# Recognizing black holes in gravitational-wave observations: Challenges in
telling apart impostors in mass-gap binaries

Sayak Datta  Inter-University Centre for Astronomy and Astrophysics, Post Bag
4, Ganeshkhind, Pune 411 007, India<EMAIL_ADDRESS>Khun Sang Phukon  Nikhef
- National Institute for Subatomic Physics, Science Park, 1098 XG Amsterdam,
Netherlands  Institute for High-Energy Physics, University of Amsterdam,
Science Park, 1098 XG Amsterdam, Netherlands  Institute for Gravitational and
Subatomic Physics, Utrecht University, Princetonplein 1, 3584 CC Utrecht,
Netherlands<EMAIL_ADDRESS>Sukanta Bose  Inter-University Centre for
Astronomy and Astrophysics, Post Bag 4, Ganeshkhind, Pune 411 007, India
Department of Physics and Astronomy, Washington State University, 1245
Webster, Pullman, Washington 99164-2814, U.S.A<EMAIL_ADDRESS>
###### Abstract

We study how by careful monitoring of the presence or absence of tidal
deformability (TD) and tidal-heating (TH) in the inspiral signal of compact
object binaries in ground-based gravitational wave (GW) detectors, one can
test if its components are black holes or not. The former property (TD) is
finite for neutron stars but vanishes for black holes (in general relativity),
whereas the latter is finite for black holes and negligible for neutron stars,
and affects the GW phase evolution of binaries in a distinctly different way.
We introduce waveform parameters that characterize the strength of tidal-
heating, and are zero when there is no horizon. We develop Bayesian methods
that use TD and TH for distinguishing the presence or absence of horizons in a
binary. This is timely owing to several claims that these stellar-mass
objects, especially, with masses heavier than those of neutron stars, may not
have a horizon but may be black hole mimickers or exotic compact objects
(ECOs). It is also astrophysically important to have the tools to test the
presence or absence of horizons in mass-gap binaries and, thereby, help detect
the heaviest neutron star or the lightest black hole. A proper accounting of
tidal-heating in binary waveform models will also be critical for an unbiased
measurement of characteristics of the equation of state of neutron stars in GW
observations of binaries containing them – or even to probe the existence of
ECOs. We show that purely based on GW waveforms it will not be possible to
discern binary horizons in the mass gap in Advanced LIGO, Virgo and KAGRA
detectors unless the binary is within a few tens of Mpc. However, third
generation ground-based detectors will be able to do so for binaries a few
hundred Mpc away.

##  I Introduction

In recent times, the discovery by LIGO-Virgo detectors of several compact
binary coalescences (CBCs) has ushered in the era of gravitational wave (GW)
astronomy Abbott  _et al._ (2019a, 2021). The LIGO-Virgo Collaboration also
observed the binary neutron (BNS) star merger GW170817 Abbott  _et al._
(2017a). These observations provided a fillip to tests of general relativity
(GR) in the strong-field regime Abbott  _et al._ (2019b, c); e.g., stringent
bounds on the mass of the graviton and violations of Lorentz invariance have
been placed Abbott  _et al._ (2016a, b, 2017b). Significantly, it has also
become possible to test the nature of the compact objects in binaries. The
deduced compactness of the components has led to the conclusion that they are
either black holes (BHs) or neutron stars (NSs). In the case of GW170817,
radius measurements were made Abbott  _et al._ (2018); De  _et al._ (2018)
that strongly disfavor them as BHs. A similar claim may be posited for the
other BNS contender GW190425 Abbott  _et al._ (2020). However, for the other
LIGO-Virgo binaries (which are much heavier than GW170817 or GW190425) Abbott
_et al._ (2019a), it remains to be conclusively proven that their components
are indeed BHs of GR and not, say, some exotic compact objects (ECOs) Yunes
_et al._ (2016); Cardoso  _et al._ (2016a); Aneesh  _et al._ (2018).

If the binaries show up with measured components masses in the mass-gap $\sim
2-5M_{\odot}$ Fryer  _et al._ (2012), then it poses the immediate challenge of
determining whether the component(s) with mass(es) in the gap are NSs or BHs.
Either occurrence will be significant, for it will either raise the maximum
known mass of a NS or lower the minimum known mass of a BH. These issues make
it imperative that methods be devised to discern compact objects with horizon
from those without. In this work we study if the presence of horizon can be
detected in binaries in the mass-gap by LIGO-Virgo. We also include in our
study the mass range $1-2M_{\odot}$ where neutron-star masses commonly occur.

Apart from NS and BH, ECOs may also occur in the same mass range. Multiple
models of ECOs have been proposed. These include Planck-scale modifications of
BH horizons Lunin and Mathur (2002); Almheiri  _et al._ (2013), gravastars,
Mazur and Mottola (2004), and boson stars Liebling and Palenzuela (2012) – to
name a few. In light of such proposals, it becomes necessary to devise
strategies to tell them apart from BHs. Several tests have been proposed to
probe the black-holeness of the compact objects in a binary. Distinguishing
binary merger remnants from BHs in the postmerger phase using echoes has
initiated rigorous modelling and search for those features in GW data Cardoso
_et al._ (2016b, a); Maggio  _et al._ (2019); Tsang  _et al._ (2020); Abedi
_et al._ (2017); Westerweck  _et al._ (2018); Cardoso and Pani (2019); Chen
_et al._ (2021); Xin  _et al._ (2021). Measurement of tidal deformability (TD)
Cardoso  _et al._ (2017); Sennett  _et al._ (2017); Maselli  _et al._ (2018);
Datta (2021) and spin-induced multipole moments Krishnendu  _et al._ (2017);
Datta and Bose (2019); Bianchi  _et al._ (2020); Mukherjee and Chakraborty
(2020); Datta and Mukherjee (2021) from the late inspiral can also be used to
test black-holeness. In this paper, we expand on past work to study how
difficult it is to perform a horizon test by using GWs emitted during the
inspiral phase of binary coalescences. For this purpose we include terms in
the binary waveform phase beyond the point-particle ones that arise due to the
material characteristics of the objects or the presence of horizons. In
particular, we introduce two new best measured horizon parameters for stellar-
mass binaries in ground-based detectors. Their precise measurement in binary
observations is useful in probing the existence of horizons in those systems.

Owing to their causal structure, BHs in GR are perfect absorbers that behave
as dissipative systems Thorne  _et al._ (1986); Damour (1982); Poisson (2009);
Cardoso and Pani (2013). The defining feature of a BH is the presence of its
horizon, which is a null surface and a one-way membrane. It is due to the
presence of the horizon that a BH in a binary absorbs energy and angular
momentum from the orbit. This phenomenon is called tidal-heating (TH) Hartle
(1973); Hughes (2001); Poisson and Will (1953). Energy loss via TH backreacts
on the binary’s evolution, resulting in a shift in the phase of the GWs
emitted by the system. Therefore, the absence of a horizon – or any kind of
change in the near horizon structure that modifies this absorption – will
leave its imprint in the phasing of GWs emitted. A careful observation thus
has the potential to measure these differences in the GW phase.

Indeed, TH has been proposed for probing the presence of horizons along with
the existence of higher dimensions and quantum effects at horizon scale
Chakraborty  _et al._ (2021); Datta (2020); Agullo  _et al._ (2021); Datta and
Phukon (2021); Sago and Tanaka (2021). Its importance in identifying horizons
of intermediate-mass and supermassive compact objects has been examined for
the space-mission LISA Datta and Bose (2019); Maselli  _et al._ (2018); Datta
_et al._ (2020). In the current work, we study its usefulness for stellar mass
binaries – of the type observable by ground-based GW detectors like LIGO,
Virgo, and KAGRA Akutsu  _et al._ (2020).

The TH of a black hole or any other star can be expressed in similar
mathematical forms if the viscosity coefficient $(\eta)$ of a BH is identified
with its mass Glampedakis  _et al._ (2014). For NS, one has $\eta_{\rm NS}\sim
10^{4}\left(\frac{\rho}{10^{14}\textrm{gm}-\textrm{cm}^{-3}}\right)^{5/4}\left(\frac{10^{8}\textrm{K}}{T}\right)^{2}\textrm{cm}^{2}\textrm{s}^{-1}$,
and for a BH, its form is $\eta_{\rm BH}\sim 8.6\times
10^{14}\left(\frac{M}{M_{\odot}}\right)\textrm{cm}^{2}\textrm{s}^{-1}$. Since
the correction in GW phase due to TH is proportional to $\eta$, for an NS that
correction is 10 orders of magnitude smaller than BH Glampedakis  _et al._
(2014). While this distinction presents an interesting prospect for
observational exploitation, as we show here the magnitude of TH for binary
black holes (BBHs) remains small and is useful for discerning the presence of
horizons for very large $M$ (as for EMRI central objects in LISA) or for
strong signals. This implies that for stellar-mass BHs, detection of TH in
LIGO-Virgo will require the binary to be within tens of Mpc, as shown below.
While the occurrence of such a golden binary is not impossible, GW
observations to date rule it as improbable. Nevertheless, for completeness of
TH analysis, we examine this case in this work. For more realistic BBH
distances, a detection of TH and its utilization for discerning horizons will
have to wait for third generation detectors. The formalism initiated here for
accounting for TH will be relevant for those detectors as well.

