). In fact, the first ever
observation of an intermediate mass black hole, GW190521 Abbott  _et al._
(2020c), hints at it being an eccentric merger of two black holes Abbott  _et
al._ (2020b) (see also Kimball  _et al._ (2021); Romero-Shaw  _et al._ (2021);
O’Shea and Kumar (2021) which also discusses other events with signs of
eccentricity apart from the event GW190521). While quasi-circular templates
should be able to detect systems with initial eccentricities $e_{0}\lesssim
0.1$, binaries with larger eccentricities would require constructing templates
including the effect of eccentricity Brown and Zimmerman (2010); Huerta and
Brown (2013). Moreover, presence of even smaller eccentricities ($e_{0}\sim$
0.01-0.05) can induce significant systematic biases in extracting source
properties Abbott  _et al._ (2017b); Favata  _et al._ (2022). Furthermore,
future ground based detectors, Cosmic Explorer (CE) McClelland  _et al._
(2016); Dwyer  _et al._ (2015); Abbott  _et al._ (2017c) and Einstein
Telescope (ET) Punturo  _et al._ (2010); Hild  _et al._ (2011), due to their
low frequency sensitivities should frequently observe systems with detectable
eccentricities Lower  _et al._ (2018); Tibrewal  _et al._ .

Even though inspiral waveforms from eccentric binary mergers involving non-
spinning compact components are sufficiently accurate Mishra  _et al._ (2015);
Moore  _et al._ (2016); Tanay  _et al._ (2016); Boetzel  _et al._ (2019);
Ebersold  _et al._ (2019); Königsdörffer and Gopakumar (2006); Moore and Yunes
(2019), waveform models including contributions from merger and ringdown
stages compared to quasi-circular versions are less developed. Numerous
efforts towards constructing eccentric inspiral-merger-ringdown waveforms
useful for data analysis purposes are underway Hinder  _et al._ (2018); Huerta
_et al._ (2017); Chen  _et al._ (2021); Chiaramello and Nagar (2020). However,
they do not include important physical effects such as spins and higher order
modes. While one can argue that since most mergers observed so far are
consistent with a zero-effective spin111Only nine of the 44 BBHs reported in
GWTC-2 catalog Abbott  _et al._ (2021a) have been identified with a positive
effective spin parameter with zero outside the 95% credible interval. Abbott
_et al._ (2019d, 2021f), models neglecting spin effects can still be
useful222Ref. O’Shea and Kumar (2021) explores correlations between binary’s
spins and eccentricity. Huerta  _et al._ (2017), modelling of higher order
modes seems necessary as far as full IMR eccentric waveform models are
concerned as Ref. Rebei  _et al._ (2019) argues and also discussed in detail
in the current work. Only recently an eccentric version of the EOB (effective-
one-body) waveforms including higher modes Ramos-Buades  _et al._ (2022)
appeared online. In the absence of reliable inspiral-merger-ringdown models
for eccentric mergers sub-optimal search methods (with little or no dependence
on signal model being searched) are used Abbott  _et al._ (2019e). While these
methods should detect binaries with arbitrary eccentricities, these are
sensitive to high mass searches (typically $\gtrsim 50M_{\odot}$) Abbott  _et
al._ (2019e), while most observed events have a mass smaller than this limit
Abbott  _et al._ (2021f, b); see for instance Fig. 3 of Divyajyoti  _et al._
(2021).

###  I.1 Summary of the Current Work

The current work assesses the parameter space where the eccentricity and
associated higher modes become crucial in GW parameter estimation. This is
illustrated through an injection analysis presented in Fig. 9 attempting
recovery of an eccentric signal with quasi-circular waveforms (see Sec. IV for
details). Both the injections and the recovery templates include the same set
of modes to avoid biases due to additional modes in target/template waveforms.
Figure 9 also shows recovery of a circular injection as a reference. Both
circular and eccentric injections correspond to a BBH system of total mass
40$M_{\odot}$, and the eccentric simulations have an orbital eccentricity of
$e_{0}\sim 0.1$ at 20Hz. Non-recovery of the injected value of the chirp mass
for the eccentric case can be interpreted as the bias induced due to the
neglect of eccentricity and associated higher modes.

The injected waveforms have been computed by matching the post-Newtonian
inspiral waveforms for individual modes Boetzel  _et al._ (2019); Ebersold
_et al._ (2019); Tanay  _et al._ (2016); Moore  _et al._ (2016) with those
extracted from eccentric numerical relativity simulations of the Simulating
eXtreme Spacetimes (SXS) collaboration Hinder  _et al._ (2018) following
extensive comparisons involving waveforms due to the two approaches performed
here (see Fig. 1). The construction of the injected waveforms (or hybrids as
we refer them through the paper) being longer in length compared to NR
simulations prove to be critical in accessing the impact of higher modes and
eccentricity in the entire mass range accessible to ground based detectors
such as LIGO and Virgo. A set of 20 eccentric hybrids have been constructed
with varying initial eccentricity in the range $0.1\lesssim e_{0}\lesssim 0.4$
and with mass ratios $q$ = $1,2,3$. All the hybrids have a start frequency of
$x_{0}$ = $0.045$ and typically have 30-40 orbital cycles before the merger.
See Table II.1 for the details.

The inadequacy of quasi-circular templates in extracting an eccentric IMR
signal including higher modes is also demonstrated through (simple) mismatch
calculations (see Fig. 3) using state-of-the-art quasi-circular waveform
families including higher modes SEOBNRv4HM Cotesta  _et al._ (2020) and
IMRPhenomXHM García-Quirós  _et al._ (2020). These mismatch plots indicate the
need for including eccentricity in gravitational waveforms for analysing BBH
systems with masses below $80M_{\odot}$, irrespective of eccentricity or mass
ratio. Additionally, as should be clear from middle and right top/bottom
panels of Fig. 3 with increasing orbit’s inclination w.r.t our line of sight,
the combined impact of eccentricity and related higher modes become more
apparent.

Figure 1: Amplitude and frequency of selected modes from an eccentric NR
simulation (SXS:BBH:1364) together with an Eccentric post-Newtonian (PN) model
are plotted. The eccentricity (mean anomaly) measured roughly $\sim 7$ orbits
before the merger at a reference frequency of $\sim 0.075$ is 0.044 (2.144)
and binary’s mass ratio ($q$) is $2$ (see Table I of Hinder  _et al._ (2018)
for details). The PN model is evolved assuming an initial eccentricity of
$e_{0}$=0.108, and mean anomaly of $l_{0}$=-2.495 (measured at an initial
frequency of $x_{0}$=0.045 using the prescription of Tanay  _et al._ (2016)).
Since the mode amplitudes and frequencies are in reasonable agreements for PN
and NR waveforms in a time window of ($1000M$, $2000M$) it should be possible
to perform hybridization in this window.

