#  Implications of supermassive neutron star for the form of equation of state
of hybrid stars

Hongyi Sun  School of Physics and Optoelectronics, South China University of
Technology, Guangzhou 510641, P.R. China  Dehua Wen111Corresponding author.
<EMAIL_ADDRESS>School of Physics and Optoelectronics, South China
University of Technology, Guangzhou 510641, P.R. China

###### Abstract

The observations of PSR J0952-0607 ($M=2.35^{+0.17}_{-0.17}~{}M_{\odot}$) and
the second object in GW190814 event ($M=2.59^{+0.08}_{-0.09}~{}M_{\odot}$)
indicate the possible existence of supermassive neutron stars. In this work,
by using the Constant-Sound-Speed (CSS) parametrization to describe the
equation of state (EOS) of quark matter, the constraints on the EOS parameters
from supermassive hybrid stars are investigated through the Maxwell and Gibbs
constructions. It is shown that to support a supermassive hybrid star (e.g.,
$M=2.5~{}M_{\odot}$), a lower transition energy density
($\varepsilon_{tran}$), a smaller energy density discontinuity
($\Delta\varepsilon$) and a higher sound speed of quark matter ($c_{sq}$) are
favored. For the constructed hybrid star EOS model, the maximum mass of the
corresponding hybrid stars will not meet the lower mass limit of the second
object in GW190814 if $\Delta\varepsilon$ takes a value higher than
$180~{}{\rm MeV~{}fm^{-3}}$. Moreover, it is confirmed that the supermassive
neutron star observation can also rule out the existence of twin stars as a
supermassive hybrid star requires a relatively small $\Delta\varepsilon$.
Finally, we give a rough estimate of the lower limit of the dimensionless
tidal deformability of neutron stars which ranges from 2 to 3.

Keywords: Supermassive hybrid stars; Phase transition; Equation of state;
Quark matter

###### pacs:

26.60.-c, 04.40.Dg, 21.65.Ef, 97.60.Jd

##  I Introduction

Neutron star is one of the objects in the universe that have the most extreme
environment. As the density of the core is far beyond the reach of terrestrial
experiments, neutron stars provide a unique way to the understanding of super
dense matter. From the surface to the interior, a neutron star is usually
divided into the outer crust, the inner crust, the outer core and the inner
core [1]. Normally, the equations of state (EOSs) below nuclear saturation
density ($\rho_{sat}$) are understood relatively well [2, 3, 4]. However, for
super dense matter, that is, matter in the inner core of neutron stars,
hyperons [7, 5, 6], free quarks [10, 9, 8], and even pion-condensation [11]
may appear, which will soften the high-density EOSs and reduce the maximum
mass and radius of neutron stars [12] (sometimes to a level that cannot meet
the observation limit of $2~{}M_{\odot}$). Overall, the high-density EOSs are
still very uncertain. Therefore, one has to rely on observations and detailed
modeling to learn more about the EOSs of super dense matter.

Recent observations of the mass-radius measurements from NICER [13, 14, 15,
16] have provided important information for improving the understanding of
neutron stars and the EOSs of dense nuclear matter [17, 18]. Besides, the
detection of the binary neutron star merger, GW170817 [19], has constrained
the tidal deformability of $1.4~{}M_{\odot}$ neutron star to
$\Lambda_{1.4}=190^{+390}_{-120}$, which in turn has been employed to further
constrain the pressure at 2$\rho_{sat}$ [19, 20]. These neutron star
properties have also been theoretically calculated using a wide range of EOSs
(moment of inertia [21, 22, 25, 23, 24], radius [20, 26, 27, 25], the
parameter of f-mode oscillation [28, 29] and gravitational binding energy [30,
31]). In addition, the discovery of supermassive compact stars is likely to
reshape our understanding of the maximum mass of neutron stars [33, 32]. In
the GW190814 event a merger of a black hole with an unknown compact object
with mass of $M=2.59^{+0.08}_{-0.09}~{}M_{\odot}$ was observed [32], and the
heaviest known galactic neutron star PSR J0952-0607 has a mass of
$M=2.35^{+0.17}_{-0.17}~{}M_{\odot}$ [33]. In light of such discoveries, the
study of supermassive compact stars can help to better understand the
possibility of the phase transition and the appearance of quark phase in
compact stars [34, 35].

It has long been noted that with the appearance of free quark matter, a new
class of compact stars composed of a deconfined quark core and a hadronic
outer shell may arise [36, 37, 38]. For better distinction, compact stars
composed of pure hadronic matter are usually called as normal neutron stars,
and the stars with a deconfined quark core and a hadronic outer shell are
referred to as hybrid stars [42, 36, 37, 40, 41, 38, 39, 43]. Many previous
works have explored the possibility of the existence of supermassive hybrid
stars with mass close to the mass of the second object in GW190814 and
investigated the macroscopic and internal microscopic properties of such
supermassive hybrid stars [44, 45, 46, 47, 48]. However, since the GW170817
event favors the soft EOSs, and the existence of supermassive hybrid stars
puts forward a requirement for the stiff EOSs, it is important to resolve this
conflict and use the GW170817 event to constrain the properties of
supermassive stars [50, 51, 52, 53, 54, 49].

In this work, we will use the Maxwell [56, 55] and Gibbs constructions [39,
57] to connect hadronic matter and quark matter, and describe quark matter by
the CSS parametrization [36]. Other quark models, such as NJL [58, 59] and MIT
bag [60] models, may be more realistic. For example, Benić $et~{}al.$
investigated the high-mass twin-stars by using NJL model with higher-order
quark interactions [59]. However, the CSS parametrization can fit these quark
models well in some conditions [61, 62], and its parameters are more
adjustable. So this ansatz provides us with the opportunity to change the
parameters and apply the mass ranges of PSR J0952-0607 and the second object
in GW190814 through a more generalized way [45]. Further, we will discuss the
mass-radius relations and tidal deformability of hybrid stars [63, 64, 18],
and then investigate the constraint of the $2.5~{}M_{\odot}$ (which conforms
to the mass range of PSR J0952-0607 and GW190814 event) hybrid star properties
[53, 34, 65, 66] on the parameters of EOSs. It is worth pointing out that
supermassive neutron stars may also appear in other forms, such as pure
nucleonic neutron stars [67, 68], quark stars [69, 70, 71], fast-spinning
neutron stars [72, 35] and dark matter admixed neutron stars [73].

This paper is organized as follows. In Sec.II, the basic formulas for the
macroscopic properties of neutron stars are briefly introduced. The
construction of hybrid star EOSs through the CSS parameterization is presented
in Sec.III. In Sec.IV, the mass-radius relations of hybrid stars are presented
and discussed. The tidal deformability of $2.5~{}M_{\odot}$ hybrid stars are
also investigated. Finally, a summary will be given in Sec.V.

##  II Basic formulas for the macroscopic properties of neutron stars

Inputting the EOSs of dense nuclear matter, the mass and radius of a neutron
star can be calculated through the Tolman-Oppenheimer-Volkoff (TOV) equation
[74, 75]

$\frac{dp}{dr}=-\frac{G\varepsilon(r)M(r)}{c^{2}r^{2}}\left(1+\frac{p(r)}{\varepsilon(r)}\right)\left(1+\frac{4\pi
r^{3}p(r)}{M(r)c^{2}}\right)\left(1-\frac{2GM(r)}{c^{2}r}\right)^{-1},$ (1)

supplemented with the differential equation of mass

$\frac{dM}{dr}=\frac{4\pi r^{2}\varepsilon(r)}{c^{2}},$ (2)

where $M(r)$, $p(r)$ and $\varepsilon(r)$ are the enclosed mass, pressure and
energy density at the radius $r$.

We solve an additional equation for the tidal deformability of neutron stars,
defined as

$\lambda=\frac{2}{3}R^{5}k_{2}.$ (3)

The second (quadrupole) tidal Love number $k_{2}$ can be solved by the
following equation [76, 77]

$\displaystyle k_{2}=$
$\displaystyle\frac{8}{5}x^{5}(1-2x)^{2}(2-y_{R}+2x(y_{R}-1))(2x(6-3y_{R}+3x(5y_{R}-8))$
(4) $\displaystyle+4x^{3}(13-11y_{R}+x(3y_{R}-2)+2x^{2}(1+y_{R}))$
$\displaystyle+3(1-2x)^{2}(2-y_{R}+2x(y_{R}-1))ln(1-2x))^{-1},$

where $x=GM/Rc^{2}$ is the compactness of neutron stars, and $y_{R}$ is
determined by solving the following differential equation

$r\frac{dy(r)}{dr}+y^{2}(r)+y(r)F(r)+r^{2}Q(r)=0$ (5)

with $F(r)$ and $Q(r)$ being the functions of $M(r)$, $p(r)$ and
$\varepsilon(r)$ [77]

$F(r)=\left[1-\frac{4\pi
r^{2}G}{c^{4}}(\varepsilon(r)-p(r))\right]\left(1-\frac{2M(r)G}{rc^{2}}\right)^{-1},$
(6) $\displaystyle r^{2}Q(r)=$ $\displaystyle\frac{4\pi
r^{2}G}{c^{4}}\left[5\varepsilon(r)+9p(r)+\frac{\varepsilon(r)+p(r)}{\partial
p(r)/\partial\varepsilon(r)}\right]$ (7)
$\displaystyle\times\left(1-\frac{2M(r)G}{rc^{2}}\right)^{-1}-6\left(1-\frac{2M(r)G}{rc^{2}}\right)^{-1}$
$\displaystyle-\frac{4M^{2}(r)G^{2}}{r^{2}c^{4}}\left(1+\frac{4\pi
r^{3}p(r)}{M(r)c^{2}}\right)^{2}\left(1-\frac{2M(r)G}{rc^{2}}\right)^{-2}.$

Once $y(r)=y_{R}$ at the radius $R$ of neutron stars is provided, the
dimensionless tidal deformability parameter $\Lambda=\lambda({GM/c^{2}})^{-5}$
can be obtained.