Another property of compact objects that leaves an imprint on GWs is tidal
deformability (TD). A body immersed in an external tidal field, such as due to
a binary companion, experiences an induced quadrupole moment. That moment is
proportional to the tidal field, and the proportionality factor is the tidal
deformability $(\lambda)$. This tidal deformation in turn affects the binary’s
orbital motion and the emitted GWs. The GW phasing carries an imprint of the
dimensionless tidal deformability $(\Lambda_{i}\equiv\lambda_{i}/m_{i}^{5})$
of the two masses $m_{i=1,2}$ Flanagan and Hinderer (2008). Material bodies,
such as NS, have substantial $\Lambda_{i}$ values Abbott  _et al._ (2018,
2020), but black holes have a vanishing value Damour and Nagar (2009);
Binnington and Poisson (2009).111See, however, Ref. Chakravarti  _et al._
(2019) for an example of a non-GR result, and Ref. Brustein and Sherf (2020)
for quantum BHs. Hence, using appropriate modeling it is possible to measure
$\Lambda_{i}$ and probe the properties of the bodies. More than TH, it is TD
that we find to have a dominating influence in recognizing the absence of
horizons in a stellar-mass binary, particularly, when the components masses
are around $1-2M_{\odot}$. TD decreases with mass, and above this range is
vanishingly small for realistic neutron star equations of state.

Mass-gap objects can be as heavy as $\sim 5M_{\odot}$. This is the reason we
analyze binaries with component masses between $1-5M_{\odot}$. Since TD has
little influence above $2M_{\odot}$ it is left to TH to help recognize the
presence of horizons. We find that it is highly improbable to do so for
binaries with component masses in the range $2-5M_{\odot}$ in the current
generation of detectors.

It has been shown that in GR the Love number vanishes for BHs, but not for
other compact objects like NSs Binnington and Poisson (2009); Landry and
Poisson (2014); Chia (2020). However, recently it has been suggested that the
Love number can be nonzero for nonaxisymmetrically perturbed rotating BHs Le
Tiec  _et al._ (2021). In the current work we take that the tidal
deformability of all BHs is zero. Thus, our results may need to be revisited
depending on how this matter gets resolved.

We begin by studying in Sec. II the TH terms that appear in the GW phase of a
binary. There we identify two horizon parameters that are best measured for
stellar-mass binary signals in ground-based detectors. There we also show how
the spin-induced quadrupole moment and tidal deformability of the binary
components influence the waveform. In Sec. III we develop the method for
weighing the evidence in data for the presence or absence of horizon,
utilizing the aforementioned horizon parameters and phase terms in a Bayesian
formalism. In Secs. IV and V we implement this formalism on a large population
of simulated binary signals in noisy data simulated with Advanced LIGO and
Advanced Virgo noise. We conclude with a discussion on future prospects in
Sec. VI.

##  II Effect of TH on binary waveforms

Consider a compact binary with component masses $m_{i}~{}~{}(i=1,2)$, total
mass $m=m_{1}+m_{2}$, and mass-ratio $q=m_{2}/m_{1}$, with $m_{2}\leq m_{1}$.
Let the dimensionless component spins be $\chi_{i}$. Under the adiabatic
approximation the orbital evolution of the binary can be quantified in the
post-Newtonian formalism with reasonable accuracy, especially, when it is far
from merger Blanchet (2014). The dynamics of the system is governed by energy
and angular momentum loss from the orbit. Usually it has a contribution
arising from taking the components as point particles (PP) and another one
originating from their finite size. The latter can be decomposed into two
parts, (i) tidal deformation of each component due to the gravitational field
of the other and (ii) the amount of energy absorbed by individual components
from the orbit, namely, tidal heating.

The dynamics of the system and, therefore, the emitted GW depends on all of
these contributions. Hence, the Fourier transformed GW waveform can be written
as

$\tilde{h}(f)=A(f)e^{i\left(\Psi_{\rm PP}+\Psi_{\rm TD}+\Psi_{\rm
TH}\right)}\,,$ (1)

where $f$ is the instantaneous GW frequency and $A(f)$ is the frequency-
dependent amplitude. The phase terms – $\Psi_{\rm PP},\Psi_{\rm TD},$ and
$\Psi_{\rm TH}$ – are the phase contributions arising from the point-particle
approximation, TD, and TH, respectively.

Since GW absorption is negligible for matter Glampedakis  _et al._ (2014), it
is reasonable to exploit evidence of TH in binary waveforms to discern the
existence of horizons Datta and Bose (2019); Maselli  _et al._ (2018). This
expectation led us to introduce the horizon parameter $H$ for extreme mass-
ratio inspirals (EMRIs) that LISA may observe Datta and Bose (2019). Till now
horizon distinguishability employing TH has been addressed primarily for LISA
sources, such as EMRIs and supermassive BH binaries. However, even in the case
of supermassive BH binaries, it is the combined tidal heating of both binary
components is what has been employed, which ignores the possibility that not
both components may have horizons (or lack them) Maselli  _et al._ (2018).
Such an approach is reasonable for initial forays in this subject but, in
general, different values of $H$ need to be considered. Here we apply the
formalism to binaries with similarly massive components primarily to target
the LIGO-Virgo population of stellar-mass binaries. As it is a broad subject,
we keep the studies with third-generation for the future.

For a near-equal-mass binary we define horizon parameters for each component,
$(H_{1},H_{2})$, such that the value of $H_{i}$ is 1 (0) when the $i$th
component has a horizon present (absent). In the case of circular orbits, the
flux of energy at the horizon can be expressed as a PN expansion Alvi (2001);
Poisson and Corrigan (2018); Poisson (2009); Nagar and Akcay (2012); Bernuzzi
_et al._ (2012); Chatziioannou  _et al._ (2016); Cardoso and Pani (2013).
Since TH signifies presence of horizon, we multiply the energy flux absorbed
by each component with the corresponding $H_{i}$. In the case of partial
absorption, one has $0<H_{i}<1$. Therefore, the absorbed flux is

$\displaystyle-\frac{dE}{dt}=$
$\displaystyle{}\frac{32}{5}\nu^{2}\frac{v^{15}}{4}\sum_{i=1}^{2}H_{i}\left(\frac{m_{i}}{m}\right)^{3}\left(1+3\chi^{2}_{i}\right)\left\\{-(\hat{L}_{N}.\hat{S}_{i})\chi_{i}\right.$
(2)
$\displaystyle\left.+2\left[1+\left(1-\chi^{2}_{i}\right)^{1/2}\right]\frac{m_{i}}{m}v^{3}\right\\}\,,$

where $\nu={m_{1}m_{2}}/{m^{2}}$ is the symmetric mass-ratio, $v$ is the
orbital velocity, and $\hat{S}_{i}$ and $\hat{L}_{N}$ are the unit vectors
along the directions of the $i$th spin and the orbital angular momentum,
respectively.

###  II.1  New waveform parameters characterizing TH

The horizon parameters $H_{1,2}$ appear in the GW phase in terms that also
include mass and spin factors. This makes them degenerate with those
parameters, in that it is more practical to measure the following effective
observables instead of $H_{1,2}$:

$\displaystyle H_{eff5}\equiv$
$\displaystyle{}\sum_{i=1}^{2}H_{i}\left(\frac{m_{i}}{m}\right)^{3}\left(\hat{L}.\hat{S}_{i}\right)\chi_{i}\left(3\chi_{i}{}^{2}+1\right)\,,$
(3a) $\displaystyle H_{eff8}\equiv$ $\displaystyle{}~{}4\pi
H_{eff5}+\sum^{2}_{i=1}H_{i}\left(\frac{m_{i}}{m}\right)^{4}\left(3\chi_{i}{}^{2}+1\right)$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\times\left(\sqrt{1-\chi_{i}{}^{2}}+1\right)\,.$
(3b)

These are analogous to the effective spin parameter $\chi_{\rm eff}$ that was
introduced Damour (2001); Racine (2008); Ajith (2011) to characterize spinning
compact binary waveforms: While the spins of the individual binary components
are themselves difficult to measure (like $H_{1,2}$ here), their combined
impact on the waveform phase, captured by $\chi_{\rm eff}$, lends itself to
more precise measurements. Dependencies of $H_{eff5}$ and $H_{eff8}$ on
component spins are shown in Fig. 1 and Fig. 2, respectively.

If the system is a binary black hole (BBH), as long as any one of the
component spins is finite both $H_{eff5}$ and $H_{eff8}$ will be nonzero. By
contrast, for the same spins a horizonless binary would have both $H_{eff5}$
and $H_{eff8}$ vanish. Therefore, it is easiest to discern between the
presence and absence of horizons in BBHs that have at least one component with
sufficiently large spin.

On the other hand, when both component spins of a BBH tend to zero, one has
$H_{eff5}\to 0$ but $H_{eff8}\neq 0$; see the inset in Fig. 2. Therefore, in
the low-spin limit $H_{eff8}$ emerges as a discriminator for the presence or
absence of horizons. Here the measurement is helped for small mass-ratio
($q$), which ensures large $H_{eff8}$.

Figure 1:  $H_{eff5}$ is plotted for a range of $\chi_{eff}$ values and for
all possible values of $m_{2}/m$. Figure 2: $H_{eff8}$ is plotted for a range
of $\chi_{eff}$ values and for all possible values of $m_{2}/m$.

It is important to note that our choice of waveforms, based on the stationary-
phase approximation (SPA) Cutler and Flanagan (1994), is for illustrative
purpose, essentially as a proof of principle that the method proposed here is
promising for identifying binary components with horizons from those without.
For making such classification in real data, it will likely be important to
use more accurate templates, such as those based on the EOB-NR formalism Husa
_et al._ (2016); Khan  _et al._ (2016); Hannam  _et al._ (2014). We will
present those results in future. Having said that, our choice of SPA-based
inspiral waveforms is a reasonable one for illustrating the power of this
method for the systems studied here.