Further, a fully analytical dominant mode ($\ell,|m|$) = (2, 2) model obtained
by matching an eccentric PN inspiral with a quasi-circular prescription for
the merger-ringdown phase calibrated against the 20 eccentric hybrids
constructed here is presented. The method used for construction of the model
and other relevant details are discussed in Sec. III. The performance of the
model can be assessed from the plots presented in Fig. 7 as well as from the
mismatch plots displayed in the left panel of Fig. 8. Since all 20 hybrids
were utilised in calibrating the model, these models are tested against an
independent family of eccentric waveforms ENIGMA Chen  _et al._ (2021). The
mismatch plots are shown in the right panel of Fig. 8.

This paper is structured as follows: In Sec. II we discuss the PN and NR
inputs as well as the method for combining them for constructing the hybrids.
Section II.4 discusses overlaps (or match maximized over time and phase
shifts) between our target hybrids and quasi-circular IMR waveforms and
eccentric templates. Section III describes the method used for construction of
the dominant mode model. In Sec. IV, we discuss systematic biases in parameter
estimation studies due to presence of eccentricity and associated higher mode
contributions. Finally, we conclude our work in Sec. V and summarize our
findings.

##  II Methodology

###  II.1 PN inspiral and NR inputs

Count | Simulation Id | $q$ | $e_{0}$ | $l_{0}$ | $N_{\rm orbs}$
---|---|---|---|---|---
1 | SXS:BBH:1132 | 1 | 0.000 | 2.852 | 53.3
2 | HYB:SXS:BBH:1355 | 1 | 0.127 | 2.739 | 40.8
3 | HYB:SXS:BBH:1356 | 1 | 0.163 | 1.606 | 40.0
4 | HYB:SXS:BBH:1357 | 1 | 0.222 | -1.020 | 36.1
5 | HYB:SXS:BBH:1358 | 1 | 0.226 | -2.937 | 35.7
6 | HYB:SXS:BBH:1359 | 1 | 0.227 | 1.850 | 36.3
7 | HYB:SXS:BBH:1360 | 1 | 0.302 | 0.730 | 31.2
8 | HYB:SXS:BBH:1361 | 1 | 0.305 | 1.146 | 31.0
9 | HYB:SXS:BBH:1362 | 1 | 0.372 | -0.726 | 25.5
10 | HYB:SXS:BBH:1363 | 1 | 0.376 | 0.385 | 25.3
11 | HYB:SXS:BBH:1167 | 2 | 0.000 | 1.308 | 48.4
12 | HYB:SXS:BBH:1364 | 2 | 0.108 | -2.495 | 46.3
13 | HYB:SXS:BBH:1365 | 2 | 0.145 | -1.116 | 44.9
14 | HYB:SXS:BBH:1366 | 2 | 0.218 | 0.096 | 39.6
15 | HYB:SXS:BBH:1367 | 2 | 0.220 | -0.964 | 40.5
16 | HYB:SXS:BBH:1368 | 2 | 0.222 | -1.553 | 40.3
17 | HYB:SXS:BBH:1369 | 2 | 0.367 | -2.489 | 28.3
18 | HYB:SXS:BBH:1370 | 2 | 0.367 | 0.754 | 28.8
19 | HYB:SXS:BBH:1221 | 3 | 0.000 | 2.461 | 56.8
20 | HYB:SXS:BBH:1371 | 3 | 0.133 | -1.757 | 52.8
21 | HYB:SXS:BBH:1372 | 3 | 0.212 | -2.101 | 48.7
22 | HYB:SXS:BBH:1373 | 3 | 0.214 | -2.655 | 48.6
23 | HYB:SXS:BBH:1374 | 3 | 0.359 | -2.953 | 35.0

Table 1: Hybrids constructed by matching NR simulations from the SXS
collaboration and available PN prescriptions for BBHs in eccentric orbits. SXS
simulation ids are retained for identification with NR simulation used in the
process of construction of the hybrids. Each hybrid starts at an averaged
orbital frequency of $x_{0}=0.045$ where eccentricity ($e_{0}$), mean anomaly
($l_{0}$) are estimated. Mass ratio ($q$) and number of orbits prior to the
merger are also listed. $N_{\rm orb}$ has been computed by taking the phase
difference between the start of the waveform and the peak of the $(2,2)$ mode
amplitude. The NR simulation, SXS:BBH:1132, is longer than the hybrids
constructed here and hence the NR data is directly used.

Spin-weighted spherical harmonic modes of the inspiral waveforms from the PN
theory, in terms of an amplitude and orbital phase, can be written as follows

$\mathpzc{h}_{\ell m}^{\mathrm{PN}}(t)={2GM\eta x\over
c^{2}D}\sqrt{\frac{16\pi}{5}}\,\hat{H}_{\ell
m}\,e^{-i\,m\varphi_{\mathrm{orb}}(t)},$ (1)

where $\hat{H}_{\ell m}$ is amplitude of a given ($\ell,m$) mode and
$\varphi_{\mathrm{orb}}(t)$ represents binary’s orbital phase. Symbols $M$ and
$D$ represent the binary’s total mass and its distance from the observer while
$\eta$ is given by the ratio of binary’s reduced mass to total mass. Unless
explicitly mentioned, we work with $G$=$c$=$1$ and set $M$=$1\,M_{\odot}$,
$D$=$1\,{\rm Mpc}$. The PN expressions for mode amplitudes ($\hat{H}_{\ell
m}$) contributing up to 3PN order for binaries with non-spinning compact
components in _quasi-circular_ orbits have been computed in Refs. Blanchet
_et al._ (1996); Arun  _et al._ (2004); Kidder  _et al._ (2007); Blanchet  _et
al._ (2008); Kidder (2008). In fact, some of the leading modes are actually
known (or can easily be computed using available inputs) with higher PN
accuracy and contribute to relevant modes at higher PN orders Marchand  _et
al._ (2020); Faye  _et al._ (2015); Henry  _et al._ (2021); Faye  _et al._
(2013). The orbital phase accurate to the 3.5PN order has been computed in
Blanchet  _et al._ (2002a, b, 2004) (see Ref. Blanchet (2006) for a review on
the subject). Expressions for mode amplitudes constituting 3PN inspiral
waveforms assuming binaries with non-spinning compact objects in _quasi-
elliptical_ orbits have been computed in Boetzel  _et al._ (2019); Ebersold
_et al._ (2019); Mishra  _et al._ (2015) by employing generalised 3PN quasi-
Keplerian representation of Damour and Deruelle (1985); Memmesheimer  _et al._
(2004). On the other hand a 3.5 PN prescription for the orbital phase for an
eccentric system has been presented in Ref. Königsdörffer and Gopakumar (2006)
and is based on phasing formulation of Damour  _et al._ (2004) and generalised
quasi-Keplerian representation of Memmesheimer  _et al._ (2004). While the
phasing of Königsdörffer and Gopakumar (2006) includes contributions due to
binary’s reactive dynamics to relative 1PN order (or absolute order of 3.5PN
of phase), Ref. Moore  _et al._ (2016) extends these results to the relative
3PN order under the assumption that binary’s initial orbital eccentricity is
small ($e_{0}\lesssim 0.2$). In another effort Tanay  _et al._ (2016) the
results of Ref. Yunes  _et al._ (2009) were extended to the 2PN order and
included eccentricity to $\mathcal{O}(e^{6})$. While the results of Ref. Moore
_et al._ (2016) should capture relativistic dynamics better (being more
accurate in PN sense) results of Ref. Tanay  _et al._ (2016) should be
applicable to systems with larger eccentricities.

