# Constraints of the maximum mass of quark stars based on post-merger
evolutions

Yurui Zhou  Department of Astronomy, School of Physics, Huazhong University of
Science and Technology, Wuhan, 430074, P. R. China  Chen Zhang  The HKUST
Jockey Club Institute for Advanced Study, The Hong Kong University of Science
and Technology, Hong Kong, P. R. China  Junjie Zhao  Henan Academy of
Sciences, Zhengzhou 450046, Henan, China  Department of Astronomy, Beijing
Normal University, Beijing 100875, China  Kenta Kiuchi  Max Planck Institute
for Gravitational Physics (Albert Einstein Institute), Potsdam, D-14476,
Germany  Center for Gravitational Physics and Quantum Information, Yukawa
Institute for Theoretical Physics, Kyoto University, Kyoto, 606-8502, Japan
Sho Fujibayashi  Frontier Research Institute for Interdisciplinary Sciences,
Tohoku University, Sendai 980-8578, Japan  Astronomical Institute, Graduate
School of Science, Tohoku University, Sendai 980-8578, Japan  Max Planck
Institute for Gravitational Physics (Albert Einstein Institute), Potsdam,
D-14476, Germany  Enping Zhou<EMAIL_ADDRESS>Department of Astronomy,
School of Physics, Huazhong University of Science and Technology, Wuhan,
430074, P. R. China

###### Abstract

We semi-analytically investigate the post-merger evolution of the binary quark
star merger. The effective-one-body method is employed to estimate the energy
and angular momentum dissipation due to gravitational waves in the inspiral
phase. Three major mechanisms of energy and angular momentum dissipation are
considered in the post-merger phase: mass outflows, neutrinos, and
gravitational waves. The proportion of each mechanism could be determined by
baryon number, energy and angular momentum conservation laws as well as the
equilibrium model for rotating quark stars. Applying this analysis to the
GW170817 event suggests two important conclusions: 1) a remnant quark star
whose mass is smaller than the maximum mass of a uniformly rotating quark star
can collapse before its rotational energy is dissipated via electromagnetic
radiation (i.e., $\displaystyle\sim 100\,\mathrm{s}$) as the angular momentum
left in the remnant quark star might not be large enough to sustain the
additional self-gravity of the supramassive quark star due to the angular
momentum dissipation of mass outflows, neutrinos and gravitational waves; 2)
considering a general quark star equation of state model, a constraint on the
maximum mass of cold and non-rotating quark stars is found as $\displaystyle
M_{\mathrm{TOV}}\lesssim 2.35^{+0.07}_{-0.17}\,M_{\odot}$, assuming a delayed
collapse occurred before a large fraction of the total rotational energy
($\displaystyle\gtrsim 10^{53}\,$erg) of the merger remnant was deposited into
the merger environment for the GW170817 event. These constraints could be
improved with future merger events, once there are more evidences on its post-
merger evolution channel or information on the amount of post-merger
gravitational wave and neutrino emissions inferred from the multi-messenger
observations.

##  I INTRODUCTION

The first direct detection of gravitational waves from an inspiraling binary
system of two neutron stars (GW170817) (LIGO Scientific Collaboration and
Virgo Collaboration, 2017) followed by a wide variety of the observations of
electromagnetic counterparts (GRB 170817A, AT 2017gfo) (Abbott  _et al._ ,
2017) have enriched our comprehension of the equation of states (EOS) of
neutron stars (NSs). Gravitational wave observations for the late inspiral
phase of binary neutron stars (BNS) can be used to constrain the tidal
deformability of NSs (Abbott  _et al._ , 2018, 2019; Annala  _et al._ , 2018;
De  _et al._ , 2018). Moreover, the electromagnetic counterparts serve as
indicators of the fate of the merger remnant, which can also be used to
constrain the EOS of merging NSs (Shibata  _et al._ , 2017a; Bauswein  _et
al._ , 2017; Margalit and Metzger, 2017; Ruiz  _et al._ , 2018; Rezzolla  _et
al._ , 2018; Shibata  _et al._ , 2019).

The property of the blue kilonova component of AT 2017gfo disfavors the case
of prompt collapse, i.e., a black hole (BH) is formed on a dynamical timescale
following the merger (Kasen  _et al._ , 2017; Margalit and Metzger, 2017),
leading to a constraint on the radius of NSs (Bauswein  _et al._ , 2017). The
case of a stable NS remnant is also disfavored, as there is no observational
evidence of significant energy injection from the remnant NS. Therefore, a
delayed collapse to BH scenario is believed to happen following the merger of
GW170817 event, and hence an upper limit of the maximum mass of non-rotating
NSs is obtained $\displaystyle M_{\mathrm{TOV}}\lesssim 2.2\,M_{\odot}$
(Margalit and Metzger, 2017; Rezzolla  _et al._ , 2018). A similar constraint
can be obtained from the observation of GRB 170817A (Ruiz  _et al._ , 2018).
However, it is pointed out that at the onset of collapse, the merger remnant
NS is not necessarily at the mass-shedding limit (Shibata  _et al._ , 2019),
which is an assumption that many previous studies applied when deriving the
constraints on $\displaystyle M_{\mathrm{TOV}}$. Employing both energy and
angular momentum conservation laws as well as results of numerical-relativity
(NR) simulations, it has been suggested that $\displaystyle M_{\mathrm{TOV}}$
could be as large as $\displaystyle\sim 2.3\,M_{\odot}$ (Shibata  _et al._ ,
2019).

Nevertheless, the EOS of NSs is currently a subject of ongoing debate,
primarily due to the complex non-perturbative QCD problems associated with it.
In addition to conventional NS models, recent studies have suggested the
possibility of the existence of deconfined quark matter in the high-density
core of massive NSs (Annala  _et al._ , 2020). Numerous efforts have been made
to detect evidence of a transition from conventional hadronic matter to quark
matter, a strong-interaction phase transition, within NSs through
astrophysical observations (Bauswein  _et al._ , 2019; Most  _et al._ , 2019).
Alternatively, the existence of quark stars (QSs), either strange quark stars
(SQSs) composed entirely of deconfined $\displaystyle u,d,s$ quarks or up-down
quark stars ($\displaystyle ud$QSs) composed entirely of $\displaystyle u,d$
quarks, has also been proposed as a potential explanation for the nature of
compact stars (Alcock  _et al._ , 1986; Drago  _et al._ , 2016a, b;
Bhattacharyya  _et al._ , 2017; Zhang, 2020; Wang  _et al._ , 2019; Ren and
Zhang, 2020), based on the conjecture that either strange quark matter (SQM)
comprising $\displaystyle u,d,s$ quarks (Bodmer, 1971; Witten, 1984) or up-
down quark matter ($\displaystyle ud$QM) comprising $\displaystyle u,d$ quarks
(Holdom  _et al._ , 2018) could represent the true ground state of the bulk
strong matter. It has been shown that interquark effects, such as color
superconductivity and QCD corrections, could significantly increase the
$\displaystyle M_{\rm TOV}$ of QSs to explain the mass-gap object
($\displaystyle M\sim 2.6M_{\odot}$) detected in GW190814 (Zhang and Mann,
2021; Roupas  _et al._ , 2021; Horvath and Moraes, 2021; Albino  _et al._ ,
2021; Miao  _et al._ , 2021; Restrepo  _et al._ , 2023; Oikonomou and
Moustakidis, 2023; Cao  _et al._ , 2022), which cannot be naturally
accommodated by NS considering NSs’ $\displaystyle M_{\rm TOV}\lesssim
2.3M_{\odot}$ as constrained by the post-merger NS evolution in GW170817 event
(Shibata  _et al._ , 2019). However, the post-merger evolutions of quark
stars, if assuming GW170817 is a merger of binary quark stars (BQS), could
also similarly set a tight upper bound on QSs’ $\displaystyle M_{\rm TOV}$,
and this essential point has been commonly overlooked in the literature.

Previous studies have shown that most of the matter ejected during and after
the merger of BQS could evaporate into nucleons and therefore BQS merger can
possibly generate a kilonova-like signal (Paulucci  _et al._ , 2017;
Bucciantini  _et al._ , 2019). Based on recent astrophysical observations,
constraints have been established on the EOS for QSs (Zhou  _et al._ , 2018;
Zhang and Mann, 2021). Efforts have also been undertaken to model the dynamics
(Bauswein  _et al._ , 2010; Zhou  _et al._ , 2022) and electromagnetic
counterparts (Bauswein  _et al._ , 2009; Li  _et al._ , 2016; Paulucci  _et
al._ , 2017; Lai  _et al._ , 2018; Bucciantini  _et al._ , 2019) associated
with BQS mergers.

It is known that uniform rotation significantly increases the maximum mass of
QSs in comparison to NSs (i.e., the ratio of the maximum mass of uniformly
rotating NSs to the maximum stable mass of non-rotating NSs $\displaystyle
M_{\mathrm{max,urot}}/M_{\mathrm{TOV}}$ is typically 1.2 for NSs, but 1.4 for
QSs). This difference may make a distinction in the scenario of the post-
merger outcome. Specifically, due to the discovery of pulsar more massive than
$\displaystyle 2\,M_{\odot}$ (Fonseca  _et al._ , 2021; Cromartie and et al.,
2020), the mass of a QS at the mass-shedding limit of rigid rotation is at
least $\displaystyle 2.8\,M_{\odot}$, which exceeds the total mass of the
system in GW170817 event ($\displaystyle M=2.74^{+0.04}_{-0.01}\,M_{\odot}$
with the $\displaystyle 90\%$ credible level (LIGO Scientific Collaboration
and Virgo Collaboration, 2017)). Therefore, for the BQS case, a long-lived QS
should be formed after the merger of the GW170817 event. However, as pointed
out in (Shibata  _et al._ , 2019), for the BNS case at the onset of collapse
to a BH the merger remnant may not contain enough angular momentum to reach
the mass-shedding limit, and hence the BQS merger remnant for the GW170817
event is not necessarily a long-lived QS if the situation is similar for the
BQS case. Therefore, analysis similar to (Shibata  _et al._ , 2019) needs to
be done for the BQS case to understand the post-merger outcome of the BQS
merger.

In this paper, we perform a similar analysis to (Shibata  _et al._ , 2019) for
the BQS scenario to investigate the post-merger evolution channels of the BQS
mergers, specifically, we focus on the case of prompt collapse and delayed
collapse. In comparison to (Shibata  _et al._ , 2019), the effective-one-body
(EOB) approach (Buonanno and Damour, 1999, 2000), which enables accurate
computation of gravitational waveforms throughout the inspiral phase, is
employed in our study to estimate the energy and angular momentum dissipation
during the inspiral stage, instead of applying the empirical relation between
the dissipated energy and angular momentum obtained through NR simulations. In
addition, a recent study (Zhang and Mann, 2021) has derived an EOS for
interacting quark matter (IQM) unifying all cases by a simple
reparametrization and rescaling, through which the number of degrees of
freedom in the QS EOS can be maximally reduced. With this unified EOS, we
systematically investigate the equilibrium models of rotating QSs and analyze
the fate of the merger remnant to establish the constraint on IQM parameter
space from observation.

This paper is organized as follows. In Sec. II, we analyze the prompt collapse
case for the BQS merger. In Sec. III, we investigate the delayed collapse
scenario. The constraint on the maximum mass of QSs from the GW170817 event is
established in Sec. IV. Finally, we summarize our work and present our
conclusions in Sec. V. Throughout this paper, we employ the unis of
$\displaystyle c=G=M_{\odot}=1$ where $\displaystyle c,\,G$ and $\displaystyle
M_{\odot}$ are the speed of light, the gravitational constant and the solar
mass, respectively.

##  II PROMPT COLLAPSE

The outcome of a BNS merger depends on the total mass of the system. A direct
gravitational collapse to a BH takes place if the total mass is beyond a
certain mass, which is called the threshold mass for prompt collapse
($\displaystyle M_{\mathrm{thres}}$) (Shibata  _et al._ , 2005; Shibata, 2005;
Shibata and Taniguchi, 2006). Therefore, $\displaystyle M_{\mathrm{thres}}$ is
an important parameter to characterize the fate of the merger remnant. Many
attempts have been made to determine $\displaystyle M_{\mathrm{thres}}$ for
the scenario of BNS mergers (Bauswein  _et al._ , 2013; Bauswein and
Stergioulas, 2017; Bauswein  _et al._ , 2020, 2021; Köppel  _et al._ , 2019;
Kiuchi  _et al._ , 2010), while only a limited number of studies for the BQS
case (Zhou  _et al._ , 2022) due to the difficulties in hydrodynamical
simulations of BQS mergers arising from the strong density discontinuity at
the stellar surface. Hence, to investigate $\displaystyle M_{\mathrm{thres}}$
of a large number of different QS EOSs, a method with relatively small
computational demands is needed. It has been suggested in (Bauswein and
Stergioulas, 2017) that whether or not prompt collapse occurs depending mostly
on the relation between the mass of the merger remnant and the maximum stable
mass ($\displaystyle M_{\mathrm{stab}}$) given the angular momentum contained
in the remnant at the time of the merger ($\displaystyle
J_{\mathrm{merger}}$). $\displaystyle M_{\mathrm{stab}}$ can be approximately
derived from differentially rotating equilibrium models Bauswein and
Stergioulas (2017); Weih  _et al._ (2018); Zhou  _et al._ (2019). Therefore,
through this semi-analytical method, one can derive $\displaystyle
M_{\mathrm{thres}}$ without time-consuming and sophisticated hydrodynamical
simulations.

