# Baryonic dense matter in view of gravitational-wave observations

Vivek Baruah Thapa, Anil Kumar and Monika Sinha
Indian Institute of Technology Jodhpur, Jodhpur 342037, India
thapa.1@iitj.ac.inanil.1<EMAIL_ADDRESS>
###### Abstract

The detection of gravitational waves (GWs) from the merger of binary neutron
star (NS) events (GW170817 and GW190425) and subsequent estimations of tidal
deformability play a key role in constraining the behaviour of dense matter.
In addition, massive NS candidates ($\sim 2M_{\odot}$) along with NICER
mass$-$radius measurements also set sturdy constraints on the dense matter
equation of state. Strict bounds from GWs and massive NS observations
constrain the theoretical models of nuclear matter comportment at large
density regimes. On the other hand, model parameters providing the highly
dense matter response are bounded by nuclear saturation properties. This work
analyses coupling parametrizations from two classes based on covariant density
functional models: non$-$linear and density$-$dependent schemes. Considering
these constraints together, we study possible models and parametrization
schemes with the feasibility of exotic degrees of freedom in dense matter
which go well with the astrophysical observations as well as the terrestrial
laboratory experiments. We show that most parametrizations with non-linear
schemes do not support the observations and experiments while density-
dependent scheme goes well with both. Astrophysical observations are well
explained if the inclusion of heavier non-strange baryons is considered as one
fraction of the dense matter particle spectrum.

###### keywords:

dense matter $-$ equation of state $-$ gravitational waves $-$ stars: neutron

††pubyear: 2021††pagerange: Baryonic dense matter in view of gravitational-
wave observations–Baryonic dense matter in view of gravitational-wave
observations

##  1 Introduction

Neutron stars (NSs), one of the most compact objects in the Universe, provide
a suitable environment to study matter at densities ranging from subsaturation
to few times nuclear saturation densities (Glendenning, 1996; Sedrakian, 2007;
Weber, 2017). The well-known strategy to comprehend dense matter behaviour
above nuclear saturation density ($n_{0}$) is understanding compact
astrophysical objects. Constraints from observation of compact stellar objects
help the understanding of dense matter above $n_{0}$. One important constraint
is the lower limit of maximum stellar mass of compact object family obtained
from several recent observations. On the high density regime, the primary
constraint on equation of state (EOS) comes from the observations of massive
compact stars with masses near $2M_{\odot}$. Observations of massive compact
objects constrain the EOS to be stiff at a higher density limit. In addition,
the recent NICER (Neutron star Interior Composition ExploreR) observations
provide the mass and radius measurements of two compact objects (PSR
J$0030+0451$ and PSR J$0740+6620$), which further constrain the EOS.

Massive compact star observations eradicate the softer EOSs failing to support
the observational lower bound on maximum mass stars. On the other hand, the
recent detection of gravitational wave (GW) emissions from binary neutron star
(BNS) inspiral of the event GW170817 (Abbott et al., 2017a, b, c) and GW190425
(Abbott et al., 2020a) by LIGO-Virgo Collaboration (LVC) imposes bounds
(although weakly) on intermediate densities of dense matter EOS. In the BNS
merger scenario, the response of a compact star to a strong gravitational
field exerted by its companion is related to the EOS via induced tidal
deformations during the inspiral phase. Recent observations of GW from binary
star coalescence directly provide the estimate of combined dimensionless tidal
deformability ($\tilde{\Lambda}$) of participating stars indicating matter to
be soft at a lower density regime.

Due to insufficient knowledge and restrictions about matter properties at
density regimes higher than $n_{0}$, it is tough and challenging to predict
the behaviour of dense matter. Several studies have been done to comprehend
the NS matter composition in phenomenological (Vautherin & Brink, 1972;
Douchin & Haensel, 2001; Bao & Shen, 2014) as well as microscopic approaches
(Akmal et al., 1998; Carlson et al., 2015; Logoteta, 2019). Phenomenological
models such as the relativistic mean-field (RMF) model (Walecka, 1974; Boguta
& Bodmer, 1977; Serot & Walecka, 1997) based on effective interactions between
baryons via meson exchange can conveniently describe various finite nuclei
properties and can be extrapolated from subsaturation to higher density
regimes (Lattimer & Prakash, 2016). Within this framework, there are a good
number of admissible phenomenological models, each with a good number of
parameter value sets. With recent observational constraints related to compact
stars it is possible to narrow down the models of dense matter EOS.

Observations of massive NSs open the likelihood of exotic matter such as
hyperons (Glendenning & Moszkowski, 1991; Bonanno & Sedrakian, 2012;
Weissenborn et al., 2012; Colucci & Sedrakian, 2013; Oertel et al., 2015;
Fortin et al., 2016; Tolos et al., 2017; Raduta et al., 2018; Li et al.,
2018a), $\Delta$-resonances (Drago et al., 2014; Cai et al., 2015; Zhu et al.,
2016; Sahoo et al., 2018; Kolomeitsev et al., 2017; Li et al., 2018b; Li &
Sedrakian, 2019; Ribes et al., 2019; Li et al., 2020a; Thapa et al., 2020) and
meson ($\pi,\bar{K},\rho$-mesons) condensations (Mannarelli, 2019; Haensel &
Proszynski, 1982; Glendenning & Schaffner-Bielich, 1999; Banik &
Bandyopadhyay, 2001; Prakash et al., 1997; Malik et al., 2021; Thapa & Sinha,
2020; Thapa et al., 2021)) in the NS interior details of which are still not
completely understood. Although hyperonization in NSs may seem to be
inescapable, it leads to softening of the EOS. Following similar energetic
reasoning employed for nucleation of strange baryons, another interesting
aspect is the onset of non-strange $\Delta$-baryons in dense NS matter. The
role of $\Delta$-resonances in NS matter is still a matter of debate (Li et
al., 2018b; Motta et al., 2020). Inclusion of $\Delta$-baryons softens the EOS
at lower matter density regimes leading to higher compactness at those
densities, consequently satisfying the GW170817 event observables (Li &
Sedrakian, 2019). Similar to other exotic degrees of freedom, the inclusion of
meson condensates also softens the EOS extensively. For recent reviews
regarding the composition of compact stars, the reader may refer to Yagi &
Yunes (2017); Chatterjee & Vidaña (2016); Sedrakian, Armen (2017); Baym et al.
(2018) and Sedrakian et al. (2021).

In this work, we employ the available constraints obtained so far from compact
object observations to narrow down the high-density matter models. We explore
the possible parametrization models based on constraints from terrestrial
experiments and astrophysical observations (viz. massive NSs, radii
estimations of NSs, GW emitted during the inspiral phase of BNS coalescence)
on dense matter EOS. To construct the EOS, we consider non-linear Walecka
(NLW) type and density-dependent (DD) meson-baryon couplings within covariant
density functional (CDF) model with exotic degrees of freedom in addition to
nucleons. The phenomenological EOS models based on density functional theories
and realistic nuclear potentials have been analysed considering matter
composition to be only nucleonic (Malik et al., 2018; Nandi et al., 2019;
Kanakis-Pegios & Moustakidis, 2020). Therefore, this work will explore the
novel aspects of CDF model parametrizations that satisfy the recent
astrophysical observable estimations considering hyperonization of dense
matter to be a viable energetic argument.

The paper is organized as follows. In Section 2, we describe a few estimated
parameters of compact stellar objects from observational constraints relevant
to this work. The CDF model formalism (NLW and DD), its extension to heavier
baryons in $\beta$-equilibrated nuclear matter, aspects of tidal deformability
of NSs and coupling parameters incorporated in this work are described in
Section 3. Section 4 provides the results and our conclusions and future
perspectives are summarized in Section 5.

Conventions: We implement the natural units $G=\hbar=c=1$ throughout the work.

##  2 Observational constraints

The understanding of dense matter above $n_{0}$ can be reasonably improved
from recent compact star observations as the constraints on lower limit of
maximum mass, observed range of mass-radius of certain compact stars and most
importantly the compactness from observed tidal deformibility. The soft EOS of
highly dense matter can be ruled out from the observations of massive stars
viz. PSR J$1614-2230$ ($M=1.908\pm 0.016~{}M_{\odot}$) (Demorest et al., 2010;
Arzoumanian et al., 2018), PSR J$0348+0432$ ($2.01\pm 0.04~{}M_{\odot}$)
(Antoniadis et al., 2013), millisecond pulsar J$0740+6620$
($2.14^{+0.20}_{-0.18}~{}M_{\odot}$ with 95$\%$ credibility (Cromartie et al.,
2020), $2.08^{+0.07}_{-0.07}$ $M_{\odot}$ with 68.3% credibility (Fonseca et
al., 2021)) and PSR J$1810+1744$ ($2.13\pm 0.04~{}M_{\odot}$ with 68$\%$
credibility) (Romani et al., 2021).