The NR simulations (describing the non-perturbative, late-time evolution of
eccentric BBH mergers) used here have been performed using the Spectral
Einstein Code (SpEC) Ossokine  _et al._ (2013); Boyle  _et al._ (2019)
developed by the SXS collaboration and are publicly available SXS . A set of
20 eccentric simulations with varying initial eccentricities ($e\leq 0.2$) and
mass ratios ($q=1,2,3$) were first presented in Hinder  _et al._ (2018).333See
Table I of Hinder  _et al._ (2018) for other relevant properties of these
simulations.

Since the NR simulations we intend to compare PN results with, include
waveforms with eccentricities as large as 0.2, we have made use of the results
of Tanay  _et al._ (2016) while comparing the phase (and the angular
frequency), due to its ability to probe larger eccentricities compared to the
ones presented in Ref. Moore  _et al._ (2016). As far as the amplitude
comparisons are concerned we make use of the PN mode amplitudes computed in
Ebersold  _et al._ (2019); Boetzel  _et al._ (2019). We present the results of
these comparisons in the next section.

Figure 2: PN-NR hybrid waveform corresponding to NR simulation SXS:BBH:1364,
an asymmetric mass binary with mass ratio $q$=$2$. The initial eccentricity of
the constructed waveform is $e_{0}=0.108$ at $x_{\rm low}=0.045$. The blue
dotted line marks the beginning of the NR waveform and the shaded grey region
$t\in(1000M,2000M)$ shows the matching window where hybridization was
performed. Overlapping hybrid and NR waveforms on the left of the matching
window, hint at the quality of hybridization performed here.

###  II.2 PN-NR comparison

Figure 1 compares the waveform data of a NR simulation (SXS:BBH:1364) and
related results from the PN theory for a number of relevant modes. The colored
lines represent the NR data, mimicked closely in first few GW cycles by the PN
results that are displayed as black lines. Close agreements between the PN and
NR waveforms in the inspiral part of the signal allow for hybridization
discussed in the next section. However, we choose to ignore the modes whose
amplitudes relative to the $\ell$=$2$, $|m|$=2 or simply 22 mode is smaller by
a factor of $10^{-3}$ as they may not be relevant for our purposes given their
small amplitudes. Additionally, we also demand that data for each mode in the
hybridization window should be relatively clean. These two conditions limit
the number of modes that are to be included in the hybrids to ($\ell,|m|$) =
(2,2), (2,1), (3,3), (3,2), (4,4), (4,3), and (5,5). Note also that the
$m$=$0$ modes (leading to non-linear GW memory; also known as DC modes) are
not included as they would not impact GW detection and parameter estimation
analyses Favata (2009); Pollney and Reisswig (2011); Ebersold  _et al._ (2019)
and are not extracted accurately in NR simulations. On the other hand, the
$m\neq 0$ mode include contributions from what are called oscillatory memory
which becomes relevant in the late stages of binary evolution Favata (2009);
Ebersold  _et al._ (2019). These are included at 3PN level in the PN inspiral
waveforms of Ebersold  _et al._ (2019) that are used in constructing the
target hybrids.

###  II.3 Construction of hybrid waveforms

Figure 3: Mismatches between quasi-circular waveforms of SEOBNRv4HM Cotesta
_et al._ (2020) (top panels)/ IMRPhenomXHM García-Quirós  _et al._ (2020)
(bottom panels) and the eccentric hybrids for three representative orbit
inclinations. The two horizontal lines report 96.5% and 99% agreement
respectively. The eccentricity values displayed with color bars are computed
at $x_{0}=0.045$ for all hybrids. Grey (dashed) vertical lines indicate the
total mass below which mismatches become $>1\%$ irrespective of binary’s
orbital eccentricity, mass-ratio and orbital inclination.

Complete inspiral-merger-ringdown (IMR) waveforms are constructed by matching
PN and NR waveforms for individual modes in a region where the PN prescription
closely mimics the NR data following the method of Ref. Varma and Ajith
(2017). These are traditionally referred to as _hybrids_ and used as targets
for modelling and data-analysis purposes. As discussed in Varma and Ajith
(2017) construction of hybrids including higher modes (in circular case) is
possible by performing at least two rotations (and a time-shift) so as to
align the frames in which PN/NR waveforms are defined.444It is assumed that
the third Euler angle can easily be fixed in the direction of binary’s total
angular momentum (See Fig. 2 and the discussions in section III C of Varma and
Ajith (2017).) We simply extend this argument to the case of eccentric orbits
assuming that the effect of marginalising over parameters such as eccentricity
and mean anomaly will not significantly affect the hybridization. The
prescription for construction of hybrids is discussed in detail in Varma and
Ajith (2017) and we reproduce some of the steps here for completeness. A
least-square minimization of the integrated difference between the GW modes
from the PN and NR in a time interval ($t_{\rm i},t_{\rm f}$), in which the
two approaches give similar results, is performed and can be defined as

$\delta=\mathrm{min}_{t_{0},\varphi_{0},\psi}\int_{t_{\rm i}}^{t_{\rm
f}}dt\sum_{\ell,m}\left|\mathpzc{h}_{\ell
m}^{\mathrm{NR}}(t-t_{0})e^{i(m\varphi_{0}+\psi)}-\mathpzc{h}_{\ell
m}^{\mathrm{PN}}(t)\,\right|.$ (2)

where the minimization is performed over a time-shift ($t_{0}$) and the two
angles ($\varphi_{0},\psi$) as discussed above. The hybrid waveforms are then
constructed by combining the NR data with the “best matched” PN waveform in
the following way:

$\mathpzc{h}_{\ell m}^{\mathrm{hyb}}(t)\equiv\,\tau(t)\,\mathpzc{h}_{\ell
m}^{\mathrm{NR}}(t-t_{0}^{\prime})\
e^{i(m\varphi_{0}^{\prime}+\psi^{\prime})}+(1-\tau(t))\,\mathpzc{h}_{\ell
m}^{\mathrm{PN}}(t),$ (3)

where ($t_{0}^{\prime}$, $\varphi_{0}^{\prime}$, $\psi^{\prime}$) are the
values of ($t_{0}$, $\varphi_{0}$, $\psi$) that minimize the integral of Eq.
(2). In the above, $\tau(t)$ is a weighting function defined by:

$\displaystyle\tau(t)\equiv\left\\{\begin{array}[]{ll}0&\textrm{if $t<t_{\rm
i}$}\\\ \frac{t-t_{\rm{i}}}{t_{\rm{f}}-t_{\rm{i}}}&\textrm{if $t_{\rm{i}}\leq
t<t_{\rm{f}}$}\\\ 1&\textrm{if $t_{\rm{f}}\leq t$.}\end{array}\right.$ (7)