In this section, we apply the method proposed in (Bauswein and Stergioulas,
2017) to determine $\displaystyle M_{\mathrm{thres}}$ for the case of BQS
mergers. In addition, the empirical relation between $\displaystyle M$ and
$\displaystyle J_{\mathrm{merger}}$ constructed through NR simulations is used
in (Bauswein and Stergioulas, 2017) to estimate the angular momentum of the
merger remnant. Compared to the empirical relation, the EOB method employed in
this paper permits a more accurate calculation of the angular momentum left in
the merger remnant, and hence a more precise $\displaystyle
M_{\mathrm{thres}}$ can be derived, which can be used to constrain QS EOSs
only relying on the measurement of the total mass of the binary and the
determination of whether a prompt collapse takes place or not in the event.

###  II.1 EOS models

In (Bauswein  _et al._ , 2013; Bauswein and Stergioulas, 2017; Bauswein  _et
al._ , 2020, 2021), various NS EOS models are constructed to derive the EOS-
insensitive relations between $\displaystyle M_{\mathrm{thres}}$ and stellar
properties. For QSs, it is known that in the simplest MIT bag model (i.e.,
with zero strange quark mass and neglecting interaction between quarks),
stellar properties of both rotating and non-rotating QSs scale with the bag
constant which is also the only parameter describing the EOS model. Therefore,
the simplest MIT bag model is not appropriate for exploring the parameter
space and deriving the EOS-insensitive relations for $\displaystyle
M_{\mathrm{thres}}$ of the BQS merger. In this paper, we employ the unified QS
EOS including the perturbative QCD corrections (Fraga  _et al._ , 2001; Alford
_et al._ , 2005) and color superconductivity (Rajagopal and Wilczek, 2001;
Lugones and Horvath, 2002; Alford  _et al._ , 1999) derived in (Zhang and
Mann, 2021), where with a simple reparametrization and rescaling, the EOS can
be expressed as (see (Zhang and Mann, 2021) for details):

$p=\frac{1}{3}(\rho-4B_{\mathrm{eff}})+\frac{4\lambda^{2}}{9\pi^{2}}\left(-1+\mathrm{sgn}(\lambda)\sqrt{1+3\pi^{2}\frac{\rho-
B_{\mathrm{eff}}}{\lambda^{2}}}\right)$ (1)

where $\displaystyle\mathrm{sgn}(\lambda)$ represents the sign of
$\displaystyle\lambda$, $\displaystyle p$ denotes the pressure,
$\displaystyle\rho$ corresponds to the total energy density and $\displaystyle
B_{\mathrm{eff}}$ is the effective bag constant as a phenomenological
representation of non-perturbative QCD vacuum effects. $\displaystyle\lambda$
is defined as:

$\lambda=\frac{\xi_{2a}\Delta^{2}-\xi_{2b}m_{s}^{2}}{\sqrt{\xi_{4}a_{4}}},$
(2)

where $\displaystyle m_{s}$ represents the finite mass of the strange quark,
$\displaystyle\Delta$ is the pairing energy gap arising from color
superconductivity and $\displaystyle a_{4}$ parametrizes the perturbative QCD
corrections. $\displaystyle\xi_{2a}$, $\displaystyle\xi_{2b}$ and
$\displaystyle\xi_{4}$ are the constant coefficients and can be expressed as:

$(\xi_{4},\xi_{2a},\xi_{2b})=\begin{cases}(((\tfrac{1}{3})^{\frac{4}{3}}+\tfrac{2}{3})^{\frac{4}{3}})^{-3},1,0)&\mathrm{2SC\
phase}\\\ (3,1,3/4)&\mathrm{2SC+s\ phase}\\\ (3,3,3/4)&\mathrm{CFL\
phase}\end{cases},$ (3)

for various phases of quark matter. Here the 2SC and 2SC+s phases both
represent the two-flavor color superconductivity phase where $\displaystyle u$
quarks pair with $\displaystyle d$ quarks, while only the latter has
(unpaired) strange quarks in composition. CFL phase denotes the color-flavor
locking phase, where $\displaystyle u,d,s$ quarks pair with each other
antisymmetrically.

By introducing the following dimensionless rescaling:

$\bar{\rho}=\frac{\rho}{4B_{\mathrm{eff}}},\
\bar{p}=\frac{p}{4B_{\mathrm{eff}}},\
\bar{\lambda}=\frac{\lambda^{2}}{4B_{\mathrm{eff}}},$ (4)

Eq. (1) can be expressed as

$\bar{p}=\frac{1}{3}(\bar{\rho}-1)+\frac{4\bar{\lambda}}{9\pi^{2}}\left(-1+\mathrm{sgn}(\lambda)\sqrt{1+\frac{3\pi^{2}}{\bar{\lambda}}(\bar{\rho}-\frac{1}{4}})\right).$
(5)

In Eq. (5), $\displaystyle\mathrm{sgn}(\lambda)\,\bar{\lambda}$ is the only
parameter governing the EOS models, and thus we can obtain different families
of QS EOS by selecting different
$\displaystyle\mathrm{sgn}(\lambda)\,\bar{\lambda}$. In addition, for every
EOS family with a certain value of
$\displaystyle\mathrm{sgn}(\lambda)\,\bar{\lambda}$, the solution scales with
$\displaystyle B_{\mathrm{eff}}$ and the properties of a QS such as the
gravitational mass and the radius can be obtained given a specific
$\displaystyle B_{\mathrm{eff}}$, just like the case of MIT bag model. In this
paper, to construct a series of representative EOSs, we explore a broad range
of values for $\displaystyle\bar{\lambda}$, ranging from $\displaystyle 0$
(indicating non-interacting QSs) to $\displaystyle 10$ (representing a high
level of strong interaction strength). In this paper, we only concentrate on
the case of the CFL phase, with the employed benchmark EOSs listed in Table 1.
Our analyses and results should also apply to the 2SC+s phase with the same
values of ($\displaystyle\mathrm{sgn}(\lambda)\,\bar{\lambda}$, $\displaystyle
B_{\mathrm{eff}}$, $\displaystyle a_{4}$), since the only difference between
these two phases is the value of $\displaystyle\xi_{2a}$, which does not
influence the relation among pressure, energy density and number density when
$\displaystyle\mathrm{sgn}(\lambda)\,\bar{\lambda}$, $\displaystyle
B_{\mathrm{eff}}$ and $\displaystyle a_{4}$ are fixed (Zhang and Mann, 2021).

Model |  $\displaystyle\Delta$ (MeV) |  $\displaystyle m_{s}$ (MeV) | $\displaystyle a_{4}$ |  $\displaystyle B_{\mathrm{eff}}^{(1/4)}$ (MeV) | $\displaystyle\mathrm{sgn}(\lambda)\,\bar{\lambda}$ |  $\displaystyle M_{\mathrm{TOV}}$ ($\displaystyle M_{\odot}$) |  $\displaystyle R_{1.4}$ (km) |  $\displaystyle R_{\mathrm{TOV}}$ (km) | $\displaystyle\Lambda_{1.4}$ | $\displaystyle f_{0}$ | $\displaystyle f_{\mathrm{TOV}}$ |  $\displaystyle f_{\mathrm{peak}}$ (kHz)
---|---|---|---|---|---|---|---|---|---|---|---|---
EOS-1 | 137.78 | 100.00 | 0.50 | 168.96 | 0.500 | 2.013 | 10.13 | 10.27 | 286 | 1.162 | 1.238 | 3.33
EOS-2 | 0.00 | 100.00 | 0.61 | 137.16 | -0.022 | 2.080 | 11.41 | 11.47 | 619 | 1.132 | 1.201 | 2.70
EOS-3 | 0.00 | 0.00 | 0.57 | 141.70 | 0.000 | 2.100 | 11.33 | 11.45 | 598 | 1.132 | 1.202 | 2.73
EOS-4 | 0.00 | 100.00 | 0.61 | 135.14 | -0.023 | 2.137 | 11.66 | 11.83 | 718 | 1.144 | 1.217 | 2.59
EOS-5 | 160.85 | 100.00 | 0.50 | 169.19 | 1.000 | 2.200 | 10.52 | 11.13 | 382 | 1.251 | 1.351 | 3.11
EOS-6 | 151.41 | 100.00 | 0.40 | 167.23 | 1.000 | 2.254 | 10.70 | 11.35 | 429 | 1.194 | 1.293 | 3.01
EOS-7 | 100.00 | 100.00 | 0.61 | 148.51 | 0.142 | 2.284 | 11.56 | 12.04 | 703 | 1.233 | 1.327 | 2.61
EOS-8 | 228.51 | 100.00 | 0.13 | 194.33 | 10.000 | 2.284 | 9.50 | 10.15 | 195 | 1.190 | 1.314 | 3.66
EOS-9 | 97.83 | 100.00 | 0.50 | 147.13 | 0.160 | 2.350 | 11.78 | 12.35 | 796 | 1.190 | 1.284 | 2.52
EOS-10 | 148.77 | 100.00 | 0.40 | 163.95 | 1.000 | 2.350 | 11.02 | 11.70 | 524 | 1.214 | 1.321 | 2.84
Table 1: The EOS of interacting QSs used in this paper, and the important
parameter of non-rotating QSs. $\displaystyle\Delta$ is the pairing energy gap
arising from color superconductivity. $\displaystyle m_{s}$ is the mass of the
strange quark, $\displaystyle a_{4}$ is the perturbative QCD correction
parameter. $\displaystyle B_{\mathrm{eff}}$ is the effective bag constant
including a phenomenological representation of non-perturbative QCD vacuum
effects. $\displaystyle\bar{\lambda}$ is the quantity that characterizes the
strength of the related strong interaction defined in (Zhang and Mann, 2021).
$\displaystyle M_{\mathrm{TOV}}$ and $\displaystyle R_{\mathrm{TOV}}$ are the
mass and radius of maximum mass non-rotating QSs. $\displaystyle R_{1.4}$ and
$\displaystyle\Lambda_{1.4}$ are the radius and the tidal deformability of 1.4
$\displaystyle M_{\odot}$ QSs. $\displaystyle f_{0}$ represents the ratio of
the baryonic mass to the gravitational mass of $\displaystyle 1.37\,M_{\odot}$
QSs. $\displaystyle f_{\mathrm{TOV}}$ is the ratio of the baryonic mass to the
gravitational mass for the maximum mass non-rotating QSs. $\displaystyle
f_{\mathrm{peak}}$ is the peak frequency in the spectrum amplitude of
gravitational waves emitted in the post-merger phase estimated by Eq. (5.4) of
Ref. (Kiuchi  _et al._ , 2020).

QSs with the EOSs listed in Table 1 are capable of reproducing $\displaystyle
2.0\,M_{\odot}$ QSs (Fonseca  _et al._ , 2021; Cromartie and et al., 2020),
and also satisfying the GW170817 constraint
$\displaystyle\Lambda_{1.4}\lesssim 800$ (LIGO Scientific Collaboration and
Virgo Collaboration, 2017; De  _et al._ , 2018), where
$\displaystyle\Lambda_{1.4}$ is the tidal deformability of a $\displaystyle
1.4\,M_{\odot}$ QS. It is worth noting that the relation between energy
density and pressure is solely dependent on
$\displaystyle(\mathrm{sgn}(\lambda)\,\bar{\lambda},B_{\mathrm{eff}})$.
Consequently, the values of $\displaystyle M_{\mathrm{TOV}}$ and
$\displaystyle\Lambda_{1.4}$ are determined solely by
$\displaystyle(\mathrm{sgn}(\lambda)\,\bar{\lambda},B_{\mathrm{eff}})$ and the
parameters encoding details of nuclear physics (i.e.,
$\displaystyle\Delta,m_{s}$ and $\displaystyle a_{4}$) listed in Table 1 can
take any values as long as
$\displaystyle(\mathrm{sgn}(\lambda)\,\bar{\lambda},B_{\mathrm{eff}})$ remain
unchanged.

###  II.2 Effective-one-body model

During the inspiral phase, the angular momentum dissipation of the binary
system is primarily due to gravitational waves. Therefore, the angular
momentum left in the merger remnant can be expressed as:

$J_{\mathrm{merger}}=J_{0}-J_{\mathrm{GW,i}},$ (6)

where $\displaystyle J_{0}$ represents the angular momentum of the binary at
the beginning of the calculation of the gravitational waves and $\displaystyle
J_{\mathrm{GW,i}}$ corresponds to the angular momentum radiated by
gravitational waves in the inspiral phase.