##  III EOS of Hybrid star

In this section we introduce how to construct the hybrid star EOSs, where a
phase transition from hadronic matter to quark matter is considered.

Figure 1:  The schematic form of the hybrid star EOSs with the CSS
parametrization and Maxwell/Gibbs construction.

For the outer crust and the inner crust, the BPS EOS [78] and NV EOS [79] are
adopted, respectively. In the outer core where the energy density of hadronic
matter is lower than the transition energy density $\varepsilon_{tran}$, two
hadronic EOSs with different stiffness, the APR3 EOS (soft) [80] and DDME2 EOS
(stiff) [81] are employed in order to explore how the stiffness of hadronic
matter effects the properties of hybrid stars. In the inner core, after the
phase transition (symbolized by the energy density discontinuity
$\Delta\varepsilon$), quark matter with energy density higher than
$\Delta\varepsilon+\varepsilon_{tran}$ is described by the CSS parametrization
[36].

For the Maxwell construction [82], the formula form of hybrid star EOSs is
given by

$\varepsilon(p)=\left\\{\begin{aligned} &\varepsilon_{h}(p),&p\leqslant
p_{tran}\\\
&\varepsilon_{tran}+\Delta\varepsilon+c_{sq}^{-2}(p-p_{tran}),&p>p_{tran},\end{aligned}\right.$
(8)

where $\varepsilon_{h}(p)$ represents the hadronic EOSs, $c_{sq}^{2}$ is the
squared sound speed of quark matter (in the following, $c_{sq}^{2}$ is in unit
of $c^{2}$), and $\varepsilon_{tran}$ $(p_{tran})$ is the transition energy
density (transition pressure). For the Gibbs construction, the phase
transition is described by a polytrope EOS $p(\rho)=K_{m}\rho^{\Gamma_{m}}$
[64, 63] to account for a mixed phase of hadron and quark. The hybrid star
EOSs with the Gibbs construction are given by

$\varepsilon(p)=\left\\{\begin{aligned} &\varepsilon_{h}(p),&p\leqslant
p_{tran}\\\
&A_{m}(p/K_{m})^{1/\Gamma_{m}}+p/(\Gamma_{m}-1),&p_{tran}<p\leqslant
p_{css}\\\
&\varepsilon_{css}+c_{sq}^{-2}(p-p_{css}),&p>p_{tran},\end{aligned}\right.$
(9)

with $\Gamma_{m}=1.03$. This choice allows us to get a strong enough softening
of the EOS, which is significantly different from the Maxwell construction.
The values of $K_{m}$ and $A_{m}$ are obtained by requiring that $p$ and
$\varepsilon$ are continuous at the transition point. $\varepsilon_{css}$ and
$p_{css}$ is the energy density and pressure at the begin of quark phase. It
is noted that the energy density discontinuity $\Delta\varepsilon$ is not
defined in Eq 9. In this case, we assign the increment of $\varepsilon(p)$
during the mixed phase to its value. The hybrid star EOSs are also illustrated
in Fig. 1.

##  IV Constraint on EOS through Macroscopic Properties of Supermassive Hybrid
Star

###  IV.1 Constraint through Mass-Radius Relations of Hybrid Stars

As we know, based on the TOV equation, each EOS corresponds to a unique mass-
radius (M-R) relation. After constructing the hybrid star EOSs by using the
APR3 and DDME2 to describe hadronic matter, the effects of energy density
discontinuity $\Delta\varepsilon$ and sound speed $c_{sq}$ on the M-R
relations of hybrid stars at different transition energy density
$\varepsilon_{tran}$ are investigated, and the results of Maxwell construction
are shown in Fig. 2,

Figure 2: The effect of energy density discontinuity $\Delta\varepsilon$
(upper panels, where $\Delta\varepsilon=0,60,120,180~{}{\rm MeV~{}fm^{-3}}$)
and the squared sound speed of quark matter $c_{sq}^{2}$ (lower panels, where
$c_{sq}^{2}=1,0.8,0.6,0.5$) on the M-R relations for hybrid stars with the
Maxwell construction, where $c_{sq}^{2}$ and $\Delta\varepsilon$ are fixed as
$c_{sq}^{2}=1$ and $\Delta\varepsilon=60~{}{\rm MeV~{}fm^{-3}}$ for the upper
panels and lower panels, respectively. The transition energy density
$\varepsilon_{tran}$ adopts $1.6~{}\varepsilon_{sat}$ (left),
$2.0~{}\varepsilon_{sat}$ (middle), $2.5~{}\varepsilon_{sat}$ (right). The
dash (dot dash) lines are the M-R relations of hybrid stars with hadronic
matter described by the DDME2 (APR3) EOS. The black (red) solid line is the
M-R relation of normal neutron stars calculated by the DDME2 (APR3) EOS. The
orange and purple shaded areas show the mass range of PSR J0952-0607 [33] and
the second object in GW190814 [32], while the magenta shaded areas are the M-R
constraint from GW170817 [20].

where the squared sound speed is fixed as $c_{sq}^{2}=1$ in the upper panels,
and the energy density discontinuity is fixed as $\Delta\varepsilon=60~{}{\rm
MeV~{}fm^{-3}}$ in the bottom panels. The effect of $\Delta\varepsilon$ on the
M-R relations of hybrid stars are also shown for the Gibbs construction in
Fig. 3. The M-R relations of normal neutron stars (without the phase
transition) are also displayed in Fig. 2 and Fig. 3 as comparison. Here we
adopt the APR3 representing the relatively soft EOS and DDME2 representing the
relatively stiff EOS to describe the hadronic part in neutron stars. It is
shown that the maximum mass of normal neutron stars supported by the
APR3/DDME2 EOS can be up to near $2.4$/$2.5~{}M_{\odot}$.

Figure 3: The effect of energy density discontinuity $\Delta\varepsilon$ on
the M-R relations for hybrid stars with the Gibbs construction. The parameters
are the same as in the upper panels of Fig. 2. The observational constraints
are also plotted.

It can be seen from Fig. 2 and Fig. 3 that both of the M-R relations of normal
neutron stars and hybrid stars meet the constraint of GW170817. For the upper
three panels of Fig. 2, $c_{sq}^{2}=1$ means that the stiffest EOS is adopted
in quark matter, then the corresponding M-R relation gives the upper limit of
stellar mass under a given set of $\Delta\varepsilon$ and
$\varepsilon_{tran}$. The upper panels show that the increase of
$\Delta\varepsilon$ will lead to a reduction of the maximum mass of hybrid
stars with the Maxwell construction. When the $\Delta\varepsilon$ takes a
relatively high value (e.g., $\Delta\varepsilon=180~{}{\rm MeV~{}fm^{-3}}$),
the maximum mass of the corresponding hybrid stars will not meet the lower
mass limit of the second object in GW190814. For the Gibbs construction (see
Fig. 3), a higher energy density discontinuity ($\Delta\varepsilon>180~{}{\rm
MeV~{}fm^{-3}}$) will induce a similar result. This means that if observations
confirm the existence of supermassive neutron stars, it will placed a well
constraint on the upper limit of $\Delta\varepsilon$ for EOS model with the
quark phase transition. In addition, the increase of $\Delta\varepsilon$ will
also lead to a reduction of the radius, which is consistent with the results
of Refs. [83, 18, 84]. However, compared to the Maxwell construction, the
Gibbs construction will result in a less reduction of radius under same
$\Delta\varepsilon$. As for the influence of transition energy density
$\varepsilon_{tran}$, we can see that with the increase of
$\varepsilon_{tran}$, the maximum mass of hybrid stars decreases.

As we know, due to the phase transition caused by the appearance of quark
matter in the core of neutron stars, a new branch with the higher central
density than that of the normal neutron stars may appear. The new branch alone
can lead to the appearance of twin stars with same mass but different radii
due to different internal constitutions [39, 18, 85]. From Fig. 2 and Fig. 3,
we can see that the M-R relations do not show the presence of twin stars. As
discussed in Ref. [38], the following three quantities, including the
transition energy density $\varepsilon_{tran}$, the sound speed $c_{sq}$ and
the energy density discontinuity $\Delta\varepsilon$, are the key factors to
determine the existence of twin stars. Here, it is the weak
$\Delta\varepsilon$ that causes the miss of twin star phenomenon [45]. The
small $\Delta\varepsilon$ also leads to the intersection of the M-R curves of
normal neutron stars and hybrid stars, which means that it is difficult to
determine the existence of the phase transition only through the M-R
relations.