The hybrids corresponding to a representative NR simulation (SXS:BBH:1364) for
all relevant modes are shown in Fig. 2. The blue dotted line marks the
beginning of the NR waveform and the shaded grey region $t\in(1000M,2000M)$
shows the matching window where hybridization was performed. Overlapping
hybrid and NR waveforms outside (on the left of) the matching window hint at
the quality of hybridization performed here. We construct IMR hybrids
corresponding to all 20 eccentric NR simulations listed in Ref. Hinder  _et
al._ (2018). These are listed in Table II.1 and the SXS simulation IDs have
been retained so as be be able to identify the hybrids with the corresponding
NR simulation. All the hybrids have a start frequency of $x_{0}$=$0.045$ and
typically have 30-40 orbital cycles before the merger. On the other hand NR
simulations used here evolve over 10-15 cycles before the merger. Being longer
in length, these hybrids should prove to be critical in assessing the impact
of higher modes and eccentricity in the entire mass range accessible to the
ground based detectors such as LIGO and Virgo. We use these hybrids to
demonstrate the impact of eccentricity as well as of higher modes in the
section that follows.

###  II.4 Impact of eccentricity and higher modes on detection and parameter
estimation: match calculations

Figure 3 displays the mismatch between a set of hybrids including the effect
of eccentricity and related higher modes constructed here, and the state-of-
the-art quasi-circular waveform families including higher modes SEOBNRv4HM
Cotesta  _et al._ (2020) (top panels) and IMRPhenomXHM García-Quirós  _et al._
(2020) (bottom panels). As discussed above our target hybrids include ($\ell$,
$|m|$)=(2, 2), (3, 3), (4, 4), (5, 5), (2, 1), (3, 2), and (4, 3) modes. The
SEOBNRv4HM Cotesta  _et al._ (2020) templates (quasi-circular) include
($\ell$, $|m|$)=(2, 2), (3, 3), (4, 4), (5, 5), and (2, 1) modes whereas those
of IMRPhenomXHM García-Quirós  _et al._ (2020) include ($\ell$, $|m|$)=(2, 2),
(3, 3), (4, 4), (2, 1), and (3, 2) modes. This means target waveforms may have
an additional mode or two depending upon which template is used for recovery.
The mismatch plots indicate the need for including eccentricity in
gravitational waveforms for analysing BBH systems with masses below
80$M_{\odot}$, irrespective of eccentricity or mass ratio for face-on systems.
The mismatches are larger for larger eccentricities. Furthermore, with
increasing inclination angle, inclusion of eccentricity in waveforms seems to
become important even for heavier systems. For instance, mismatches are $>1\%$
for systems lighter than 90$M_{\odot}$(160$M_{\odot}$) for inclination angle
of $\pi/4$ ($\pi/2$). Additionally, as should be clear from middle and right
top/bottom panels of Fig. 3, with increasing orbit’s inclination w.r.t our
line of sight, the combined impact of eccentricity and higher modes becomes
more apparent.

##  III A dominant mode model

Figure 4: PN-NR hybrid waveform corresponding to NR simulation SXS:BBH:1364,
an asymmetric mass binary with mass ratio $q$=$2$, which has been used for
constructing the models. The initial eccentricity of the constructed waveform
is $e_{0}=0.108$ at $x_{\rm low}=0.045$. The eccentric inspiral waveform used
for constructing the target hybrids for modelling are presented in Tanay  _et
al._ (2016).

Figure 5: Amplitude and frequency model produced by combining an eccentric
inspiral waveform with a circular merger-ringdown waveform. (Left panel) The
amplitude transitions smoothly from the inspiral to the merger-ringdown inside
the shaded region, ending at $t_{\rm match}$. (Right panel) The frequency
transitions smoothly from inspiral to merger-ringdown, starting at $t_{\rm
match}$, and ending at $30M$ before merger.

Here we develop a fully analytical dominant ($\ell,|m|$) = (2, 2) mode model
obtained by matching an eccentric PN inspiral with a quasi-circular
prescription for the merger-ringdown phase, the model is calibrated against a
set of 20 eccentric hybrids constructed here. The method used for construction
of the hybrids is discussed in Sec. II. We use another set of hybrids here
(constructed by simply matching waveforms of Tanay  _et al._ (2016) with NR
simulations of Hinder  _et al._ (2018)) as only a dominant mode target is
required. A graphical representation of this hybrid corresponding to the
simulation SXS:BBH:1364 is shown in Fig. 4. Overlapping hybrid and NR
waveforms outside (on the left of) the matching window hint at the quality of
hybridization performed here. The inspiral part of the model is the waveform
presented in Tanay  _et al._ (2016) while the quasi-circular merger-ringdown
part is described by the waveform from EOB family discussed in Cotesta  _et
al._ (2020). Below we discuss the construction of the model as well as present
the analytical prescription for the same.

###  III.1 Time-shift

As described in Sec. II.3, the process of hybridization involves minimization
over a time shift. So when producing the amplitude model, we first perform a
time shift of the inspiral waveform relative to the circular IMR waveform,
because we do not know the exact time to merger. This is done by first setting
the merger time for the circular IMR waveform to zero and then time sliding
the eccentric inspiral about the merger. We obtain a numerical estimate of the
time shift for each target hybrid and denote it by $t_{\rm{shift}}$. Once the
time shift is performed, the amplitude and frequency model is generated using
the prescription as discussed in III.2 and III.3 respectively.

###  III.2 Amplitude model

Figure 6: Numerical fits are mapped into the physical parameter space for
eccentric systems characterised by binary’s eccentricity, mean anomaly at a
reference frequency and the mass ratio parameter $q$ or $\eta$ depending upon
the model. Circles represent the numerical data points while crosses represent
the value returned by the best-fit model.

As shown in the plots in Figure 1, the waveforms tend to circularize near
merger.555See also the discussion around Figure 3 of Hinder  _et al._ (2018)
which clearly shows all NR simulations become circular $30M$ before the
merger. Hence, in order to model this effect we can suitably join the
eccentric inspiral to the circular IMR at an appropriate time
$t_{\rm{match}}$. The amplitude model is obtained by joining the eccentric
inspiral with the circular IMR using a transition function over a fixed time
interval of $500M$ which ends at $t_{\rm{match}}$. Given a target hybrid, we
start with a trial choice of $t_{\rm{match}}$ roughly $500M_{\odot}$ before
the merger and produce the amplitude model as given below