In this paper, we just focus on the case of quasi-circular, nonspinning
coalescences. In this case, given the properties of the binary, $\displaystyle
J_{0}$ can be obtained through the EOB dynamics for a specific initial orbit
frequency or a specific initial separation (Buonanno and Damour, 2000). When
the initial condition is determined, the gravitational waveform until the time
of the merger can be generated through the EOB approach. Moreover, the
radiated angular momentum can be calculated as:

$\displaystyle\displaystyle
J_{\mathrm{GW,i}}=\frac{1}{16\pi}\sum\limits_{(l,m)}\int_{t_{0}}^{t_{\mathrm{merger}}}dt^{\prime}m\Im[r^{2}h_{lm}\left(t^{\prime}\right)\dot{h}_{lm}^{*}\left(t^{\prime}\right)],$
(7)

where $\displaystyle h_{lm}$ is the gravitational waveform generated via the
EOB method, $\displaystyle r$ represents the radius where $\displaystyle
h_{lm}$ is extracted, $\displaystyle t^{\prime}$ corresponds to the time from
the start of the simulation ($\displaystyle t_{0}$) to the merger
($\displaystyle t_{\mathrm{merger}}$), $\displaystyle\Im$ is the imaginary
part of a complex number and $\displaystyle(l,m)$ represents the different
modes of gravitational waves. The $\displaystyle x$ and $\displaystyle y$
components of angular momentum loss are not included because they turn out to
have a negligible effect on the computation of $\displaystyle
J_{\mathrm{GW,i}}$ (Damour  _et al._ , 2012; Bernuzzi  _et al._ , 2012). As
shown in (Dietrich  _et al._ , 2017; Kiuchi  _et al._ , 2020), most of the
energy radiated in a BNS merger is contained in the (2,2) mode of
gravitational waves. Therefore, for simplicity, only the dominant (2,2) mode
is considered in this paper.

In the present paper, we employ the state-of-the-art time-domain EOB waveform
model, specifically SEOBNRv4T (Hinderer  _et al._ , 2016; Steinhoff  _et al._
, 2016), which is available in the LSC Algorithm Library (LIGO Scientific
Collaboration, 2018). SEOBNRv4T incorporates dynamic tides and accounts for
the spin-induced quadrupole moment effects. This model aligns with the latest
BNS numerical simulations, extending up to the merger stage (Dietrich  _et
al._ , 2017), and has been currently implemented for LIGO data analysis Abbott
_et al._ (2019). Similar to SEOBNRv4T, Other accurate gravitational waveform
models like TEOBResumS (Bernuzzi  _et al._ , 2015a; Nagar  _et al._ , 2018;
Akcay  _et al._ , 2019), NRTidalv2 (Dietrich  _et al._ , 2019), NRTidalv3
(Abac  _et al._ , 2024) can also be used to obtain $\displaystyle
J_{\mathrm{merger}}$. The comparison among these gravitational waveform models
(Abac  _et al._ , 2024) shows that the deviation of waveform appears only a
few orbits before the merger and the dissipation in these few orbits is minor
compared to the dissipation in the whole inspiral phase, and thus the
systematic difference among these models is insignificant for calculating
$\displaystyle J_{\mathrm{merger}}$. In the rest of this paper, we only show
results calculated through SEOBNRv4T model and compare them with the results
obtained by NR simulations to estimate the error of our method.

Given an EOS, we first set the mass components ($\displaystyle M_{1}$,
$\displaystyle M_{2}$) of the binary and set the initial orbit frequency, and
then calculate the tidal deformability of both NSs
($\displaystyle\Lambda_{1}$, $\displaystyle\Lambda_{2}$). After that, we can
use SEOBNRv4T model to generate the gravitational waveforms, and
$\displaystyle J_{\mathrm{GW,i}}$ can be calculated through Eq. (7). As
mentioned above, given the initial orbit frequency, $\displaystyle J_{0}$ can
also be derived through Eqs. (4.5) and (4.8) in (Buonanno and Damour, 2000).
Finally, $\displaystyle J_{\mathrm{merger}}$ can be obtained through Eq. (6).

To evaluate the accuracy of our method, we make a comparison among the angular
momentum at the moment of merger obtained by the EOB method, NR simulations,
and empirical relations. Here we employ the results of NR simulations in
(Bernuzzi  _et al._ , 2016) and the empirical relation Eq. (3.1) proposed in
(Shibata  _et al._ , 2019). Table 2 lists the value of $\displaystyle
J_{\mathrm{merger}}$ of different binary parameters derived via different
methods. It can be seen that the values of $\displaystyle J_{\mathrm{merger}}$
obtained through the EOB method and NR simulations differ by approximately
1$\displaystyle\%$, while the deviation between empirical relation and NR
simulations is about 5$\displaystyle\%$. Therefore, through the EOB method,
$\displaystyle J_{\mathrm{merger}}$ can be obtained accurately without relying
on the time-consuming NR simulations.

EOS, references |  $\displaystyle M_{1}$ |  $\displaystyle M_{2}$ |  $\displaystyle J_{\mathrm{merger}}^{\mathrm{EOB}}$ |  $\displaystyle J_{\mathrm{merger}}^{\mathrm{NR}}$ | $\displaystyle J_{\mathrm{merger}}^{\mathrm{empirical}}$
---|---|---|---|---|---
DD2, (Hempel and Schaffner-Bielich, 2010; Typel  _et al._ , 2010) | 1.4 | 1.2 | 5.904 | 5.996 | 5.944
1.365 | 1.25 | 5.981 | 6.066 | 6.154
1.35 | 1.35 | 6.33 | 6.379 | 6.633
1.44 | 1.39 | 6.853 | 6.888 | 7.274
SFHo, (Steiner  _et al._ , 2013) | 1.4 | 1.2 | 5.699 | 5.726 | 5.807
1.365 | 1.25 | 5.773 | 5.792 | 6.010
1.35 | 1.35 | 6.101 | 6.109 | 6.478
1.44 | 1.39 | 6.595 | 6.615 | 7.104
Table 2: The values $\displaystyle J_{\mathrm{merger}}$ obtained through
different methods. The units of mass and angular momentum are $\displaystyle
M_{\odot}$ and $\displaystyle GM^{2}_{\odot}/c$, respectively. Here
$\displaystyle M_{1}$ and $\displaystyle M_{2}$ denote the individual
isolation masses, $\displaystyle J_{\mathrm{merger}}^{\mathrm{EOB}}$,
$\displaystyle J_{\mathrm{merger}}^{\mathrm{NR}}$ and $\displaystyle
J_{\mathrm{merger}}^{\mathrm{empirical}}$ are the angular momentum at the
moment of merging calculated by the EOB method, NR simulations and the
empirical relation respectively.

###  II.3 Threshold mass of prompt collapse

Given the total mass and angular momentum of the merger remnant, the maximum
stable mass needs to be calculated to determine the threshold mass for prompt
collapse. Similar to the previous study Bauswein and Stergioulas (2017); Weih
_et al._ (2018); Zhou  _et al._ (2019), we estimate $\displaystyle
M_{\mathrm{stab}}$ for given angular momentum through the differentially
rotating equilibrium models.

The relation between $\displaystyle M_{\mathrm{stab}}$ and $\displaystyle
J_{\mathrm{merger}}$ can be determined through the “turning point” theorem of
Friedman, Ipser, and Sorkin (Friedman  _et al._ , 1988). According to the
research, for rigidly rotating stars, along a sequence with a constant angular
momentum and varying mass and central energy density
($\displaystyle\rho_{\mathrm{c}}$), a secular instability occurs at the
maximum mass, corresponding to the turning point of the $\displaystyle
M-\rho_{\mathrm{c}}$ curve (see Figure 1 for turning points). In addition, as
shown in (Weih  _et al._ , 2018), the turning point criterion can be seen as a
first approximation for determining $\displaystyle M_{\mathrm{stab}}$ for
differentially rotating stars.

Figure 1:  Gravitational mass versus central density diagram of uniformly
rotating QSs with various fixed values of angular momentum $\displaystyle J$
for the EOS-3. The lower and upper solid curves show the sequences of non-
rotating QSs (labelled by $\displaystyle J=0$) and uniformly rotating QSs at
mass-shedding limits. The dashed curve represents the sequence of the turning
points, where QSs are marginally stable to gravitational collapse. QSs located
on the lower density side of this curve are stable, while those on the higher
density side are unstable.

The differentially rotating QSs have been investigated in (Zhou  _et al._ ,
2019), where it has been indicated that the details of the rotational profile
in the merger remnant only have little impact on the relation between the
maximum stable mass and the angular momentum of the remnant. Therefore, for
simplicity, we adopt the widely used differential rotation law known as the
$\displaystyle j$-constant law (Komatsu  _et al._ , 1989),

$j(\Omega)=A^{2}(\Omega_{\mathrm{c}}-\Omega),$ (8)

where $\displaystyle A$ is a constant parameter in the model, $\displaystyle
j:=u^{t}u_{\phi}$ is the specific angular momentum,
$\displaystyle\Omega:=u^{\phi}/u^{t}$ is the angular velocity as measured by
an observer at infinity, and $\displaystyle\Omega_{\mathrm{c}}$ is the angular
velocity at the center of the star. We choose
$\displaystyle\hat{A}:=A/r_{\mathrm{e}}=$1.5, 2 and 3 respectively to estimate
the differences of various rotation laws, where $\displaystyle r_{\mathrm{e}}$
is the equatorial radius of the star.

Figure 2: Deriving the threshold mass to collapse by our equilibrium models.
The dashed line corresponds to the relation between the maximum stable mass
$\displaystyle M_{\mathrm{stab}}$ and the angular momentum $\displaystyle J$
of differentially rotating QSs with different rotation laws. The blue solid
line represents the relation between the total binary ADM mass $\displaystyle
M$ and the angular momentum of the remnant at the time of merger computed by
the SEOBNRv4T model $\displaystyle J_{\mathrm{merger}}^{\mathrm{EOB}}$ for the
equal-mass case. The intersection of these two lines determines the threshold
mass to collapse calculated from our equilibrium models.

We first employ EOS-3 the same EOS as used in NR simulations (Zhou  _et al._ ,
2021, 2022). RNS code (Stergioulas and Friedman, 1995) is used to construct
the equilibrium models of differentially rotating QSs. The relation between
$\displaystyle M_{\mathrm{stab}}$ and $\displaystyle J$ of differentially
rotating QSs with the $\displaystyle j$-constant law with
$\displaystyle\hat{A}=$ 1.5, 2 and 3 calculated by RNS code and the relation
between $\displaystyle M$ and $\displaystyle J_{\mathrm{merger}}$ computed by
the EOB method are displayed in Figure 2. The point of intersection between
the solid and dashed lines in Figure 2 can be used to provide an estimation of
the threshold mass of the prompt collapse in BQS mergers
($\displaystyle\approx 3.10^{+0.06}_{-0.06}\,M_{\odot}$). $\displaystyle
M_{\mathrm{thres}}$ of this EOS has been determined through NR simulations
(Zhou  _et al._ , 2022) to range from $\displaystyle 3.05\,M_{\odot}$ to
$\displaystyle 3.10M_{\odot}$. The two results differ only by
$\displaystyle\sim 2\%$, which may be attributed to the assumptions of
axisymmetry and stationarity, and the uncertainty in the differential rotation
law used in constructing the equilibrium models. Compared to the results in
(Bauswein and Stergioulas, 2017), it can be found that with the help of the
EOB method, the deviation of $\displaystyle M_{\mathrm{thres}}$ derived by the
equilibrium models decreases from $\displaystyle\sim 5\%$ to
$\displaystyle\sim 2\%$.

Moreover, as discussed in (Zhang and Mann, 2021; Zhou  _et al._ , 2022), both
$\displaystyle M_{\mathrm{TOV}}$ and $\displaystyle M_{\mathrm{thres}}$ scale
with the bag constant in the same way (see Eq. (4) in (Zhou  _et al._ ,
2022)). The quantities used in this approach ($\displaystyle
M_{\mathrm{stab}}$ and $\displaystyle J$ for rotating QSs) can also scale with
the bag constant (Bhattacharyya  _et al._ , 2016). Therefore, the consistency
between the results obtained from this approach and NR simulations is not
coincidental and the validation of this approach is independent of the choice
of the bag constant. Consequently, this semi-analytical method allows the
study of a large quantity of different QS EOSs.

Figure 3: $\displaystyle M_{\mathrm{thres}}$ for different QS EOSs obtained by
different methods. The blue squares are $\displaystyle M_{\mathrm{thres}}$
calculated by the semi-analytical method with the $\displaystyle j$-constant
law with $\displaystyle\hat{A}=2.0$. The red circles represent $\displaystyle
M_{\mathrm{thres}}$ calculated by the empirical relation Eq. (9). The red
error bars indicate the maximum deviation between Eq. (9) and the results of
NR simulations of the BNS merger. The black triangle is $\displaystyle
M_{\mathrm{thres}}$ calculated by the NR simulations and the black error bar
indicates the uncertainty range of $\displaystyle M_{\mathrm{thres}}$ obtained
by NR simulations due to the finite number of simulations. The numbers
attached in each data denote the EOSs listed in Table 1. The black horizontal
dashed line represents the total binary ADM mass of GW170817 with the error
bar indicated by the grey-dashed area.

Except for EOS-3, there have been no current NR simulations to investigate
$\displaystyle M_{\mathrm{thres}}$ for other QS EOSs. Here we calculate
$\displaystyle M_{\mathrm{thres}}$ for QS EOSs listed in Table 1 through the
semi-analytical method and derive the EOS-insensitive relation between
$\displaystyle M_{\mathrm{thres}}$ and stellar properties.