GW observations also can constrain well the models of highly dense matter as
already stated. For low-spin prior systems from GW170817 event, it is
estimated that the combined dimensionless tidal deformability
($\tilde{\Lambda}$) parameter value has an upper and lower bounds of $900$
(TaylorF2 model) (Abbott et al., 2017a) and $400$ (AT2017gfo event) (Radice et
al., 2018) respectively. Reanalysis of the GW170817 data by LVC has set new
limits as $110\leq\tilde{\Lambda}\leq 720$ (PhenomPNRT model) (Abbott et al.,
2019). Another estimation of $\tilde{\Lambda}$ based on the viability of
chiral effective field theory results provides the limit to be in the range
$80\leq\tilde{\Lambda}\leq 570$ (Tews et al., 2018). In addition, an
ameliorated analysis of GW170817 event data implementing identical EOS for
both the compact stars producing rational waveforms provides a limit on the
dimensionless tidal deformability ($\Lambda$) for a $1.4M_{\odot}$ NS to be in
the range $70\leq\Lambda_{1.4}\leq 580$ with 90$\%$ credibility (Abbott et
al., 2018). Raithel et al. (2018) reported that the $\tilde{\Lambda}\leq 800$
constraint implies the radius of primary compact object to be $<13$ km.

Another GW observation (GW190814) by the LIGO Livingston detector (LVC)
inferred to be from a coalescence of a black-hole (BH) and lighter compact
object appendage with mass of the latter to be
$2.59^{+0.08}_{-0.09}~{}M_{\odot}$ (Abbott et al., 2020b) which falls in the
‘mass-gap’. The nature of lighter companion is still not resolved (Most et
al., 2020; Tews et al., 2021; Sedrakian et al., 2020; Li et al., 2020b;
Dexheimer et al., 2021; Bombaci et al., 2020; Fattoyev et al., 2020).

Along with this, we get good information about mass-radius relation of compact
stars from NICER observations. This space mission recently provided adequate
information to estimate the mass-radius of PSR J$0030+0451$ to be in the range
of $1.44^{+0.15}_{-0.14}$ $M_{\odot}$, $13.02^{+1.24}_{-1.06}$ km (Miller et
al., 2019) and $1.34^{+0.15}_{-0.16}$ $M_{\odot}$, $12.71^{+1.14}_{-1.19}$ km
(with 68.3$\%$ credibility) (Riley et al., 2019) respectively. Latest estimate
of mass-radius of PSR J$0740+6620$ by NICER is in the range of
$2.072^{+0.067}_{-0.066}$ $M_{\odot}$, $12.39^{+1.30}_{-0.98}$ km (Riley et
al., 2021) and $2.08\pm 0.07$ $M_{\odot}$, $13.71^{+2.62}_{-1.50}$ km (with
68$\%$ credibility) (Miller et al., 2021) respectively. Several works (Zhang &
Li, 2021; Biswas, 2021; Pang et al., 2021; Raaijmakers et al., 2021;
Somasundaram & Margueron, 2021; Li et al., 2021a) have been performed based on
different analyses on astrophysical observations inclusive of the latest NICER
measurements to extract new information regarding dense matter EOS.

An estimation for tidal deformability from NICER observations of PSR
J$0030+0451$ jointly with GW170817 event provides $240\leq\Lambda_{1.4}\leq
730$ with radius range as $R_{1.4}=12.1^{+1.2}_{-0.8}$ km (Jiang et al.,
2020). Another recent analysis of the same PSR J$0030+0451$ data (NICER)
reveals $R_{1.4}=12.32^{+1.09}_{-1.47}$ km (Landry et al., 2020). The radius
constraint on the $1.4M_{\odot}$ NSs from the GW170817 event has been derived
to be in the range $10.5\leq R_{1.4}/\text{km}\leq 13.4$ (Abbott et al., 2019)
while Malik et al. (2018) provide the radius limit to be $11.82\leq
R_{1.4}/\text{km}\leq 13.72$. Considering similar low-spin prior systems for
the GW190425 event, an upper bound of $600$ (PhenomPv2NRT model) has been
placed on $\tilde{\Lambda}$ and the radius upper limit is derived as $R<15$ km
(Abbott et al., 2020a).

From the analysis of GW170817 event data, bounds on matter pressure ranges are
derived at $2n_{0}$ and $6n_{0}$ to be $3.5^{+2.7}_{-1.7}\times 10^{34}$ dyn-
cm-2 and $9.0^{+7.9}_{-2.6}\times 10^{35}$ dyn-cm-2 respectively (Abbott et
al., 2018). Recent results obtained from GW170817 event by Bayesian analysis
suggest matter pressure at $2n_{0}$ to be $\sim 3.81^{+1.18}_{-2.32}\times
10^{34}$ dyn/cm2 and $133\leq\Lambda_{1.4}\leq 686$ (Li et al., 2021b). The
analysis of GW190425 event data reveals core matter density of the primary
component involved to be in the range $3\leq n/n_{0}\leq 6$ and matter
pressure to be in $10^{35}\leq P(3-6~{}n_{0})\leq 8\times 10^{35}$ dyn/cm2
range (Abbott et al., 2020a).

We take into consideration the source properties at $90\%$ credible intervals
with low-spin posterior distributions from PhenomPNRT (Abbott et al., 2019)
and PhenomPv2NRT (Abbott et al., 2020a) waveform models for GW170817 and
GW190425, respectively, to evaluate various GW observables. Based on recent
observable estimations of GW events, in addition to setting radius bounds on
the NSs involved in GW170817 and GW190425 events, we also find limits on
compactness parameter for a $1.4~{}M_{\odot}$ NS.

##  3 Formalism

###  3.1 EOS model

In this section, we briefly discuss the NLW and DD CDF models implemented to
contrive the EOS in this work. The dense matter possible constituents
considered here are nucleons ($N\equiv n,p$), hyperons
($Y\equiv\Lambda^{0},\Sigma^{\pm,0},\Xi^{-,0}$) and $\Delta$-resonances
($\Delta\equiv\Delta^{++},\Delta^{+},\Delta^{-},\Delta^{0}$) alongside leptons
($l\equiv e^{-},\mu^{-}$) to maintain $\beta$-equilibrium. The interaction
between non-strange baryons are described via the exchange of isoscalar-scalar
$\sigma$, isoscalar-vector $\omega$, and isovector-vector $\rho$ mesons. For
the hyperonic sector interactions, an additional hidden strangeness isoscalar-
vector $\phi$ meson is taken into consideration. In general, the total
Lagrangian density describing the baryon-meson interactions is given by
(Glendenning & Schaffner-Bielich, 1999; Banik & Bandyopadhyay, 2001; Li et
al., 2018b)

$\displaystyle\mathcal{L}$ $\displaystyle=\sum_{b\equiv
N,Y}\bar{\psi}_{b}(i\gamma_{\mu}D^{\mu}_{(b)}-m^{*}_{b})\psi_{b}+\sum_{l}\bar{\psi}_{l}(i\gamma_{\mu}\partial^{\mu}-m_{l})\psi_{l}$
(1)
$\displaystyle+\sum_{\Delta}\bar{\psi}_{\Delta\nu}(i\gamma_{\mu}D^{\mu}_{(\Delta)}-m^{*}_{\Delta})\psi^{\nu}_{\Delta}+\frac{1}{2}(\partial_{\mu}\sigma\partial^{\mu}\sigma-
m_{\sigma}^{2}\sigma^{2})$
$\displaystyle-\frac{1}{4}\omega_{\mu\nu}\omega^{\mu\nu}+\frac{1}{2}m_{\omega}^{2}\omega_{\mu}\omega^{\mu}-\frac{1}{4}\boldsymbol{\rho}_{\mu\nu}\cdot\boldsymbol{\rho}^{\mu\nu}+\frac{1}{2}m_{\rho}^{2}\boldsymbol{\rho}_{\mu}\cdot\boldsymbol{\rho}^{\mu}$
$\displaystyle-\frac{1}{4}\phi_{\mu\nu}\phi^{\mu\nu}+\frac{1}{2}m_{\phi}^{2}\phi_{\mu}\phi^{\mu}-\text{U}(\sigma)$

where, the last term $\text{U}(\sigma)$ corresponds to the self-interactions
of scalar mesons only for NLW CDF models.
$\psi_{b},~{}\psi_{l},~{}\psi_{\Delta}^{\nu}$ represent Dirac-fields of the
baryon-octet, leptons and Schwinger-Rarita fields of $\Delta$-quartet
respectively. $m_{b},~{}m_{l}$, and $m_{\Delta}$ denote the bare masses of
baryon octet, leptons and $\Delta$-quartet species respectively. The covariant
derivative mentioned in eq.-(1) is given by

$D_{\mu(j)}=\partial_{\mu}+ig_{\omega j}\omega_{\mu}+ig_{\rho
j}\boldsymbol{\tau}_{j3}\cdot\boldsymbol{\rho}_{\mu}+ig_{\phi j}\phi_{\mu}$
(2)

with $j$ denoting the baryon octet ($b$) and $\Delta$-quartet ($\Delta$). The
coupling constants are represented by $g_{pj}$ with index-$p$ being the
considered mesons. $\boldsymbol{\tau}_{j3}$ is the iso-spin projection of
third component of isovector-vector meson fields. The scalar self-interaction
term for NLW CDF model is given by

$\text{U}(\sigma)=\frac{1}{3}g_{2}\sigma^{3}+\frac{1}{4}g_{3}\sigma^{4}$ (3)

where $g_{2}$ and $g_{3}$ are the coefficients of self-interactions.