We deduce the GW phase involving TH by using Eq. (2.7) of Ref. Tichy  _et al._
(2000) (see Isoyama and Nakano (2018) for the details). We find the phase
shift due to the associated horizon absorption to be

$\displaystyle\Psi_{\rm TH}=$
$\displaystyle{}\frac{3}{128\nu}\left(\frac{1}{v}\right)^{5}\left[-\frac{10}{9}v^{5}H_{eff5}\left(3\log\left(v\right)+1\right)\right.$
(4) $\displaystyle-\frac{5}{168}v^{7}H_{eff5}\left(952\nu+995\right)$
$\displaystyle\left.+\frac{5}{9}v^{8}\left(3\log\left(v\right)-1\right)(-4H_{eff8}+H_{{eff5}}\psi_{\text{SO}})\right]\,,$

where

$\begin{split}\psi_{\text{SO}}\equiv~{}&\frac{1}{6}\big{[}\big{(}-56\nu-73\sqrt{1-4\nu}+73\big{)}\big{(}\hat{L}.\hat{S}_{1}\big{)}\chi_{1}\\\
&+\big{(}-56\nu+73\sqrt{1-4\nu}+73\big{)}\big{(}\hat{L}.\hat{S}_{2}\big{)}\chi_{2}\big{]}\,.\end{split}$
(5)

Note that $H_{eff5}$ and $H_{eff8}$ arise at different PN orders in the phase.

###  II.2 New waveform parameters characterizing quadrupole moment

The two bodies in a coalescing compact binary can have spin. In case of a
nonzero spin a body would develop a spin-induced quadrupole moment. The
leading order contribution arises due to the mass quadrupole moment
$(M_{2(i)})$ of both bodies, $(i=1,2)$, at 2 PN order. If the bodies are BHs,
then $M_{2(i)}=-\chi^{2}_{(i)}m_{(i)}^{3}$. If they are NSs or ECOs that
moment may be modified as $M_{2(i)}=-\kappa_{(i)}\chi_{(i)}^{2}m_{(i)}^{3}$.
Measuring the quadrupole moment $\kappa$ from observations can be used to
probe the nature of the compact objects Krishnendu  _et al._ (2017, 2019a);
Datta and Bose (2019); Krishnendu and Yelikar (2019). Since in a binary the
quadrupole moments $\kappa_{i}$ of both the bodies contribute at the similar
order, they are degenerate. Usually, a combination of $\kappa_{1}$ and
$\kappa_{2}$ are used for the measurement Krishnendu  _et al._ (2017). Here we
define a new effective parameter $\tilde{Q}$ as follows:

$\tilde{Q}m^{4}=-\kappa_{+}S_{\ell}^{2}+S_{\ell}\Sigma_{\ell}(\kappa_{-}-\delta\kappa_{+})+\frac{\Sigma_{\ell}^{2}}{2}(\delta\kappa_{-}-\kappa_{+}+2\nu\kappa_{+})\,,$
(6)

where $\kappa_{+}=\kappa_{1}+\kappa_{2}$, $\kappa_{-}=\kappa_{1}-\kappa_{2}$,
$\delta=\frac{m_{1}-m_{2}}{m}$,
$S_{\ell}=\sum_{i}\hat{L}.\hat{S}_{i}\chi_{i}m_{i}^{2}$,
$\Sigma_{\ell}=m(\hat{L}.\hat{S}_{2}\chi_{2}m_{2}-\hat{L}.\hat{S}_{1}\chi_{1}m_{1})$.
Then the phase can be expressed as Bohé  _et al._ (2015)

$\psi_{QM}=-\frac{25}{32}\frac{\tilde{Q}}{\nu}\frac{1}{v}\,.$ (7)

Note that once the phase has been expressed in terms of $\tilde{Q}$, it is not
necessary for $\tilde{Q}$ to be limited to Eq. (6): It is straightforward to
incorporate other models of $M_{2}$ into it, such as for boson stars Ryan
(1997); Cardoso and Pani (2019). We will use Eq. (7) in our modeling of phase
due to nonzero quadrupole moment. In Refs. Krishnendu  _et al._ (2019b);
Narikawa  _et al._ (2021), observational constraint on spin induced quadrupole
moment has been found. Individual measurement suffers from broad posterior
distribution, pointing towards low measurability with current detectors. Our
results below are consistent with these observations.

##  III Bayes factors for horizon discrimination

In the presence of a GW signal strain $h(t,\boldsymbol{\theta})$,
characterized by parameters $\boldsymbol{\theta}$, the detector strain time-
series can be modeled as $d(t)=n(t)+h(t,\boldsymbol{\theta})$, where $n(t)$
denotes the detector’s noise. In the presence of a GW signal
$h(t,\boldsymbol{\theta})$, described by a model $\mathcal{H}$, the likelihood
of the data is Cutler and Flanagan (1994):

$P\left(d|\boldsymbol{\theta},\mathcal{H}\right)\propto\exp{\left[-\frac{1}{2}\langle
d-h(\boldsymbol{\theta})|d-h(\boldsymbol{\theta})\rangle\right]}\,,$ (8)

under the assumption of Gaussian and stationary-detector noise. The angular
bracket in Eq. (8) defines a noise-weighted inner product between two real
time-series $a(t),\,b(t)$, and is given as

$\langle
a,b\rangle=4\Re\int_{f_{low}}^{f_{high}}df\frac{\tilde{a}^{*}(f)\tilde{b}(f)}{S_{n}(f)},$
(9)

where $S_{n}(f)$ is the one-sided power spectral density (PSD) of the detector
noise, and $f_{low}$ and $f_{high}$ are the low-frequency cutoff and high-
frequency cutoff, respectively Cutler and Flanagan (1994); Allen  _et al._
(2012). Using the inner product, one can also define the signal-to-noise ratio
(SNR) $\rho$ for the template $h(t,\boldsymbol{\theta})$ as

$\rho=\frac{\langle d|h\rangle}{\sigma},$ (10)

where $\sigma=\sqrt{\langle h|h\rangle}$ is the template normalization.

We will assume that noncolocated detectors on the globe have uncorrelated
noise; hence, the combined likelihood is given as Veitch  _et al._ (2015),

$P\left(\mathbf{d}|\boldsymbol{\theta},\mathcal{H}\right)=\prod_{i}^{N}P\left(d_{i}|\boldsymbol{\theta},\mathcal{H}\right)\,,$
(11)

where $\mathbf{d}\in\\{d_{1},d_{2},\cdots,d_{N}\\}$ represents combined data
from all $N$ detectors. Using the coherent network likelihood function,
posterior probability density can be written as

$P\left(\boldsymbol{\theta}|\mathbf{d},\mathcal{H}\right)=\frac{P\left(\mathbf{d}|\boldsymbol{\theta},\mathcal{H}\right)P\left(\boldsymbol{\theta}|\mathcal{H}\right)}{P\left(\mathbf{d}|\mathcal{H}\right)}\,,$
(12)

where $P\left(\boldsymbol{\theta}|\mathcal{H}\right)$ is the prior probability
density function or prior of the parameters $\boldsymbol{\theta}$. In the
denominator, $P\left(\mathbf{d}|\mathcal{H}\right)$ is the marginalized
posterior probability density over all parameters $\boldsymbol{\theta}$, and
is also known as the evidence for the model $\mathcal{H}$. The evidence
$P\left(d|\mathcal{H}\right)$ serves as a normalization constant of the
posterior probability for $\mathcal{H}$. The evidence computed for two
competing models or hypotheses can be used to determine which one is favored
by the data. In this work, we compute Bayes factors for simulated signals to
compare two hypotheses, namely,

  1. 1.

The horizon hypothesis $\mathcal{H}_{\textrm{BBH}}$: Signal carries imprints
of horizon absorption and spin-induced quadrupole moment,

  2. 2.

The no-horizon hypothesis $\mathcal{H}_{\textrm{BNS}}$: Signal has no imprint
of horizon absorption, but has TD and spin-induced quadrupole moment.

In Bayesian model selection, we compute the Bayes factor,

$\textrm{BF}=\frac{P\left(\mathbf{d}|\mathcal{H}_{\textrm{BBH}}\right)}{P\left(\mathbf{d}|\mathcal{H}_{\textrm{BNS}}\right)}\,.$
(13)

If the Bayes factor is greater than some preset threshold, i.e.,
$\textrm{BF}>\textrm{BF}_{\textrm{Th}}$ then the hypothesis
$\mathcal{H}_{\textrm{BBH}}$ is preferred over the other hypothesis
$\mathcal{H}_{\textrm{BNS}}$ in the data. Moreover, we use the Dynesty sampler
Speagle (2020), as implemented in the Bilby package Ashton  _et al._ (2019);
Romero-Shaw  _et al._ (2020), to compute the posterior probability densities
for our simulated signals. We use a likelihood function marginalized over time
$t_{c}$ and phase $\phi_{c}$ at coalescences of binaries Farr ; Thrane and
Talbot (2019) and distance $d_{L}$ Singer and Price (2016); Singer  _et al._
(2016), thus removing the need for sampling those parameters without affecting
the posterior probability densities in the parameters of interest. The
posterior probability densities for these parameters can be reconstructed
analytically from the full set of posterior samples Thrane and Talbot (2019).

The posteriors of some of the parameters for the $\mathcal{H}_{\textrm{BBH}}$
hypothesis are shown in Fig. 3. To compute them, we considered the signal
integration in a frequency range such that it ends at $f_{\rm ISCO}$, while
the duration of the signal is 16s. $f_{\rm ISCO}$ is the instantaneous GW
frequency at the ISCO of the binary Kidder  _et al._ (1992); Blanchet (2002).
In practice, it may be possible to begin the signal integration at a frequency
as low as 10Hz, which is what aLIGO design targets. Similarly, when waveform
modeling is available to accurately incorporate TH beyond the ISCO, the upper
frequency cutoff will also be raised. Both these changes will improve
parameter estimation as well as Bayes-factor based model discrimination.