Similar to $\displaystyle M_{\mathrm{TOV}}$ and $\displaystyle
M_{\mathrm{thres}}$, the radius of the non-rotating maximum mass QS
($\displaystyle R_{\mathrm{TOV}}$) can also scale with the bag constant.
Therefore, $\displaystyle M_{\mathrm{thres}}$ can be fitted with
$\displaystyle M_{\mathrm{TOV}}$ and $\displaystyle R_{\mathrm{TOV}}$, and a
bag constant independent relation can be obtained. The empirical relation
among these quantities for NSs has been derived in (Bauswein  _et al._ , 2021)
(No.18 fit) as:

$M_{\mathrm{thres}}=0.497M_{\mathrm{TOV}}+0.1789R_{\mathrm{TOV}}-0.004,$ (9)

where $\displaystyle M_{\mathrm{thres}}$ and $\displaystyle M_{\mathrm{TOV}}$
are in units of $\displaystyle M_{\odot}$ and $\displaystyle R_{\mathrm{TOV}}$
is in units of km. The maximum deviation between this fit and $\displaystyle
M_{\mathrm{thres}}$ obtained by NR simulations is $\displaystyle
0.078\,M_{\odot}$. Employing QS EOSs listed in Table 1, we calculate
$\displaystyle M_{\mathrm{thres}}$ for QSs through the empirical relation Eq.
(9) and the semi-analytical method, and the results are demonstrated in Figure
3.

As shown in Figure 3, it can be seen that $\displaystyle M_{\mathrm{thres}}$
obtained through the empirical relation Eq. (9) and the semi-analytical method
agree well. Although the last term in Eq. (9) is a constant that can not scale
with the bag constant, it is so small that it can be neglected. Therefore, the
empirical relation Eq. (9) is also valid for the BQS merger and can be used to
constrain QS EOSs. With the assumption of no prompt collapse in GW170817
(Bauswein  _et al._ , 2017; Margalit and Metzger, 2017; Kasen  _et al._ ,
2017), the constraint on the parameter space of QS EOSs can be established and
the results are plotted in Figure 9. In addition, it can also be seen in
Figure 3 that $\displaystyle M_{\mathrm{thres}}$ of the selected QS EOSs are
larger than the total binary ADM mass of GW170817, and hence prompt collapse
is unlikely to take place for these EOSs. The specific collapse behavior of
the merger remnant with these EOSs will be investigated in Sec. IV.

##  III DELAYED COLLAPSE

The merger of BNS with $\displaystyle M<M_{\mathrm{thres}}$ leads to the
formation of a massive NS. If the mass of the remnant NS is greater than the
maximum stable mass of uniformly rotating NSs, it has to be supported by
differential rotation and is called a hypermassive NS (HMNS) (Baumgarte  _et
al._ , 2000; Kastaun and Galeazzi, 2015; Hanauske  _et al._ , 2017). For the
case where the mass of the remnant NS is in the range of $\displaystyle
M_{\mathrm{TOV}}$ to $\displaystyle M_{\mathrm{max,urot}}$, the remnant NS is
called a supramssive NS (SMNS) (Cook  _et al._ , 1994; Lasota  _et al._ ,
1996). Both the HMNS and the SMNS are transiently stable NS and eventually
undergo delayed collapse to BH due to the angular momentum redistribution and
dissipation. The collapse of the HMNS remnant could be triggered by the
angular momentum loss due to gravitational waves or through the angular
momentum transport process in a timescale of $\displaystyle\sim 100\,$ms
Kiuchi  _et al._ (2018); Fujibayashi  _et al._ (2018, 2020), while the SMNS
remnant can live in a spin-down timescale which is much longer than the HMNS
case.

For the BQS case, NR simulations in (Zhou  _et al._ , 2022) have indicated
that the gap between $\displaystyle M_{\mathrm{thres}}$ and $\displaystyle
M_{\mathrm{max,urot}}$ is much smaller than the BNS case, which may lead to
differences in the scenario of the post-merger product. Specifically, a
supramassive remnant of the BQS merger could collapse to a BH within several
times the dynamic timescale if the angular momentum contained in the remnant
QS is not large enough to reach the mass-shedding limit (Zhou  _et al._ ,
2022). Thus, analyzing the angular momentum dissipation process is necessary
for understanding the evolution of the merger remnant for the BQS merger. In
this section, we consider the angular momentum dissipation process in the
post-merger stage and employ both energy and angular momentum conservation
laws to investigate the post-merger outcome for the BQS case.

###  III.1 Equilibrium models of rotating QSs

In the delayed collapse case, an at least transiently stable QS is formed
after the merger. We determine the stability and the stellar properties of the
remnant rotating QS by constructing the equilibrium models. However, the
rotation profile of the remnant QS is uncertain, because the remnant QS
changes from a differential rotation state to a rigid rotation state due to
the angular momentum redistribution process (Shibata  _et al._ , 2017b;
Fujibayashi  _et al._ , 2018). For the merger remnant of GW170817, at the
onset of collapse to a BH, whether a rigid rotation state was reached or not
is still unclear. Nevertheless, our constraints of $\displaystyle
M_{\mathrm{TOV}}$ just based on the exclusion of the very long-lived
($\displaystyle\sim 100\,\mathrm{s}$)/stable remnant that injects a large
amount of its kinetic energy into the outflows from the observation results of
the GW170817 event (Margalit and Metzger, 2017; Shibata  _et al._ , 2017a,
2019), and such remnant would achieve a rigid rotation state
($\displaystyle\sim 100\,$ms (Kiuchi  _et al._ , 2018; Fujibayashi  _et al._ ,
2018, 2020)) before collapse to a BH. Therefore, to exclude QS EOSs with which
the merger remnant is very long-lived or stable, it is enough to just focus on
the uniform rotating case. In the rest of this paper, we employ the uniform
rotation profile to construct the axisymmetric equilibrium models for the
rotating QSs through the RNS code (Stergioulas and Friedman, 1995) and derive
the relation between $\displaystyle M_{\mathrm{stab}}$ and $\displaystyle J$
by the turning point criterion.

Figure 4 displays the relation between the scaled maximum stable mass
$\displaystyle\left(M_{\mathrm{stab}}/M_{\mathrm{TOV}}\right)$ and the scaled
angular momentum $\displaystyle\left(c\,J/G\,M_{\mathrm{TOV}}^{2}\right)$ for
uniformly rotating QSs and NSs. The mass and angular momentum for QSs can be
rescaled by $\displaystyle B_{\mathrm{eff}}$, and hence the relation is solely
dependent on $\displaystyle\mathrm{sgn}(\lambda)\,\bar{\lambda}$. Figure 4
demonstrates that as $\displaystyle\mathrm{sgn}(\lambda)\,\bar{\lambda}$
increases, the scaled maximum stable mass decreases, given a scaled angular
momentum. This is because the QS with larger
$\displaystyle\mathrm{sgn}(\lambda)\,\bar{\lambda}$ is more compact (see
Figure 1 in (Zhang and Mann, 2021)), and thus the gravitational field inside
the star is stronger. Therefore, stronger centrifugal force is needed to
sustain the additional self-gravity of the supramassive QS. In addition, it
can be seen in Figure 4 that the $\displaystyle M_{\mathrm{stab}}-J$ relation
is similar between NSs and QSs. Therefore, the value of $\displaystyle
M_{\mathrm{max,urot}}/M_{\mathrm{TOV}}$ is larger for QSs mainly because at
the mass-shedding limit more angular momentum is contained in QSs, but for a
given $\displaystyle J$, the difference in $\displaystyle M_{\mathrm{stab}}$
is not significant (within $\displaystyle 2\%$).

Figure 4: The relation between the scaled maximum stable ADM mass and the
scaled angular momentum of uniformly rotating QSs and NSs. Solid lines of
different colors correspond to various values of
$\displaystyle\mathrm{sgn}(\lambda)\,\bar{\lambda}$, an EOS parameter of QSs.
The black dashed line represents the universal relation of NSs in Ref.
(Bozzola  _et al._ , 2018), where all the EOSs used in the study are within a
$\displaystyle 1.6\%$ error bar. Figure 5: The relation between $\displaystyle
f_{0}/f_{\mathrm{MS}}$ and $\displaystyle J$ of uniformly rotating QSs, where
$\displaystyle f_{0}$ is the ratio of the baryonic mass to the gravitational
mass for binaries of masses $\displaystyle 1.37\,M_{\odot}$ and $\displaystyle
1.37\,M_{\odot}$, and $\displaystyle f_{\mathrm{MS}}$ represents the ratio of
the baryonic mass to the gravitational mass at the turning points (a
marginally stable state) for constant angular momentum sequences for uniformly
rotating QSs. Different EOSs are shown with different colors. The grey-shaded
region is the same relation for NSs with various piecewise polytrope EOSs
(Read  _et al._ , 2009) by which $\displaystyle
2.0\,M_{\odot}<M_{\mathrm{TOV}}<2.4\,M_{\odot}$ and
$\displaystyle\Lambda_{1.35}<1000$.  Figure 6: The relation between
$\displaystyle f_{0}/f_{\mathrm{MS}}$ and $\displaystyle M_{\mathrm{TOV}}$ for
non-rotating QSs, where $\displaystyle f_{0}$ is the ratio of the baryonic
mass to the gravitational mass for binaries of masses $\displaystyle
1.37\,M_{\odot}$ and $\displaystyle 1.37\,M_{\odot}$, and $\displaystyle
f_{\mathrm{MS}}$ represents the ratio of the baryonic mass to the
gravitational mass for marginally stable stars. Solid lines correspond to the
relation between $\displaystyle f_{0}/f_{\mathrm{MS}}$ and $\displaystyle
M_{\mathrm{TOV}}$ for EOSs with the same
$\displaystyle\mathrm{sgn}(\lambda)\,\bar{\lambda}$ but different
$\displaystyle B_{\mathrm{eff}}$. Different colors correspond to various
values of $\displaystyle\mathrm{sgn}(\lambda)\,\bar{\lambda}$. The grey-shaded
region is the same relation for NSs with various piecewise polytrope EOSs
(Read  _et al._ , 2009) by which $\displaystyle
2.0\,M_{\odot}<M_{\mathrm{TOV}}<2.4\,M_{\odot}$ and
$\displaystyle\Lambda_{1.35}<1000$.

In the delayed collapse case, a QS surrounded by a torus is formed after the
merger, and the mass ejection processes would occur. The baryonic masses
($\displaystyle M_{\ast}$) of the remnant QS, the torus and the ejecta can be
related through the baryon number conservation equation. From the time of the
merger to the onset of the remnant QS collapse to a BH, the baryon number
conservation equation can be expressed as:

$M_{\ast f}=M_{\ast}-M_{\ast\mathrm{out}}-M_{\ast\mathrm{eje}},$ (10)

where $\displaystyle M_{\ast f}$ represents the baryonic mass of the remnant
QS at the onset of collapse to a BH, $\displaystyle M_{\ast}$ denotes the
baryonic mass of the binary system at its formation, $\displaystyle
M_{\ast\mathrm{out}}$ is the baryonic mass of the torus surrounding the
remnant BH at its formation, $\displaystyle M_{\ast\mathrm{eje}}$ corresponds
to the baryonic mass of the ejecta at the time of BH formation. Here we
conventionally set the baryonic mass per baryon number for the torus, the
ejecta and the QS equal to the atomic mass unit ($\displaystyle
m_{u}=931.494\,\mathrm{MeV}/c^{2}$) and the baryon number conservation can be
written as the baryonic mass conservation.

To relate the baryonic mass to the gravitational mass, we define
$\displaystyle f_{0}:=M_{\ast}/M$ and $\displaystyle f_{\mathrm{MS}}:=M_{\ast
f}/M_{f}$. According to (Friedman  _et al._ , 1988), a secular instability
occurs at the turning point (a marginally stable state) of the $\displaystyle
M-\rho_{\mathrm{c}}$ curve with constant $\displaystyle J$. Hence,
$\displaystyle f_{\mathrm{MS}}$ can be obtained by calculating the ratio of
the baryonic mass to the gravitational mass at the turning point, and the
baryon number conservation equation can be further expressed as:

$\begin{split}M_{\ast\mathrm{out}}+M_{\ast\mathrm{eje}}\approx&\,0.11\,M_{\odot}\left(\frac{f_{\mathrm{MS}}}{1.2}\right)\\\
&\times\left[\left(\frac{f_{0}/f_{\mathrm{MS}}}{0.925}\right)\left(\frac{M}{2.8\,M_{\odot}}\right)-\left(\frac{M_{f}}{2.5\,M_{\odot}}\right)\right].\end{split}$
(11)

From Eq. (11), it can be found that $\displaystyle f_{0}/f_{\mathrm{MS}}$
deeply affects the sum of the masses of the torus and the ejecta. One percent
increase of $\displaystyle f_{0}/f_{\mathrm{MS}}$ results in
$\displaystyle\sim 0.02\,M_{\odot}$ increase in the sum of the masses of the
torus and the ejecta, for the fixed $\displaystyle M_{f}$ and $\displaystyle
M$.