With the monotonic increase in baryon chemical potentials interior to NSs, the
nucleonic matter may well transform to heavier strange and non-strange baryons
leading to new hadronic degrees of freedom. The conditions which are necessary
to maintain strong $\beta$-equilibrium between different particle species
without strangeness being conserved are (Banik & Bandyopadhyay, 2001; Drago et
al., 2014)

$\displaystyle\mu_{e}$
$\displaystyle=\mu_{n}-\mu_{p}=\mu_{\mu},\quad\mu_{\Sigma^{+}}=\mu_{\Delta^{+}}=\mu_{p},$
(4) $\displaystyle\mu_{\Sigma^{-}}$
$\displaystyle=\mu_{\Xi^{-}}=\mu_{\Delta^{-}}=\mu_{n}+\mu_{e},\quad\mu_{\Delta^{++}}=\mu_{p}-\mu_{e},$
$\displaystyle\mu_{\Sigma^{0}}$
$\displaystyle=\mu_{\Xi^{0}}=\mu_{\Lambda^{0}}=\mu_{\Delta^{0}}=\mu_{n}$

with $\mu_{j}$ denoting chemical potential of the $j$-th baryon and defined as

$\displaystyle\mu_{j}=\sqrt{p_{F_{j}}^{2}+m_{j}^{*2}}+\Sigma_{B}.$ (5)

Here, $\Sigma_{B}=\Sigma^{0}+\Sigma^{r}$ represents the vector self energies
with $\Sigma^{0}=g_{\omega j}\omega_{0}+g_{\phi j}\phi_{0}+g_{\rho
j}\boldsymbol{\tau}_{j3}\rho_{03}$ and $\Sigma^{r}$ denotes the rearrangement
term (present in DD CDF models only) necessary to maintain thermodynamic
consistency which is given by

$\displaystyle\Sigma^{r}$ $\displaystyle=\sum_{b}\left[\frac{\partial
g_{\omega b}}{\partial n}\omega_{0}n_{b}-\frac{\partial g_{\sigma b}}{\partial
n}\sigma n_{b}^{s}+\frac{\partial g_{\rho b}}{\partial
n}\rho_{03}\boldsymbol{\tau}_{b3}n_{b}\right.$ (6)
$\displaystyle\left.+\frac{\partial g_{\phi b}}{\partial
n}\phi_{0}n_{b}\right]+\sum_{\Delta}(\psi_{b}\longrightarrow\psi_{\Delta}^{\nu}),$

with $n^{s}=\langle\bar{\psi}\psi\rangle$ and
$n=\langle\bar{\psi}\gamma^{0}\psi\rangle$ denoting the scalar and vector
(number) densities respectively.

Two additional constraints, viz. charge neutrality and global baryon number
conservation, are administered while evaluating the EOS self-consistently. The
re-arrangement term considered in DD CDF model contributes explicitly to the
pressure term. For details, the readers may refer to Hofmann et al. (2001).

###  3.2 Tidal deformability

The compact stars in a binary system experience tidal deformations due to the
gravitational fields of their respective companions. These tidal effects can
be quantified in terms of tidal deformability parameter ($\lambda$) defined as
the ratio of induced mass quadrupole moment $Q_{ij}$ to external perturbing
tidal field $\mathcal{E}_{ij}$ (Hinderer, 2008; Flanagan & Hinderer, 2008;
Hinderer et al., 2010),

$\lambda=-\frac{Q_{ij}}{\mathcal{E}_{ij}}=\frac{2}{3}k_{2}R^{5},$ (7)

where,

$\displaystyle k_{2}=$
$\displaystyle\frac{8C^{5}}{5}(1-2C^{2})[2+2C(y-1)-y]\cdot\\{2C[6-3y+3C(5y-8)]$
(8) $\displaystyle+4C^{3}[13-11y+C(3y-2)+2C^{2}(1+y)]+3(1-2C^{2})\cdot$
$\displaystyle[2-y+2C(y-1)]\log(1-2C)\\}^{-1}$

with $C=M/R$ being the compactness parameter, $M$ and $R$ being mass and
radius of the star respectively and $k_{2}$, the EOS dependent tidal Love
number. $y=y(R)$ is the function obtained after solving the differential
equation (Binnington & Poisson, 2009; Damour & Nagar, 2010; Kumar et al.,
2017),

$r\frac{dy(r)}{dr}+y(r)^{2}+y(r)F(r)+r^{2}Q(r)=0,$ (9)

where the functions are

$F(r)=\frac{r-4\pi r^{3}[\varepsilon(r)-P(r)]}{r-2M(r)},$ (10) $\displaystyle
Q(r)=$ $\displaystyle\frac{4\pi
r[5\varepsilon(r)+9P(r)+\frac{\varepsilon(r)+P(r)}{\partial
P(r)/\partial\varepsilon(r)}]}{r-2M(r)}-4\left[\frac{M(r)+4\pi
r^{3}P(r)}{r^{2}(1-2M(r)/r)}\right].$ (11)

Table 1: The nuclear properties of the CDF models at respective $n_{0}$. CDF Model | $n_{0}$ | $-E_{0}$ | $K_{0}$ | $E_{\text{sym}}$ | $L_{\text{sym}}$ | $K_{\text{sym}}$
---|---|---|---|---|---|---
|  | (fm-3) | (MeV) | (MeV) | (MeV) | (MeV) | (MeV)
| GM1 | 0.153 | $16.30$ | 300.00 | 32.50 | 93.857 | 17.91
| GM2 | 0.153 | $16.30$ | 300.00 | 32.50 | 89.289 |  $-$ 11.98
| GM3 | 0.153 | $16.30$ | 240.00 | 32.50 | 89.627 |  $-$ 6.46
| NL3 | 0.148 | $16.29$ | 271.76 | 37.40 | 118.317 | 100.53
NLW | NL3-II | 0.149 | $16.28$ | 272.15 | 37.70 | 119.563 | 103.19
| NL-SH | 0.146 | $16.346$ | 355.36 | 36.10 | 113.654 | 79.81
| NL-RA1 | 0.1466 | $16.15$ | 285.00 | 36.10 | 115.305 | 95.57
| NL3* | 0.150 | $16.31$ | 258.27 | 38.68 | 122.71 | 105.73
| GMT | 0.145 | $16.30$ | 281.00 | 36.90 | 112.796 | 63.04
| DD1 | 0.1487 | $16.021$ | 240.00 | 31.60 | 55.949 |  $-$ 95.24
| DD2 | 0.149065 | $16.02$ | 242.70 | 32.73 | 54.966 |  $-$ 93.24
| DD-ME1 | 0.152 | $16.20$ | 244.50 | 33.10 | 55.370 |  $-$ 101.07
| DD-ME2 | 0.152 | $16.14$ | 250.89 | 32.30 | 51.253 |  $-$ 87.31
DD | PKDD | 0.149552 | $16.267$ | 262.181 | 36.79 | 90.139 |  $-$ 80.56
| TW99 | 0.153 | $16.247$ | 240.00 | 33.39 | 55.309 |  $-$ 124.68
| DDV | 0.151 | $16.097$ | 240.00 | 33.589 | 71.463 |  $-$ 93.97
| DDF | 0.1469 | $16.024$ | 223.10 | 31.60 | 55.919 |  $-$ 139.66
| DD-MEX | 0.152 | $16.14$ | 267.059 | 32.269 | 49.576 |  $-$ 71.47

Due to paramount dependence on the stellar radius, $\lambda$ imposes stringent
constraints on dense matter EOS. $\lambda$ or, equivalently $k_{2}$ provides
the ease of induced deformation estimate of bulk matter. This parameter is
evaluated self-consistently alongside the Tolman-Oppenheimer-Volkov (TOV)
equations (Glendenning, 1996). Another dimensionless quantity $\Lambda$ is
much more expedient as it relates $\lambda$ with $C$ through the relation

$\Lambda=\frac{\lambda}{M^{5}}=\frac{2}{3}\frac{k_{2}}{C^{5}}.$ (12)

GW signal encodes information regarding deformation of both compact objects in
the binary system as the weighted tidal deformability ($\tilde{\lambda}$) and
is given by (Hinderer, 2008; Hinderer et al., 2010)

$\tilde{\lambda}=\frac{1}{26}\left[\frac{M_{1}+12M_{2}}{M_{1}}\lambda_{1}+\frac{M_{2}+12M_{1}}{M_{2}}\lambda_{2}\right]$
(13)

where $\lambda_{1}$ and $\lambda_{2}$ are the tidal deformabilities
corresponding to stars with masses $M_{1}$ and $M_{2}$ respectively. In order
to relate to $\Lambda$, we incorporate the combined dimensionless tidal
deformability ($\tilde{\Lambda}$) defined as (Favata, 2014)

$\displaystyle\tilde{\Lambda}$
$\displaystyle=32\frac{\tilde{\lambda}}{(M_{1}+M_{2})^{5}}$ (14)
$\displaystyle=\frac{16}{13}\frac{(M_{1}+12M_{2})M_{1}^{4}\Lambda_{1}+(M_{2}+12M_{1})M_{2}^{4}\Lambda_{2}}{(M_{1}+M_{2})^{5}}.$

###  3.3 Coupling parameters

In this work, we adopt the NLW model with GM1, GM2, GM3 (Glendenning &
Moszkowski, 1991), NL3, NL3-II (Lalazissis et al., 1997), NL-SH (Sharma et
al., 1993), NL-RA1 (Rashdan, 2001), NL3* (Lalazissis et al., 2009), GMT (Pal
et al., 2000) parametrizations and the DD CDF model with DD1 (Typel, 2005),
DD2 (Typel et al., 2010), DD-ME1 (Nikšić et al., 2002), DD-ME2 (Lalazissis et
al., 2005), PKDD (Long et al., 2004), TW99 (Typel & Wolter, 1999), DDV (Typel
& Alvear Terrero, 2020), DDF (Klähn et al., 2006), DD-MEX (Taninah et al.,
2020) parametrizations for meson-baryon couplings.

