# Color-flavor locked strange stars with a mirror-dark-matter core and the
observations of compact stars

S.-H. Yang,11footnotetext: Corresponding author. and C.-M. Pi

###### Abstract

We investigate the structure and the tidal deformability of the color-flavor
locked strange stars with a mirror-dark-matter core. Assuming the stars in the
GW170817 event have a mirror-dark-matter core, the observations of the central
compact object within the supernova remnant HESS J1731-347 and the compact
objects in the GW190814 and GW170817 events could be explained simultaneously
with a pairing gap much smaller than 200 MeV, while a pairing gap larger than
about 200 MeV must be employed without the consideration of a mirror-dark-
matter core. More importantly, we find that for the case of the quartic
coefficient $a_{4}<0.589$, if the mirror-dark-matter core in the compact stars
of GW170817 is large enough (e.g., the mass fraction of the mirror-dark-matter
core is larger than 22.8% for $a_{4}=0.55$), the minimum allowed value of the
pairing gap could be less than 46.5 MeV (i.e., one half of the value of the
strange quark mass which is taken as 93 MeV in this paper), which leads to the
result that all astrophysical observations mentioned above could be satisfied
without violating the conformal bound or the recently proposed positive trace
anomally bound.

##  1 Introduction

Strange stars (SSs) made of strange quark matter (SQM), which consists of up
($u$), down ($d$) and strange ($s$) quarks and electrons, ought to exist in
the universe [1, 2, 3, 4, 5, 6]. This is a consequence of the hypothesis
proposed in refs. [7, 8, 9, 10, 11] that SQM may be the true ground state of
baryonic matter. It is also suggested that at sufficiently high density,
quarks of different color and flavor form Cooper pairs with the same Fermi
momentum, and the ground state of quantum chromodynamics with three flavors is
the color-flavor locked (CFL) phase [12, 13, 14, 15]. Thus, SSs might be made
of SQM in the CFL phase.

Recently, the mass of the central compact object within the supernova remnant
HESS J1731-347 is measured to be $M=0.77_{-0.17}^{+0.20}M_{\odot}$ [16]. Such
a small mass implies that it could be an SS rather than a neutron star (NS)
[17, 18, 20, 21, 22, 19, 23, 24, 25], since it is less than $1.17\,M_{\odot}$
which is suggested to be the minimum mass of an NS within the supernova
remnant [26]. What’s more, the HESS J1731-347 compact object might be a SS in
the CFL phase considering its high temperature [18, 19].

CFL SSs are also employed to explain the high mass (2.5-2.67 $M_{\odot}$ [27])
of GW190814’s secondary component [28, 29, 30, 31, 32, 20], since they could
support a larger maximum mass than SSs composed of unpaired SQM [33, 34]
222The compact object in the pulsar binary PSR J0514-4002E with a mass of 2.09
to 2.71 $M_{\odot}$ [35] could also be a CFL SS in this sense.. Oikonomou and
Moustakidis [20] found that both the central compact object within the
supernova remnant HESS J1731-347 and the compact object in the GW190814 event
could be CFL SSs. They also found that these objects can be explained by CFL
SSs without violating the conformal bound or the recently proposed positive
trace anomally bound ($\left\langle\Theta\right\rangle_{\mu_{B}}\geq 0$ [36]).
However, when they investigate the agreement of the CFL SSs with the tidal
deformability observation of GW170817 [37, 38] additionally, it is found that
the pairing gap ($\Delta$) must be extremely large ($>$ 200 MeV, such large
pairing gap was also employed in refs. [29, 30, 39]) and the conformal bound
or the positive trace anomally bound must be violated.

In this paper, we propose an explanation to all of the above-mentioned
astrophysical observations which supposes that these compact stars are SSs in
the CFL phase, and assumes that a mirror-dark-matter (MDM) core exists in the
CFL SSs in GW170817 event. We will show that in this scenario, the
observations of the central compact object within the supernova remnant HESS
J1731-347 and the compact objects in GW190814 and GW170817 events could be
explained simultaneously without the use of the extremely large pairing gap
($\Delta>$ 200 MeV). More importantly, we find that the conformal bound or the
positive trace anomally bound could be satisfied.