Before setting up signals simulation for BF computations, it is worthwhile to
examine through computationally inexpensive, even if approximate, means how
precisely the horizon parameters would be measurable in mass-gap binaries.
Such a computation is afforded by the Fisher information matrix (FIM), as
defined below. We estimate how large the noise-limited errors are of the
horizon parameters ${\bm{\vartheta}}$, by modeling the measured values after
the maximum likelihood estimators MLEs Helstrom (1995). Owing to noise, the
MLE will fluctuate about the respective true values, i.e.,
$\hat{{\bm{\vartheta}}}={\bm{\vartheta}}+\delta{\bm{\vartheta}}$, where
$\delta{\bm{\vartheta}}$ is the random error. The extent of these fluctuations
is estimated by the elements of the variance-covariance matrix,
$\gamma^{ab}=\overline{\delta{\bm{\vartheta}}^{a}\delta{\bm{\vartheta}}^{b}}$
Helstrom (1995), which is bounded by the signal via the Cramer-Rao inequality,
namely,

$\left\lVert\mathbf{\gamma}\right\rVert\geq\left\lVert\mathbf{\Gamma}\right\rVert^{-1}\;,$
(14)

where $\mathbf{\Gamma}$ is the FIM:

$\displaystyle\Gamma_{ab}$
$\displaystyle=\left\langle{\partial_{a}\tilde{h}({\bm{\vartheta}}),\partial_{b}\tilde{h}({\bm{\vartheta}})}\right\rangle\,,$
$\displaystyle\equiv
4\Re\int_{f_{low}}^{f_{high}}d{f}\,\frac{\partial_{a}\tilde{h}^{*}(f;{\bm{\vartheta}})\;\partial_{b}\tilde{h}(f;{\bm{\vartheta}})}{S_{n}(f)}\,.$
(15)

Above, $\partial_{a}$ is the partial derivative with respect to the parameter
$\vartheta^{a}$. Therefore,
$\Delta\vartheta^{a}\equiv(~{}{\overline{\delta\vartheta^{a}\,\delta\vartheta^{a}}}~{})^{-1/2}=\Gamma_{aa}^{-1/2}$
gives the lower bound on the root-mean-square error in the estimate of
$\vartheta^{a}$. The two are equal in the limit of large SNR Helstrom (1995).
The error estimates listed here are the $\Delta\vartheta^{a}$ obtained from
the FIM.

When one computes $\Gamma_{ab}$ for the binary parameters one typically finds
that its offdiagonal terms are nonzero, which implies that there are
covariances among the parameter errors. It is, however, possible to mitigate
those covariances for a different set of parameters. In the two-dimensional
parameter subspace of $(H_{1},H_{2})$, we find that $(H_{eff5},H_{eff8})$ are
such parameters.

One can also use FIM to deduce errors $(\Delta H_{eff5},\Delta H_{eff8})$ in
the new horizon parameters for our binaries of interest. This is how we
estimate that for a mass-gap binary at a distance of 10Mpc to a few tens of
Mpc, it is possible to measure $(H_{eff5},H_{eff8})$ to a few tens of percent
in a three detector LIGO-Virgo network with the aforementioned noise PSD. A
similar FIM calculation for the third generation detector Einstein Telescope
shows that the same measurement precision is achievable even when the same
mass-gap BBH is pushed out to a few 100 Mpc. As mentioned above, in spite of
the weak effect of TH in mass-gap binaries in current detectors, for the
completeness of the waveforms used in our simulations we continue to retain
the TH terms in their phases. The impact of those terms for third generation
detectors and binaries not limited to the mass-gap will be studied elsewhere.

##  IV Priors

The distributions and ranges of parameter priors of the simulated binary
waveforms used in our Bayesian model selection studies are listed in Table 1.
The possible values of $H_{eff5}$ and $H_{eff8}$ are shown in Fig. 1 and Fig.
2, respectively.

In Fig. 3, we show the posterior probability distributions of various
parameters of a BBH injected signal obtained from a Bayesian analysis. The
luminosity distance $(d_{L})$, chirp mass $(\mathcal{M})$, mass ratio $(q)$,
and effective spin $(\chi_{eff})$ are well measured. The estimation recovers
the injected values. Comparatively $H_{eff5}$ and $H_{eff8}$ are poorly
measured. Although we recover the injected values, and the posterior is
certainly different from the flat prior, the error is large. This is expected
as TH is a higher-order effect.

Figure 3: This parameter estimation corner plot shows the posterior probabilities of the parameters of a BBH injection. The injected values are indicated by the orange lines. The posteriors show that the measurement of the new parameters introduced here, $H_{eff5}$ and $H_{eff8}$, will add very little bias to the estimation of standard intrinsic parameters of compact binaries. Those parameters are estimated as precisely as expected from past studies.  Table 1: Choice of priors in our Bayesian analysis of simulated signal injections.  Parameter | Distribution | Range | Boundary condition | Units
---|---|---|---|---
Chirp mass ($\mathcal{M}$) | Uniform | [1.0, 4.5] | – | $\textrm{M}_{\odot}$
Mass ratio ($q$) | Uniform | [0.5, 1.0] | – | –
Spin of primary object $(\chi_{1})$ | Uniform | [0.00, 0.99] | Reflective | –
Spin of secondary object $(\chi_{2})$ | Uniform | [0.00, 0.99] | Reflective | –
Tidal deformability of primary object $(\Lambda_{1})$ | Uniform | [0.0, 500] | – | –
Tidal deformability of secondary object $(\Lambda_{2})$ | Uniform | [0.0, 800] | – | –
2.5 PN horizon parameter ($H_{eff5}$) | Uniform | [-4.0, 4.00] | – | –
4 PN horizon parameter ($H_{eff8}$) | Uniform |  [-45, 45.0] | – | –
Luminosity distance ($d_{L}$) | Uniform |  [10.0, 500] | – | Mpc
Right ascension (RA) | Uniform | [0.0, $2\pi$] | Periodic | radian
Declination (DEC) | Cosinusoidal | [$-\pi/2$, $+\pi/2$] | – | radian
Phase at reference frequency $(\phi_{0})$ | Uniform |  [0.0, $2\pi$] | Periodic | radian
Polarization angle $(\psi)$ | Uniform |  [0, $\pi$] | Periodic | radian
Line-of-sight angle $(\theta_{JN})$ | Sinusoidal |  [0, $\pi$] | – | radian
Effective parameter for spin-induced quadrupole moment $\tilde{Q}$ | Uniform | [$-2$, 0.0] | – | –

##  V Simulation and Results

###  V.1 Properties of the simulated sources

We now quantify how successfully one can discriminate between a BBH signal
from a BNS one in noisy data. For this signal model selection test we
simulated a population of 1250 binaries, which are distributed uniformly in
comoving volume between 50Mpc to 250Mpc. Component masses were taken to be
$m_{1,2}\in~{}[1-5]M_{\odot}$ and spins chosen to be aligned or antialigned
with the orbital angular momentum, and with dimensionless magnitude
$\chi_{1,2}\in~{}[0,0.9]$. For model selection we constructed two families of
waveforms – both for signals (for adding in simulated noisy data) and
templates (for matched-filtering that data) – namely: (a) TaylorF2 (TF2),
modified with TD contribution (TidalTF2) for representing horizonless
components with nonzero TD. Here, the GW phase is devoid of any contribution
from $H_{eff5}$ or $H_{eff8}$; (b) HeatedTaylorF2 (HTF2), which is TF2 but
with additional phase terms arising from TH, as described in Eq. (4). We have
included the effect of the spin-induced quadrupole moment appropriately in
both cases via the phase term in Eq. (7). We used the Akmal, Pandharipande,
and Ravenhall (APR) equation of state (EOS) Akmal  _et al._ (1998) for this
purpose Pappas and Apostolatos (2012) to model the new effective parameter
$\tilde{Q}$ introduced in Eq. (7). The injected values of $\tilde{Q}$ have the
range $[-1.60,0]$. From Ref. Pappas and Apostolatos (2012) we constructed the
values of $\kappa_{i}$ of the $i$th body of mass $m_{i}$ and spin $\chi_{i}$.
From these values we find the corresponding value of $\tilde{Q}$ using Eq.
(6), which is used for injection.

Using the aforementioned waveform models we performed simulated signal
injection studies in simulated colored-Gaussian data of two LIGO detectors
(Hanford and Livingston) with aLIGO zero-detuned high-power (ZDHP) noise
power-spectral density Shoemaker . To keep computational costs manageable we
limited all our signals (and the filtering and parameter estimation) to only
16sec, and till the innermost stable circular orbit (ISCO). In one study, the
sources are taken to be CBCs, with BHs as components. Hence, the injected
waveform used is HTF2. We then performed a Bayesian analysis to measure the
parameters of these sources with both TidalTF2 and HTF2 templates and compared
the natural-log of their Bayes factor $\ln{\mathrm{BF}}$, utilizing the
definition in Eq. 13, for the same “horizon” injections to test if such an
analysis has the power to identify the true signal model.

In Fig. 4 we plot the $\ln{\rm BF}$ with respect to $\chi_{eff}$ in the
$x$-axis and $\mathcal{M}$ in color. Each point in this figure represents an
HTF2 injection. The fact that for a large majority of them the values of
$\ln{\rm BF}$ are positive, suggests that for this injection set model
selection strongly favors horizon injections. In an ideal case, all of the
points should be above the $\ln{\rm BF}=0$ line. Deviation from this
expectation for a minority of the injections is due to their low SNRs or
similarity of their signals with the TidalTF2 waveforms for the same
parameters (as will be explored in more detail below). With longer duration
waveforms and higher SNRs this result should get somewhat better, for lower
masses. The origin of the high values is likely due to a combination of TD,
quadrupole moment, which will be discussed below.