Figure 5 shows the relation between $\displaystyle f_{0}/f_{\mathrm{MS}}$ and
$\displaystyle J$ for QSs and NSs, where $\displaystyle f_{0}$ is the ratio of
the baryonic mass to the gravitational mass for binaries of masses
$\displaystyle 1.37\,M_{\odot}$ and $\displaystyle 1.37\,M_{\odot}$ inferred
from GW170817 with the symmetric binary assumption 111It is known that for QSs
described by IQM models, their masses scale with $\displaystyle
1/\sqrt{B_{\mathrm{eff}}}$. In other words, for different $\displaystyle
B_{\mathrm{eff}}$, $\displaystyle f=M_{*}/M$ remains unchanged if the masses
of the model are multiplied by $\displaystyle\sqrt{B_{\mathrm{eff}}}$.
Therefore, $\displaystyle f_{\mathrm{MS}}$ is independent of $\displaystyle
B_{\mathrm{eff}}$ as we are considering the TOV maximum mass solution, and the
$\displaystyle M_{\mathrm{TOV}}$ of QSs indeed scales as $\displaystyle
1/\sqrt{B_{\mathrm{eff}}}$. However, $\displaystyle f_{0}$ is dependent on
$\displaystyle B_{\mathrm{eff}}$, since we are considering models with a fixed
mass of $\displaystyle 1.37\,M_{\odot}$, instead of a scaled mass.. It can be
seen that this relation is similar for NSs and QSs, and thus the relation
among $\displaystyle M_{\ast\mathrm{out}}+M_{\ast\mathrm{eje}}$,
$\displaystyle M$ and $\displaystyle M_{f}$ is also similar for NSs and QSs.
Moreover, we fix $\displaystyle\mathrm{sgn}(\lambda)\,\bar{\lambda}$ and
change $\displaystyle B_{\mathrm{eff}}$ to derive the relation between
$\displaystyle f_{0}/f_{\mathrm{MS}}$ and $\displaystyle M_{\mathrm{TOV}}$ for
non-rotating QSs, for which $\displaystyle f_{\mathrm{MS}}=f_{\mathrm{TOV}}$
because the turning point for the non-rotating case corresponds to the maximum
mass spherical configuration. As shown in Figure 6, the increases of
$\displaystyle M_{\mathrm{TOV}}$ and
$\displaystyle\mathrm{sgn}(\lambda)\,\bar{\lambda}$ lead to the decrease of
$\displaystyle f_{0}/f_{\mathrm{MS}}$. Therefore, for the fixed $\displaystyle
M_{\mathrm{TOV}}$, $\displaystyle M$ and $\displaystyle M_{f}$, the value of
$\displaystyle M_{\ast\mathrm{out}}+M_{\ast\mathrm{eje}}$ for QS EOSs with
larger $\displaystyle\mathrm{sgn}(\lambda)\,\bar{\lambda}$ would be smaller.
This trend matches the expectation from a simple qualitative picture: the one-
to-one positive correlation between $\displaystyle\bar{M}_{\text{TOV}}$
(defined as $\displaystyle M_{\text{TOV}}\sqrt{4B_{\text{eff}}}$) and
$\displaystyle\bar{\lambda}$ (Zhang and Mann, 2021) indicates that a larger
$\displaystyle\text{sgn}(\lambda)\bar{\lambda}$ for a given $\displaystyle
M_{\text{TOV}}$ maps to a larger bag constant, which characterizes the inward
pressure from the QCD vacuum differences that render the QSs more bounded and
thus smaller $\displaystyle M_{\ast\text{out}}+M_{\ast\text{eje}}$.

###  III.2 Dissipation mechanisms and conservation equations

After the merger, the remnant NS is highly deformed and has a non-axisymmetric
configuration, and hence quasi-periodic gravitational waves are emitted from
the remnant NS (Bauswein and Janka, 2012; Bauswein  _et al._ , 2012; Takami
_et al._ , 2014; Shibata and Taniguchi, 2006; Kiuchi  _et al._ , 2009;
Sekiguchi  _et al._ , 2011a, b; Bernuzzi  _et al._ , 2015b; Oechslin and
Janka, 2007; Stergioulas  _et al._ , 2011; Hotokezaka  _et al._ , 2013; Takami
_et al._ , 2015; Dietrich  _et al._ , 2017). Copious neutrinos are also
emitted from the remnant NS because, after the merger, strong shocks are
generated and heat the remnant NS (Eichler  _et al._ , 1989; Ruffert  _et al._
, 1997; Ruffert and Janka, 2001; Rosswog and Liebendörfer, 2003; Sekiguchi
_et al._ , 2015; Palenzuela  _et al._ , 2015; Foucart  _et al._ , 2016a, b; Wu
_et al._ , 2017; George  _et al._ , 2020; Kullmann  _et al._ , 2022; Cusinato
_et al._ , 2021; Radice  _et al._ , 2022; Espino  _et al._ , 2024; Fujibayashi
_et al._ , 2018, 2020). Moreover, a fraction of the matter obtains sufficient
specific angular momentum to result in a torus or ejecta through the angular
momentum transport process (Hotokezaka  _et al._ , 2013a, b; Fujibayashi  _et
al._ , 2018, 2020; Kiuchi  _et al._ , 2023). Although electromagnetic
radiation like magnetic dipole radiation also dissipates the energy and the
angular momentum of the remnant NS, it only has a minor effect unless the
remnant is very long-lived ($\displaystyle\sim 100\,\mathrm{s}$). Such a very
long-lived remnant would have injected a large fraction of its total
rotational kinetic energy ($\displaystyle\sim 10^{53}\,$erg) into the ejecta,
which is disfavored by the observational results of GW170817 (Margalit and
Metzger, 2017; Shibata  _et al._ , 2019), and thus the dissipation due to
electromagnetic radiation is not taken into account in this paper. Similar to
the BNS merger, these dissipation processes also take place in the BQS
scenario (Bauswein  _et al._ , 2009, 2010; Zhou  _et al._ , 2021; Zhou and et
al., ). Therefore, the energy and the angular momentum conservation equations
for the BQS merger from the moment of the merger to the onset of the collapse
can be expressed as:

$\displaystyle\displaystyle
M_{f}=M_{\mathrm{merger}}-E_{\mathrm{GW,p}}-E_{\nu}-M_{\mathrm{out}}-M_{\mathrm{eje}},$
(12) $\displaystyle\displaystyle
J_{f}=J_{\mathrm{merger}}-J_{\mathrm{GW,p}}-J_{\nu}-J_{\mathrm{out}}-J_{\mathrm{eje}},$
(13)

where $\displaystyle M_{f}$ and $\displaystyle J_{f}$ are the gravitational
mass and angular momentum of the remnant QS at the onset of the collapse to a
BH, respectively; $\displaystyle M_{\mathrm{merger}}$ and $\displaystyle
J_{\mathrm{merger}}$ denote the gravitational mass and angular momentum at the
time of the merger; $\displaystyle E_{\mathrm{GW,p}}$ and $\displaystyle
J_{\mathrm{GW,p}}$ represent the energy and angular momentum carried away by
gravitational waves in the post-merger phase; $\displaystyle E_{\mathrm{\nu}}$
and $\displaystyle J_{\mathrm{\nu}}$ are the energy and angular momentum
dissipated through neutrino emission; $\displaystyle M_{\mathrm{out}}$ and
$\displaystyle J_{{\mathrm{out}}}$ correspond to the gravitational mass and
angular momentum of the torus surrounding the remnant BH at its formation;
$\displaystyle M_{\mathrm{eje}}$ and $\displaystyle J_{{\mathrm{eje}}}$ are
the gravitational mass and angular momentum of the ejecta at the time of BH
formation. In this paper, it is assumed that the internal, kinetic, and
gravitational binding energy of the matter outside the merger remnant is
significantly smaller compared to its baryonic mass, and thus we do not
distinguish the baryonic mass and the gravitational mass of the matter outside
the merger remnant (i.e., $\displaystyle
M_{\ast\mathrm{out}}=M_{\mathrm{out}}$ and $\displaystyle
M_{\ast{\mathrm{eje}}}=M_{\mathrm{eje}}$).

To solve the baryon number, the energy and the angular momentum conservation
equations in the post-merger phase, we first need to calculate the baryonic
mass, the gravitational mass and the angular momentum of the remnant QS at the
time of the merger which serve as the initial properties of the remnant QS.
The baryon number is conserved in the inspiral phase, and hence we can
calculate the baryonic mass of the binary at the time of merger by solving the
TOV equation, given the mass components and the EOS. $\displaystyle
J_{\mathrm{merger}}$ can be accurately computed by the EOB method as shown in
Sec. II. Similar to $\displaystyle J_{\mathrm{merger}}$, $\displaystyle
M_{\mathrm{merger}}$ can also be obtained via the EOB method. The energy
carried away by gravitational waves during the inspiral phase ($\displaystyle
E_{\mathrm{GW,i}}$) can be calculated via an integration of the gravitational
waveform generated by the EOB method:

$E_{\mathrm{GW,i}}=\frac{1}{16\pi}\sum\limits_{(\ell,m)}\int_{t_{0}}^{t_{\mathrm{merger}}}dt^{\prime}r^{2}\dot{h}_{\ell
m}\left(t^{\prime}\right)\dot{h}_{\ell m}^{*}\left(t^{\prime}\right).$ (14)

Therefore, $\displaystyle M_{\mathrm{merger}}$ can be expressed as:

$M_{\mathrm{merger}}=M_{0}-E_{\mathrm{GW,i}},$ (15)

where $\displaystyle M_{0}$ is the energy of the binary at the beginning of
the calculation of gravitational wave, and can be derived through Eqs. (2.4)
and (4.3) in (Buonanno and Damour, 2000).

At the onset of collapse, the configuration of the remnant QS can be estimated
by the equilibrium configuration at the turning point of the $\displaystyle
M-\rho_{\mathrm{c}}$ curve with constant $\displaystyle J$, where the secular
instability occurs (Friedman  _et al._ , 1988). Therefore, the baryonic mass,
the gravitational mass and the angular momentum of the remnant QS at the onset
of collapse, which serve as the final properties of the remnant QS, can be
calculated as long as one of these three quantities is given.

As discussed above, given $\displaystyle M_{f}$ (or $\displaystyle J_{f}$ and
$\displaystyle M_{\ast f}$) we can obtain the dissipated baryonic mass,
gravitational mass and angular momentum. Furthermore, there are three main
dissipation mechanisms, the gravitational radiation, the neutrino emission and
the hydrodynamical angular momentum transport process, which leads to the
formation of the torus and the ejecta, and obviously, they dissipate the
baryonic mass, the energy and the angular momentum in different proportions.
Therefore, in principle, we can determine the amount of dissipation of the
three dissipation mechanisms via the three conservation equations, which is
equivalent to solving a linear system with three unknowns. However, before
solving this linear system, we should know the proportion of the dissipated
baryonic mass, energy and angular momentum for each mechanism, which serves as
the coefficient of each unknown and has been obtained through NR simulations
as summarized in (Shibata  _et al._ , 2019). Here we briefly introduce them.

For gravitational waves in the post-merger phase, the peak frequency,
$\displaystyle f_{\mathrm{peak}}$, in the spectrum amplitude corresponds to
the frequency of the $\displaystyle f$–mode oscillation of the remnant massive
NS (Bauswein and Janka, 2012; Hotokezaka  _et al._ , 2013a; Rezzolla and
Takami, 2016), and it is reported to correlate with the tidal deformability
(Rezzolla and Takami, 2016; Kiuchi  _et al._ , 2020), and the approximate
expression for $\displaystyle J_{\mathrm{GW,p}}$ can be expressed as:

$\begin{split}J_{\mathrm{GW,p}}&\approx\frac{E_{\mathrm{GW,p}}}{\pi
f_{\mathrm{peak}}}\\\ &\approx 9.5\times
10^{48}\,\mathrm{erg\,s}\left(\frac{E_{\mathrm{GW,p}}}{0.05\,M_{\odot}}\right)\left(\frac{f_{\mathrm{peak}}}{3.0\,\mathrm{kHz}}\right)^{-1}.\end{split}$
(16)

This relation has been confirmed by NR simulations (Kiuchi  _et al._ , 2020),
with a relative error $\displaystyle\lesssim 1\%$. The value of $\displaystyle
f_{\mathrm{peak}}$ can be determined using the relation provided in Eq. (5.4)
of Ref. (Kiuchi  _et al._ , 2020). Furthermore, this relation is consistent
with the BQS merger simulations in (Zhou and et al., ), and hence we employ
this relation for the BQS scenario.

For neutrino emission, as the emitter (remnant QS) undergoes rotation, the
neutrinos carry away the angular momentum of the remnant QS. Therefore,
$\displaystyle J_{\nu}$ can be approximately written as $\displaystyle
J_{\nu}\approx(2/3)R_{\mathrm{MQS}}^{2}\Omega E_{\nu}$ (Baumgarte and Shapiro,
1998), and correspondingly

$\begin{split}J_{\nu}\approx&3.0\times
10^{48}\,\mathrm{erg\,s}\left(\frac{E_{\nu}}{0.1\,M_{\odot}c^{2}}\right)\left(\frac{R_{\mathrm{MQS}}}{15\,\mathrm{km}}\right)^{2}\\\
&\times\left(\frac{\Omega}{10^{4}\,\mathrm{rad/s}}\right),\end{split}$ (17)

where $\displaystyle R_{\mathrm{MQS}}$ and $\displaystyle\Omega$ represent the
equatorial circumferential radius and angular velocity of the remnant massive
QS. This relation is consistent with the NR simulation results within a factor
of 2 (Fujibayashi  _et al._ , 2020). In addition, comparing Eq. (17) with Eq.
(16), we find that neutrino emission serves as a secondary mechanism for the
dissipation of angular momentum, and thus the uncertainty in neutrino emission
only has a minor impact on the results.