From the lower three panels in Fig. 2, we can see that for hybrid star EOSs
with energy density discontinuity fixed as $\Delta\varepsilon=60~{}{\rm
MeV~{}fm^{-3}}$ and with transition energy density $\varepsilon_{tran}$ higher
than $1.6~{}\varepsilon_{sat}$, the corresponding hybrid stars will not meet
the lower mass limit of the second object in GW190814 as $c_{sq}^{2}$ reduces
to 0.6. When considering the Gibbs construction, it leads to the same result.
This means that if a supermassive neutron star is confirmed in the future, it
will constrain the lower limit of quark matter‘s sound speed.

###  IV.2 Constraint through Tidal Deformability of Supermassive Hybrid Stars
($M~{}=~{}$2.5$~{}M_{\odot}$)

Figure 4: The mass-tidal deformability (M-$\Lambda$) relations of hybrid stars
with the Maxwell construction at different energy density discontinuity
$\Delta\varepsilon$ (upper panels) and squared sound speed of quark matter
$c_{sq}^{2}$ (lower panels), where the $c_{sq}^{2}$ and $\Delta\varepsilon$
are fixed as $c_{sq}^{2}=1$ and $\Delta\varepsilon=60~{}{\rm MeV~{}fm^{-3}}$
for the upper panels and lower panels, respectively. The $\varepsilon_{tran}$
adopts the same values as in Fig. 2. For simplicity, only the APR3 EOS is
considered here, and the black solid line is the M-$\Lambda$ relation of the
corresponding normal neutron stars. The constraints from PSR J0952-0607 [33],
the second object in GW190814 [32] and GW170817 [20] are also plotted.

The influence of the energy density discontinuity $\Delta\varepsilon$ (upper
panels) and sound speed $c_{sq}$ (lower panels) on the tidal deformability of
hybrid stars with the Maxwell construction are shown in Fig. 4. As shown in
the upper panels, at a relatively low transition energy density (e.g.,
$\varepsilon_{tran}=1.6~{}\varepsilon_{sat}$), the decrease of
$\Delta\varepsilon$ will obviously increase the tidal deformability, and when
the $\Delta\varepsilon$ approaches zero with the squared sound speed of quark
matter fixed at $c_{sq}^{2}=1$, the tidal deformability will exceed the
constraint of GW170817. This result indicates that the observation of GW170817
will place some constraints on the upper limit of $c_{sq}$ and the lower limit
of $\Delta\varepsilon$ and $\varepsilon_{tran}$. The tidal deformability of
hybrid stars with the Gibbs construction is slightly greater than that with
the Maxwell construction, which can also be inferred from their different
impacts on the radius. However, due to the small difference in tidal
deformability caused by the two phase transition structures, the observation
of GW170817 will impose the similar constraints on the parameters of hybrid
star EOSs with the Gibbs construction.

From the upper panels ($c_{sq}^{2}=1$), it is interesting to note that even
though different M-$\Lambda$ curves correspond to different maximum masses as
the $\Delta\varepsilon$ changes, the maximum mass hybrid stars all correspond
to the similar values of tidal deformability. Moreover, the lower panels show
that the decrease of $c_{sq}^{2}$ will increase the tidal deformability of the
maximum mass hybrid stars, while the $\varepsilon_{tran}$ does not have a
significant impact on the tidal deformability of the maximum mass hybrid
stars. In addition, although Fig. 4 only shows the results for APR3 EOS with
the Maxwell construction, the calculations for DDME2 EOS and Gibbs
construction give the similar results. So the results in the upper panels of
Fig. 4 provide a useful clue to the lower limit of the tidal deformability for
the maximum mass hybrid stars. A rough estimate of the lower limit of the
dimensionless tidal deformability for the maximum mass hybrid stars ranges
from 2 to 3. In fact, this can also be seen as the lower limit for all neutron
stars since the tidal deformability of the maximum mass neutron stars is the
smallest in the neutron star sequences.

Figure 5:  The tidal deformability as a function of energy density
discontinuity $\Delta\varepsilon$ at different $c_{sq}^{2}$
($c_{sq}^{2}=1,0.9,0.8,0.7,0.6,0.5$), where the stellar mass is fixed as 2.5
$M_{\odot}$. The solid and dash lines represent the results of Maxwell and
Gibbs constructions, respectively. $\varepsilon_{tran}$ adopts the same values
as in Fig. 2. The DDME2 (APR3) EOS is taken to describe hadronic matter in the
upper (bottom) panels.

In order to illustrate how the tidal deformability of supermassive hybrid
stars is affected by the relative parameters, the tidal deformability
$\Lambda$ as a function of the energy density discontinuity
$\Delta\varepsilon$ at different $\varepsilon_{tran}$ and $c_{sq}^{2}$ for the
DDME2 EOS model (upper panels) and APR3 EOS model (lower panels) is drawn in
Fig. 5, where the stellar mass is fixed as 2.5 $M_{\odot}$. It is easy to see
that for a hybrid star with fixed mass, its tidal deformability with different
phase transition structures will both decrease with the decrease of sound
speed $c_{sq}$ and with the increase of transition energy density
$\varepsilon_{tran}$ or energy density discontinuity $\Delta\varepsilon$,
which can be understood by the M-R relations in Fig. 2 and Fig. 3, that is, a
smaller $c_{sq}$, a higher $\varepsilon_{tran}$, or a larger
$\Delta\varepsilon$ corresponds to a smaller stellar radius for supermassive
hybrid stars and further gives a smaller tidal deformability. It is worth
noting that for the same energy density discontinuity, the tidal deformability
of 2.5 $M_{\odot}$ hybrid star with the Gibbs construction is greater (i.e.,
the dash lines are higher than the solid lines in Fig. 5), which is due to the
fact that the hybrid star EOSs with the Gibbs construction are stiffer than
that with the Maxwell construction under the same parameters.

Comparing the upper and lower graphes of Fig. 5, one can see that if the
observed dimensionless tidal deformability of 2.5 $M_{\odot}$ neutron star is
at around $\Lambda_{2.5}\sim 35$, which can be detected under the precision
($\sigma_{\tilde{\Lambda}}<20$) of Einstein telescope [86, 87], then for both
the phase transition structures, a smaller tidal deformability of 2.5
$M_{\odot}$ neutron star (e.g., $\Lambda_{2.5}\lesssim 15$) will be excluded
in an ideal scenario, and only a smaller energy density discontinuity
($\Delta\varepsilon<75~{}\textrm{MeV fm}^{-3}$, which becomes smaller for a
higher $\varepsilon_{tran}$ and for the Maxwell construction), a lower
transition energy density ($\varepsilon_{tran}<2.5~{}\varepsilon_{sat}$) and a
higher squared sound speed ($c_{sq}^{2}>0.5$) are possible in hybrid stars.
Conversely, for a small tidal deformability, such as $\Lambda_{2.5}\sim 10$,
even the third-generation gravitational detectors cannot reach the required
precision, so it is hard to make effective constraints on the EOS parameters
when the tidal deformability of supermassive neutron stars is small.

In Fig. 5, it also can directly show the ranges of parameters
$\Delta\varepsilon$, $c_{sq}$ and $\varepsilon_{tran}$ that can support 2.5
$M_{\odot}$ hybrid star. For example, a higher transition energy density
requires a higher sound speed and a smaller energy density discontinuity. In
addition, a softer EOS (e.g., APR3) will narrow the ranges of parameters in
order to support 2.5 $M_{\odot}$ hybrid star.

##  V summary

The observations of PSR J0952-0607 ($M=2.35^{+0.17}_{-0.17}~{}M_{\odot}$) and
the second object in GW190814 event ($M=2.59^{+0.08}_{-0.09}~{}M_{\odot}$)
indicate the possible existence of supermassive neutron stars. The possibility
of the existence of supermassive neutron stars provides an opportunity to
understand the high density matter from a new angle. In this work, the
Constant-Sound-Speed (CSS) parametrization is adopted to describe the EOS of
quark matter for the hybrid star EOS, and the Maxwell/Gibbs construction is
used to connect hadronic and quark phase. By using the constructed EOSs, the
constraints on the EOS parameters through the properties of supermasive hybrid
stars are probed. The main conclusions of this work are summarized as follows.

  1. (1)

It is shown that to support a supermassive hybrid star (e.g.,
$M=2.5~{}M_{\odot}$), a lower transition energy density
($\varepsilon_{tran}$), a smaller energy density discontinuity
($\Delta\varepsilon$) and a higher sound speed of quark matter ($c_{sq}$) are
favored. For the constructed hybrid star EOSs model with the Maxwell and Gibbs
constructions, the maximum mass of the corresponding hybrid stars will not
meet the lower mass limit of the second object in GW190814 if the
$\Delta\varepsilon$ takes a value higher than $180~{}{\rm MeV~{}fm^{-3}}$. If
observations confirm the existence of supermassive neutron stars, it will
place a well constraint on the upper limit of $\Delta\varepsilon$ for the
hybrid star EOSs.