Figure 4: The logarithms of Bayes factors are plotted vs $\chi_{eff}$ above,
in support of HTF2, when the injection is also HTF2. The color bar represents
the chirp masses of injections.

###  V.2 Assessing the statistical significance of the horizon discriminator

In the preceding section, we found that barring a small subset the model
selection returns positive $\ln{\rm BF}$ values for the injected sources,
which is tantamount to saying that the observations favor the true signal
model, namely, HTF2 here. However, for a small subset, with negative ln BF
values, the wrong signal model (TidalTF2) is preferred. This raises the
possibility that the opposite can also happen, i.e., some TidalTF2 injections,
searched with both types of templates, may return $\ln{\rm BF}$ values
favoring the HTF2 signal model. As with any statistical analysis, it also
becomes important to interpret quantitatively the probability with which the
nature of those sources will be identified correctly. BBH injection studies
enable one to do precisely that. However, it is also important to assess the
probability with which the nature of that source will be misidentified. For
example, if the value of $\ln{\mathrm{BF}}$ turns out to be 5 for a BBH
signal, it is important to interpret that value in terms of how probable it is
to be identified correctly as of BBH origin (the true hypothesis,
$\mathcal{H}_{\rm BBH}$) and incorrectly from a BNS (the wrong hypothesis,
$\mathcal{H}_{\rm BNS}$), in noisy data. As we discuss next, the former
probability can be assessed from the above study of BBH injections, whose
$\ln{\mathrm{BF}}$ values form the foreground distribution
$p\left(\ln\mathrm{BF}\mid\mathcal{H}_{\textrm{BBH}}\right)$, which is the
probability distribution of the ln BF values given that the
$\mathcal{H}_{\textrm{BBH}}$ hypothesis is true, i.e., the (injected) signals
belong to the HTF2 model. On the other hand, to assess how probable it is for
the TidalTF2 signals to be misidentified as HTF2, we also study injections of
horizonless signals generated using the TidalTF2 waveform model; these
$\ln{\mathrm{BF}}$ values form the background distribution
$p\left(\ln\mathrm{BF}\mid\mathcal{H}_{\textrm{BNS}}\right)$ when the
hypothesis being tested for a detected event is that it is from a BBH.

To obtain the foreground distribution corresponding to HTF2 signals, we
compute the BF values for HTF2 injections using Eq. (13). We plot the
distribution of the ln BF for these values in red in Fig. 5. To construct the
background distribution for the same signals, we compute the BF values – but
now for TidalTF2 injections – using Eq. (13). The distribution of the ln BF
for these values is plotted in black in the same figure. The samples of
$\ln\mathrm{BF}$ from the foreground and background distributions are used to
estimate the efficiency with which BBHs can be identified in GW events and
assign a statistical significance to each identification.

Figure 5:  The red curve represents the foreground distribution
$p\left(\ln\mathrm{BF}\mid\mathcal{H}_{\textrm{BBH}}\right)$ of
$\ln{\mathrm{BF}}$ obtained from simulated BBH signal injections, with
$m_{i}\in[1,2]M_{\odot}$, in simulated Gaussian noise of Advanced LIGO. The
blue vertical line denotes threshold log Bayes factors
$\ln{\mathrm{BF}}_{\mathrm{threshold}}$ corresponding to $2\sigma$
significance. The black curve depicts the background distribution
$p\left(\ln\mathrm{BF}\mid\mathcal{H}_{\textrm{BNS}}\right)$ obtained from BNS
injections.

In Fig. 5, we show the estimated foreground and background distributions of
the $\ln{\rm BF}$ for a subpopulation of the injected binaries discussed in
Fig. 4 and the preceding subsection. This subpopulation includes only those
binaries that have component masses in the range
$m_{1,2}\in[1-2]\mathrm{M}_{\odot}$. As noted earlier, the foreground
distribution is constructed by injecting HTF2 and calculating the Bayes factor
in favor of HTF2. The background distribution is estimated by calculating the
Bayes’ factor in support of HTF2 for the TidalTF2 injections. We use the
background distribution
$p\left(\ln\mathrm{BF}\mid\mathcal{H}_{\textrm{BNS}}\right)$ to compute the
false-detection-probability (FDP) of a BBH claim, given that a BNS signal is
actually present in the data.222The FDP is the frequentist p-value of
hypothesis testing Pearson (1992); Neyman and Pearson (1992). The FDP is
computed from $p\left(\ln\mathrm{BF}\mid\mathcal{H}_{\textrm{BNS}}\right)$ for
a measured $\ln\mathrm{BF}$-value $\ln\mathrm{BF}_{\mathrm{measured}}$ as
follows:

$\mathrm{FDP}=1-\int_{-\infty}^{\ln\mathrm{BF}_{\mathrm{measured}}}p\left(\ln\mathrm{BF}\mid\mathcal{H}_{\textrm{BNS}}\right)d\ln\mathrm{BF}.$
(16)

If the FDP is sufficiently low, then it is less likely that the event is
consistent with the $\mathcal{H}_{\mathrm{BNS}}$ hypothesis. Often the FDP
values are converted to equivalent significance levels, e.g.,
$\mathrm{n}\,\sigma$ deviation of a Gaussian random process. From the
background distribution of $\ln\mathrm{BF}$, we can compute the threshold
Bayes’ factor $\ln\mathrm{BF}_{\mathrm{threshold}}$ corresponding to a certain
statistical significance.

In our analysis, each GW signal is injected in a 16-second-long simulated
colored-Gaussian data of the two Advanced LIGO detectors. In Fig. 5, the
foreground distribution
$p\left(\ln\mathrm{BF}\mid\mathcal{H}_{\textrm{BBH}}\right)$ and the
background distribution
$p\left(\ln\mathrm{BF}\mid\mathcal{H}_{\textrm{BNS}}\right)$ are shown in red
and black colors, respectively. The blue vertical line denotes the threshold
value $\ln{\mathrm{BF}}_{\mathrm{threshold}}$ corresponding to $2\sigma$
significance. Above that $2\sigma$ threshold, the areas under the foreground
and the background curves are $\sim 0.17$ and $\sim 0.04$, respectively.
Therefore, around $17\%$ of BBH signals will have ln BF greater than that
threshold, and will be correctly identified as BBHs with a significance at the
$2\sigma$ level. However, there is a 4$\%$ chance that a signal from a BNS
will be mischaracterized as a BBH at the same level of significance. We notice
that there is a significant overlap between the foreground and background
distributions, which is expected to decrease somewhat with longer waveforms
and more sensitive detectors.

In Fig. 6 we perform the same exercise for heavier component masses, namely,
$m_{i}>2M_{\odot}$. In this mass range interestingly we find that irrespective
of the type of the source injected, model selection prefers HTF2. This is
because for those masses neither the TD (and $\tilde{Q}$) phase terms in
TidalTF2 nor the TH ones in HTF2 are large enough to induce phase difference
between the TidalTF2 and the HTF2 waveforms that is significant enough to tell
them apart. In fact, both waveforms are very similar to the point-particle
waveform there. While with increasing total mass TH would eventually become
large, nevertheless in the mass-gap it is weak enough to tell such binaries
apart at realistic distances in the current generation of detectors.

We further test the conclusion above with a study of any systematics that may
be induced in the estimation of parameters when the wrong waveform is used to
search for a signal. For this purpose, we focus on the most precisely measured
binary parameter, namely, the chirp mass ${\cal M}$. In Fig. 7 we investigate
these systematics. When the injection is TidalTF2 but parameter estimation
employs TidalTF2, on the one hand, and HTF2, on the other hand. We plot the
values of $\mathcal{M}$ so measured in Fig. 7. If there were no systematics,
then the measured values should be highly correlated between TidalTF2 and HTF2
measurements. In an ideal case they should fall along the diagonal line,
barring a spread owing to detector noise. In the figure we find this behavior
as expected. The measured $\mathcal{M}$ values fall exactly on the diagonal
line. This implies that the measured values are highly correlated, which
implies the absence of systematics.

The fact that even the presence of the TH terms in the phase of HTF2 waveforms
are not able to effect any discriminatory power may not be suprising but is of
special significance. While this may be a disappointing result for prospects
of characterizing the nature of compact objects in the mass gap, it is
important to note that this conclusion is reached with the TH terms in phase.
In retrospect, this is not surprising since Fisher studies point to the same
conclusion. These studies, with full waveforms, also indicate that such a
distinction is possible in third-generation detectors for binaries within a
few hundred Mpc. Bayesian studies with longer waveform simulations for those
detectors are computationally expensive and will be pursued elsewhere.

Figure 6:  Same as Fig. 5, but for BBH injections with component masses
$m_{i}\in[2,5]M_{\odot}$. The red curve represents the foreground distribution
$p\left(\ln\mathrm{BF}\mid\mathcal{H}_{\textrm{BBH}}\right)$ of
$\ln{\mathrm{BF}}$ obtained from BBH injection runs in the simulated Gaussian
noise of Advanced LIGO. The black curve depicts the background distribution
$p\left(\ln\mathrm{BF}\mid\mathcal{H}_{\textrm{BNS}}\right)$ of
$\ln{\mathrm{BF}}$ obtained from BNS injections.  Figure 7:  For TidalTF2
signal injections, we do parameter estimation with both TidalTF2 and HTF2
waveforms and plot the measured values of the chirp mass $\mathcal{M}$ here.
The range of the binary component masses range is $m_{1,2}\in[1,5]M_{\odot}$.
In the absence of systematics the measured values should be highly correlated
between TidalTF2 and HTF2 measurements, and should fall along the diagonal
line. In the presence of noise, however, we expect some scatter around that
line – as evidenced here. As expected, the measured $\mathcal{M}$ values shown
above are nicely clustered around the diagonal. This shows that any biases, if
present, are much less significant than the statistical errors.