$\mathcal{A}_{22}^{\mathrm{model}}(t)\equiv\,\tau_{\rm{a}}(t)\,\mathcal{A}_{22}^{\mathrm{IMR}}(t)\
+(1-\tau_{\rm{a}}(t))\,\mathcal{A}_{22}^{\mathrm{inspiral}}(t),$ (8)

where, $\tau_{\rm{a}}(t)$ is defined as

$\displaystyle\tau_{\rm{a}}(t)\equiv\left\\{\begin{array}[]{ll}0&\textrm{if
$t<t_{\rm{i}}$}\\\ \frac{t-t_{\rm{i}}}{t_{\rm{f}}-t_{\rm{i}}}&\textrm{if
$t_{\rm{i}}\leq t<t_{\rm{f}}$}\\\ 1&\textrm{if $t_{\rm{f}}\leq
t$.}\end{array}\right.$ (12)

we set $t_{\rm{i}}=t_{\rm{match}}-500M$ and $t_{\rm{f}}=t_{\rm{match}}$ as the
bounds of the time interval over which the two waveforms are joined. Figure 5
demonstrates the process. The grey region is the time interval ending at
$t_{\rm{match}}$ where the inspiral and circular IMR is joined. After the
amplitude model is obtained for a particular choice of trial $t_{\rm{match}}$,
we combine it with the target hybrid phase to obtain the polarizations, and
then calculate the match with the target hybrid. We then change the trial
choice of $t_{\rm{match}}$ by $5M$, bringing it closer to the merger and
repeat the process of producing the amplitude model, and calculating the
match. This variation of $t_{\rm{match}}$ is done till roughly $30M$ before
merger. We thus obtain a set of match values for varying $t_{\rm{match}}$ and
pick the one which has the highest value of match. The corresponding amplitude
$t_{\rm{match}}$ is the numerical estimate for a particular target hybrid. We
obtain numerical estimates using the same process for all the 20 target
hybrids.

###  III.3 Frequency model

Figure 7: (Top panel) Amplitude and frequency of the dominant mode model
(constructed out of stitching an inspiral and a merger-ringdown model) plotted
together with target hybrids for comparison for three representative low
eccentricity simulations. (Bottom panel) One of the polarizations, obtained by
combining the amplitude and the frequency model shown in the top panel for
$q=3$ case, is shown as a visual proof of the quality of the model being
presented.

For the frequency model, we follow a similar procedure as described in Sec.
III.2 with the only difference being the duration of the time interval where
the inspiral frequency is joined with the circular IMR frequency. Once again,
similar to the amplitude model procedure, we determine an appropriate
$t_{\rm{match}}$ for joining the inspiral frequency with the circular IMR
frequency. However, the time interval where the two are joined starts at
$t_{\rm{match}}$ and ends at a time close to $30M$666This choice is motivated
by the fact that all NR simulations necessarily circularise $30M$ before the
mergerHinder  _et al._ (2018). before merger. Just like the amplitude model,
we start with the choice of a trial value of $t_{\rm{match}}$ roughly $6000M$
before merger and obtain the frequency model as given below

$\omega_{22}^{\mathrm{model}}(t)\equiv\,\tau_{\rm{a}}(t)\,\omega_{22}^{\mathrm{IMR}}(t)\
+(1-\tau_{\rm{a}}(t))\,\omega_{22}^{\mathrm{inspiral}}(t),$ (13)

where, $\tau_{\rm{a}}(t)$ is as defined in Eq. (12) with the difference being,
$t_{\rm{i}}=t_{\rm{match}}$ and $t_{\rm{f}}\lesssim-30M$. Figure 5
demonstrates the process. Once the frequency model is obtained for the choice
of trial $t_{\rm{match}}$, we calculate the phase by integrating the frequency
model. This is then combined with the amplitude model obtained for the same
target hybrid to produce the polarizations and a match with the target hybrid
is calculated. We then change the trial choice of $t_{\rm{match}}$ by $1M$,
bringing it closer to the merger and repeat the process of producing the
frequency model, and calculating the match. Once again we do this variation
till roughly $30M$ before merger to obtain a set of match values for varying
$t_{\rm{match}}$ and pick the one which has the highest value of match. The
corresponding value of frequency $t_{\rm{match}}$ is the numerical estimate
for a particular target hybrid. We obtain numerical estimates for all the 20
target hybrids using the same process.

###  III.4 Analytical model

Figure 8: (Left panel) Mismatches between set of eccentric hybrids presented
in Table II.1 and the dominant mode model presented in Sec. III. (Right panel)
Mismatches between the dominant mode model of ENIGMA Chen  _et al._ (2021) and
the model presented here. The two horizontal lines report 96.5% and 99%
agreement, respectively. The eccentricity values displayed with color bars are
computed at $x_{0}=0.045$ for all hybrids.

We have described the procedure of producing (numerical) time domain model
fits for the dominant mode model, where we used a set of 20 eccentric hybrids
as targets to calibrate our model. For each hybrid we obtained a numerical
estimate for $t_{\rm{shift}}$, amplitude $t_{\rm{match}}$, and frequency
$t_{\rm{match}}$. In order to be able to generate waveforms for an arbitrary
configuration these numerical fits need to be mapped into the physical
parameter space for eccentric systems characterised by binary’s eccentricity,
mean anomaly at a reference frequency and the mass ratio parameter. In this
section we determine a functional form by performing analytical fits to these
numerical estimates. The fitted functions obtained are of the form

$\displaystyle t_{\rm{shift}}\,(q,e,l)$
$\displaystyle=\sum_{\alpha,\beta,\gamma,\delta}A_{\alpha\beta\gamma\delta}\,e^{\alpha}\,q^{\beta}\,\cos(\gamma\,l\,+\,\delta\,e\,l\,+\,a_{\alpha\beta\gamma\delta})$
(14)

for time shift, where
$A_{\alpha\beta\gamma\delta}=a_{\alpha\beta\gamma\delta}=0$ for
$\alpha+\beta>4$ and/or $\gamma+\delta>1$, and $A_{\alpha 010}=A_{\alpha
001}=A_{0\beta 10}=A_{0\beta 01}=A_{00\gamma\delta}=a_{\alpha\beta 00}=0$.

$\displaystyle t_{\rm{match}}\,(\eta,e,l)$
$\displaystyle=\sum_{\alpha,\beta,\gamma,\delta}B_{\alpha\beta\gamma\delta}\,e^{\alpha}\,\eta^{\beta}\,\cos(\gamma\,l\,+\,\delta\,e\,l\,+\,b_{\alpha\beta\gamma\delta})$
(15)

for amplitude, where
$B_{\alpha\beta\gamma\delta}=b_{\alpha\beta\gamma\delta}=0$ for
$\alpha+\beta>4$ and/or $\gamma+\delta>1$, and $B_{\alpha 010}=B_{\alpha
001}=B_{0\beta 10}=B_{0\beta 01}=B_{00\gamma\delta}=b_{\alpha\beta 00}=0$.