  2. (2)

As supermassive hybrid star requires a relatively small $\Delta\varepsilon$,
and a small $\Delta\varepsilon$ do not support the existence of twin stars,
thus the supermassive neutron star observation may also rule out the
possibility of the existence of twin stars, which is consistent with what has
been found in previous [45].

  3. (3)

A rough estimate of the lower limit of the dimensionless tidal deformability
of neutron stars ranges from 2 to 3. It is shown that accurate observations on
supermassive neutron stars with relatively large tidal deformability can
provide effective constraints on the phase transition that softens EOS and its
parameters $\Delta\varepsilon$, $\varepsilon_{tran}$ and $c_{sq}$. However,
for supermassive neutron stars with small tidal deformability, even the third-
generation gravitational detectors cannot reach the required precision.

The phase transition may occur in other forms, such as hadron-quark crossover
[88, 89, 90, 91], which can stiffen the EOS in the phase transition region and
bring different effects to hybrid stars. In future investigation, the
macroscopic properties of supermassive hybrid stars can be further exploited
to constrain the parameters of the crossover phase.

Supermassive neutron stars contain both the extreme properties of neutron
stars and the extreme properties of dense matter. If the observations confirm
the existence of supermassive neutron stars, it will open a new window to
enrich the theory of nuclear physics, gravity theory and astrophysics.

##  VI acknowledgement

This work is supported by NSFC (Grant No. 11975101) and Guangdong Natural
Science Foundation (Grants No. 2022A1515011552 and No. 2020A151501820).

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&=\int_{\mathbb{R}^2} J_{\sqrt{\varepsilon}}(x-y)x^2\rho_0(y)\,\dd y\dd x\nonumber\\
&=\int_{\mathbb{R}}\Big(\int_{\mathbb{R}}x^2J_{\sqrt{\varepsilon}}(x-y) \,\dd x\Big)\rho_0(y)\,\dd y.
\end{align}
For the integrand in \eqref{A.8}, we have
\int_{\R}x^2J_{\sqrt{\varepsilon}}(x-y)\,\dd x=\int_{\R}(s+y)^2J_{\sqrt{\varepsilon}}(s)\,\dd s
=\int_{\R}(s^2+2sy+y^2)J_{\sqrt{\varepsilon}}(s)\,\dd s.
\int_{\R}sJ_{\sqrt{\varepsilon}}(s)\,\dd s=0,\qquad
\int_{\R}s^2J_{\sqrt{\v}}(s)\,\dd s=\int_{\R}s^2\frac{1}{\sqrt{\varepsilon}}J(\frac{s}{\sqrt{\varepsilon}})\,\dd s
=\v\int_{\R}t^2J(t)\,\dd t\leq C\v,
then, for \eqref{A.8}, we have
\int_{\R^2}J_{\sqrt{\varepsilon}}(y)x^2\rho_0(x-y)\,\dd y\dd x\leq C\Big(\int_{\R}y^2\rho_0(y)\,\dd y+\v\Big).

Therefore, we obtain \eqref{A.5}:
\int_{\R} x^2|\tilde{\rho}^{\v}_0(x)|\,\dd x
\leq C\Big(\int_{\R} x^2|\rho_0(x)|\,\dd x+\v+\v^{\frac{2}{2\alpha-1}}\Big)
\leq C\Big(\int_{\R} x^2|\rho_0(x)|\,\dd x+\v\Big).

\medskip
\noindent{\emph{Step 4.}} We now prove \eqref{A.6} by three three steps.

\medskip
\noindent{\emph{Step 4(i).}} Since $M_2<\infty$, then $x^2 \rho_0(x)\in L^1$.  By the standard properties of mollifier $J_{\sqrt{\v}}$,
we have
\int_{\R} x^2\big|\big((\rho_0\mathbf{I}_{[-b+1,b-1]})\ast J_{\sqrt{\v}}\big)(x)-x^2\rho_0(x)\big|\,\dd x \rightarrow 0
\qquad\text{ as $\varepsilon\rightarrow 0^+$.}

\smallskip
\noindent{\emph{Step 4(ii).}} We write
\begin{align*}
&x^2\big(\big((\rho_0\mathbf{I}_{[-b+1,b-1]})\ast J_{\sqrt{\v}})(x)-\rho_0(x)\big)\\
&=x^2\big(\big((\rho_0\mathbf{I}_{[-b+1,b-1]})\ast J_{\sqrt{\v}})(x)-\big((x^2\rho_0\mathbf{I}_{[-b+1,b-1]})\ast J_{\sqrt{\v}}\big)(x)\\
&\quad+\big((x^2\rho_0\mathbf{I}_{[-b+1,b-1]})\ast J_{\sqrt{\v}}\big)(x)-x^2\rho_0(x).
\end{align*}
%we have
\begin{align*}
&\int_{\R} x^2\big|\big((\rho_0\mathbf{I}_{[-b+1,b-1]})\ast J_{\sqrt{\v}})(x)-\rho_0(x)\big|\,\dd x\nonumber\\
&\leq \int_{\R} x^2\big|\big((\rho_0\mathbf{I}_{[-b+1,b-1]})\ast J_{\sqrt{\v}}\big)(x)
-\big((x^2\rho_0\mathbf{I}_{[-b+1,b-1]})\ast J_{\sqrt{\v}}\big)(x)\big|\,\dd x\nonumber\\
&\quad+\int_{\R} x^2\big|\big((\rho_0\mathbf{I}_{[-b+1,b-1]})\ast J_{\sqrt{\v}})(x)-\rho_0(x)\big|\,\dd x:=I_1+I_2.
\end{align*}
It follows from Step 4(i) that $I_2\rightarrow0$ as $\varepsilon\rightarrow 0^+$.

Now, we prove that $I_1\rightarrow0$  as $\varepsilon\rightarrow 0^+$.
We write
\begin{align*}
I_1&=\int_{\R}\Big|x^2\int_{\R}(\rho_0\mathbf{I}_{[-b+1,b-1]})(y)J_{\sqrt{\v}}(x-y)\,\dd y-\int_{\R}y^2(\rho_0\mathbf{I}_{[-b+1,b-1]})(y)J_{\sqrt{\v}}(x-y)\,\dd y\Big|\,\dd x\nonumber\\
&=\int_{\R}\Big|\int_{\R}(x^2-y^2)(\rho_0\mathbf{I}_{[-b+1,b-1]})(y)J_{\sqrt{\v}}(x-y)\,\dd y\Big|\,\dd x\nonumber\\
&\leq\int_{\R}\Big(\int_{\R}|x^2-y^2|J_{\sqrt{\v}}(x-y)\,\dd x\Big)|\rho_0(y)|\,\dd y.
\end{align*}
When $|y|<1,$ we have
\begin{align*}
\int_{\R}|x^2-y^2|J_{\sqrt{\v}}(x-y)\,\dd x&=\int_{|x-y|<\sqrt{\v}}|x^2-y^2|J_{\sqrt{\v}}(x-y)\,\dd x\nonumber\\
&\leq C\int_{|x-y|<\sqrt{\v}}|x-y|(|x|+|y|)J_{\sqrt{\v}}(x-y)\,\dd x\leq C\sqrt{\v}.
\end{align*}
When $|y|\geq1,$ we have
\begin{align*}
\int_{\R}|x^2-y^2|J_{\sqrt{\v}}(x-y)\,\dd x
&=|y|^2\int_{\R}\big|1-(\frac{x}{y})^2\big|J_{\sqrt{\v}}(x-y)\,\dd x\nonumber\\
&=|y|^2\int_{\R}\big|1-(1+\frac{s}{y})^2\big|J_{\sqrt{\v}}(s)\,\dd s\nonumber\\
&=|y|^2\int_{|s|<\sqrt{\v}}\big|1-(1+\frac{s}{y})^2\big|J_{\sqrt{\v}}(s)\,\dd s
\leq C\sqrt{\v}|y|^2.
\end{align*}
Since $M_2<\infty$, we conclude that
\begin{align*}
\int_{\R} x^2\big|\big((\rho_0\mathbf{I}_{[-b+1,b-1]})\ast J_{\sqrt{\v}}\big)(x)-\rho_0(x))\big|\,\dd x\rightarrow0
\qquad\text{ as $\varepsilon\rightarrow0^+$}.
\end{align*}