##  VI Conclusion

We have developed a method to search for and characterize TH in the inspiral
phase of a binary. We have defined two new parameters that capture the effect
of TH in the inspiral waveform. These parameters are robust enough that even
partial absorption can be modeled with them – something we will pursue in
detail in the future. To test for the presence of horizon we performed model
selection using the Bayes factor. We constructed two sets of waveforms, one
for BBHs, which incorporates TH but no TD, and the other for binaries of
horizonless compact objects, which does not include TH but has TD. We also
defined a new effective parameter for the quadrupole moment, namely
$\tilde{Q}$, which has been added in both waveform models appropriately.

We showed that for $m_{i}>2M_{\odot}$ it was not possible to distinguish
between the two models. We did so by employing the Bayes factor in a full
Bayesian analysis with simulated injections of both types of signals. We also
checked our results with a Fisher analysis and found that in this mass range
it will be hard to test the presence of horizon.

It remains to be seen if it is possible to find better result with golden
binaries. It is obvious that with increased SNR the error will decrease,
resulting in better measurement of $H_{eff5}$ and $H_{eff8}$. We can estimate
the error reduction in such a case. Assuming the sources like in Fig. 3 at
$d_{L}\sim 50$Mpc and taking account of the whole signal duration we can
estimate the error reduction. Taking into account of these two, we find that
the SNR will increase by a factor of $\sim 3.1$. Hence, the estimation in such
case would result in $H_{eff5}\sim 0.06\pm 0.66$ and $H_{eff8}\sim-1.88\pm
7.66$. Hence, with golden binaries even with advanced LIGO detectors, it is
possible to find some meaningful constraints on the sources.

An immediate continuation of the current work will be to construct better and
more complete waveform models than TidalTF2 and HTF2 that can be used for more
precise parameter estimation and more accurate model selection for real
signals in contemporaneous GW detector data. Since the waveforms used here
were limited to a short part (16s) of the late inspiral phase, it is likely
that utilizing more complete waveforms may improve the ability to distinguish
BBH and BNSs signals. Another problem we plan to address is the challenge
posed by mixed binaries (NSBH) in discerning the presence of horizons.
Thirdly, future generation detectors may allow enough precision so that proper
discrimination of NSBH binaries as well as the horizon parameter with
intermediate values $(0<H_{i}<1)$ may be realizable, thereby, affording the
possibility of probing the existence of ECOs, such as stellar-mass gravastars,
boson stars, etc. Datta (2020); Datta  _et al._ (2020).

## Acknowledgments

We thank Samanwaya Mukherjee for providing useful inputs which helped us
express our results better. It is a pleasure to thank N. V. Krishnendu, Andrea
Maselli and Paolo Pani for useful discussions. We would also like to thank
Richard Brito and Otto Hannuksela for carefully reading the manuscript and
providing helpful inputs, and Bhaskar Biswas, Soumak Maitra and Niladri Paul
for useful comments. We gratefully acknowledge the use of the IUCAA computing
cluster, Sarathi, and the computational resources provided by the LIGO
Laboratory (CIT) and supported by National Science Foundation Grants
PHY-0757058 and PHY-0823459. SD would like to thank University Grants
Commission (UGC), India, for financial support for a senior research
fellowship. KSP acknowledges support of the Netherlands Organisation for
Scientific Research (NWO). This work was done with partial support provided by
the Tata Trusts. This paper has been assigned LIGO Document Number
LIGO-P2000115.

We would like to thank all of the essential workers who put their health at
their risk during the COVID-19 pandemic, without whom we would not have been
able to complete this work.

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However, their simulations of the stripping of UGC 7695 assume constant
properties for the ICM, whereas if it has traversed over 450 kpc through the
cluster, then the surrounding density will have changed significantly during
the course of its interaction. It is also possible that the gas that formed
BC4 and BC5 was the first material to be stripped, before the ram pressure was
sufficient to strip the main disk. We note that the metallicities of BC4 and
BC5 correspond to the most metal-rich measurements of UGC 7695. This may be an
indication that their seed gas originated in a particularly metal-rich region,
such as gas that has been enriched by recent supernovae and pushed far from
the plane of the galaxy. We therefore suggest that despite the young age
estimate of the ram pressure stripping event in Boselli et al. (2021b), UGC
7695 is still consistent with being the origin of these two BCs.

Finally, we note that in the ALFALFA data cube the H i emission appears to
extend slightly from UGC 7695 in the approximate direction of BC4 and BC5, but
unfortunately this emission cannot be reliably traced as it becomes blended
with Galactic H i emission at the same velocity. Given the morphology,
orientation, and location of UGC 7695, along with its estimated metallicity,
this is by far the best candidate for the origin of that material that formed
BC4 and BC5.


11institutetext:  Institute for Big Data Analytics, Dalhousie University,
Halifax, Canada
22institutetext: Institute for Computer Science, Polish Academy of Sciences,
Warsaw, Poland
33institutetext: Memorial University of Newfoundland, St. John’s, Canada
44institutetext: Qom University of Technology, Iran
55institutetext: Shiraz University, Iran
66institutetext: Amirkabir University of Technology, Iran
66email: {nader.zare<EMAIL_ADDRESS>
66email: {arefsayareh<EMAIL_ADDRESS>
66email: {arad.firouzkouhi<EMAIL_ADDRESS>
66email<EMAIL_ADDRESS>66email<EMAIL_ADDRESS>
# Cyrus2D base: Source Code Base for RoboCup 2D Soccer Simulation League

Nader Zare 11 Omid Amini 44 Aref Sayareh 55 Mahtab Sarvmaili 11 Arad
Firouzkouhi 66 Saba Ramezani Rad 66 Stan Matwin 1122 Amilcar Soares 33

###### Abstract

Soccer Simulation 2D League is one of the major leagues of RoboCup
competitions. In a Soccer Simulation 2D (SS2D) game, two teams of 11 players
and one coach compete against each other. Several base codes have been
released for the RoboCup soccer simulation 2D (RCSS2D) community that have
promoted the application of multi-agent and AI algorithms in this field. In
this paper, we introduce ”Cyrus2D Base”, which is derived from the base code
of the RCSS2D 2021 champion. We merged Gliders2D base V2.6 with the newest
version of the Helios base. We applied several features of Cyrus2021 to
improve the performance and capabilities of this base alongside a Data
Extractor to facilitate the implementation of machine learning in the field.
We have tested this base code in different teams and scenarios, and the
obtained results demonstrate significant improvements in the defensive and
offensive strategy of the team.

111NOTICE: This is an accepted article to be published by Lecture Notes in
Artificial Intelligence (LNCS/LNAI) series by Springer-Verlag in the
proceedings of the RoboCup International Symposium.

###### Keywords:

2D Soccer Simulation RoboCup Base Code.

##  1 Introduction

Soccer is one of the most popular team-based sports in the world. This is a
multi-player, real-time, strategic, and partially observable game in which
players of each team should cooperate to score more goals. In addition to the
cooperative strategy, the players should manage different tactical and
technical strategies against their opponent. Designing and implementing this
game in a good, realistic graphical simulation environment and encouraging
researchers to develop fully autonomous players with human-like skills creates
complex challenges for A.I. research. Hence, soccer is considered an exciting
environment for developing A.I. and robotic algorithms to solve real-world
challenges. The importance of soccer as a game and as a challenging domain for
testing the A.I. and machine learning algorithms led to an overreaching vision
of a robotic team competing against the best human team by 2050[1].

The world Cup Robot Soccer Initiative was founded to create a realistic
environment similar to real soccer that encourages researchers to employ
Robotic and A.I. for solving wide ranges of problems[2]. The first RoboCup was
held during the IJCAI-97 [3], and it offered three competition tracks: real
robot league, software robots, and expert robot competition. Among them, the
Soccer Simulation 2D league (SS2D) [4] provides a wide range of research
challenges such as autonomous decision-making, communication and coordination,
tactical planning, collective behaviour and teamwork, opponent modelling and
behavior predicting [5, 6, 7, 8, 9, 10, 11, 12].

In this league, the RoboCup Soccer Simulation Server (RCSSServer) executes and
manages a 2D soccer game between two teams of eleven autonomous software
programs(agents). It holds the complete knowledge of the game, such as the
exact position of every element in the game and their movements. The game
further relies on the communication between the server and each agent. Each
player receives relative and noisy information about the environment, and
based on its logic and algorithms, the agent produces basic commands (like
dashing, turning, or kicking) to influence the environment. A visual example
of the game is shown in Figure 1. Another key component of this league is the
base code222For simplicity , throughout this paper we will use the ”base” term
instead of base code. of agents that is responsible for communicating with the
server, handling the noisy partial observability of the game, modeling the
server world, and making multi-agent decisions throughout the game. Due to the
complexity of these tasks, designing an operational base code can
astonishingly accelerates the RSS2D teams’ progress.

Over the past years, many teams have contributed to the RCSS2D community by
releasing their bases, which are mentioned below. One of the first bases was
from Carnegie Mellon University, a.k.a ”CMUnited” in 2001 [13, 14], then a
windows-based team was released by ”TsinghuAeolus” [15] in 2002. The release
of ”UvA Trilearn” base [16] in 2003, helped many teams worldwide.
”Brainstormers” [17], ”WrightEagle” [18] and ”Marlik” [19] released their team
codes in 2005, 2011 and 2012 respectively. ”HELIOS-Base or Agent2D” has been
released by the ”HELIOS” team from AIST Information Technology Research
Institute [20, 21]; this is the most important, most relevant, and most
frequently used publicly available source code release in soccer simulation
2d. It has been considered as the base code for many prosperous teams such as
Cyrus2d [24, 25, 26] and Glider2d[27, 29]. Later, ”Cyrus2D” and ”Gliders2D”
released their 2014 [22] and 2019 [23] bases respectively that are based on
agent2d.