$\displaystyle t_{\rm{match}}\,(\eta,e,l)$
$\displaystyle=\sum_{\alpha,\beta,\gamma,\delta}C_{\alpha\beta\gamma\delta}\,e^{\alpha}\,\eta^{\beta}\,\cos(\gamma\,l\,+\,\delta\,e\,l\,+\,c_{\alpha\beta\gamma\delta})$
(16)

for frequency, where
$C_{\alpha\beta\gamma\delta}=c_{\alpha\beta\gamma\delta}=0$ for
$\alpha+\beta>5$ and/or $\gamma+\delta>1$, and $C_{\alpha 010}=C_{\alpha
001}=C_{0\beta 10}=C_{0\beta 01}=C_{00\gamma\delta}=c_{\alpha\beta 00}=0$. The
values for the coefficients $A_{\alpha\beta\gamma\delta}$,
$B_{\alpha\beta\gamma\delta}$, $C_{\alpha\beta\gamma\delta}$,
$a_{\alpha\beta\gamma\delta}$, $b_{\alpha\beta\gamma\delta}$, and
$c_{\alpha\beta\gamma\delta}$ obtained by performing a fit to the numerical
values are tabulated in Tables II.1, II.1, and II.1. Figure 6 shows
comparisons between the numerically obtained values for $t_{\rm{shift}}$,
amplitude $t_{\rm{match}}$, and frequency $t_{\rm{match}}$, with the values
predicted by our analytical fits. The predictions are within $\pm 12M$ of the
numerical estimates for time shift, within $\pm 7M$ for amplitude
$t_{\rm{match}}$, and within $\pm 51M$ for frequency $t_{\rm{match}}$. We show
amplitude and frequency comparison between the target hybrids and our models
for three cases along with the full waveform for the $q$=$3$ case in Fig. 7.

The performance of the model can be assessed from the plots presented in Fig.
7 as well as from the mismatch plots displayed in the left panel of Fig. 8.
Since all 20 hybrids were utilised in calibrating the model, these models are
tested against an independent waveform ENIGMA Chen  _et al._ (2021). For this
comparison we choose to sample a parameter space which is not identical to the
calibration set (hybrids). We choose to generate both the target (ENIGMA) and
the template (dominant eccentric model) by randomly sampling values of a
reference eccentricity ($e_{0}$), mass ratio ($q$) and reference mean anomaly
($l_{0}$) in range $0\lesssim e_{0}\lesssim 0.2$, $1\lesssim q\lesssim 3$, and
$-\pi\leq l_{0}\leq\pi$ respectively. The mismatch plot obtained is shown in
the right panel of Fig. 8.

##  IV PE Systematics

Figure 9: Chirp-mass recovery of circular and eccentric injections using
quasi-circular waveforms is shown. Thick-black line denotes the injected value
of the chirp-mass parameter and the dashed lines denote 90% credible
intervals. The orange (olive) posteriors denote measurement of the chirp mass
for the injected circular (eccentric) signal. Both circular and eccentric
injections correspond to a BBH of total mass 40$M_{\odot}$, and the eccentric
simulations have an orbital eccentricity of $e_{0}\sim 0.1$ at 20Hz. Non-
recovery of the injected value of the chirp mass for the eccentric case can be
interpreted as the bias induced due to the neglect of eccentricity and
associated higher modes.

Figure 9 shows 90% error bounds in the measurement of binary’s chirp mass
($\mathcal{M}$) and are based on a Bayesian analysis performed on simulated
data containing injections (hybrids) with distinct mass ratios ($q$=$1,2,3$).
The total mass of injected signals is assumed to be fixed at 40$M_{\odot}$ and
luminosity distance ($d_{L}$) at 410 Mpc. The value of eccentricity at 20 Hz
(starting frequency of the analysis) is $e_{0}\sim 0.1$. These are assumed to
have non-spinning components and inclined at $30^{\circ}$ w.r.t the binary’s
orbital angular momentum (also the line of sight). Angular parameters giving
binary’s location [$\theta$ and $\phi$ (sky angles)], as well as the
polarisation angle ($\psi$) are chosen arbitrarily with values: $\pi/6$,
$\pi/4$, and $\pi/3$ respectively. Unlike the match computations of Sec. II.4,
injections here include ($\ell$, $|m|$)=(2, 2), (3, 3), (4, 4), (2, 1), (3, 2)
modes. These are then recovered using circular higher mode waveform
IMRPhenomXHM García-Quirós  _et al._ (2020) that contains all the above
mentioned modes. Additional $(4,3)$ and $(5,5)$ present in the hybrids have
been dropped to have same set of modes in injection and the recovery waveforms
to avoid misinterpretation of the results. The aim here is to observe if the
bias in the parameter estimates is purely due to the combined impact of
eccentricity and associated higher order modes. The injected parameter values
are indicated by vertical black line in the figure while the recovery is shown
by posteriors with 90% error bounds (vertical dashed lines). An injection is
assumed to be recovered if the injected value falls within the 90% bounds. The
posteriors in orange denote injections with circular simulations
(SXS:BBH:1132, HYB:BBH:1167, HYB:BBH:1221) while those in olive green denote
injections with eccentric simulations (HYB:BBH:1356, HYB:BBH:1364,
HYB:BBH:1371). It is clear from the figure that eccentric injections are not
recovered with the quasi-circular waveform. This indicates that the presence
of residual eccentricity of the order $e_{0}\sim 0.1$ and related higher modes
in systems entering the ground based detectors will lead to significant biases
in recovering source parameters. This observation should motivate including
the effect of orbital eccentricity in dominant and other higher modes in
waveforms from compact binary mergers.

##  V Discussion

We started by comparing high accuracy PN inspiral waveform results for compact
binaries in eccentric orbits of Ebersold  _et al._ (2019); Boetzel  _et al._
(2017); Tanay  _et al._ (2016) with SXS NR data for eccentric BBH mergers
presented in Hinder  _et al._ (2018). Figure 1 compares the waveform data for
one particular data set. Based on this comparison we select a set of modes
which will be included in the target IMR model (hybrids). The hybrids are
constructed for ($\ell$, $|m|$)=(2, 2), (3, 3), (4, 4), (5, 5), (2, 1), (3,
2), (4, 3) modes. The hybridization procedure is discussed in Sec. II.3. We
use these hybrids to show the impact of eccentricity and associated higher
modes by computing the mismatch with the state-of-the-art quasi-circular
waveforms such as SEOBNRv4HM Cotesta  _et al._ (2020) and IMRPhenomXHM García-
Quirós  _et al._ (2020). The mismatches are shown in Fig. 3. In Sec. III we
develop a fully analytical dominant mode ($\ell,|m|$) = (2, 2) model obtained
by matching an eccentric PN inspiral with a quasi-circular prescription for
the merger-ringdown phase calibrated against a set of eccentric hybrids.
Figure 5-6 demonstrate the procedure while Fig. 7-8 display the performance of
the model developed here. Finally, in Sec. IV, with an injection analysis we
demonstrate it will not be possible to ignore the presence of eccentricity and
related higher modes while recovering GW signals using current state-of-the-
art quasi-circular waveforms including the effect of higher order modes.

##  VI Acknowledgments

We thank Prayush Kumar for useful comments on the manuscript. We thank K G
Arun and Guillaume Faye for useful discussions and Sumit Kumar for help with
the parameter estimation pipeline (pycbc_inference). We thank the authors of
Ref. Chen  _et al._ (2021) for providing a code implementation of the ENIGMA
waveform model, and Prayush Kumar for technical discussions about the use of
ENIGMA. We are thankful to the SXS Collaboration for making a public catalog
of numerical-relativity waveforms and in particular Ian Hinder and Harald
Pfeiffer for clarifying our doubts at various stages of the project. T.R.C
thanks the members of the gravitational wave group at the Department of
Physics, IIT Madras for organizing the weekly journal club sessions and the
insightful discussions. The authors are grateful for computational resources
provided by the LIGO Laboratory and supported by National Science Foundation
Grants PHY-0757058 and PHY-0823459. The authors also acknowledge the
workstation ‘powehi’ in the Department of Physics, IIT Madras which has been
used to perform most of the parameter estimation runs shown in this paper.
This document has LIGO preprint number LIGO-P2200106.