\smallskip
\noindent{\emph{Step 4(iii).}} Using the properties of the mollifier, we have
\begin{align*}
&\big\|\big((\rho_0\mathbf{I}_{[-b+1,b-1]})^{\frac{1}{\beta}}\ast J_{\sqrt{\v}}+\v e^{-x^2}\big)^{\beta}
-(\rho_0\mathbf{I}_{[-b+1,b-1]})\ast J_{\sqrt{\v}}\big\|_{L^1}\nonumber\\
&\leq\big\|\big((\rho_0\mathbf{I}_{[-b+1,b-1]})^{\frac{1}{\beta}}\ast J_{\sqrt{\v}}+\v e^{-x^2}\big)^{\beta}
&\quad+\big\|(\rho_0\mathbf{I}_{[-b+1,b-1]})-(\rho_0\mathbf{I}_{[-b+1,b-1]})\ast J_{\sqrt{\v}}\big\|_{L^1}
\rightarrow0 \qquad \text{ as $\varepsilon\rightarrow0^+$}.
\end{align*}

We now prove
\begin{align*}
\int_{\R}x^2\big|\big((\rho_0\mathbf{I}_{[-b+1,b-1]})^{\frac{1}{\beta}}\ast J_{\sqrt{\v}})(x)
+\v e^{-x^2}\big)^{\beta}-\rho_0(x)\big|\,\dd x\rightarrow 0\qquad \text{ as $\varepsilon\rightarrow0^+$}.
\end{align*}
Notice first that
\begin{align}\label{A.188}
&\int_{\R} x^2\big|\big(((\rho_0\mathbf{I}_{[-b+1,b-1]})^{\frac{1}{\beta}}\ast J_{\sqrt{\v}})(x)+\v e^{-x^2}\big)^{\beta}
-\rho_0(x)\big|\,\dd x\nonumber\\
&\leq\int_{\R}x^2\big|\big(((\rho_0\mathbf{I}_{[-b+1,b-1]})^{\frac{1}{\beta}}\ast J_{\sqrt{\v}})(x)+\v e^{-x^2}\big)^{\beta}
-\big((\rho_0\mathbf{I}_{[-b+1,b-1]})\ast J_{\sqrt{\v}}\big)(x)\big|\,\dd x^1\nonumber\\
&\quad+\int_{\R}x^2\big|\big((\rho_0\mathbf{I}_{[-b+1,b-1]})\ast J_{\sqrt{\v}}\big)(x)-\rho_0(x)\big|\, \dd x
\rightarrow0\qquad \,\, \text{ as $\varepsilon\rightarrow0^+$}.
\end{align}
As in
Step 4(ii),
we can prove for the second term on the right-hand side of \eqref{A.188} that
\int_{\R} x^2\big|\big((\rho_0\mathbf{I}_{[-b+1,b-1]})\ast J_{\sqrt{\v}}\big)(x)-\rho_0(x)\big|\,\dd x\rightarrow 0
\qquad \text{ as $\varepsilon\rightarrow0^+$}.
For the first term in \eqref{A.188}, we have
\begin{align*}
&\big((\rho_0\mathbf{I}_{[-b+1,b-1]})^{\frac{1}{\beta}}\ast J_{\sqrt{\v}}+\v e^{-x^2}\big)^{\beta}
-(\rho_0\mathbf{I}_{[-b+1,b-1]})\ast J_{\sqrt{\v}}\nonumber\\
&=\big((\rho_0\mathbf{I}_{[-b+1,b-1]})^{\frac{1}{\beta}}\ast J_{\sqrt{\v}}+\v e^{-x^2}\big)^{\beta}
-\big((\rho_0\mathbf{I}_{[-b+1,b-1]})^{\frac{1}{\beta}}\ast J_{\sqrt{\v}}\big)^{\beta}\nonumber\\
&\quad+\big((\rho_0\mathbf{I}_{[-b+1,b-1]})^{\frac{1}{\beta}}\ast J_{\sqrt{\v}}\big)^{\beta}
-(\rho_0\mathbf{I}_{[-b+1,b-1]})\ast J_{\sqrt{\v}}.
\end{align*}

Notice that
\begin{align*}
&\Big|\big((\rho_0\mathbf{I}_{[-b+1,b-1]})^{\frac{1}{\beta}}\ast J_{\sqrt{\v}}+\v e^{-x^2}\big)^{\beta}
-\big((\rho_0\mathbf{I}_{[-b+1,b-1]})^{\frac{1}{\beta}}\ast J_{\sqrt{\v}}\big)^{\beta}\Big|\nonumber\\
&\leq\beta\v e^{-x^2}\big((\rho_0\mathbf{I}_{[-b+1,b-1]})^\frac{1}{\beta}\ast J_{\sqrt{\v}}+\v e^{-x^2}\big)^{\beta-1},
\end{align*}
which leads directly to
\begin{align*}
\lim_{\v\rightarrow0+}
\int_{\R} x^2\big|\big(((\rho_0\mathbf{I}_{[-b+1,b-1]})^{\frac{1}{\beta}}\ast J_{\sqrt{\varepsilon}})(x)+\v e^{-x^2}\big)^{\beta}
-\big(((\rho_0\mathbf{I}_{[-b+1,b-1]})^{\frac{1}{\beta}}\ast J_{\sqrt{\v}})(x)\big)^{\beta}\big|\,\dd x=0.
\end{align*}
Thus, it remains to prove
\lim_{\v\rightarrow0+}
\int_{\R} x^2\big|\big(((\rho_0\mathbf{I}_{[-b+1,b-1]})^{\frac{1}{\beta}}\ast J_{\sqrt{\varepsilon}})(x)\big)^{\beta}
-\rho_0(x)\big|\,\dd x=0.

Since $M_2<\infty$, then, for any $\varepsilon>0$, there exists $N>1$ such that
\begin{align*}
\int_{|x|>N-1}x^2\rho_0(x)\,\dd x<\varepsilon.
\end{align*}
Using the H\"{o}lder inequality and the fact that $\int_{\R}J_{\sqrt{\varepsilon}}(y)\,\dd y=1$ for $0<\v<1$,
we have
\begin{align*}
&\int_{|x|>N}x^2\Big(\int_{\R}(\rho_0\mathbf{I}_{[-b+1,b-1]})^{\frac{1}{\beta}}(x-y)J_{\sqrt{\v}}(y)\,\dd y\Big)^{\beta}\,\dd x\nonumber\\
&\leq \int_{|x|>N}x^2\int_{\R}\rho_0(x-y)J_{\sqrt{\v}}(y)\,\dd y\dd x=\int_{\R}\Big(\int_{|x|>N}x^2\rho_0(x-y)\,\dd x\Big)J_{\sqrt{\v}}(y)\,\dd y\nonumber\\
&=\int_{\R}\Big(\int_{|y+\xi|>N}(y^2+\xi^2+2y\xi)\rho_0(\xi)\,\dd \xi\Big)J_{\sqrt{\v}}(y)\,\dd y\nonumber\\
&\leq2\int_{|y|<\sqrt{\v}}\Big(\int_{|y+\xi|>N}(\v^2+\xi^2)\rho_0(\xi)\,\dd\xi\Big)J_{\sqrt{\v}}(y)\,\dd y< 2(\v^2+\v)\rightarrow 0
\qquad \text{ as $\varepsilon\rightarrow0^+$}.\nonumber
\end{align*}
Therefore, it suffices to show
\begin{align}\label{AA}
\lim_{\v\rightarrow0+}\int_{|x|\leq N}x^2\Big|\Big(\int_{\R}(\rho_0\mathbf{I}_{[-b+1,b-1]})^{\frac{1}{\beta}}(x-y)J_{\sqrt{\v}}(y)\,\dd y\Big)^{\beta}-\rho_0(x)\Big|\,\dd x=0.
\end{align}

Note that
\begin{align*}
&\int_{|x|\leq N}x^2\Big|\Big(\int_{\R}(\rho_0\mathbf{I}_{[-b+1,b-1]})^{\frac{1}{\beta}}(x-y)J_{\sqrt{\v}}(y)\,\dd y\Big)^{\beta}-\rho_0(x)\Big|\,\dd x\nonumber\\
&\leq N^2\int_{|x|<N}\Big(\int_{|y|<\sqrt{\v}}\rho_0^{\frac{1}{\beta}}(x-y)J_{\sqrt{\v}}(y)\,\dd y\ \Big)^{\beta}-\rho_0(x)\Big|\,\dd x.
\end{align*}
Using H\"{o}lder's inequality, we have
\begin{align*}
\Big(\int_{|y|<\sqrt{\varepsilon}}\rho_0^{\frac{1}{\beta}}(x-y)J_{\sqrt{\v}}(y)\,\dd y\Big)^{\beta}
&\leq \int_{|y|<\sqrt{\v}}\rho_0(x-y)J_{\sqrt{\v}}(y)\,\dd y\nonumber\\
&\leq \sup_{0<\varepsilon<1}\int_{|y|<\sqrt{\v}}\rho_0(x-y)J_{\sqrt{\v}}(y)\,\dd y:=\Phi^{\ast}\rho_0(x).
\end{align*}

Notice that
\Phi^{\ast}\rho_0(x)\leq C\|J\|_{L^1}\mathcal{M}\rho_0(x),
where $\mathcal{M}$ is the Hardy-Littlewood maximal operator and is of strong $(p,p)$ type for $1<p<\infty$.
Then, using Lebesgue's dominated convergence theorem and
\lim_{\v\rightarrow0^+}\big((\rho_0\mathbf{I}_{[-b+1,b-1]})^{\frac{1}{\beta}}\ast J_{\sqrt{\v}})(x)
we conclude \eqref{AA}, which implies \eqref{A.6}.