In this paper we are planning to describe and release a more advanced base
that is called Cyrus2D in three consecutive versions. We have followed the
incremental strategy of evolving base code proposed by [23, 30] to exemplify
the impact of different approaches and to trace their functionalities. In the
first version(v0.0) we combined the newest release of Helios and Gliders bases
with some modifications on their parameters. In the second version(v1.0), we
have developed this base code to include three A.I.-based components of
Cyrus2D that were successfully implemented in this team. Finally, in the third
version(v1.1) we took advantage of Pass Prediction Deep Neural Network module
for unmarking decision-making. The performance of these versions went on the
rigorous evaluations against the Agent2D, Glider2D bases and the obtained
outcomes (number of scored and received goals, and the winning rate) proved
the prevalence of our base code. The rest of the paper is organized as
follows: Section 2 we will define the foundation of our base (version zero),
in the next section we will explain the deployment of three ideas(Blocking
Strategy, Offensive Risk Evaluation, Simple Unmarking Strategy) on the Cyrus2D
v0.0 which results in Cyrus2D v1.0. In section 4, we will present the idea of
using Pass Prediction in Unmarking Strategy(Cyrus2D v1.1). In the next
section, we will compare Cyrus2D base with other Soccer Simulation 2D bases
against best three teams in RoboCup 2021. Finally, we talk about our future
works.

Figure 1: A: Soccer Simulation 2D League. B: The evolution of Helios2D,
Glider2D and Cyrus2D base codes

##  2 Cyrus2D Base Version 0.0

One of the most popular SS2D bases is the Helios Base (agent2d) V3.11 which
was released in 2010[20, 21]. This base includes several components such as
librcsc-4.0.0, soccerwindow2-5.0.0 and fedit2-2.0.0. Gliders and Fractals, who
use the Agent2D base, won the championship of RoboCup 2016 and 2019,
respectively[27, 28, 29]. They also released a simplified version of their
teams called Gliders2d base[23, 30]. It is an advanced version of Helios base
v3.11 with improved formation, passing behavior, and stamina management. It
employs a modified version of the Marlik team[19] blocking algorithm, and few
unique strategies specifically designed for each team. On the other hand,
Helios has started improving its base and components such as librcsc based on
the new versions of C++ from 2019[31, 32]. In this paper we start by
introducing the first version of Cyrus2D base (V0.0). It is established by
rewriting the newest version of the Agent2D by merging the latest Gliders2D
base (see Figure1[B]). This base code is fully compatible with the latest
version of librcsc, but the blocking algorithm and tuning parameters of the
Gliders2D base are removed. The Cyrus2D base is released in the Cyrus team
repository and will be updated to be compatible with the rcssserver and
librcsc333https://github.com/Cyrus2D/Cyrus2DBase. In order to enhance the
functionality of this base, we have implanted three simplified functionalities
of Cyrus on this base and we introduce them as the consecutive versions of
Cyrus2D base. In the following sections, we will describe these ideas.

##  3 Cyrus2D Base Version 1.0

###  3.1 Blocking Strategy 444This algorithm is implemented in
src/bhv_basic_block.cpp

As the environment of SS2D is highly dynamic and unpredictable, an innovative
defensive strategy can increase the winning chance of the team. To establish
the defensive strategies, we need to understand defensive actions and how
players can cooperatively perform to minimize the risk of receiving the goal.
Blocking and marking are two main defensive actions that prevent the opposing
team from controlling the ball and playing with it. Blocking stops the
progress of the opponent’s ball holder on the field, and marking prevents the
passing of the ball to the opposing team players. Therefore, when one of our
agents tries to block the ball holder, the other players should choose to mark
the opponent players. In the Cyrus2D base, we implemented multi-agent blocking
decision-making. The blocking function or ”Blocking Simulator” is called when
the opponent owns the ball. It simulates the dribbling behavior of the
opponent ball holder called the ”dribbling curve” and then finds a position
that one of our players can arrive in, before arrival of the opponent’s ball
holder and (our) players. To simulate the dribbling curve, it predicts the
first position of the ball that the opponent’s player can kick the ball. In
the next step, it predicts the following ball positions of dribbling behavior.
The dribbling speed is considered 0.7 m/s. To find the dribbling direction, we
evaluate ten positions around the ball position using the reversed formulation
of ”Field Evaluator” in Helios base. To improve the performance of the
Blocking algorithm, we implemented some conditions to prevent players from
using extra stamina or going far from their home position.

###  3.2 Offensive Risk Evaluation 666This algorithm is implemented in
src/chain_action/action_chain_graph.cpp

To score more goals, the team’s ball holder must move the ball towards the
opponent’s goal area, and a final striker must shoot the ball towards the
goal. Dribbling and passing are examples of possible actions that can lead the
ball towards the goal. Henceforth, the ball holder must choose the best action
between the possible passes and dribbles. For this purpose, we need to
scrutinize our base code and improve the implementation of the offensive
strategy.

The Agent2D base has a decision-making algorithm called _Chain-Action_ , which
uses a modified version of Breadth-First Search to decide an action for the
ball owner in an action graph tree. The Chain-Action has _action generator_
modules such as Pass-Generator, Short-Generator, and Dribble-Generator. An
_action generator_ module receives a state of the game and then generates all
possible actions in that state. The Chain-Action also includes a simple
_predictor module_ that receives a state and an action; then, it generates a
new state. It simulates the possible outcome of the game after applying the
received action[26].

After predicting a new state, Chain-Action evaluates the state based on the
ball position using a module called _Field-Evaluator_. This module receives a
state and uses the $X$ coordinate of the ball and its distance to the
opponent’s goal to measure its value. To expand the tree to the next level,
the chain-action chooses a node with the maximum value. An example of this
procedure is shown in Figure 2.

Figure 2: Example of Chain-Action. A multi-branch tree search is performed.
Each edge presents an action and each node corresponds to a state instance.

We improved the Field-Evaluator module by including a term that is subtracted
to its calculation algorithms which is called _Offensive Risk Evaluation
(ORE)_. This value is calculated based on the minimum number of cycles that
the opponent players need to reach the ball in the input state of Field-
Evaluator.

The Field-Evaluator first calculates the minimum number of cycles $c$ that the
opponent player needs to reach the ball. Then it uses an array with seven
elements where the n-Th element would be the ORE term if the opponent reaches
the ball in n-Th-cycles. To populate this array, we took advantage of the
genetic algorithm888We reduced the array size to seven because our GA
algorithm with several settings found that the eighth and following cells of
the best arrays will be 0..

For our task, the genetic representation is a list of seven values. A solution
must be an array of seven values between 0 to 50, in descending order, as
opponents closer to the ball are more dangerous. The fitness function is the
average goal difference of Cyrus2D base in 100 games against random opponents
from 10 teams of RoboCup competitions in 2021. To initialize the first
population, we randomly generated 100 solutions. After evaluating with the
fitness function, we generate 80 new children, from 160 parents which are
selected randomly with probability based on their fitness score. After cross-
over, we update the new children to possible solutions by making sure each
value is less than or equal to the value before it. In the next step, some of
their genes can be mutated with a low random probability. The mutation is done
in a manner that the mutated solution is still considered possible.

If the generated solution is not in descending order, to preserve the validity
of the solution, we replace the first occurrence of illegal value with a value
lower than the previous element for example if the generated solution looks
like solution = [10 18 5 4 3 2 1], the validity procedure transforms it to
fixed solution = [10 9 5 4 3 2 1]. Afterwards, we create the new population of
$100$ by selecting $20$ of the best chromosomes of the previous generation and
adding the $80$ new children.

We repeat this process until the population converges or until $100$
iterations are evaluated.

###  3.3 Unmarking Strategy 999This algorithm is implemented in
src/bhv_unmark.cpp

Unmarking is the player’s ability to move, avoid being marked, and relocate
himself in a space where he could receive a pass from the ball possessor.

In the unmarking algorithm, a player who wants to unmark is called the
”unmarker”, and the player who will pass the ball to the unmarker is called
the ”passer”. The passer player can be a player who owns the ball or does not
have ball possession at the moment, but it is possible to be a ball possessor
in the future. An unmarking Strategy identifies the passer, and after
identifying the passer, the unmarker should find a position to receive a pass
from the passer in future cycles. An effective unmarking should consider the
actions of other agents and the cooperation between them.

Cyrus2D base version 1.0 includes a simple unmarking strategy. In this
algorithm, all players do unmarking for the ball possessor to receive a pass
from him. After identifying the passer, the unmarker simulates ten targets in
ten directions around him according to its previous movement. After generating
100 targets, it ignores targets close to teammate or opponent players and
targets far from its home position. The home position is the target position
of a player that is calculated based on team’s formation. The next step
simulates eight lead passes from the passer to itself in every target to find
which target it can receive a pass. A pass has a score calculated using the
”Field Evaluation” formula in the Helios base. The score of each target is
calculated based on the scores of possible received passes in the position and
the minimum distance of opponent players to the position. Eventually, the
target with the maximum score will be selected as the unmarking target.

##  4 Cyrus2D Base Version 1.1 111111This algorithm is implemented in
src/bhv_unmark.cpp, src/data_extractor/DEState.cpp and
src/data_extractor/offensive_data_extractor.cpp

In Cyrus2D base Version 1.0, we improved the offensive strategy, using a novel
Blocking behavior, and a simplified Unmarking strategy. In this section, we
will explain the improvement on the Unmarking strategy using a module called
Pass Prediction. The Pass Prediction module includes a trained DNN, that
receives a state of the game, and identifies which player will be the pass
receiver in that state. This module enables us to generate a tree that assigns
a passer to each player in the future cycles of the game. To generate a data
set for training the DNN, we employed Data Extractor module.