Figure 1: Symmetry energy coefficient ($E_{\text{sym}}$) and its corresponding
slope parameter ($L_{\text{sym}}$) for all the parametrization models
considered in this work. The shaded region represents the current empirical
range in $E_{\text{sym}}-L_{\text{sym}}$ plane following recent experimental
and microscopic model calculations (Oertel et al., 2017).

Comparison of experimental data (Oertel et al., 2017) from finite nuclei and
heavy-ion collisions with different microscopic model calculations have
provided bounds on nuclear saturation properties of symmetric nuclear matter
(SNM):

  1. 1.

Incompressibility, $210$ MeV $\leq K_{0}(n_{0})\leq 280$ MeV,

  2. 2.

Symmetry energy coefficient, $28.5$ MeV $\leq E_{\text{sym}}(n_{0})\leq 34.9$
MeV,

  3. 3.

Slope parameter of $E_{\text{sym}}$, $30.6$ MeV $\leq
L_{\text{sym}}(n_{0})\leq 86.8$ MeV.

Table-1 displays the nuclear saturation properties obtained for models with
different CDF parametrizations considered in this work with $E_{0}$, the
saturation energy. The bounds on curvature of symmetry energy,
$K_{\text{sym}}(n_{0})=-111.8\pm 71.3$ MeV (Mondal et al., 2017),
$-85^{+82}_{-70}$ MeV (Baillot d’Etivaux et al., 2019) and $-102^{+71}_{-72}$
MeV (Zimmerman et al., 2020) deduced from nuclear data and various NS
observations provide additional constraint on dense matter EOS.

(a)  (b)

Figure 2: The family of solutions of TOV equations for matter composed of,
left-hand panel: only nucleons and right-hand panel: $\Delta$-admixed
hypernuclear matter alongside leptons to maintain $\beta$-equilibrium. The
solid curves denote the $M$-$R$ curves for different density-dependent model
parametrizations, while the dot-dashed curves denote the cases with non-linear
model parametrizations. The astrophysical constraints from GW190814 (Abbott et
al., 2020b), PSR J$1810+1744$ (Romani et al., 2021), PSR J$0030+0451$ (Miller
et al., 2019; Riley et al., 2019; Jiang et al., 2020; Landry et al., 2020),
PSR J$0740+6620$ (Riley et al., 2021; Miller et al., 2021) are represented by
the shaded regions.

The density-dependent meson-nucleon couplings implemented in DD CDF model are
defined as (Typel & Wolter, 1999; Nikšić et al., 2002),

$g_{iN}(n)=g_{iN}(n_{0})f_{i}(n/n_{0})\quad\quad\text{for }i=\sigma,\omega$
(15)

where, $n$ is the baryon number density and

$f_{i}(n/n_{0})=a_{i}\frac{1+b_{i}(n/n_{0}+d_{i})^{2}}{1+c_{i}(n/n_{0}+d_{i})^{2}}.$
(16)

In the case of $\rho$-meson couplings, the functional is defined as

$g_{\rho N}(n)=g_{\rho N}(n_{0})e^{-a_{\rho}(n/n_{0}-1)}.$ (17)

For the coefficient values in equations (15)-(17), the readers may refer to
Typel (2005); Typel et al. (2010); Nikšić et al. (2002); Lalazissis et al.
(2005); Long et al. (2004); Typel & Wolter (1999); Typel & Alvear Terrero
(2020); Klähn et al. (2006) and Taninah et al. (2020). These coefficients are
associated with different DD CDF model parametrizations and fitted to
reproduce various nuclei properties. The hidden strangeness meson $\phi$ does
not couple with nucleons, so $g_{\phi N}=0$.

The meson-hyperon and meson-$\Delta$ couplings are considered similar to the
density-dependence footing in the case of nucleons in DD CDF model. In the
case of the meson-hyperon vector couplings, we implement the SU(6) symmetry
and quark counting rule (Schaffner et al., 1994). For the scalar meson-hyperon
couplings, we consider the optical potentials of $\Lambda,\Sigma$ and
$\Xi$-hyperons in SNM to be $-28,+30$ and $-14$ MeV, respectively, at nuclear
saturation (Feliciello & Nagae, 2015; Gal et al., 2016). The optical potential
depths $U_{\Xi}^{(N)}(n_{0})=-18$ MeV and $U_{\Lambda}^{(N)}(n_{0})=-30$ MeV
(Schaffner-Bielich & Gal, 2000; Gomes et al., 2015) are also widely
implemented in dense matter studies. Recently Friedman & Gal (2021) reported
an attractive optical potential depth of $\Xi$-hyperons in SNM to be
$\gtrsim-20$ MeV. For a more recent review on the aspects of strangeness in
dense matter, the reader may refer to Tolos & Fabbietti (2020).

Due to scarce information regarding the $\Delta$-nucleon interactions, we
treat the meson-$\Delta$ resonances couplings as parameters. Nakamura et al.
(2010); Koch & Ohtsuka (1985); Wehrberger et al. (1989) have reported the data
to constrain meson-$\Delta$ baryon couplings at $n_{0}$ based on pion-nucleus
scattering, electron scattering on nuclei and excitation studies of
$\Delta$-quartet experiments. Recent reviews (Drago et al., 2014; Klähn et
al., 2006; Kolomeitsev et al., 2017) on this aspect have reported the
$\Delta$-potential ($V_{\Delta}$) in nuclear medium to be in the range
$-30~{}\text{MeV}+V_{N}\leq V_{\Delta}\leq V_{N}$ ($V_{N}$ being the nucleon
potential), the values of factor $x_{\sigma\Delta}-x_{\omega\Delta}$ to be
between $0$ and $0.2$ with $x_{\sigma\Delta}=g_{\sigma\Delta}/g_{\sigma N}$
and $x_{\omega\Delta}=g_{\omega\Delta}/g_{\omega N}$. Many works (Drago et
al., 2014; Kolomeitsev et al., 2017; Li et al., 2018b; Ribes et al., 2019;
Chen et al., 2007; Cai et al., 2015; Raduta, 2021) have considered the ranges
for $x_{\omega\Delta}\in[0.6-1.2]$ and $x_{\rho\Delta}\in[0.5-3.0]$. For
recent development regarding $\Delta$-potential in dense matter, the reader
may refer to Cozma & Tsang (2021). In the ensuing discussion, we will consider
$x_{\omega\Delta}=1.10$, $x_{\rho\Delta}=1.00$ for the vector meson-$\Delta$
baryon couplings and $x_{\sigma\Delta}=1.20$ for the scalar coupling.
$\Delta$-quartet being non-strange baryons does not couple with $\phi$-meson,
so $g_{\phi\Delta}=0$.

##  4 Constraints on dense matter models

Next, we sort out the different EOSs with various possible compositions and
parametrizations based on terrestrial experimental and stellar observational
values. From table-1 it can be noticed that all the models reproduce $n_{0}$
and $E_{0}$ in the correct range of empirical values. However, among the NLW
parametrizations considered in this work, GM1 and GM2 cannot reproduce the
empirical range of $K_{0}$ at $n_{0}$ mentioned in Section 3.3, while all the
DD CDF models considered in this work satisfy the empirical range of $K_{0}$
at $n_{0}$. On the other hand, the empirical range of symmetry energy
coefficient is satisfied by GM1, GM2 and GM3 parameter sets among NLW models,
while in the case of DD models, all except PKDD parametrization lies within
the bounds of this particular saturation property. The current empirical bound
on $L_{\text{sym}}$ is satisfied by all DD parametrization models considered
in this work except PKDD parametrization. In addition, the current empirical
bounds on $K_{\text{sym}}$ are satisfied by GM2, GM3 (among NLW models) and
all DD parameter sets considered in this work. Fig.-1 displays the parameter
sets which are compatible with the current bounds on $E_{\text{sym}}$ and
$L_{\text{sym}}$. All the DD models (except PKDD) considered in this work
satisfy the current bounds on nuclear saturation properties.

Table 2: Summary of TOV results evaluated from parameter sets considered in
this work (pure $N$-matter, refer to left-hand panel of fig.-2). Fulfillment
of the mass-radius constraints from various astrophysical observations are
marked by $+(-)$.