The velocity of the torus surrounding the remnant QS and the ejecta should be
of the order of their escape velocity, and thus the angular momentum carried
away by them can be approximately expressed as $\displaystyle
J_{\mathrm{out}}\approx
M_{\mathrm{out}}\sqrt{GM_{\mathrm{MQS}}R_{\mathrm{out}}}$ and $\displaystyle
J_{\mathrm{eje}}\approx
M_{\mathrm{eje}}\sqrt{GM_{\mathrm{MQS}}R_{\mathrm{eje}}}$, i.e.,

$\begin{split}J_{\mathrm{out}}\approx&5.8\times
10^{48}\,\mathrm{erg\,s}\left(\frac{M_{\mathrm{out}}}{0.05\,M_{\odot}}\right)\left(\frac{R_{\mathrm{out}}}{100\,\mathrm{km}}\right)^{1/2}\\\
&\times\left(\frac{M_{\mathrm{MQS}}}{2.6\,M_{\mathrm{\odot}}}\right)^{1/2},\end{split}$
(18)

$\begin{split}J_{\mathrm{eje}}\approx&6.9\times
10^{48}\,\mathrm{erg\,s}\left(\frac{M_{\mathrm{eje}}}{0.05\,M_{\odot}}\right)\left(\frac{R_{\mathrm{eje}}}{140\,\mathrm{km}}\right)^{1/2}\\\
&\times\left(\frac{M_{\mathrm{MQS}}}{2.6\,M_{\mathrm{\odot}}}\right)^{1/2},\end{split}$
(19)

where $\displaystyle{M_{\mathrm{MQS}}}$ represents the gravitational mass of
the remnant massive QS, $\displaystyle R_{\mathrm{out}}$ is the typical radius
of the torus surrounding the remnant QS, and $\displaystyle R_{\mathrm{eje}}$
is the typical location where ejection occurs. Both gravitational mass and
equatorial circumferential radius for the remnant QS at the onset of collapse
can be estimated by constructing the equilibrium models with the turning point
criterion for the given angular momentum or total baryonic mass. The NR
simulations for the BNS merger have shown that $\displaystyle
R_{\mathrm{out}}$ would experience an increase from $\displaystyle\sim
50\,\mathrm{km}$ to $\displaystyle\sim 200\,\mathrm{km}$ due to the long-term
viscous evolution of the torus, as well as the transport of angular momentum
from the remnant NS (Fujibayashi  _et al._ , 2018; Kiuchi  _et al._ , 2023).
It has been suggested that the dynamics of the BQS merger and the BNS merger
are similar (Zhou and et al., ). Therefore, the evolution of the torus may not
make much difference and we use the same relation as in (Shibata  _et al._ ,
2019), where it is simply set that $\displaystyle
R_{\mathrm{out}}=R_{\mathrm{eje}}=140\,\mathrm{km}$. In addition,
$\displaystyle J_{\mathrm{out}}$ mainly depends on $\displaystyle
M_{\mathrm{out}}$ instead of $\displaystyle R_{\mathrm{out}}$ ($\displaystyle
J_{\mathrm{out}}\propto M_{\mathrm{out}}$, while $\displaystyle
J_{\mathrm{out}}\propto\sqrt{R_{\mathrm{out}}}$), and thus the value of
$\displaystyle R_{\mathrm{out}}$ and $\displaystyle R_{\mathrm{eje}}$ do not
make a great effect on our analysis.

##  IV APPLICATION TO GW170817 EVENT

As mentioned in Sec. I, for the GW170817 event, the prompt collapse is not
likely to take place due to the very presence of an electromagnetic
counterpart (Margalit and Metzger, 2017; Kasen  _et al._ , 2017; Bauswein  _et
al._ , 2017)222Although it is suggested in (Kiuchi  _et al._ , 2019) that the
luminosity of AT 2017gfo could be explained by models that undergo prompt
collapse, blue kilonova component of AT 2017gfo also disfavors the case of
prompt collapse as discussed in these studies.. Therefore, the constraint on
the threshold mass for prompt collapse can be made $\displaystyle
M_{\mathrm{thres}}\gtrsim 2.74\,M_{\odot}$. On the other hand, a very long-
lived ($\displaystyle\sim 100\,\mathrm{s}$)/stable remnant that injects a
large number of its kinetic energy into the outflows is also disfavored
because, with such strong energy injection, the ultraviolet-optical-infrared
counterparts for GW170817 would be much brighter than the observational
results (Margalit and Metzger, 2017; Shibata  _et al._ , 2017a, 2019)333A
long-lived NS as merger remnant for the GW170817 event can not be completely
ruled out, and this scenario has been investigated in (Li  _et al._ , 2018; Ai
_et al._ , 2018; Piro  _et al._ , 2019; DuPont and MacFadyen, 2024). However,
in this paper, our constraint on QS EOSs is only based on the predominant
scenario proposed in (Margalit and Metzger, 2017; Shibata  _et al._ , 2017a,
2019).. Therefore, our analysis in this Sec. III can be directly applied to
the GW170817 event to understand the post-merger evolution of this event and
set the constraint on QS EOSs.

Model | $\displaystyle M_{\mathrm{merger}}$ | $\displaystyle J_{\mathrm{merger}}$ |  $\displaystyle M_{f}/M_{\mathrm{TOV}}$ |  $\displaystyle M_{f}$ |  $\displaystyle J_{f}$ |  $\displaystyle E_{\mathrm{GW,p}}$ |  $\displaystyle E_{\mathrm{GW}}$ | $\displaystyle E_{\nu}$ | $\displaystyle M_{\mathrm{out}}+M_{\mathrm{eje}}$
---|---|---|---|---|---|---|---|---|---
EOS-1 | 2.695 | 6.192 | 1.269 | 2.550 | 4.812 | 0.022 | 0.067 | 0.073 | 0.05
… | … | … | … | … | … | 0.10
… | … | … | … | … | … | 0.15
… | … | … | … | … | … | 0.20
EOS-2 | 2.700 | 6.438 | 1.226 | 2.550 | 4.434 | 0.046 | 0.086 | 0.054 | 0.05
1.206 | 2.508 | 4.139 | 0.025 | 0.065 | 0.067 | 0.10
1.186 | 2.466 | 3.844 | 0.006 | 0.046 | 0.078 | 0.15
1.186 | 2.466 | 3.844 | 0.006 | 0.046 | 0.078 | 0.15
… | … | … | … | … | … | 0.20
EOS-3 | 2.700 | 6.424 | 1.213 | 2.547 | 4.335 | 0.050 | 0.091 | 0.052 | 0.05
1.193 | 2.505 | 4.037 | 0.030 | 0.070 | 0.065 | 0.10
1.173 | 2.463 | 3.737 | 0.010 | 0.051 | 0.076 | 0.15
… | … | … | … | … | … | 0.20
EOS-4 | 2.701 | 6.488 | 1.190 | 2.542 | 4.117 | 0.059 | 0.099 | 0.049 | 0.05
1.170 | 2.501 | 3.816 | 0.041 | 0.080 | 0.059 | 0.10
1.151 | 2.459 | 3.512 | 0.022 | 0.062 | 0.069 | 0.15
1.131 | 2.418 | 3.204 | 0.005 | 0.045 | 0.078 | 0.20
EOS-5 | 2.697 | 6.274 | 1.143 | 2.516 | 3.746 | 0.080 | 0.123 | 0.052 | 0.05
1.126 | 2.477 | 3.445 | 0.057 | 0.101 | 0.062 | 0.10
1.108 | 2.439 | 3.137 | 0.036 | 0.080 | 0.072 | 0.15
1.091 | 2.400 | 2.817 | 0.016 | 0.060 | 0.080 | 0.20
EOS-6 | 2.697 | 6.311 | 1.112 | 2.505 | 3.356 | 0.098 | 0.141 | 0.044 | 0.05
1.094 | 2.464 | 3.015 | 0.078 | 0.121 | 0.055 | 0.10
1.076 | 2.424 | 2.657 | 0.060 | 0.103 | 0.063 | 0.15
1.058 | 2.383 | 2.274 | 0.044 | 0.087 | 0.070 | 0.20
EOS-7 | 2.701 | 6.478 | 1.102 | 2.516 | 3.164 | 0.100 | 0.140 | 0.034 | 0.05
1.085 | 2.477 | 2.832 | 0.083 | 0.123 | 0.040 | 0.10
1.067 | 2.438 | 2.479 | 0.067 | 0.107 | 0.045 | 0.15
1.050 | 2.399 | 2.099 | 0.053 | 0.093 | 0.048 | 0.20
EOS-8 | 2.692 | 6.078 | 1.079 | 2.465 | 2.992 | 0.127 | 0.175 | 0.050 | 0.05
1.060 | 2.422 | 2.571 | 0.108 | 0.156 | 0.061 | 0.10
1.042 | 2.379 | 2.094 | 0.094 | 0.142 | 0.069 | 0.15
1.023 | 2.336 | 1.519 | 0.087 | 0.135 | 0.069 | 0.20
EOS-9 | 2.701 | 6.521 | 1.066 | 2.505 | 2.599 | 0.121 | 0.160 | 0.024 | 0.05
1.049 | 2.465 | 2.192 | 0.108 | 0.147 | 0.028 | 0.10
1.032 | 2.425 | 1.731 | 0.097 | 0.136 | 0.029 | 0.15
1.015 | 2.385 | 1.151 | 0.091 | 0.130 | 0.025 | 0.20
EOS-10 | 2.699 | 6.380 | 1.059 | 2.489 | 2.493 | 0.136 | 0.177 | 0.024 | 0.05
1.042 | 2.449 | 2.061 | 0.121 | 0.162 | 0.029 | 0.10
1.024 | 2.409 | 1.552 | 0.112 | 0.153 | 0.028 | 0.15
1.008 | 2.368 | 0.829 | 0.119 | 0.160 | 0.018 | 0.20
Table 3: The properties ($\displaystyle M_{f}/M_{\mathrm{TOV}}$,
$\displaystyle M_{f}$, $\displaystyle J_{f}$, $\displaystyle
E_{\mathrm{GW,p}}$, $\displaystyle E_{\mathrm{GW}}$, $\displaystyle E_{\nu}$,
$\displaystyle M_{\mathrm{out}}+M_{\mathrm{eje}}$) of the potential states of
uniformly rotating QSs at the onset of collapse in the GW170817 event are
shown, considering various EOSs. The units of the mass, energy and angular
momentum are $\displaystyle M_{\odot}$, $\displaystyle M_{\odot}c^{2}$ and
$\displaystyle GM_{\odot}^{2}/c$, respectively. The chosen values of
$\displaystyle M_{\mathrm{out}}+M_{\mathrm{eje}}$ are $\displaystyle
0.05\,M_{\odot}$, $\displaystyle 0.10\,M_{\odot}$, $\displaystyle
0.15\,M_{\odot}$, $\displaystyle 0.20\,M_{\odot}$. The values of
$\displaystyle M_{\mathrm{merger}}$ and $\displaystyle J_{\mathrm{merger}}$
are calculated by EOB method. ‘…’ means only solutions with negative values of
$\displaystyle E_{\mathrm{GW,p}}$ are obtained.

We apply the analysis in Sec. III to the GW170817 event and search for
solutions that satisfy the energy and angular momentum conservation laws for
the selected QS EOSs. We fix the total gravitational mass of binary
$\displaystyle M=2.74\,M_{\odot}$. The mass ratio $\displaystyle M_{1}/M_{2}$
only has little effect on the final results (Shibata  _et al._ , 2019), and
thus we only show the results with $\displaystyle
M_{1}=M_{2}=1.37\,M_{\odot}$. As discussed in III.2, given a series of
$\displaystyle M_{f}$, the values of $\displaystyle E_{\mathrm{GW,p}}$,
$\displaystyle E_{\nu}$ and $\displaystyle M_{\mathrm{out}}+M_{\mathrm{eje}}$
can be calculated, and thus the relation between them can also be obtained. To
determine whether or not the solutions are reasonable, we should check whether
the obtained solution range could satisfy the observation results and the
constraints placed by NR simulations. In this paper, we employ the NR
simulation results in (Zappa  _et al._ , 2018), where the value of the total
energy radiated by gravitational waves throughout the inspiral to the post-
merger phases ($\displaystyle E_{\mathrm{GW}}$) can be constrained as:

$E_{\mathrm{GW}}:=E_{\mathrm{GW,i}}+E_{\mathrm{GW,p}}\lesssim 0.13\pm
0.01\,M_{\odot}c^{2}\left(\frac{M}{2.8\,M_{\odot}}\right).$ (20)

The uncertainty is due to the finite grid resolution and the finite sample
size. Although this relation is obtained through NR simulations for BNS
merger, due to the similarity between the dynamics of the BQS merger and the
BNS merger (Zhou and et al., ), we can employ this relation for the BQS
merger. Only QS EOSs with which self-consistent solutions (i.e., three
conservation equations could be solved with reasonable values of
$\displaystyle E_{\mathrm{GW}}$, $\displaystyle E_{\nu}$ and $\displaystyle
M_{\mathrm{out}}+M_{\mathrm{eje}}$) exist can pass our consistency checks.

Figure 7: The relation between $\displaystyle E_{\mathrm{GW,p}}$,
$\displaystyle E_{\nu}$ and $\displaystyle M_{\mathrm{out}}+M_{\mathrm{eje}}$
for BQS of mass $\displaystyle 1.37\,M_{\odot}$ and $\displaystyle
1.37\,M_{\odot}$ with various QS EOSs. The blue and orange solid lines
represent the energy radiated by gravitational waves and neutrinos,
respectively. The horizontal dashed line represents the upper limits of the
energy radiated by gravitational waves in the post-merger phase given by Eq.
(20) with the error bar indicated by the red shaded area.