###  4.1 Data Extractor

As in real soccer, passing is one of the possible actions that can lead the
ball to the goal. Predicting the pass target player, from the point of view of
the ball possessor, has many benefits in defensive and offensive algorithms.
In this paper, the ball possessor is the player who can kick the ball in the
current cycle or receive the ball in the future cycles.

To predict the behavior of (our) ball possessor, we were required to create
the dataset of game states from this player point of view. For this purpose,
we embedded a Data Extractor module in each one of the players and then we
recorded the features of game states and their corresponding label. The label
shows the uniform number of the player who is target of the best pass[25, 26,
12].

To generate a data set, our player (ball holder) feeds the state of the game
and its selected pass receiver uniform number to the Data Extractor when the
ball is in its kickable area. After that it saves the features and the label
in a CSV file. Later this dataset will be used to train the Pass Prediction
model.

###  4.2 Unmarking Strategy with help of Pass Prediction Module

To improve the Unmarking Strategy and sketching the flow of the game, each
player of Cyrus2D base tries to simulate a tree that includes the probable
passes and their outcoming states. Each node of the tree contains a state of
the game where one of our players is the ball owner in that state. The edges
from the current state shows a probable pass in the future. The root node of
the tree is the first state of the game where one of our players can kick the
ball. A player can not be ball possessor in more than one node of the tree. To
create the tree, the unmarker feeds the state of the root node to the Pass
Prediction module, and receives the probability of players for receiving a
pass. This module includes a trained DNN that can receives features of the
game generated by Data Extractor and gives probability of players to receive a
pass in the given state. In the next step, the unmarker selects two passes
with the maximum probability higher than a limit and inserts them into a list
called Pass List. Then after, it pops the pass with highest probability from
the Pass List. Next, it simulate the outcome that it send the outcome state to
Pass Prediction module and eventually insert best passes from the outcome of
Pass Prediction module. This procedure continues until the number of tree
nodes is equal to ten or there is not any pass in the Pass List. Figure 3
shows an overview of the Unmarking Decisioning. After termination of this
procedure, the umarker agent looks for its corresponding node in the tree,
then it chooses the parent player as the ”Passer” for the unmarking procedure
in order to receive the pass from that player in the future.

Figure 3: Overview of the Unmark Decisioning algorithm. The left tree shows
the result states and their points. The Bold circles (full, dotted, and long
dashed) in the right tree present the selected best nodes from the Candidate
list. The dashed circle one indicates the node that the unmarker is the ball
owner in its state, and its parent node is the dotted circle.

##  5 Results

###  5.1 Training DNN for Cyrus2D v1.1 131313All scripts for training are
available in scripts/training_unmark

For generating a data-set for training the ”Pass Prediction DNN”, we ran 500
games against Helios Base v3.11 and newest version, Gliders2D base v1.6 and
v2.6, Cyurs 2021, Helios 2021, and YuShan 2021. We obtained total 1,429,032
data instances. We split them into two subsets, 85% for training and 15% for
testing. The prediction model (DNN) has three layers of 128, 64, 32 and 11
neurons, with RELU activation function and a softmax function at the last
layer. The validation accuracy of the trained neural network on the test data
was 68.1%. We used Python TensorFlow Keras library[33] for training the model,
and we implemented a library called CppDnn[34] to use the trained model in
C++. The CppDnn is a C++ library powered by Eigen[37]; this library creates a
deep neural network model by reading the weights of a trained DNN model.

To evaluate the impact of the implemented features and algorithms and
comparing the Cyrus2D base with the HELIOS base(Ag) and Gliders2d Base(G2D),
we ran X-number games between two versions of HELIOS base(3.1.1 / newest), two
versions of Gliders2d Base (1.6 / 2.6) , six versions of Cyrus2D base (C2D0 =
Cyrus2D zero, C2DB = Cyrus2D zero base with Blocking Strategy, C2DR = Cyrus2D
zero base with ORE, C2DU = Cyrus2D zero base with UnMarking Strategy, C2DV1.0
= Cyrus2D version one with all of the previous features and C2DV1.1 = V1.0
with pass prediction) against Helios bases[20], Glider2d Base(v2.6) and three
of the best teams in RoboCup (Helios2021 [35], YuShan[36], Cyrus2021 [25]).

Table 1 shows the expected winning rate of all version of Cyrus against
opponent teams. The winning rate is calculated by $num\\_wins/(num\\_games-
num\\_draws)$. Tables 2 presents the average number of our scored goals and
conceded goals respectively.

The results demonstrate Cyrus2D base v1.1 prevalence over other released
bases. For instance the Cyrus2D base wins Helios and Gliders2D bases in more
than 99% and 84% of games respectively. The average win-rate of Cyrus2D
against best three RoboCup teams is 3.76 (0.2% to 3.8%) percent higher than
the winning rate of Helios base against those teams, and 2.86 (2.9% to 3.8%)
percent higher than Gliders2D base.

Table 1: Win Rate Team |  H2D 3.11 |  H2D new |  G2D 2.6 |  Cyrus21 |  Helios21 |  YuShan21 | Average
---|---|---|---|---|---|---|---
H2D 3.11 | – | 26.2 | 3.5 | 0.0 | 0.0 | 0.0 | 4.9
H2D new | 73.8 | – | 9.8 | 0.2 | 0.0 | 0.0 | 14.0
G2D 1.6 | 95.5 | 85.4 | 30.3 | 0.4 | 0.0 | 0.2 | 35.3
G2D 2.6 | 96.5 | 90.2 | – | 1.9 | 0.0 | 1.0 | 31.6
C2D 0.0 | 100.0 | 98.1 | 78.5 | 7.2 | 0.0 | 4.3 | 48.0
C2D B | 99.6 | 97.2 | 77.0 | 6.4 | 0.0 | 3.6 | 47.3
C2D R | 99.4 | 97.4 | 81.0 | 7.9 | 0.2 | 6.3 | 48.7
C2D U | 100.0 | 99.0 | 80.8 | 5.6 | 0.2 | 5.8 | 48.6
C2D 1.0 | 99.3 | 98.6 | 79.9 | 8.6 | 0.3 | 4.6 | 48.6
C2D 1.1 | 99.8 | 99.6 | 84.1 | 5.8 | 0.8 | 4.9 | 49.1
Table 2: Goals Scored (Goals Conceded) Team |  H2D 3.11 |  H2D new |  G2D 2.6 |  Cyrus21 |  Helios21 |  YuShan21 | Average
---|---|---|---|---|---|---|---
H2D 3.11 | — | 1.6(2.7) | 0.5(3.0) | 0.2(6.2) | 0.1(13.0) | 0.2(7.5) | 0.4(5.4)
H2D new | 2.7(1.6) | — | 0.7(2.3) | 0.3(5.9) | 0.1(11.1) | 0.3(6.4) | 0.7(4.5)
G2D 1.6 | 3.5(0.7) | 2.6(1.0) | 0.8(1.4) | 0.5(5.3) | 0.1(6.5) | 0.2(4.8) | 1.3(3.3)
G2D 2.6 | 3.0(0.5) | 2.3(0.7) | — | 0.5(3.5) | 0.1(5.5) | 0.2(3.8) | 1.0(2.3)
C2D 0.0 | 4.3(0.2) | 2.8(0.3) | 1.1(0.4) | 0.6(2.8) | 0.2(3.6) | 0.3(1.9) | 1.6(1.5)
C2D B | 4.2(0.2) | 2.9(0.2) | 1.1(0.4) | 0.6(2.7) | 0.2(3.8) | 0.3(2.4) | 1.6(1.6)
C2D R | 4.1(0.2) | 2.9(0.3) | 1.1(0.4) | 0.6(2.6) | 0.2(3.8) | 0.3(1.7) | 1.5(1.5)
C2D U | 4.4(0.2) | 3.2(0.3) | 1.3(0.5) | 0.6(2.9) | 0.2(4.0) | 0.4(2.1) | 1.7(1.7)
C2D 1.0 | 4.8(0.2) | 3.6(0.2) | 1.3(0.4) | 0.7(2.6) | 0.2(3.9) | 0.4(2.5) | 1.8(1.6)
C2D 1.1 | 4.4(0.2) | 3.2(0.2) | 1.2(0.4) | 0.6(2.8) | 0.2(3.8) | 0.3(2.3) | 1.7(1.6)

##  6 Conclusion

In this paper, we aimed to introduce three versions of Cyrus2D base code and
their particular features. The first version of Cyrus2D base was created by
combining the latest release of Helios Agent2D and Gliders2D bases. For this
version, we removed some of the fine tuned parameters. In the next version,
Cyrus2D v1.0, we have upgraded the Blocking, and offensive strategy by using
the Offensive Risk Evaluation and unmarking behavior. In the Cyrus2D v1.1 we
improved the unmarking behavior using the Pass Prediction. To evaluate the
performance of Cyrus2D, we ran 500 games against Gliders2D, Helios base, and
best three teams in RoboCup 2021. The obtained results shows significant
improvement on win-rate, scored goals and conceded goals. For our future work,
we are planning to enhance the Cyrus2D base in terms of chain action movement
prediction, and marking by using multi-agent decision-making.

#### Acknowledgements

We acknowledge the support of the Natural Sciences and Engineering Research
Council of Canada (NSERC). We thank the HELIOS and Gliders teams for their
code bases and extraordinary contributions to the SS2D league.

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