[b]  CDF Model PSR J$0030+0451$ PSR J$0740+6620$ PSR J$1810+1744$ GW190814
(secondary)∗∗ GM1 $+$ $+$ $+$ $-$ GM2 $+$ $+$ $-$ $-$ GM3 $+$ $-$ $-$ $-$ NL3
$-$ $+$ $+$ $+$ NLW NL3-II $-$ $+$ $+$ $+$ NL-SH $-$ $+$ $+$ $+$ NL-RA1 $-$
$+$ $+$ $+$ NL3* $-$ $+$ $+$ $+$ GMT $-$ $+$ $+$ $+$ DD1 $+$ $+$ $+$ $-$ DD2
$+$ $+$ $+$ $-$ DD-ME1 $+$ $+$ $+$ $-$ DD-ME2 $+$ $+$ $+$ $-$ DD PKDD $+$ $+$
$+$ $-$ TW99 $+$ $-$ $-$ $-$ DDV $+$ $-$ $-$ $-$ DDF $+$ $-$ $-$ $-$ DD-MEX
$+$ $+$ $+$ $+$

  * ∗∗

The nature of secondary component of GW190814 is still in tension

The mass-radius ($M$-$R$) relations corresponding to the various
parametrizations for only nucleonic matter composition, are shown in left
panel of fig.-2 obtained by solving the TOV equations for spherically
symmetric, non-rotating stars. For the crustal region, we consider the Baym et
al. (1971) EOS. The transition from crust to the core is modelled in a way
that is thermodynamically consistent following Fortin et al. (2016). We
consider the recent massive NS observation (PSR J$1810+1744$) as the lower
bound for maximum mass configurations. If we consider pure nucleonic matter,
all the EOSs considered in this work except a few satisfy the lower bound
constraint of maximum mass as evident from the left-hand panel of fig.-2. GM2
($M_{\text{max}}\sim 2.08~{}M_{\odot}$), GM3 ($2.02~{}M_{\odot}$) among NLW
type and TW99 ($2.07~{}M_{\odot}$), DDV ($1.93~{}M_{\odot}$), DDF
($1.96~{}M_{\odot}$) among DD type models fail to satisfy this mass
constraint. However, the constraints obtained from NICER observations (PSR
J$0030+0451$) are seen to be not satisfied by the NLW models satisfying mass
constraint, except the GM1 parametrization, while all the DD type models
satisfying the mass constraint satisfy this constraint too. Moreover, these
models satisfy the recent NICER results of PSR J$0740+6620$ simultaneously.
Here, PSR J$0740+6620$ observation suggests the dense matter at higher
densities is repulsive enough to produce large radii of heavier NSs. Most of
the NLW models not satisfying the NICER observations of PSR J$0030+0451$ are
observed to satisfy this recent constraint with a wider radii range. However,
even though they are observed to fulfil the constraint from Miller et al.
(2021), they fail to fulfil another NICER measurement of PSR J$0030+0451$ from
Miller et al. (2019); Riley et al. (2019); Jiang et al. (2020); Landry et al.
(2020). Stiffer EOSs ($N$-matter) obtained from NLW type parametrizations
(except GM1, GM2, GM3) and DD-MEX (DD type) yield $M_{\text{max}}\geq
2.5~{}M_{\odot}$ satisfying the GW190814 event secondary component’s mass
constraint. Although this constraint is not unequivocal as the nature of
secondary compact object is not found to be NS explicitly (Sedrakian et al.,
2020; Li et al., 2020b). The TOV results involving different parametrizations
with pure $N$-matter are summarized in table-2. As a result of EOS softening
due to the inclusion of hyperons and $\Delta$-quartet, the only NLW model EOS
with the GM1 parametrization, satisfying all mass-radius constraints, produce
stars that fail to fulfil the lower bound constraint of maximum mass as shown
in the right-hand panel of fig.-2. Except PKDD ($M_{\text{max}}\sim
1.93~{}M_{\odot}$), DD1 ($2.05~{}M_{\odot}$) and DD2 ($2.06~{}M_{\odot}$)
parametrizations all other parametrizations from DD CDF models which satisfy
the joint constraints from PSR J$1810+1744$, PSR J$0740+6620$ and PSR
J$0030+0451$ with pure $N$-matter satisfy the mass constraints even after
softening due to hyperonization. The incorporation of $\Delta$-quartet reduces
the radii of NSs, enhancing their respective compactness parameter.

Figure 3: The variation of matter pressure as a function of energy density
(EOS) for $\beta$-equilibrated matter composed of only nucleons and leptons.
The solid curves denote the EOSs for different density-dependent model
parametrizations, while the dot-dashed curves denote the cases with non-linear
model parametrizations. The constraints on matter pressure evaluated from
GW170817 event are denoted by the vertical lines and provided in sec.-1.
Figure 4: Combined dimensionless tidal deformability as a function of mass-
ratio ($q$) considering a fixed chirp mass, $\mathcal{M}=1.188~{}M_{\odot}$
(GW170817 event). Left panel: nucleonic matter, right panel:
$\Delta$-resonances and baryon octet matter. The solid, dot-dashed lines
depicts the DD and NLW type parametrizations respectively. The
parametrizations yielding soft EOSs are not shown in the right panel. The
horizontal dotted lines denote bounds on $\tilde{\Lambda}$ (with $90\%$
credibility, low-spin priors). The vertical dotted lines represent the mass-
ratio, $q=0.73$ boundary (Abbott et al., 2019). Figure 5: Similar to fig.-4
but considering a fixed chirp mass, $\mathcal{M}=1.43~{}M_{\odot}$ (GW190425
event) with pure nucleonic matter. The solid, dot-dashed lines depicts the DD
and NLW type parametrizations respectively. $\tilde{\Lambda}$ constraints are
similar to fig.-4. The vertical dotted lines represent the mass-ratio,
$q=0.80$ boundary (Abbott et al., 2020a). Figure 6: Tidal deformability
parameters $\Lambda_{1}$, $\Lambda_{2}$ corresponding to the binary
components, $M_{1},~{}M_{2}$ for GW170817 event with matter composition as,
left panel: pure nucleonic matter, right panel: $\Delta$-baryon admixed
hypernuclear and assuming a fixed chirp mass, $\mathcal{M}=1.188~{}M_{\odot}$.
The solid curves denote the DD type parametrizations while the NLW type are
denoted by dash-dotted curves. The dotted, short-dashed curves denote the
$\tilde{\Lambda}\sim 900$ (TaylorF2), $720$ (PhenomPNRT) upper bounds at
$90\%$ confidence level (low-spin priors) respectively (Abbott et al., 2017a;
Abbott et al., 2019). The diagonal long-dashed line marks
$\Lambda_{1}=\Lambda_{2}$ case.

We now look into the pressure bounds in lower and higher matter density
regimes derived from GW170817 event data. Fig.-3 shows that all the EOSs with
pure nuclear matter satisfying mass-radius constraints also satisfy both the
pressure bounds. It is to be noted that among NLW models, GM1, GM2 and GM3
satisfy only the lower bound at $n\sim 2n_{0}$ while they fail to satisfy the
bound at $n\sim 6n_{0}$. It has to be kept in mind that the estimated bound at
$6n_{0}$ is more than the central pressures of compact stars involved in
GW170817 event.

We have evaluated $\tilde{\Lambda}$ with the range of primary and secondary
masses $1.36-1.60~{}M_{\odot}$ and $1.17-1.36~{}M_{\odot}$ respectively to
provide the chirp mass,
$\mathcal{M}=(M_{1}M_{2})^{3/5}(M_{T})^{-1/5}=1.188~{}M_{\odot}$ where the
total mass $M_{T}=M_{1}+M_{2}$, is in the range $2.73-2.78~{}M_{\odot}$ for
GW170817 event. In this work, we have considered the source properties to be
circumscribing within $90\%$ credible intervals. Fig.-4 shows the
$\tilde{\Lambda}$ variation with mass-ratio parameter ($q$). $\tilde{\Lambda}$
is found to be almost independent of the mass asymmetry factor $q$ (refer to
table-3 for numerical results). In the left panel, curves are for pure
nucleonic matter and in the right panel, they are for $\Delta$-admixed
hypernuclear matter. For pure nucleonic matter, only GM2, GM3 models lie
within the observational $\tilde{\Lambda}\sim 900$ bound among the NLW CDF
models. However, they do not satisfy the lower bound constraint of maximum
mass. In DD CDF models, all parametrizations considered in this work follow
the upper bound of $900$. The stringent limit of $\tilde{\Lambda}\sim 720$ is
observed to be satisfied by only TW99, DDV, DDF models which do not satisfy
lower bound constraint for maximum mass. With $\Delta$-baryons coming into the
picture, the models fulfil the latter stringent upper bound on
$\tilde{\Lambda}$ as shown from the right panel of the figure. Since the
effect of hyperon inclusion on $\tilde{\Lambda}$ is similar to that of
nucleons for the NSs with mass bounds obtained from GW170817 event, so they
are not shown in fig.-4.