###  IV.1 Results with different $\displaystyle M_{\mathrm{TOV}}$

We first study the influence of $\displaystyle M_{\mathrm{TOV}}$ on the
results. EOS-1 – 7 have different $\displaystyle M_{\mathrm{TOV}}$ from
$\displaystyle 2.0\,M_{\odot}$ to $\displaystyle 2.28\,M_{\odot}$, while the
values of $\displaystyle\mathrm{sgn}(\lambda)\bar{\lambda}$ only vary from
$\displaystyle-0.02$ to $\displaystyle 1$. Solutions that consistently satisfy
the conservation laws of energy and angular momentum are shown in Table 3,
where the values of $\displaystyle M_{\mathrm{out}}+M_{\mathrm{eje}}$ are
fixed to $\displaystyle 0.05\,M_{\odot}$, $\displaystyle 0.10\,M_{\odot}$,
$\displaystyle 0.15\,M_{\odot}$ and $\displaystyle 0.20\,M_{\odot}$. The
relation among $\displaystyle E_{\mathrm{GW,p}}$, $\displaystyle E_{\nu}$ and
$\displaystyle M_{\mathrm{out}}+M_{\mathrm{eje}}$ is illustrated in Figure 7.
It can be observed from the Table 3 and the Figure 7 that an increase in
$\displaystyle M_{\mathrm{out}}+M_{\mathrm{eje}}$ leads to a decrease in
$\displaystyle E_{\mathrm{GW,p}}$. This is because the amount of angular
momentum loss to trigger collapse is certain, and hence as $\displaystyle
M_{\mathrm{out}}+M_{\mathrm{eje}}$ increases, more angular momentum is carried
away by the torus and ejecta, and the value of $\displaystyle
E_{\mathrm{GW,p}}$ decreases. In addition, as $\displaystyle M_{\mathrm{TOV}}$
increases, more angular momentum dissipation is needed to trigger the
collapse. Therefore, the value of $\displaystyle E_{\mathrm{GW,p}}$ increases
with the increase of $\displaystyle M_{\mathrm{TOV}}$ for the fixed
$\displaystyle M_{\mathrm{out}}+M_{\mathrm{eje}}$.

It can also be seen in Table 3 and Figure 7 that for EOS-1, 2, 3, 4, 5, for
which the maximum mass ranges from $\displaystyle 2.01$ to $\displaystyle
2.20\,M_{\odot}$, solutions that consistently meet our constraint can be
found, and thus we have no reason to exclude these EOSs. In addition, As shown
in Table 3, the value of $\displaystyle M_{f}/M_{\mathrm{TOV}}$ for these EOSs
is in the range of $\displaystyle 1.1\lesssim M_{f}/M_{\mathrm{TOV}}\lesssim
1.3$ and is smaller than 1.4, which is the typical value of $\displaystyle
M_{\mathrm{max,urot}}/M_{\mathrm{TOV}}$. Therefore, for these EOSs, the
angular momentum left in the remnant QS is not large enough to reach the mass-
shedding limit, and a supramassive remnant of the BQS merger could also
collapse to a BH in a short timescale (before its kinetic energy is dissipated
via electromagnetic radiation) due to the angular momentum dissipation.

As $\displaystyle M_{\mathrm{TOV}}$ increases, the dissipation of angular
momentum required to trigger the collapse increases. For EOS-6, 7, for which
$\displaystyle 2.252\,M_{\odot}\lesssim M_{\mathrm{TOV}}\lesssim
2.284M_{\odot}$, solutions can also be obtained. However, a significant
angular momentum dissipation through gravitational radiation or torus and
ejecta is required. To determine whether this efficient dissipation of angular
momentum is possible or not, NR simulations must be performed and the details
of such NR simulations will be presented in another paper (Zhou and et al., ).

###  IV.2 Results with the same $\displaystyle M_{\mathrm{TOV}}$ and the upper
limit of $\displaystyle M_{\mathrm{TOV}}$

The value of $\displaystyle\mathrm{sgn}(\lambda)\bar{\lambda}$ also influences
the relation among $\displaystyle E_{\mathrm{GW,p}}$, $\displaystyle E_{\nu}$
and $\displaystyle M_{\mathrm{out}}+M_{\mathrm{eje}}$ while keeping
$\displaystyle M_{\mathrm{TOV}}$ constant. For EOS-7 and EOS-8,
($\displaystyle M_{\mathrm{TOV}}\approx 2.284\,M_{\odot}$), the great
difference in $\displaystyle\mathrm{sgn}(\lambda)\bar{\lambda}$ leads to
significantly different results. For EOS-7, with smaller
$\displaystyle\mathrm{sgn}(\lambda)\bar{\lambda}$, we can easily find the
self-consistent solutions, while for EOS-8 solutions can only be marginally
found. The same trend is also found in the comparison of EOS-9 and EOS-10,
($\displaystyle M_{\mathrm{TOV}}\approx 2.35\,M_{\odot}$). However, the
difference in $\displaystyle\mathrm{sgn}(\lambda)\bar{\lambda}$ of EOS-9, 10
is not as significant as EOS-7,8, leading to a minor difference in the
results. It can be seen that for EOS-9, the self-consistent solutions can be
found within the uncertainty range, while for EOS-10 solutions can not be
found because an unfeasibly high level of angular momentum dissipation is
required to trigger the collapse.

To find the upper limit of $\displaystyle M_{\mathrm{TOV}}$ for the whole
parameter space of QS EOSs, we need to calculate the results for EOSs with the
same $\displaystyle M_{\mathrm{TOV}}$ but different
$\displaystyle\mathrm{sgn}(\lambda)\bar{\lambda}$. For a given $\displaystyle
M_{\mathrm{TOV}}$, if self-consistent solutions can not be found for any value
of $\displaystyle\mathrm{sgn}(\lambda)\bar{\lambda}$, and then EOSs with this
$\displaystyle M_{\mathrm{TOV}}$ can be excluded. In addition, EOSs with
larger $\displaystyle M_{\mathrm{TOV}}$ can also be excluded because, as
discussed before, EOS with larger $\displaystyle M_{\mathrm{TOV}}$ is more
difficult to satisfy the constraint given by Eq. (20).

From the results of EOS-7–10, it can be seen that given a $\displaystyle
M_{\mathrm{TOV}}$, EOSs with smaller
$\displaystyle\mathrm{sgn}(\lambda)\bar{\lambda}$ are easier to find results
satisfying the constraint. This is because as shown in Figure 6, for fixed
$\displaystyle M_{\mathrm{TOV}}$ the decrease of
$\displaystyle\mathrm{sgn}(\lambda)\bar{\lambda}$ leads to the increase of
$\displaystyle f_{0}/f_{\mathrm{MS}}$, which leads to the increase of
$\displaystyle M_{\mathrm{out}}+M_{\mathrm{eje}}$ for fixed $\displaystyle
M_{f}$ and $\displaystyle M$. Therefore, for the same energy dissipation, the
proportion of the dissipation through the torus and the ejecta is larger for
EOS-7 and EOS-9, while for EOS-8 and EOS-10 more energy should be dissipated
through gravitational waves and neutrinos, making $\displaystyle
E_{\mathrm{GW}}$ exceed the upper limits given by Eq. (20). Therefore, to
satisfy the constraint given by Eq. (20), an upper limit of
$\displaystyle\mathrm{sgn}(\lambda)\bar{\lambda}$ can be obtained for a given
$\displaystyle M_{\mathrm{TOV}}$.

Figure 8: Deriving the upper limit of $\displaystyle M_{\mathrm{TOV}}$ of QSs
that can pass our consistency checks and satisfy
$\displaystyle\Lambda_{1.4}\lesssim 800$ at the same time. The blue solid line
represents the relation between $\displaystyle M_{\mathrm{TOV}}$ and the
minimum $\displaystyle E_{\mathrm{GW,i}}+E_{\mathrm{GW,p}}$ to trigger the
collapse derived by our semi-analytical method with the assumption that the
total binary ADM mass $\displaystyle M=2.74\,M_{\odot}$, and the radius of the
torus $\displaystyle R_{\mathrm{out}}=140\,$km for various EOSs with
$\displaystyle\Lambda_{1.4}=800$ (We vary the value of $\displaystyle
M_{\mathrm{out}}+M_{\mathrm{eje}}$ from $\displaystyle 0$ to $\displaystyle
0.2\,M_{\odot}$ to find the minimum value of $\displaystyle E_{\mathrm{GW}}$,
which serve as the minimum $\displaystyle E_{\mathrm{GW,i}}+E_{\mathrm{GW,p}}$
to trigger the collapse.). The blue shaded area indicates the error arising
from the uncertainties of $\displaystyle M$ ($\displaystyle
2.74^{+0.04}_{-0.01}\,M_{\odot}$) and $\displaystyle R_{\mathrm{out}}$
($\displaystyle 140^{+60}_{-90}\,$km). The horizontal dashed line shows the
upper limits of $\displaystyle E_{\mathrm{GW}}$ given by Eq. (20) with the
assumption of $\displaystyle M=2.74\,M_{\odot}$. The red shaded area provides
the error resulting from the uncertainties of $\displaystyle M$ and Eq. (20).
The intersection of these two lines determines the upper limit of
$\displaystyle M_{\mathrm{TOV}}$ of QSs that can pass our consistency checks
and satisfy $\displaystyle\Lambda_{1.4}\lesssim 800$ at the same time.

It is suggested in (Zhang and Mann, 2021) that the most tight lower limits of
$\displaystyle\mathrm{sgn}(\lambda)\bar{\lambda}$ are given by the constraint
$\displaystyle\Lambda_{1.4}\lesssim 800$ from GW170817. Therefore, for a given
$\displaystyle M_{\mathrm{TOV}}$, the upper limit and the lower limit of
$\displaystyle\mathrm{sgn}(\lambda)\bar{\lambda}$ can be obtained by Eq. (20)
and $\displaystyle\Lambda_{1.4}\lesssim 800$ respectively, and thus a definite
upper limit of $\displaystyle M_{\mathrm{TOV}}$ can be derived by looking for
the critical $\displaystyle M_{\mathrm{TOV}}$ with which the upper limit of
$\displaystyle\mathrm{sgn}(\lambda)\bar{\lambda}$ given by the scheme
demonstrated here is equal to the lower limit of it given by
$\displaystyle\Lambda_{1.4}\lesssim 800$. It is equivalent to finding an EOS
with which $\displaystyle\Lambda_{1.4}=800$ and Eq. (20) can be marginally
satisfied.

Thus we scan over different $\displaystyle M_{\mathrm{TOV}}$ but all with
$\displaystyle\Lambda_{1.4}=800$ to yield a definite answer. We vary the value
of $\displaystyle M_{\mathrm{out}}+M_{\mathrm{eje}}$ from $\displaystyle 0$ to
$\displaystyle 0.2\,M_{\odot}$ to find the minimum value of $\displaystyle
E_{\mathrm{GW,i}}+E_{\mathrm{GW,p}}$ to trigger the collapse and compare it to
Eq. (20) to determine the upper limit of $\displaystyle M_{\mathrm{TOV}}$. In
addition, the main uncertainties in our analysis are also considered. We vary
the total binary ADM mass and the radius of the torus in their ranges of
uncertainty and calculate the results ($\displaystyle
M=2.74^{+0.04}_{-0.01}\,M_{\odot}$ with the $\displaystyle 90\%$ credible
level (LIGO Scientific Collaboration and Virgo Collaboration, 2017),
$\displaystyle R_{\mathrm{out}}$ varies from $\displaystyle 50\,$km to
$\displaystyle 200\,$km due to the long-term viscous evolution as well as the
angular momentum transport process (Fujibayashi  _et al._ , 2018, 2020)).
Figure 8 shows the main results: the blue solid line represents the relation
between $\displaystyle M_{\mathrm{TOV}}$ and the minimum $\displaystyle
E_{\mathrm{GW,i}}+E_{\mathrm{GW,p}}$ to trigger the collapse with the
assumption that $\displaystyle M=2.74\,M_{\odot}$ and $\displaystyle
R_{\mathrm{out}}=140\,$km, the horizontal red dashed line shows the upper
limits of $\displaystyle E_{\mathrm{GW,i}}+E_{\mathrm{GW,p}}$ given by Eq.
(20) with the assumption that $\displaystyle M=2.74\,M_{\odot}$ and the shaded
areas indicate the uncertainties in $\displaystyle
E_{\mathrm{GW,i}}+E_{\mathrm{GW,p}}$ associate with the uncertainties of
$\displaystyle M$ and $\displaystyle R_{\mathrm{out}}$. The intersection of
these two lines determines the critical $\displaystyle M_{\mathrm{TOV}}$
($\displaystyle\approx 2.35^{+0.07}_{-0.17}\,M_{\odot}$), and EOS with this
critical $\displaystyle M_{\mathrm{TOV}}$ and $\displaystyle\Lambda_{1.4}=800$
can marginally satisfy the constraint given by Eq. (20). Therefore, it is
illustrated that EOSs with $\displaystyle M_{\mathrm{TOV}}\gtrsim
2.35^{+0.07}_{-0.17}\,M_{\odot}$ can not pass our consistency checks and
satisfy the constraint $\displaystyle\Lambda_{1.4}\lesssim 800$ at the same
time. Note that in our analysis, we only vary $\displaystyle
M_{\mathrm{out}}+M_{\mathrm{eje}}$ from 0 to $\displaystyle 0.2\,M_{\odot}$,
but $\displaystyle 0.2\,M_{\odot}$ is not a solid upper limit of
$\displaystyle M_{\mathrm{out}}+M_{\mathrm{eje}}$, and it is feasible
especially for the unequal-mass case (Kiuchi  _et al._ , 2019, 2024). However,
for EOSs with $\displaystyle
M_{\mathrm{TOV}}>2.35^{+0.07}_{-0.17}\,M_{\odot}$, if $\displaystyle
M_{\mathrm{out}}+M_{\mathrm{eje}}>0.2\,M_{\odot}$, such large mass loss would
result in a remnant less massive than its $\displaystyle M_{\mathrm{TOV}}$ and
becomes a stable remnant (see the lowest row in Table 3, when $\displaystyle
M_{\mathrm{out}}+M_{\mathrm{eje}}=0.2\,M_{\odot}$, $\displaystyle
M_{f}/M_{\mathrm{TOV}}\approx 1.008$ is very close to 1, and thus for
$\displaystyle M_{\mathrm{out}}+M_{\mathrm{eje}}>0.2\,M_{\odot}$ a stable
remnant would be formed). Therefore, $\displaystyle M_{\mathrm{TOV}}$ has to
be smaller than $\displaystyle 2.35^{+0.07}_{-0.17}\,M_{\odot}$, as the very
long-lived ($\displaystyle\sim 100\,\mathrm{s}$)/stable remnant can be
excluded from the observation results of the GW170817 event (Margalit and
Metzger, 2017; Shibata  _et al._ , 2019).