\medskip
\noindent{\emph{Step 5.}} We now prove \eqref{A.62}. We first note that
\begin{align*}
&\int_{\R}\Big|\tilde{\rho}^{\v}_0(x)(W\ast\tilde{\rho}^{\v}_0)(x)-\rho_0(x)(W\ast\rho_0)(x)\Big|\,\dd x\nonumber\\
&\leq \int_{\R}|\tilde{\rho}^{\v}_0(x)-\rho_0(x)||(W\ast\tilde{\rho}^{\v}_0)(x)|\,\dd x
+\int_{\R}\rho_0(x)|(W\ast(\tilde{\rho}_0-\rho))(x)|\,\dd x:=I+II.
\end{align*}
\begin{align*}
&|(W\ast\tilde{\rho}^{\v}_0)(x)|=\Big|\int_{\R}W(x-y)\tilde{\rho}^{\v}_0(x)\,\dd x\Big|
\leq \int_{\R}\big(x^2+y^2+\frac{1}{2}\big)\tilde{\rho}^{\v}_0(y)\,\dd y,\\
&\tilde{\rho}^{\v}_0(x)=\Big(\int_{\R}((\rho_0\mathbf{I}_{[-b+1,b-1]})(x-\bar{y}))^{\alpha-\frac{1}{2}}J_{\sqrt{\v}}(\bar{y})\,\dd \bar{y} +\v e^{-x^2}\Big)^{\beta}\nonumber\\
&\qquad\,\,\leq C\Big(\int_{\R}(\rho_0\mathbf{I}_{[-b+1,b-1]})^{\frac{1}{\beta}}(x-\bar{y})J_{\sqrt{\v}}(\bar{y})\,\dd \bar{y}\Big)^{\beta}
+C\big(\v e^{-x^2}\big)^{\beta},
\end{align*}
we have
\begin{align}\label{A.27}
&\int_{\R}\big(x^2+y^2+\frac{1}{2}\big)\tilde{\rho}^{\v}_0(y)\,\dd y\nonumber\\
&=(x^2+\frac{1}{2})\int_{\R}\tilde{\rho}^{\v}_0(y)\,\dd y+\int_{\R}y^2\tilde{\rho}^{\v}_0(y)\,\dd y\nonumber\\
&\leq C(x^2+\frac{1}{2})\Big(\int_{\R}\Big(\int_{\R}(\rho_0\mathbf{I}_{[-b+1,b-1]})^{\frac{1}{\beta}}(y-\bar{y})J_{\sqrt{\v}}(\bar{y})\,\dd \bar{y}\Big)^{\beta}\,\dd y+\int_{\R}(\v e^{-y^2})^{\beta}\,\dd y\Big)\nonumber\\
&\quad+ C\int_{\R}\Big(\int_{\R}|y|^{\frac{2}{\beta}}(\rho_0\mathbf{I}_{[-b+1,b-1]})^{\frac{1}{\beta}}(y-\bar{y})J_{\sqrt{\v}}(\bar{y})\,\dd \bar{y}\Big)^{\beta}\,\dd y+ C\int_{\R}y^2(\v e^{-y^2})^{\beta}\,\dd y\nonumber\\
&\leq C(x^2+\frac{1}{2})\Big(\int_{\R}\Big(\int_{\R}\rho_0(y-\bar{y})(J_{\sqrt{\v}}(\bar{y}))^{\beta}\,\dd y\Big)^{\frac{1}{\beta}}\,\dd \bar{y}\Big)^{\beta}+\v^{\beta}\Big)\nonumber\\
&\quad+ C\Big(\int_{\R}\Big(\int_{\R}|y|^2\rho_0(y-\bar{y})(J_{\sqrt{\v}}(\bar{y}))^{\beta}\,\dd y\Big)^{\frac{1}{\beta}}\,\dd \bar{y}\Big)^{\beta}+C \v^{\beta}\nonumber\\
&\leq CM^{\beta}(x^2+\frac{1}{2})+C\v^{\beta}(x^2+\frac{1}{2})+C\int_{\R}\Big(|\bar{y}|^{\frac{2}{\beta}}M+M_2\Big)J_{\sqrt{\v}}(\bar{y})\,\dd \bar{y}\nonumber\\
&\leq C(x^2+1).
\end{align}

Using \eqref{A.27}, we have
\begin{align}
I&=\int_{\R}|\tilde{\rho}^{\v}_0(x)-\rho_0(x)||(W\ast\tilde{\rho}^{\v}_0)(x)|\,\dd x\nonumber\\
&\leq C\int_{\R}(x^2+1)|\tilde{\rho}^{\v}_0(x)-\rho_0(x)|\,\dd x\rightarrow 0\qquad \text{ as $\varepsilon\rightarrow0^+$}.\label{A.28}
\end{align}

\begin{align*}
\big|(W\ast(\tilde{\rho}^{\v}_0(x)-\rho_0))(x)\big|
&=\Big|\int_{\R}W(x-y)(\tilde{\rho}^{\v}_0(y)-\rho_0(y))\,\dd y\Big|\nonumber\\
&\leq \int_{\R}(x^2+y^2+1)|\tilde{\rho}^{\v}_0(y)-\rho_0(y)|\,\dd y\nonumber\\
&=(x^2+1)\int_{\R}|\tilde{\rho}^{\v}_0(y)-\rho_0(y)|\,\dd y+\int_{\R}y^2|\tilde{\rho}^{\v}_0(y)-\rho_0(y)|\,\dd y,
\end{align*}
\begin{align}\label{A.29}
II&=\int_{\R}\rho_0(x)\big|(W\ast(\tilde{\rho}_0-\rho))(x)\big|\,\dd x\nonumber\\
&\leq\int_{\R}(x^2+1)\rho_0(x)\,\dd x\int_{\R}|\tilde{\rho}^{\v}_0(y)-\rho_0(y)|\,\dd y+\int_{\R}\rho_0(x)\,\dd x+
\int_{\R}y^2|\tilde{\rho}^{\v}_0(y)-\rho_0(y)|\,\dd y\nonumber\\
&=(M_2+M)\|\tilde{\rho}^{\v}_0-\rho_0\|_{L^1}+M\|y^2(\tilde{\rho}^{\v}_0(y)-\rho_0(y))\|_{L^1}\rightarrow 0
\qquad \text{ as $\varepsilon\rightarrow0^+$}.
\end{align}

Combining \eqref{A.28} and \eqref{A.29}, we conclude \eqref{A.62}.

\medskip
\noindent{\emph{Step 6.}} We now prove \eqref{A.63}.
For $\tilde{W}:=W+\frac{1}{2}$, then $0\leq\tilde{W}(x-y)\leq x^2+y^2+1.$
\hat{\rho}_0(x):=\Big(\int_{\R}(\rho_0\mathbf{I}_{[-b+1,b-1]})^{\alpha-\frac{1}{2}}(x-y)J_{\sqrt{\v}}(y)\,\dd y\Big)^{\beta}.
Then we have
\begin{align}\label{A.291}
\tilde{\rho}^{\v}_0(x)\leq C\big(\hat{\rho}_0(x)+(\v e^{-x^2})^{\beta}\big).
\end{align}
It follows from \eqref{A.291} that
\begin{align}\label{A.30}
&\int_{\R}\tilde{\rho}^{\v}_0(x)(W\ast\tilde{\rho}^{\v}_0)(x) \,\dd x
\leq\int_{\R}\tilde{\rho}^{\v}_0(x)(\tilde{W}\ast\tilde{\rho}^{\v}_0)(x) \,\dd x\nonumber\\
&\leq C\int_{\R}(\hat{\rho}_0(x)+\v^{\beta}e^{-\beta x^2})\Big(\int_{\R}\tilde{W}(x-y)\hat{\rho}_0(y)\,\dd y+\v^{\beta}\int_{\R}\tilde{W}(x-y)e^{-\beta y^2}\,\dd y\Big)\,\dd x\nonumber\\
&=C\int_{\R}\hat{\rho}_0(x)\int_{\R}\tilde{W}(x-y)\hat{\rho}_0(y)\,\dd y \dd x+C\v^{\beta}\int_{\R}\hat{\rho}_0(x)\int_{\R}\tilde{W}(x-y)e^{\beta y^2}\,\dd y \dd x\nonumber\\
&\quad+C\v^{\beta}\int_{\R}e^{-\beta x^2}\int_{\R}\tilde{W}(x-y)\hat{\rho}_0(y)\,\dd y\dd x+C\v^{2\beta}\int_{\R}e^{-\beta x^2}\int_{\R}\tilde{W}(x-y)e^{-\beta y^2}\,\dd y\dd x\nonumber\\
\end{align}