Figure 7: Effective tidal deformability variation with radius of the binary’s primary component for GW170817 and GW190425 events. The solid, dotted, stripped squares represent the DD parametrizations with pure nucleonic, baryon octet, $\Delta$-admixed hypernuclear matter respectively. The solid circles denote the NLW parametrizations with nucleonic matter case. The quadratic correlations for the GW170817, GW190425 cases are given by short-dashed and dash-dotted curves respectively. The horizontal dotted lines represent bounds on $\tilde{\Lambda}$ similar to figs.-4,5. The vertical short-dashed lines mark the points where the quadratic fits intersect $\tilde{\Lambda}$ bounds.  Table 3: Observational properties of various CDF models with different matter compositions. Here, $q$ represents the mass ratio of the secondary component ($M_{2}$) to primary one ($M_{1}$) involved in GW event. $C_{1.4}$ denotes the compactness parameter for a $1.4M_{\odot}$ NS. Matter | CDF Model | $M_{\text{max}}$ | $R_{1.4}$ | $\Lambda_{1.4}$ | $C_{1.4}$ | $\tilde{\Lambda}~{}(q=0.8)$ | $\tilde{\Lambda}~{}(q=1.0)$
---|---|---|---|---|---|---|---
composition |  | ($M_{\odot}$) | (km) |  |  | GW170817 | GW190425 | GW170817 | GW190425
| GM1 | 2.387 | 13.939 | 966.34 | 0.148 | 1109.39 | 355.77 | 1121.63 | 358.93
| GM2 | 2.079 | 13.468 | 743.19 | 0.153 | 870.59 | 246.12 | 871.51 | 244.33
| GM3 | 2.019 | 13.146 | 607.88 | 0.157 | 724.64 | 189.05 | 728.79 | 184.81
| NL3 | 2.774 | 14.430 | 1222.72 | 0.143 | 1394.86 | 489.88 | 1410.87 | 495.58
| NL3-II | 2.773 | 14.408 | 1217.59 | 0.143 | 1381.14 | 483.69 | 1401.64 | 494.33
| NL-SH | 2.799 | 14.630 | 1368.94 | 0.141 | 1547.36 | 554.41 | 1563.07 | 565.93
| NL-RA1 | 2.785 | 14.490 | 1278.05 | 0.143 | 1440.78 | 508.58 | 1465.28 | 518.21
Pure | NL3* | 2.762 | 14.355 | 1181.41 | 0.144 | 1341.70 | 465.63 | 1358.48 | 476.51
Nucleonic | GMT | 2.662 | 14.355 | 1177.91 | 0.144 | 1342.24 | 456.92 | 1359.93 | 462.36
Matter | DD1 | 2.410 | 13.126 | 678.92 | 0.157 | 772.75 | 259.42 | 784.91 | 261.88
| DD2 | 2.418 | 13.133 | 683.97 | 0.157 | 778.13 | 261.79 | 787.99 | 265.77
| DD-ME1 | 2.443 | 13.086 | 672.37 | 0.158 | 765.14 | 261.09 | 776.04 | 266.31
| DD-ME2 | 2.483 | 13.146 | 706.08 | 0.157 | 798.93 | 276.48 | 811.16 | 281.47
| PKDD | 2.328 | 13.461 | 750.66 | 0.154 | 869.05 | 273.56 | 874.03 | 275.28
| TW99 | 2.076 | 12.245 | 402.46 | 0.169 | 474.17 | 134.40 | 474.57 | 133.73
| DDV | 1.929 | 12.360 | 398.67 | 0.167 | 485.52 | 117.40 | 479.76 | 112.95
| DDF | 1.956 | 11.871 | 311.618 | 0.174 | 372.22 | 94.59 | 375.27 | 91.84
| DD-MEX | 2.556 | 13.293 | 773.49 | 0.156 | 869.28 | 309.58 | 883.91 | 316.32
| DD1 | 2.039 | 13.124 | 680.44 | 0.158 | 777.19 | 262.33 | 783.48 | 264.47
| DD2 | 2.046 | 13.132 | 684.61 | 0.157 | 784.18 | 265.08 | 793.09 | 267.20
Hypernuclear | DD-ME1 | 2.075 | 13.086 | 673.16 | 0.158 | 773.05 | 265.19 | 778.05 | 267.71
Matter | DD-ME2 | 2.115 | 13.146 | 707.53 | 0.157 | 808.16 | 277.39 | 813.83 | 282.33
| PKDD | 1.943 | 13.444 | 744.97 | 0.154 | 866.23 | 269.07 | 870.23 | 273.08
| DD-MEX | 2.186 | 13.293 | 773.49 | 0.156 | 873.62 | 312.38 | 884.72 | 319.36
| DD1 | 2.052 | 12.254 | 398.11 | 0.169 | 462.63 | 137.06 | 465.33 | 138.56
| DD2 | 2.059 | 12.260 | 402.03 | 0.169 | 466.22 | 139.86 | 469.16 | 140.69
$\Delta$-admixed | DD-ME1 | 2.085 | 12.320 | 418.19 | 0.168 | 485.03 | 146.52 | 486.65 | 148.15
Hypernuclear | DD-ME2 | 2.126 | 12.400 | 444.20 | 0.167 | 514.21 | 160.55 | 517.89 | 163.58
Matter | PKDD | 1.934 | 12.832 | 501.68 | 0.161 | 617.18 | 144.53 | 602.05 | 136.74
| DD-MEX | 2.198 | 12.588 | 503.30 | 0.164 | 577.19 | 189.24 | 586.07 | 192.52
Figure 8: Similar to fig.-7 but for secondary components of GW170817 and
GW190425 events. The solid, dotted, stripped squares represent the DD
parametrizations with pure nucleonic, baryon octet, $\Delta$-admixed
hypernuclear matter respectively. The solid circles denote the NLW
parametrizations with nucleonic matter case. The quadratic correlations for
the GW170817, GW190425 cases are given by short-dashed and dash-dotted curves
respectively. $\tilde{\Lambda}$ constraints are similar to fig.-7.

Similar to fig.-4, the effective tidal deformability as a function of mass-
asymmetry factor $q$ corresponding to the GW190425 event with a fixed chirp
mass, $\mathcal{M}=1.43~{}M_{\odot}$ for all the parametrizations considered
in this work is shown in fig.-5 with only nucleonic matter. In this case,
masses of the two compact stars are varied in the ranges $1.60\leq
M_{1}/M_{\odot}\leq 1.87$ (primary) and $1.46\leq M_{2}/M_{\odot}\leq 1.69$
(secondary) (Abbott et al., 2020a). Weak dependece of $\tilde{\Lambda}$ on $q$
can be inferred. It is observed that all the parametrizations satisfy the
upper bound constraint on $\tilde{\Lambda}$ provided by GW190425 event data.
Consequently this GW event does not provide enough information to put strict
limits on constraining dense matter EOSs. Inclusion of $\Delta$-quartet will
decrease $\tilde{\Lambda}$ compared to pure nucleonic case.

We next evaluate the tidal deformabilities ($\Lambda_{1}$, $\Lambda_{2}$) of
binary components involved in GW170817 event with different matter
compositions. For the evaluation of $\Lambda_{1}$ and $\Lambda_{2}$, we
consider $\mathcal{M}=1.188~{}M_{\odot}$ where $M_{T}=2.73-2.78~{}M_{\odot}$.
The masses of the two components are varied in $1.36\leq M_{1}/M_{\odot}\leq
1.60$ (primary) and $1.17\leq M_{2}/M_{\odot}\leq 1.36$ (secondary) ranges
(Abbott et al., 2017a). From the left panel of fig.-6, it is observed that for
pure nucleonic matter the NLW CDF model parametrization GM1 do not lie within
the $90\%$ probability contours of $\tilde{\Lambda}\sim 900$, although GM2 and
GM3 do, while in case of DD CDF models, all the parametrizations lie inside
these $\tilde{\Lambda}\sim 900$ contours. In the case of matter composition as
$NY$, the tidal deformability is quite similar to the ones with pure nucleonic
matter and hence not shown in fig.-6. In case of $\Delta$-resonance admixed
hypernuclear matter, for all the relevant EOSs $\Lambda_{1}$, $\Lambda_{2}$
falls well even within the $\tilde{\Lambda}\sim 720$ probability contour
(obtained from recent reanalysis) as shown in the right panel of the figure.
Table-3 provides the numerical estimates of various observational properties
of different CDF models considering matter composition to be purely nucleonic,
hypernuclear and $\Delta$-resonance admixed hypernuclear matter.

Figure 9: Dimensionless tidal deformability ($\Lambda$) as a function of the
NS mass ($M$) for CDF model parametrizations considered in this work (with
$N$, $NY\Delta$-matter). DD, NLW type models with $N$-matter are represented
similar to fig.-2,3. While DD models with $NY\Delta$-matter are represented by
long-dashed curves. The horizontal dotted lines denote the upper bounds on
$\Lambda_{1.4}=800,580$ obtained from Abbott et al. (2017a) and recent
reanalysis Abbott et al. (2018) respectively (GW170817 event). The vertical
range denotes joint constraints from NICER observations (PSR J$0030+0451$)
$\&$ GW170817 event (Jiang et al., 2020) and another obtained implementing
Bayesian analysis (Li et al., 2021b) for a $1.4~{}M_{\odot}$ NS. The various
astrophysical constraints are similar as in fig.-2.