##  Appendix A Details of parameter estimation study

Figure 10: Corner plot for $q$=$1$, where circular (SXS:BBH:1132) simulation
is shown in orange, and eccentric (HYB:SXS:BBH:1356) simulation is shown in
dark green. Histograms on the diagonal show marginalized 1D posteriors,
whereas the contours denote the joint 2D posteriors for various parameters.
The vertical dashed lines in 1D histograms mark 90% credible intervals, and
dotted lines in orange show the prior. The black lines mark the injected
values for various system parameters. Figure 11: Corner plot for $q$=$2$,
where circular (HYB:SXS:BBH:1167) simulation is shown in orange, and eccentric
(HYB:SXS:BBH:1364) simulation is shown in dark green. Histograms on the
diagonal show marginalized 1D posteriors, whereas the contours denote the
joint 2D posteriors for various parameters. The vertical dashed lines in 1D
histograms mark 90% credible intervals, and dotted lines in orange show the
prior. The black lines mark the injected values for various system parameters.
Figure 12: Corner plot for $q$=$3$, where circular (SXS:BBH:1221) simulation
is shown in orange, and eccentric (HYB:SXS:BBH:1371) simulation is shown in
dark green. Histograms on the diagonal show marginalized 1D posteriors,
whereas the contours denote the joint 2D posteriors for various parameters.
The vertical dashed lines in 1D histograms mark 90% credible intervals, and
dotted lines in orange show the prior. The black lines mark the injected
values for various system parameters.

In this section, we include the corner plots for all the injections which have
been performed in order to highlight the correlations between various binary
parameters. The parameters chosen for corner plots are chirp mass
($\mathcal{M}_{c}$), total mass ($M$), symmetric mass ratio ($\eta$),
luminosity distance ($d_{L}$) and binary’s inclination angle w.r.t the line of
sight ($\iota$). Figures 10, 11 and 12 show the corner plots for $q=1,2,$ and
$3$ respectively (see also O’Shea and Kumar (2021)). These plots clearly show
the correlations among various mass parameters ($\mathcal{M}_{c}$, $M$, and
$\eta$) and strong correlations between luminosity distance $d_{L}$ and
inclination angle ($\iota$). It is interesting to note that the symmetric mass
ratio $\eta$ has an upper limit of $\eta\leq 0.25$ (by definition), thus for
$q$=$1$, since $\eta=0.25$, the posterior hits the prior boundary on the
right. Now, because of the correlations between chirp mass ($\mathcal{M}_{c}$)
and $\eta$, this translates into slight shifting of the posterior which
results in the injection value not coinciding with the median of the
posterior. This is observed both in Figs.9-10. This feature also appears when
we inject a signal using IMRPhenomXHM and recover using the same waveform,
thus indicating that this is solely because of the correlations between $\eta$
and $\mathcal{M}_{c}$ and not because of differences in the injected and
recovery waveform. The slight shift in $q$=$3$ posterior of circular injection
(HYB:SXS:BBH:1221) though can be attributed to the slight differences in the
injected and recovered signal. Another interesting feature of these plots is
that while the mass parameters show a shift in the posterior with eccentric
injections, the distance and inclination posteriors are still able to recover
the injected values even for eccentric injections. This hints at the fact that
while the eccentricity parameter is strongly correlated with the mass
parameters of the binary, it is not so with the case of extrinsic parameters
like luminosity distance and inclination angle (see also O’Shea and Kumar
(2021)).

$\alpha\beta\gamma\delta$ | $A_{\alpha\beta\gamma\delta}$ | $a_{\alpha\beta\gamma\delta}$
---|---|---
$1110$ | $-1781.3755$ | $23.049324$
$2201$ | $8601.0938$ | $12.644587$
$3110$ | $24719.217$ | $35.365563$
$1210$ | $218.37381$ | $-2.5655030$

Table 2: Table of coefficients for the analytical expression of
$t_{\rm{shift}}$ in Eq. (14). All other coefficients not included in the table
are zero.

$\alpha\beta\gamma\delta$ | $B_{\alpha\beta\gamma\delta}$ | $b_{\alpha\beta\gamma\delta}$
---|---|---
$1110$ | $8589.8002$ | $220.81235$
$2210$ | $-155537.19$ | $-1123.6840$
$3101$ | $57429.241$ | $2257.6694$

Table 3: Table of coefficients for the analytical expression of amplitude
$t_{\rm{match}}$ in Eq. (15). All other coefficients not included in the table
are zero.

$\alpha\beta\gamma\delta$ | $C_{\alpha\beta\gamma\delta}$ | $c_{\alpha\beta\gamma\delta}$
---|---|---
$1110$ | $-317485.702$ | $123.07063$
$2210$ | $5.7669647\times 10^{6}$ | $883.39202$
$3101$ | $1.0347922\times 10^{7}$ | $3823.0109$
$3201$ | $5.2523026\times 10^{7}$ | $16116.188$

Table 4: Table of coefficients for the analytical expression of frequency
$t_{\rm{match}}$ in Eq. (16). All other coefficients not included in the table
are zero.

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[a,b]André Sopczak

# Outreach Modules for New Particle Searches Using the ATLAS Forward Proton
Detector and for Higgs Boson Physics

###### Abstract

We present two modules as part of the Czech Particle Physics Project (CPPP).
These modules are intended as learning tools in masterclasses aimed at high-
school students (aged 15 to 18). The first module is dedicated to the
detection of an Axion-Like-Particle (ALP) using the ATLAS Forward Proton (AFP)
detector. The second module focuses on the reconstruction of the Higgs boson
mass using the Higgs boson golden channel with four leptons in the final
state. The modules are accessible at http://cern.ch/cppp.

##  1 Introduction

Outreach is an important part of CERN’s mission. An audience that is
particularly important consists of high-school students who might be
interested in pursuing further studies in the Science, Technology, Engineering
and Mathematics (STEM) fields. A set of online materials is presented that can
be used in masterclasses to teach high-school students (15 to 18 years old)
about particle physics at CERN. These tools are part of the wider Czech
Particle Physics Project (CPPP).

Two modules have been developed, each having two main components: an
introduction which explains the basic physics involved and a simulation which
allows users to learn about physics research interactively. The first module
takes students through the process of finding an Axion-Like Particle (ALP)
using the ATLAS Forward Proton (AFP) detector. The second shows students how
to determine the mass of the Higgs boson by studying the so-called golden
channel ($H\to ZZ\to 4e$). Details of the implementation of the modules are
given in Refs. [1, 2].