Using H\"{o}lder's inequality, we have
\begin{align*}
\hat{\rho}_0(x)&=\Big(\int_{\R}(\rho\mathbf{I}_{[-b+1,b-1]})^{\alpha-\frac{1}{2}}(x-y)J^{\frac{1}{\beta}}_{\sqrt{\v}}(y)J^{\frac{1}{\beta'}}_{\sqrt{\v}}(y)\,\dd y\Big)^{\beta}\nonumber\\
&\leq\Big(\int_{\R}\rho_0(x-y)J_{\sqrt{\v}}(y)\,\dd y\Big)\Big(\int_{\R}J_{\sqrt{\v}}(y)\,\dd y\Big)^{\frac{\beta}{\beta'}}\leq\int_{\R}\rho_0(x-y)J_{\sqrt{\v}}(y)\,\dd y,
\end{align*}
where we have used $\frac{1}{\beta}+\frac{1}{\beta'}=1,$ and the property of mollifier: $\int_{\R}J_{\sqrt{\v}}(y)\,\dd y=1.$
\int_{\R}\hat{\rho}_0(x)\,\dd x\leq\int_{\R}\int_{\R}\rho_0(x-y)J_{\sqrt{\v}}(y)\,\dd x\dd y=M,
\begin{align*}
\int_{\R}x^2\hat{\rho}_0(x)\,\dd x
&\leq\int_{\R}x^2\int_{\R}\rho_0(x-y)J_{\sqrt{\v}}(y)\,\dd y\dd x=\int_{\R}\Big(\int_{\R}x^2\rho_0(x-y)\,\dd x\Big)J_{\sqrt{\v}}(y)\,\dd y\nonumber\\
&=\int_{\R}\Big(\int_{\R}(y+\bar{x})^2\rho_0(\bar{x})\,\dd \bar{x}\Big)J_{\sqrt{\v}}(y)\,\dd y\nonumber\\
&\leq2\int_{\R}(y^2M+M_2)J_{\sqrt{\v}}(y)\,\dd y\leq C,
\end{align*}
where we have used $\int_{\R}y^2J_{\sqrt{\v}}(y)\,\dd y\leq\v.$

Now, we estimate \eqref{A.30} term by term.
\begin{align*}
\mathcal{T}_1&\leq C\int_{\R}\hat{\rho}_0(x)\int_{\R}(x^2+y^2+1)\hat{\rho}_0(y)\,\dd y\dd x\nonumber\\
&\leq CM\int_{\R}\hat{\rho}_0(x)(x^2+1)\,\dd x+C\int_{\R}\hat{\rho}_0(x)M_2\,\dd x\leq C(M,M_2).\\
\mathcal{T}_2&\leq C\v^{\beta}\int_{\R}\hat{\rho}_0(x)\int_{\R}(x^2+y^2+1)e^{-\beta y^2}\,\dd y\dd x\leq C\v^{\beta}\int_{\R}\hat{\rho}_0(x)(x^2+1)\,\dd x\leq C\v^{\beta}.\\
\mathcal{T}_3&\leq
C\v^{\beta}\int_{\R}e^{-\beta x^2}\int_{\R}(x^2+y^2+1)\hat{\rho}_0(y)\,\dd y\dd x\leq C\v^{\beta}\int_{\R}e^{-\beta x^2}(x^2M+M_2+M)\,\dd x\leq C\v^{\beta}.\\
\mathcal{T}_4&\leq
C\v^{2\beta}\int_{\R}e^{-\beta x^2}\int_{\R}(x^2+y^2+1)e^{-\beta x^2}\,\dd y\dd x\leq C\v^{2\beta}.
\end{align*}
Combining these estimates together, we conclude \eqref{A.63}.

\medskip
Now we define $\bar{\rho}_{0}^{\v}(x)$ by
\begin{align}\label{A.40}
\bar{\rho}_{0}^{\v}(x)
\end{align}

\begin{remark}
We have multiplied by a cut-off function $\mathbf{I}_{[-b,b]}(x)$ in \eqref{A.40}
such that $(W\ast\bar{\rho}_{0}^{\v})
(x)$ is meaningful.
Otherwise, when $|y|>b$, $\bar{\rho}_{0}^{\v}(y)=b^{-\frac{1}{\gamma}}$
so that $(W\ast\bar{\rho}_{0}^{\v})(x)\sim\int_{|y|>b}W(x-y)\,\dd y \, b^{-\frac{1}{\gamma}}=\infty$.
\end{remark}

Then we can obtain the following lemma.
\begin{lemma}\label{lemA.6}
Let $b=\v^{-p}, p>\frac{\gamma}{\gamma-\alpha},$  and $q\in\{1,\gamma,2\alpha-1\}.$
The smooth function $\bar{\rho}_{0}^{\v}(x)$ defined in \eqref{A.40} satisfies the
following properties: There exists $C_0>0$ independent of $\v$ but may depend on $(\mathcal{E}_0, M, M_2, \gamma,\alpha)$ such that
\begin{align*}
&\bar{\rho}^{\v}_0(b)=\bar{\rho}^{\v}_0(-b)\leq C_0 b^{-\frac{1}{\gamma}},\\
&\int^{b}_{-b} |\tilde{\rho}_0^{\v,b}(x)-\bar{\rho}_0^\v(x)|^q\,\dd x
+\int^{b}_{-b} \Big| (\tilde{\rho}_{0}^{\v}(x))^{\alpha-\frac{1}{2}}
-(\bar{\rho}_{0}^{\v}(x))^{\alpha-\frac{1}{2}}\Big|^{\frac{2q}{2\alpha-1}} \,\dd x\rightarrow0 \qquad\mbox{as $\v\rightarrow 0^+$},\\
&\int^b_{-b}\bar{\rho}^{\v}_0(x)\,\dd x\rightarrow\int_{\R}\rho_0(x)\,\dd x=M\qquad
\text{ as } \v\rightarrow 0^+,\\
&\v^2\int^b_{-b}\Big|\big((\bar{\rho}_0^{\v}(x))^{\alpha-\frac{1}{2}}\big)_x\Big|^2 \,\dd x
\leq C_0\v ,
\\
&\int^{b}_{-b} x^2|\tilde{\rho}_0^{\v}(x)-\bar{\rho}_0^\v(x)|\,\dd x\rightarrow 0
\qquad\,\,\mbox{as $\v\rightarrow 0^+,$}\\
&\int^b_{-b}|\bar{\rho}^{\v}_0(x)W\ast\bar{\rho}^{\v}_0(x)-\tilde{\rho}^{\v}_0(x)W\ast\tilde{\rho}^{\v}_0(x)|\,\dd x\rightarrow 0 \qquad\,\,\text{ as $\varepsilon\rightarrow0^+$}.
\end{align*}
\end{lemma}

\smallskip
Since $\dis \int^b_{-b} \tilde{\rho}_{0}^{\v}(x)\,\dd x\neq M$ in general, we define
\begin{align}\label{A.23-1}
\rho_0^{\v}(x):=\frac{M }{\int^b_{-b} \bar{\rho}_{0}^{\v}(x)\,\dd x}
\bar{\rho}_{0}^{\v}(x).
\end{align}
Combining \eqref{A.23-1} with Lemmas \ref{lemA.2} and \ref{lemA.6}, we have

\begin{lemma}
Let $q \in\{1,\gamma,2\alpha-1\}$ and $\frac{2}{3}<\alpha\leq1.$
The smooth function $\rho_{0}^{\v}(x)$ defined in \eqref{A.23-1} satisfies the following properties: There exists $C_0>0$
independent of $\v\in (0,1]$ but may depend on $(\mathcal{E}_0, M, M_2, \gamma,\alpha)$ such that
\begin{align*}
&(\rho^{\v}_0(b))^{\gamma}\, b=(\rho^{\v}_0(-b))^{\gamma}\, b\leq C_0,\\
&\int^b_{-b} \big|\rho_0^{\v}(x)-\rho_0(x)\big|^q\, \dd x+\int^b_{-b} \Big| (\rho_{0}^{\v}(x))^{\alpha-\frac{1}{2}}
-(\rho_{0}(x))^{\alpha-\frac{1}{2}}\Big|^{\frac{2q}{2\alpha-1}} \,\dd x\rightarrow0 \qquad\mbox{as $\varepsilon\rightarrow0^+$},\\
&\v^2\int^b_{-b}\Big|\big((\rho^{\v}_0(x))^{\alpha-\frac{1}{2}}\big)_x\Big|^2 \dd x
\leq C_0\v,
\\
&\int^{b}_{-b} x^2|\rho_0^{\v}(x)-\rho_0(x)|\,\dd x\rightarrow 0\qquad\mbox{as $\v\rightarrow 0^+,$}\\
&\int^b_{-b}|\rho^{\v}_0(x)(W\ast\rho^{\v}_0)(x)-\rho_0(x)(W\ast\rho_0)(x)|\,\dd x\rightarrow 0
\qquad\text{ as $\varepsilon\rightarrow0^+$}.
\end{align*}
\end{lemma}

\medskip

\subsection{The initial velocity} Next, we construct the approximate initial data for the velocity. We first define
\begin{align}
(x-y)\,J_{\sqrt{\varepsilon}}(y)\,\dd y,\label{A.31}
\end{align}
for which the following properties are valid.
\begin{lemma}\label{lemA.5}
Let $\frac{2}{3}<\alpha\leq1$. Then $\bar{u}^\v_0(x)$ defined in \eqref{A.31}  satisfies
\begin{align*}
&{\rm supp}\, \bar{u}^{\v}_0(x) \subset \{x\in \R\, :\,-b+1\leq x\leq b-1\},\\[1.5mm]
&\int_{\mathbb{R}}\rho_0^\v(x) |\bar{u}_0^\v(x)|^2\,\dd x
\equiv\int_{\mathbb{R}}\frac{|m_0(x)|^2}{\rho_0(x)}\,\dd x
\qquad\,\,\,\mbox{for any $\v\in (0,1]$}, \\[1.5mm]
&\lim_{\v\rightarrow0^+} \|\rho^\v_0 \bar{u}^\v_0-m_0\|_{L^1(\mathbb{R})}=0.
\end{align*}
\end{lemma}

The proof of Lemma \ref{lemA.5} is similar to that in \cite[Lemma A.8]{Chen2023} and \cite[Lemma 5.4]{He},
so we omit it for brevity.