Next we attempt to restrict radius of compact stars by evaluating
$\tilde{\Lambda}$ with particular star combinations for both the events
considering different parametrization models. For the event GW170817, we have
taken $M_{1}=1.50~{}M_{\odot}$ and $M_{2}=1.24~{}M_{\odot}$ and for the event
GW190425 we have taken $M_{1}=1.70~{}M_{\odot}$ and $M_{2}=1.59~{}M_{\odot}$
where, $M_{1}$, $M_{2}$ correspond to primary and secondary components
respectively. We plot the values of $\tilde{\Lambda}$ with radius of the
primary star for respective EOSs in fig.-7. Tight correlations between
$\tilde{\Lambda}$ and $R(M_{1})$ are given by the fits

$\displaystyle\tilde{\Lambda}^{\text{(GW170817)}}_{\text{fit}}$
$\displaystyle=7571-1423R(M_{1})+68.93(R(M_{1}))^{2},$
$\displaystyle\tilde{\Lambda}^{\text{(GW190425)}}_{\text{fit}}$
$\displaystyle=2638-501.2R(M_{1})+24.34(R(M_{1}))^{2},$

with maximum deviations,
$(|\tilde{\Lambda}_{\text{fit}}-\tilde{\Lambda}|/\tilde{\Lambda})\sim
2.65\%,~{}7.45\%$,
$\chi^{2}=\sum_{i}^{\mathcal{N}}\left[(\tilde{\Lambda}_{\text{fit}}^{i}-\tilde{\Lambda}^{i})^{2}/\tilde{\Lambda}^{i}\right]=4.16,~{}7.16$
with $\mathcal{N}=30$ EOS models and coefficients of determination,
$\mathcal{R}^{2}=1-SS_{\text{res}}/SS_{\text{total}}$ $\sim 0.999,~{}0.996$
for GW170817, GW190425 event respectively. Here,
$SS_{\text{res}}=\sum_{i}(\tilde{\Lambda}_{i}-\tilde{\Lambda}_{i}^{\text{fit}})^{2}$,
$SS_{\text{total}}=\sum_{i}(\tilde{\Lambda}_{i}-\bar{\tilde{\Lambda}}_{i})^{2}$
are sum of squares of the residual errors and squared error of the mean line
respectively. The point where the curves cross $\tilde{\Lambda}$ bounds
corresponds to limits on primary component’s radius and hence the EOSs. The
figure shows that the upper bound on $\tilde{\Lambda}\sim 900$ results in
radius $\leq 13.44$ km, which excludes certain NLW models except for GM2 and
GM3 EOSs. On the other hand, if we consider the upper bound on
$\tilde{\Lambda}\sim 720$, the radius bound $\leq 12.98$ km not only excludes
all the NLW EOSs but also matter without $\Delta$ resonances with DD
parametrizations. From the lower bound of $\tilde{\Lambda}\sim 400$, the lower
bound on radius $\geq 11.89$ km which excludes dense matter composed of only
nucleons with DDF (DD type) parametrization. From the observation of GW190425,
the upper bound on $\tilde{\Lambda}\sim 600$ provides the upper bound on the
primary component’s radius $\leq 15.00$ km, which gives no limit on the EOSs.
Similar to fig.-7, effective tidal deformability as a function of secondary
component’s radius in GW170817 and GW190425 events is shown in fig.-8. The
correlation fits between $\tilde{\Lambda}$ and $R(M_{2})$ are given by

$\displaystyle\tilde{\Lambda}^{\text{(GW170817)}}_{\text{fit}}$
$\displaystyle=6249-1251R(M_{2})+63.62(R(M_{2}))^{2},$
$\displaystyle\tilde{\Lambda}^{\text{(GW190425)}}_{\text{fit}}$
$\displaystyle=2783-527.5R(M_{2})+25.48(R(M_{2}))^{2},$

with maximum deviations of $\sim 17.76\%,~{}19.49\%$,
$\chi^{2}=54.49,~{}17.44$ and $\mathcal{R}^{2}\sim 0.983,~{}0.988$ for
GW170817 and GW190425 events respectively. The upper bounds on radius for the
secondary components in GW170817 case are estimated to be $13.37$ km
($\tilde{\Lambda}\sim 900$) and $12.94$ km ($\tilde{\Lambda}\sim 720$) while
the lower bound is evaluated to be $11.99$ km ($\tilde{\Lambda}\sim 400$). In
case of GW190425 case, radius of the secondary component, $R_{1.59}\leq 14.99$
km ($\tilde{\Lambda}\sim 600$). From both the correlations it can be inferred
that the radius bounds on NSs involved in GW events are approximately $12\leq
R_{*}/\text{km}\leq 13$ and $R_{*}\leq 15$ km in GW170817 and GW190425 events
respectively.

Table 4: Threshold densities denoted by $n_{u}^{Y}$ (in units of $n_{0}$) for onset of hyperons in hypernuclear dense matter for various DD CDF models. $n_{1.4}^{c(Y)}$ represents the central number density for a $1.4M_{\odot}$ NS with hypernuclear matter composition. $\Lambda_{1.4}^{N}$, $\Lambda_{1.4}^{NY}$ are the dimensionless tidal deformability of a $1.4M_{\odot}$ NS with nucleonic and hypernuclear matter respectively. CDF Model | $n_{1.4}^{c(Y)}~{}(n_{0})$ | $n_{u}^{Y}~{}(n_{0})$ | $\Lambda_{1.4}^{N}/\Lambda_{1.4}^{NY}$
---|---|---|---
DD1 | 2.42 | 2.26 | 0.998
DD2 | 2.40 | 2.25 | 0.999
DD-ME1 | 2.33 | 2.23 | 0.999
DD-ME2 | 2.25 | 2.21 | 0.998
PKDD | 2.48 | 2.15 | 1.01
DD-MEX | 2.13 | 2.15 | 1.00

Fig.-9 depicts the dimensionless tidal deformability parameter as a function
of NS mass evaluated from the DD and NLW type models with matter composition
to be nucleonic and $\Delta$-commixed baryon octet. It is observed that among
the NLW CDF models, GM2 and GM3 parametrizations fulfil the
$\Lambda_{1.4}=800$ upper bound (with $N$-matter), while they fail to satisfy
the recent $\Lambda_{1.4}=580$ bound. Other NLW parametrizations produce
larger radii NSs and, as a result, are more inclined to be easily deformable
since $\lambda\sim R^{5}$ (i.e. higher tidal deformability values). Hence no
NLW model considered in this work satisfy the mass and tidal deformability
constraints simultaneously. In the case of DD CDF models (with $N$-matter),
all the models satisfy the upper bound constraint on $\Lambda_{1.4}=800$.
However, except TW99, DDV and DDF parametrizations none of them fulfil the
upper bound ($\Lambda_{1.4}=580$) obtained from reanalysis of GW170817 event
data. No coupling parameter set considered in this work is seen to satisfy the
more strict constraints of $\Lambda_{1.4}=580$ and maximum mass simultaneously
with pure nucleonic matter. Another joint constraint from NICER (PSR
J$0030+0451$) and GW170817 data sets an upper bound on $\Lambda_{1.4}=730$.
Recent constraint on $\Lambda_{1.4}$ obtained from Bayesian analysis provides
an upper bound of $686$. DD CDF models (DD1, DD2, DD-ME1, DD-ME2) are observed
to satisfy these criteria inclusive with the lower bound on $M_{\text{max}}$
(see table-3 for numerical results). It should be noted, as evident from
table-4 appearance of heavier baryons are inevitable in a $1.4$ $M_{\odot}$
star with all DD CDF parametrizations except DD-MEX.

Figure 10: Correlation between weighted average tidal deformability and tidal
deformability for $1.4~{}M_{\odot}$ NSs. The short-dashed line denotes the
best fit for the EOS data sets. The bounds on ${\tilde{\Lambda}}\sim 900,720$
(upper), $400$ (lower) corresponding to GW170817 event are represented by the
horizontal dotted lines similar to fig.-7. The vertical short-dashed lines
mark the points where the linear fit intersects $\tilde{\Lambda}$ bounds.
Parametrization models involved are similar as in fig.-7.

Fig.-10 displays the tight correlation between weighted average
$\tilde{\Lambda}$ and $\Lambda_{1.4}$ tidal deformability for GW170817 event
data. In this case, we have considered $M_{1}=1.40~{}M_{\odot}$,
$M_{2}=1.33~{}M_{\odot}$ with $\mathcal{M}=1.1878~{}M_{\odot}$. The tight
linear correlation between $\tilde{\Lambda}$ and $\Lambda_{1.4}$ is given by

$\displaystyle\tilde{\Lambda}^{\text{(GW170817)}}_{\text{fit}}$
$\displaystyle=16.28+1.138\Lambda_{1.4},$

with maximum deviation of $\sim 3.11\%$, $\chi^{2}=2.39$ and
$\mathcal{R}^{2}\sim 0.999$. The upper bounds on $\Lambda_{1.4}$ are deduced
to be $777,~{}619$ corresponding to $\tilde{\Lambda}\sim 900,~{}720$
respectively. While the lower bound on $\Lambda_{1.4}$ based on
$\tilde{\Lambda}\sim 400$ (AT2017gfo) is estimated to be $337$. The upper
bound on $\tilde{\Lambda}$ favours the DD parametrizations as evident from
fig.-10.

Figure 11: Dimensionless tidal deformability as a function of compactness
parameter ($C$) for an isolated NS (with $N$-matter) for the CDF
parametrizations implemented in this work. The bounds on ${\Lambda}\sim
800,580$ corresponding to GW170817 event are represented by the horizontal
dotted lines similar to fig.-9.