##  2 Czech Particle Physics Project

The Czech Particle Physics Project (CPPP) is planned as a wide-ranging
outreach project with a home page which lets the user easily access different
modules. Currently, it contains the ALP and Higgs boson golden channel
modules, as well as a module that allows users to search a database of
publications with automated update and categorization related to the
experimental Higgs boson research [3]111Following a feasibility study by
Martin Kupka [4], implemented by Peter Zacik [1].. The CPPP is expected to be
expanded with new modules added to the home page in the structure that is
provided.

##  3 Finding an ALP using the AFP detector

###  3.1 ALP and AFP

Axion-Like-Particles (ALP) are potential dark matter candidates that are
predicted by some extensions of the Standard Model [5]. These new particles
could be detected with the ATLAS detector at CERN. The ATLAS Forward Proton
(AFP) detector is used to enhance the sensitivity for a discovery. The AFP
detector is located on either side of the ATLAS main detector and it is able
to detect protons that come from the central $pp$ interaction point but are
deviated from the axis of the beampipe. Two incoming protons could pass
closely by each other instead of strongly interacting in which case they will
interact through their electromagnetic fields and be deflected. This can
produce a pair of photons. If ALPs exist the photons could scatter through an
ALP ($\gamma\gamma\to ALP\to\gamma\gamma$), such that the scattered photons
are detected in the central detector while the deflected protons are detected
in the AFP detector. There are also diphoton events without an ALP being
involved which are referred to as background events and can be largely removed
by the two calculated energy loss values from the diphoton system and the
deflected protons. If the scattering happened through an ALP the calculated
energy loss should match. This criterion allows us to increase the ratio of
signal-to-background events and thus isolate the signal better. The fact that
the information from an additional detector enhances the sensitivity to find a
new particle is an important learning aspect.

###  3.2 Introduction page

Before obtaining access to the simulation, the user reads through an
introduction page which has five sections. This introductory page contains all
the information the user needs to appreciate the simulation. It should also
stimulate further reading about CERN, LHC and the ATLAS detector.

###  3.3 User experience

The main components of the interactive event display are:

  1. 1.

Control panel: allows the user to interact with the simulation

  2. 2.

Event counters: counter for the total number of events fired and counter for
auto-fired events

  3. 3.

Inner ATLAS central detector

  4. 4.

ATLAS central detector side view with AFP detectors on either side: animation
of the protons being fired

  5. 5.

Energy loss matching histograms: the energy loss calculated from the AFP
detectors in red and from the ATLAS central detector in green

  6. 6.

Invariant mass histograms: left histogram for all events, right histogram only
for events where the energy loss values are matched (as decided by the user,
or automatically in auto-fire mode)

  7. 7.

Access to admin page: password protected

Once the user loads the simulation page, a succession of post-it notes on the
screen will guide through the steps needed to run the simulation.

After going through the analysis process manually at least five times, the
user unlocks the auto-fire feature. This allows the user to generate a large
number of events automatically. In general, around 100 events are needed to
observe a clear signal in the simulated data.

Once the user has gone through the full auto-fire mode, a quiz pops-up which
asks the user to locate the signal. If this is done correctly an explanation
is given and the user is free to continue with the simulation. If a wrong
answer is given, the user is encouraged to continue firing more events and
collect more statistics.

###  3.4 Admin page

The website has a password protected admin page. This page allows an
administrator to modify the parameters of the simulation. Changes on this page
are sent to the server and are applied globally, thus anyone on the website
will have the new parameters. A possible extension of the project could be
that individual user groups can define their own Admin settings.

##  4 Higgs boson module

The Higgs boson golden channel is one of the research modes of the Higgs boson
[6]. In this channel, the Higgs boson decays into a pair of $Z$ bosons (real
or virtual). Each $Z$ boson then decays into a pair of light leptons (electron
or muon). The signal for the Higgs boson is four leptons with an invariant
mass equal to the Higgs boson mass. The main background to this signal is the
production of di-$Z$ bosons directly from the proton collision decaying to
leptons. For simplicity, the module focuses only on the four-electron final
state.

This process has been observed with the ATLAS detector [6]. The signal peaks
at the Higgs boson mass (125 GeV). The background covers a wide range of
invariant masses and peaks at the $Z$ boson mass around 91 GeV. In this
module, the user learns how to identify the four leptons by applying a
selection on the lepton transverse momentum, and thus the user will reproduce
the ATLAS distribution with the $Z$ and Higgs boson mass peaks.

The introduction page, user experience and admin page are similarly structured
as in the ALP with AFP module. The interactive event display for this module
is shown in Figure 1.

Figure 1: Screenshot of the interactive event display in the Higgs boson
golden channel module.

##  5 Conclusions

Two new outreach modules have been developed and implemented for the Czech
Particle Physics Project. They are accessible at http://cern.ch/cppp, and are
ready to serve students, teachers, and the interested public. Learning goals
are: giving school students a sense of the fun with science, technology,
engineering and mathematics, awakening interest in CERN, LHC and ATLAS
physics, understanding basic concepts in particle physics, acquiring initial
knowledge about elementary particles and how to detect them, becoming aware
that additional detectors help to separate signal and background events,
improving the understanding that increasing data statistics leads to new
observations and discoveries.

Feedback on the developed modules is very welcome and will be used in future
updates of the modules. Further modules on related fields of research as well
as modules on other research directions can be efficiently created and
maintained using the structural setup in this project.

##  6 Acknowledgements

The author would like to thank the International Particle Physics Outreach
Group (IPPOG) for fruitful discussions, and the students Martin Kupka, Antoine
Vauterin and Peter Zacik for their dedication. The grant support from the
"Fondu celoškolských aktivit pro rok 2022" Czech Technical University in
Prague is gratefully acknowledged. The project is also supported by the
Ministry of Education, Youth and Sports of the Czech Republic under the
project number LTT 17018.

## References

  * [1] P. Zacik, Implementation of a Portal Dedicated to Higgs Bosons for Experts and the General Public, May, 2021.  https://cds.cern.ch/record/2774895, presented 15 June 2021.
  * [2] A. Vauterin and A. Sopczak. 22nd IPPOG meeting, 17-19 Nov. 2021 Available online https://indico.cern.ch/event/1084892, 2021.
  * [3] A. Sopczak, A Portal Dedicated to Higgs Bosons for Experts and the General Public, PoS (2022). these proceedings.
  * [4] M. Kupka, Feasibility Study of Portal to Provide Knowledge about Higgs Boson to General Public and Experts, June, 2020.  https://cds.cern.ch/record/2722144, presented 23 June 2020.
  * [5] C. Baldenegro, Searching for Axion-like Particles with Proton Tagging at the LHC, Acta Phys. Pol. B Proc. Suppl. 12 (2019) 799–804.
  * [6] ATLAS Collaboration, Measurement of inclusive and differential cross sections in the $H\rightarrow ZZ^{*}\rightarrow 4\ell$ decay channel in $pp$ collisions at $\sqrt{s}$ = 13 TeV with the ATLAS detector, JHEP 10 (2017) 132–180.