In conclusion, for the EOSs with $\displaystyle M_{\mathrm{TOV}}\lesssim
2.35^{+0.07}_{-0.17}\,M_{\odot}$, it is possible to pass our consistency
checks and satisfy the constraint $\displaystyle\Lambda_{1.4}\lesssim 800$
simultaneously, which is similar to the results for the BNS scenario in
(Shibata  _et al._ , 2019). Although the value of $\displaystyle
M_{\mathrm{max,urot}}/M_{\mathrm{TOV}}$ is larger for QSs, the angular
momentum left in the remnant QS is always not large enough to reach the mass-
shedding limit, and thus even the mass of remnant QS is smaller than
$\displaystyle M_{\mathrm{max,urot}}$, the collapse can still take place
before its kinetic energy is dissipated via electromagnetic radiation
($\displaystyle\sim 100\,\mathrm{s}$). On the other hand, despite the fact
that interquark effects could significantly increase the $\displaystyle
M_{\mathrm{TOV}}$ of QSs, QS EOSs with large $\displaystyle M_{\mathrm{TOV}}$
are disfavored by the electromagnetic counterparts of GW170817 event. It
should be noted that the accurate evolution pictures of post-merger and the
exact values of $\displaystyle E_{\mathrm{GW,p}}$, $\displaystyle E_{\nu}$ and
$\displaystyle M_{\mathrm{out}}+M_{\mathrm{eje}}$ and the collapse behavior of
the merger remnant can only be determined by NR simulations (e.g., (Zhou  _et
al._ , 2022; Zhou and et al., )), and our analysis can only offer relatively
conservative constraint.

###  IV.3 Constraints on the parameter space of QS EOSs

Figure 9: Astrophysical constraints on the parameter space in the
($\displaystyle B_{\mathrm{eff}},\bar{\lambda}^{1/4}$) plane (only positive
$\displaystyle\lambda$ space is considered in this figure). The
$\displaystyle\Lambda_{1.4}=800$, $\displaystyle M_{\mathrm{TOV}}=2.08,\
2.35\,M_{\odot}$ and $\displaystyle M_{\mathrm{thres}}=2.74\,M_{\odot}$ lines
are plotted with different colors, which correspond to the GW170817 constraint
($\displaystyle\Lambda_{1.4}\lesssim 800$ (LIGO Scientific Collaboration and
Virgo Collaboration, 2017; De  _et al._ , 2018)), the maximum pulsar mass
constraint from PSR J0740+6620 ($\displaystyle M_{\mathrm{TOV}}\gtrsim
2.08\,M_{\odot}$ (Fonseca  _et al._ , 2021; Cromartie and et al., 2020)) and
two constraints established by our semi-analytical method from the
electromagnetic counterparts of GW170817 ($\displaystyle
M_{\mathrm{TOV}}\lesssim 2.35\,M_{\odot}$ and $\displaystyle
M_{\mathrm{thres}}\gtrsim 2.74\,M_{\odot}$). The regions to the right of the
orange and green solid lines satisfy the constraints
$\displaystyle\Lambda_{1.4}\lesssim 800$ and $\displaystyle
M_{\mathrm{TOV}}\lesssim 2.35\,M_{\odot}$, respectively. The regions to the
left of the blue and black solid lines satisfy the constraints $\displaystyle
M_{\mathrm{thres}}\gtrsim 2.74\,M_{\odot}$ and $\displaystyle
M_{\mathrm{TOV}}\gtrsim 2.08\,M_{\odot}$, respectively. The grey-shaded region
indicates the allowed parameter space.

In this paper, constraints on $\displaystyle M_{\mathrm{TOV}}$
($\displaystyle\lesssim 2.35\,M_{\odot}$) and $\displaystyle
M_{\mathrm{thres}}$ ($\displaystyle\gtrsim 2.74\,M_{\odot}$) are established
from the assumption that a delayed collapse occurred before the kinetic energy
of the merger remnant is dissipated via electromagnetic radiation for the
GW170817 event. Combining them with the current astrophysical constraints from
GW170817 ($\displaystyle\Lambda_{1.4}\lesssim 800$ (LIGO Scientific
Collaboration and Virgo Collaboration, 2017; De  _et al._ , 2018)) and PSR
J0740+6620 ($\displaystyle M_{\mathrm{TOV}}\gtrsim 2.08\,M_{\odot}$ (Fonseca
_et al._ , 2021; Cromartie and et al., 2020)), we can well constrain the
parameter space of QS EOSs and the resultant constrained parameter space is
shown in Figure 9, where the solid lines with different colors represent the
different constraints and the blue line, $\displaystyle
M_{\mathrm{thres}}=2.74\,M_{\odot}$ line, is calculated by Eq. (9), which is
solidified by our semi-analytical calculations of equilibrium models and can
be corroborated and refined by future NR simulations like (Zhou  _et al._ ,
2021). The GW190425 (LIGO Scientific Collaboration and Virgo Collaboration,
2020) event also makes constraint on the parameter of QS EOSs, but the
constraint from it is looser compared to GW170817 constraint (see Figure 3 in
(Zhang and Mann, 2021)), and thus it is not displayed in Figure 9.

As shown in Figure 9, for $\displaystyle\bar{\lambda}^{1/4}\gtrsim 0.6$
($\displaystyle\bar{\lambda}^{1/4}\gtrsim 1.5$), the lower (upper) bounds of
$\displaystyle B_{\mathrm{eff}}$ are determined by $\displaystyle
M_{\mathrm{TOV}}\lesssim 2.35\,M_{\odot}$ ($\displaystyle
M_{\mathrm{thres}}\gtrsim 2.74\,M_{\odot}$). On the other hand, for
$\displaystyle\bar{\lambda}^{1/4}\lesssim 0.6$
($\displaystyle\bar{\lambda}^{1/4}\lesssim 1.5$), the lower (upper) bounds of
$\displaystyle B_{\mathrm{eff}}$ are determined by
$\displaystyle\Lambda_{1.4}\lesssim 800$ ($\displaystyle
M_{\mathrm{TOV}}\gtrsim 2.08\,M_{\odot}$). It should be noted that only
positive $\displaystyle\lambda$ space is considered in Figure 9. The parameter
space of QS EOSs with negative $\displaystyle\lambda$ has been well-
constrained by $\displaystyle\Lambda_{1.4}\lesssim 800$ and $\displaystyle
M_{\mathrm{TOV}}\gtrsim 2.08\,M_{\odot}$ (Zhou  _et al._ , 2018) and the
investigation in this paper do not impose a further constraint on it.
Therefore, the investigation of the post-merger evolution for the GW170817
event can make tight constraints on QS EOSs with the large
$\displaystyle\mathrm{sgn}(\lambda)\bar{\lambda}$.

##  V CONCLUSIONS AND DISCUSSIONS

We semi-analytically investigate the post-merger evolution of the BQS merger.
With the help of the EOB method and considering the angular momentum
dissipation process, we obtain the angular momentum left in the merger remnant
at the time of the merger. The amounts of three main dissipation mechanisms in
the post-merger phase are determined by the conservation laws of baryon
number, energy and angular momentum. In addition, we determine the stability
of the remnant QS by constructing the equilibrium models of rotating QSs.
Through this semi-analytical method, we investigate both prompt collapse and
delayed collapse cases and set an upper bound on QSs’ $\displaystyle
M_{\mathrm{TOV}}$ through the GW170817 event.

Assuming GW170817 originated from a BQS merger, we analyze the post-merger
evolution for GW170817 in the BQS scenario. Our analysis suggests that a QS
with $\displaystyle M_{\mathrm{TOV}}>2.0\,M_{\odot}$, leading to
$\displaystyle M_{\mathrm{max,urot}}>2.8\,M_{\odot}$, can collapse to a BH
before its kinetic energy and angular momentum are dissipated via
electromagnetic radiation ($\displaystyle\sim 100\,\mathrm{s}$) because due to
the angular momentum dissipation of mass outflows, neutrinos and gravitational
waves, the angular momentum left in the remnant QS might not be large enough
to sustain the additional self-gravity of the supramassive QS. Moreover, we
also systematically investigate the influence of $\displaystyle
M_{\mathrm{TOV}}$ and $\displaystyle\mathrm{sgn}(\lambda)\bar{\lambda}$ on the
results of our consistency checks. Finally, an upper limit of $\displaystyle
M_{\mathrm{TOV}}$ is obtained, and only QS EOSs with $\displaystyle
M_{\mathrm{TOV}}\lesssim 2.35^{+0.07}_{-0.17}\,M_{\odot}$ can pass our
consistency checks and satisfy $\displaystyle\Lambda_{1.4}\lesssim 800$
simultaneously. Therefore, although considering interquark effects can make
$\displaystyle M_{\mathrm{TOV}}$ of QSs very large, these QS EOSs with large
$\displaystyle M_{\mathrm{TOV}}$ are unlikely to pass the constraint of the
electromagnetic counterparts of the GW170817 event. It should be noted that
our derived constraints in this paper are valid for the EOSs based on the IQM
model (Zhang and Mann, 2021) in the one-family scenario, while other cases of
different quark models (such as quark-mass density-dependent model (Fowler
_et al._ , 1981; Chakrabarty  _et al._ , 1989; Peng  _et al._ , 2000; Xia  _et
al._ , 2014) and NJL model (Buballa, 2005; Yuan  _et al._ , 2022; Yuan and Li,
2024; Xia, 2024)) or different scenarios (such as the two-family scenario
(Berezhiani  _et al._ , 2003; Bombaci  _et al._ , 2004; Drago  _et al._ ,
2004)) are not considered, but analyses could be done in a similar way.

Although there are many unknowns in the electromagnetic counterparts
associated with BQS mergers, our constraint $\displaystyle
M_{\mathrm{TOV}}\lesssim 2.35\,M_{\odot}$ is only based on the exclusion of
the very long-lived ($\displaystyle\sim 100\,\mathrm{s}$)/stable remnant that
injects a large amount of its kinetic energy into the outflows from the
observation results of the GW170817 event (Margalit and Metzger, 2017; Shibata
_et al._ , 2017a, 2019). In comparison, the assumption that the short
$\displaystyle\gamma$-ray burst is powered by the spinning BH-torus system
(Ruiz  _et al._ , 2018) may be inappropriate considering the simulations in
(Kiuchi  _et al._ , 2024; Bamber  _et al._ , 2024), where it is indicated that
SMNS and HMNS may be able to serve as the central engine of the short
$\displaystyle\gamma$-ray burst. Additionally, as discussed in (Shibata  _et
al._ , 2019) and the current paper, the assumption that the remnant NS/QS is
at the mass-shedding limit is also improper, and thus for both NSs and QSs,
$\displaystyle M_{\mathrm{TOV}}$ can only be weakly constrained as
$\displaystyle M_{\mathrm{TOV}}\lesssim 2.35\,M_{\odot}$. Moreover, our
analysis can also connect to the multi-messenger observations. In this paper,
we only employ the constraint of the energy of gravitational waves obtained by
NR simulations. In the future, with observations of post-merger gravitational
waves and neutrinos, the values of $\displaystyle E_{\mathrm{GW,p}}$ and
$\displaystyle E_{\mathrm{\nu}}$ can be further constrained by observations.
As a result, the constraints for EOSs of compact stars can be better imposed
and the central engine of the short $\displaystyle\gamma$-ray burst can be
understood more deeply.

###### Acknowledgements.

We thank the members of the Relativistic AstroParticle Physics group at
Huazhong University of Science and Technology and the Computational
Relativistic Astrophysics division at Max Planck Institute for Gravitational
Physics (Potsdam) for their helpful discussions. E. Z. acknowledges the
support of the National SKA Program of China No. 2020SKA0120300 and NSFC Grant
No. 12203017. C. Z. is supported by the Jockey Club Institute for Advanced
Study at The Hong Kong University of Science and Technology. J. Z. is
supported by the NSFC Grant No. 12147177 and the “LiYun” Postdoctoral
Fellowship of Beijing Normal University. This work was in part supported by
the Grant-in-Aid for Scientific Research (grant Nos. 23H01172, 23K25869) of
Japan MEXT/JSPS.

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