We have shown the dependence of $\Lambda$ on compactness of the star for
different parametrizations (with $N$-matter) in fig.-11. From the figure, it
is clear that the dependence of $\Lambda$ on compactness is almost independent
of EOSs. This relates with the result from Maselli et al. (2013). Then in
fig.-12, we plot the $\Lambda$ for an isolated NS of mass $1.4M_{\odot}$ which
shows a general trend with almost all EOSs. In order to find lower bound on
the compactness parameter of a $1.4~{}M_{\odot}$ NS ($C_{1.4}$), we compute
the correlations between $\Lambda_{1.4}$ and $C_{1.4}$ as
$\Lambda_{1.4}/10^{4}=2.094-22.67C_{1.4}+62.2C^{2}_{1.4}$. The maximum
deviation estimated is $\sim 11.94\%$ with $\chi^{2}=41.03$, $\mathcal{R}^{2}$
corresponding to $0.989$. The stiffer NLW type parametrization models produce
compact stars with larger mass and radius leading to higher tidal
deformabilities. Obviously, these stiffer EOSs do not fulfil the upper limit
of $\Lambda_{1.4}\leq 800$. Softer parametrizations GM2 and GM3 satisfy the
upper limit of $\Lambda$. On the other hand, DD type parametrizations
considered in this work generate compact stars with $\Lambda_{1.4}\leq 800$.
Inclusion of $\Delta$-resonances reduces $R_{1.4}$ resulting in increase of
$C_{1.4}$ keeping $\Lambda$ less than its upper limit. From fig.-12, it can be
inferred that for a $1.4~{}M_{\odot}$ NS, the lower bound in compactness is
$0.153(0.154)$ corresponding to $\Lambda_{1.4}\sim 800(777)$ and it is
$0.160(0.159)$ following the stringent upper bound of $\Lambda_{1.4}\sim
580(619)$. It also shows that the upper bound in compactness for a
$1.4~{}M_{\odot}$ NS is $0.173$ following the lower bound $\Lambda_{1.4}\sim
337$. The points which lie away from the correlation fit are from GM2, GM3,
PKDD coupling models. Based on the derived bounds of $C_{1.4}$, the range of
$R_{1.4}$ is found to be $11.95-13.00$ km ($C_{1.4}\sim 0.159-0.173$) and
$11.95-13.42$ km ($0.154-0.173$). The estimated bounds on $R_{1.4}$ satisfy
the range $11.5\leq R_{1.4}/\text{km}\leq 13.6$ as reported in Li & Steiner
(2006) with the latter constrained from terrestrial experimental data.

Figure 12: Correlation between tidal deformability and compactness parameter
for $1.4~{}M_{\odot}$ NSs. The short-dashed line denotes the best fit for the
EOS data sets. The bounds on ${\Lambda_{1.4}}$ corresponding to GW170817 event
are represented by the horizontal dotted lines similar to figs.-9,11 in
addition to ones deduced from fig.-10. The vertical short-dashed lines mark
the points where the quadratic fit intersects $\Lambda_{1.4}$ bounds.
Parametrization models involved are similar as in fig.-7.

In addition to heavier baryons, meson such as (anti)kaon condensations may
also come into the picture with rising energy density towards the interior of
NSs. Thapa & Sinha (2020); Thapa et al. (2021) reported that these meson
condensations affect the lowering of maximum mass NS configurations and do not
have any significant reduction in radii of $\sim 1.4~{}M_{\odot}$ NSs compared
to pure hadronic matter cases.

##  5 Conclusions and Outlook

The BNS mergers provide intriguing information to constrain various
theoretical formulations of dense nuclear matter EOSs. In this work, we have
analysed two schemes of coupling parametrizations and tried to constrain
purely nucleonic, hypernuclear, and $\Delta$-baryon admixed hypernuclear
matter EOSs within the CDF framework model via employing GW and other
astrophysical observations.

Confronting the coupling parametrizations considered in this work with the
recent bounds on nuclear saturation properties infers that the NLW parameter
sets fail to satisfy these empirical ranges while DD coupling models (except
PKDD) are seen to fulfil these bounds. In addition, imposing the lower limit
of maximum mass constraint rules out the relatively softer EOSs viz. GM2, GM3
among NLW models and DDF, DDV, TW99 among DD CDF models. On the other hand,
very stiff NLW EOSs, except GM1 fail to satisfy the measured radius range from
observation of PSR J$0030+0451$. In the core of NSs, with enhanced matter
density, the appearance of heavier baryons are inevitable. Their appearance
softens the matter, excluding some more parametrization viz. DD1, DD2, PKDD
among DD CDF models and only admissible NLW model parametrization GM1. Even
with pure nucleonic matter, the parametrization GM1 is ruled out because it is
not as soft as to provide $\tilde{\Lambda}\leq 900$ from GW170817 event.
Hence, none of the parameter sets within NLW model considered in this work can
satisfy all the observational constraints simultaneously. The upper bound on
$\tilde{\Lambda}\leq 900$ allow all the parametrizations of DD CDF models,
although the strict upper limit $\tilde{\Lambda}\leq 720$ does not allow any
parametrization with pure nuclear matter maintaining lower bound constraints
of maximum mass. However, the appearance of heavier non-strange baryons allows
all the DD CDF parametrizations to lie within the stringent upper limit.

The advent of heavier baryons (hyperons) leads to softening of EOSs, which in
turn reduce the NS maximum masses values by $\sim 0.38~{}M_{\odot}$ than that
of pure nucleonic cases. The effect on dimensionless tidal deformability
($\Lambda_{1.4}$) due to the incorporation of hyperons is very marginal. It
can be seen that the central number densities of $1.4M_{\odot}$ NSs are quite
close to the onset of hyperons. This attributes to the marginal changes in
$\Lambda_{1.4}$. However, the appearance of $\Delta$-resonances in dense
matter is worthy of mentioning. From fig.-9, it can be seen that the value of
$\Lambda_{1.4}$ lies above the stringent $580$ upper bound for the stiffer DD
CDF EOSs for pure nucleonic matter. With the advent of $\Delta$-quartet,
$\Lambda_{1.4}$ reduces sufficiently and moves below $580$ for all stiff DD
CDF parametrizations considered in this work. This relates with the results
from Li & Sedrakian (2019) and Raduta (2021). $\Delta$-baryon admixed
hypernuclear NSs are observed to have smaller radius values in comparison to
the NSs with only baryon octet particle spectrum (refer to table-3). This is
due to the early onset of $\Delta^{-}$ baryon, which relatively softens the
EOSs at lower densities.

Abbott et al. (2021) recently reported the observation of GWs from two NS-BH
coalescences (GW200105 and GW200115). The masses of the NSs involved are
deduced to be $1.9^{+0.3}_{-0.2}~{}M_{\odot}$, $1.5^{+0.7}_{-0.3}~{}M_{\odot}$
for the GW200105 and GW200115 events respectively. However, no constraints on
tidal deformation of the secondary components have been reported. The coupling
parametrizations considered in this work satisfy the NS mass constraints set
by these recent GW events.

Based on the coupling data sets contemplated in this work and observed strict
restrictions on $\tilde{\Lambda}$, a radius range of the NSs involved in
GW170817 event is deduced to be around $12\leq R_{*}/\text{km}\leq 13$ and
that with GW190425 event’s data is found to be $R_{*}\leq 15.00$ km. The lower
bound on $\tilde{\Lambda}$ considered in this work is evaluated from the
electromagnetic counterpart (kilonova) signal in GW170817 event. The strong
correlation between weighted average $\tilde{\Lambda}$ and $\Lambda_{1.4}$
sets bounds as $337\leq\Lambda_{1.4}\leq 619$ corresponding to
$400\leq\tilde{\Lambda}\leq 720$. Furthermore, similar analysis with strict
bounds on $\Lambda_{1.4}$ translate to compactness parameter value of a
$1.4~{}M_{\odot}$ NS to be in the range $0.159\leq C_{1.4}\leq 0.173$. This
yields $R_{1.4}$ in the range $11.95-13.00$ km.

Another inference from the GW observational constraints is the possible
existence of quark matter in the interior of NSs. Many works (Nandi & Char,
2018; Gomes et al., 2019; Paschalidis et al., 2018; Mariani et al., 2019;
Nandi & Pal, 2021; Rather et al., 2021) reported hybrid star configurations
consistent with the strict upper bounds on $\tilde{\Lambda}$, $\Lambda_{1.4}$
and massive NS observations. Since the knowledge regarding high-density quark
matter is still inadequate, we could explore and constrain the parameter space
of the functionals involved in describing the quark matter behaviour at such
density regimes based on GW events and recent NICER data. Further analysis on
the aspect is beyond the scope of this work and will be discussed in future
studies.

## Acknowledgements

The authors thank the anonymous referee for constructive comments that
significantly contributed to enhancing the manuscript’s quality. The authors
acknowledge financial support from Science and Engineering Research Board,
Department of Science and Technology, Government of India through Project No.
EMR/2016/006577 and Ministry of Education, Government of India. VBT and MS are
also thankful to Sarmistha Banik for fruitful discussions.

## Data Availability

Data sharing not applicable to this article as no data sets were generated
during this study.

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