\smallskip
We still need to modify $\bar{u}_{0}^{\v}(x)$ so that it satisfies the stress-free boundary condition \eqref{2.5}
at $x=\pm b$. To this end, we define a cut-off function $S=S(z)\in C^\infty(\mathbb{R})$ that satisfies
\begin{align}\label{A.19-1}
0\quad   &\mbox{if $z\in(-\infty,0)$},\\
1\quad  &\mbox{if $z\in(2,\infty)$},\\
\mbox{monotonically  increasing}  &\mbox{if $z\in [0,2]$}.
\end{cases}
\end{align}
Then we define
\begin{equation}\label{A.37}
\int^b_x\frac{P(\rho^{\v}_0(z))}{\mu(\rho^{\v}_0(z))}\,\dd z-\frac{1}{\varepsilon}\,S(-4(x-(b-\frac{1}{2})))
\int^{-b}_{x}\frac{P(\rho^{\v}_0(z))}{\mu(\rho^{\v}_0(z))}\,\dd z.
\end{equation}

It is direct to check that  $u_{0}^{\v}(x)\in C^{\infty}([-b,b])$ satisfies
the following boundary conditions{\rm :}
\begin{align*}
u_{0}^{\v}(\pm b)=0,\qquad \big(P((\rho^{\v}_0(x)))-\v\mu(\rho^{\v}_0(x))(u^{\v}_0(x))_x\big)\big|_{x=\pm b}=0.
\end{align*}
A direct calculation following \eqref{A.37} implies
\begin{align}\label{A.1111}
&\int^b_{-b}\Big|\sqrt{\rho_{0}^{\v}(x)}u_{0}^{\v}(x)-\sqrt{\rho_{0}^{\v}(x)}\bar{u}_{0}^{\v}(x)\Big|^2 \,\dd x\nonumber\\
&\leq\frac{C}{\varepsilon^2}\Big(\int^{-b+\frac{1}{2}}_{-b}\rho^{\v}_0(x)\big|S(-4x-4b+2)\int^x_{-b}(\rho^{\v}_0(z))^{\gamma-\alpha}\,\dd z\big|^2\,\dd x\nonumber\\
&\qquad\quad +\int^b_{b-\frac{1}{2}}\rho^{\v}_0(x)\big|S(4x-4b+2)\int^b_x(\rho^{\v}_0(z))^{\gamma-\alpha}\,\dd z\big|^2 \,\dd x\Big)\nonumber\\
&\leq\frac{C}{\v^2}\Big(\int^b_{b-\frac{1}{2}}b^{-\frac{1}{^{\gamma}}}\big|\int^b_xb^{-\frac{\gamma-\alpha}{\gamma}}\,\dd z\big|^2\,\dd x+\int^{-b+\frac{1}{2}}_{-b}b^{-\frac{1}{^{\gamma}}}\big|\int^{x}_{-b}b^{-\frac{\gamma-\alpha}{\gamma}}\,\dd z\big|^2\,\dd x\Big)\nonumber\\
&\leq\frac{C}{\v^2}b^{-\frac{2(\gamma-\alpha)+1}{\gamma}}\leq C\v^{{\frac{2(\gamma-\alpha)+1}{\gamma}p}-2}\rightarrow 0 \qquad\text{ as } \varepsilon\rightarrow 0^+,
\end{align}
for $p>\frac{\gamma}{\gamma-\alpha},$ where we have used \eqref{A.19-1} and $b=\v^{-p}.$

Therefore, similarly to those as in [12, 28], using Lemma \ref{lemA.5} and \eqref{A.1111},
we conclude
\begin{lemma}\label{lemA.7}
For fixed $\v>0$,
\begin{align*}
&\lim_{\varepsilon\rightarrow0^+}\int^b_{-b}\big|\sqrt{\rho_{0}^{\v}(x)}u_{0}^{\v}(x)\big|^2\,\dd x
=\int_{\mathbb{R}} \big|\frac{m_0(x)}{\sqrt{\rho_0(x)}}\big|^2\,\dd x,\\
&\lim_{\varepsilon\rightarrow0^+}\int^b_{-b}|\rho^{\v}_0(x)u^{\v}_0(x)-m_0(x)|\,\dd x=0.
\end{align*}
\end{lemma}

\smallskip

In the previous argument, we have already defined the cut-off for the density. Now,
with $\rho^{\v}_0(x)$ and $u_0^{\v}(x)$ defined respectively in \eqref{A.23-1} and \eqref{A.37},
we also need to define the cut-off for the velocity function. For $b\gg1$,  define
\begin{align}\label{A.61}
\big(\rho^{\v}_0, u^{\v}_0\big)(x):=\big(\rho_{0}^{\v}(x), u^{\v}_0(x)\big) \mathbf{I}_{[-b,b]}(x).
\end{align}
Then, collecting all the above estimates, we have the following results.

\begin{lemma}\label{propA.1}
Let $\frac{2}{3}<\alpha\leq1,$ $b=\v^{-p},$ $p>\frac{\gamma}{\gamma-\alpha}$,
$q\in \{1,\gamma\}$, and $\v\in(0,1].$
Let $\big(\rho^{\v}_0, u^{\v}_0\big)(x)$ be the functions defined in \eqref{A.61}
so that $(\rho^{\v}_0, u^{\v}_0)\in C^\infty([-b,b]).$
\begin{enumerate}
\item[\rm (i)]  For all $\v\in(0,1]$,
\begin{align*}
&\,\, \, b^{\frac{1}{\gamma}}\rho^{\v}_0(-b)=b^{\frac{1}{\gamma}}\rho^{\v}_0(b)\leq C_0,\quad u^{\v}_0(\pm b)=0,
\quad \big(P(\rho^{\v}_0(x)-\v\mu(\rho^{\v}_0(x)))(u^{\v}_0(x)))_x\big)\big|_{x=\pm b}=0,\\
&\,\,\,\int^b_{-b}\rho^{\v}_0(x)\,\dd x=M, \qquad \mathcal{E}^{\v}_0+\v^{-1}\mathcal{E}^{\v}_1\le C_0,
\end{align*}
where $C_0>0$  is a constant independent of $\v\in (0,1]$ but may depend on $(\mathcal{E}_0, M, M_2, \gamma, \alpha)$.

\smallskip
\item[\rm (ii)]
As $\varepsilon\rightarrow 0^+$,
\begin{align*}
& (\mathcal{E}_0^{\v}, \mathcal{E}_1^{\v})\rightarrow (\mathcal{E}_0, 0), \\
& \int^{\infty}_{-\infty}x^2\rho^{\v}_0(x)\,\dd x\rightarrow \int^{\infty}_{-\infty}x^2\rho_0 (x) \,\dd x=M_2, \\
&(\rho_0^{\v}, m_0^{\v})\rightarrow (\rho_0, m_0)(x) \qquad\,
\mbox{\rm in $L^q_{\rm loc}(\R)\times L^1_{\rm loc}(\R)$},
\end{align*}
where $\mathcal{E}_0^{\v}$ and $\mathcal{E}_1^{\v}$ are defined in \eqref{3.5} and \eqref{3.6}, respectively.
\end{enumerate}
\end{lemma}

\medskip
\noindent{\bf Acknowledgments.}
JAC, GQC, and DF acknowledge support by EPSRC grant EP/V051121/1. JAC was also partially supported by the Advanced Grant Nonlocal-CPD (Nonlocal PDEs for Complex Particle Dynamics: Phase Transitions, Patterns and Synchronization) of the European Research Council Executive Agency (ERC) under the European Union Horizon 2020 research and innovation programme (grant agreement No. 883363), and by EPSRC grant EP/T022132/1. GQC was also partially supported by EPSRC grants EP/L015811/1 and EP/V008854. DF was also partially supported by the National Natural Sciences Foundation of China Grant 12001045. The research of EZ was supported by the EPSRC Early Career Fellowship  EP/V000586/1.

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\end{document}