Compact stars might contain a dark-matter core made of self-interacting dark
matter [40, 41, 42, 43]. NSs and SSs with a dark-matter core have been studied
extensively [44, 46, 45, 47, 48, 49, 50, 51, 52, 55, 56, 53, 54, 58, 60, 57,
59, 61, 62, 67, 63, 65, 66, 64, 70, 68, 71, 69, 72, 73, 74, 76, 75, 80, 77,
89, 85, 79, 78, 82, 81, 86, 83, 84, 87, 88, 90, 91, 92, 94, 93, 95, 96, 97,
98, 99, 100, 101], and the case of a MDM core has also been studied [102, 103,
105, 104, 107, 106, 108, 109, 110, 111]. Technically, this study is an
extension of our earlier paper [106] in which SSs (made of unpaired SQM) with
a MDM core are investigated.

In this paper, we will study the structure and the tidal deformability of the
CFL SSs with a MDM core and explain the mass and radius of the HESS J1731-347
compact object, the mass of GW190814’s secondary component and the tidal
deformability of GW170817 simultaneously. The paper will be arranged as the
followings. In section 2, we briefly review the equation of state (EOS) of the
CFL SQM and MDM. In section 3, numerical results and discussions are
presented. Finally, the summary is given in section 4.

##  2 Equation of state of the CFL SQM and MDM

For unpaired SQM, the grand canonical potential per unit volume can be written
as [112, 113, 114, 115]

$\Omega_{\rm{free}}=\sum_{i=u,d,s,e}\Omega_{i}^{0}+\frac{3\mu_{b}^{4}}{4\pi^{2}}(1-a_{4})+B_{\rm{eff}},$
(2.1)

where $\Omega_{i}^{0}$ is the grand canonical potential for $u$, $d$, $s$
quarks and electrons described as ideal relativistic Fermi gases. In our
calculation, we choose the strange quark mass to be $m_{s}=93$ MeV [116] while
the masses of the up and down quarks and electrons are set to be zero. The
second term at the right hand side of eq. (2.1) accounts for the perturbative
quantum chromodynamics (QCD) corrections due to gluon mediated quark
interactions to $O(\alpha_{s}^{2})$ [117, 118, 112, 113]. The quartic
coefficient $a_{4}$ represents the degree of quark interaction correction in
perturbative QCD, and $a_{4}=1$ corresponds to no QCD corrections.
$\mu_{b}=(\mu_{u}+\mu_{d}+\mu_{s})/3$ is the baryon chemical potential, where
$\mu_{i}$ ($i=u,d,s$) are the chemical potentials for each species of quarks.
$B_{\rm{eff}}$ is the effective bag constant which includes non-perturbative
QCD effects in a phenomenological way.

At large densities, such as in compact star interiors, up, down and strange
quarks are assumed to undergo pairing and form the so-called CFL phase [12].
For the CFL SQM, the grand canonical potential per unit volume can be written
as [119, 112]

$\Omega_{\rm{CFL}}=\Omega_{\rm{free}}-\frac{3}{\pi^{2}}\Delta^{2}\mu_{b}^{2},$
(2.2)

where $\Delta$ is the pairing gap for the CFL phase, which lacks an accurate
calculation and is within a typical range of $10-100\,\rm{MeV}$ [118]
(however, $\Delta>$ 200 MeV is also employed by some authors [29, 30, 20,
39]). In this paper, we will treat $\Delta$ as a free parameter.

In the CFL phase of SQM, the pairing locks the Fermi momenta of all the quarks
to a single value, and requires the number densities of up, down, and strange
quarks to be equal [120]. Thus, when color and electric neutrality are
imposed, there are no electrons in the CFL phase. The number density of each
species of quarks is given by

$n_{i}=-\frac{\partial\Omega_{\rm{CFL}}}{\partial\mu_{i}},$ (2.3)

which turns out to be

$n_{u}=n_{d}=n_{s}=\frac{1}{\pi^{2}}[\nu^{3}-(1-a_{4})\mu_{b}^{3}+2\Delta^{2}\mu_{b}],$
(2.4)

where $\nu$ is the common Fermi momentum [119]

$\nu=2\mu_{b}-\left(\mu_{b}^{2}+\frac{m_{s}^{2}}{3}\right)^{1/2}.$ (2.5)

The baryon number density is given by

$n_{b}=\frac{1}{3}(n_{u}+n_{d}+n_{s})=n_{u}=n_{d}=n_{s}.$ (2.6)

The pressure of the CFL SQM is given by

$p_{Q}=-\Omega_{\rm{CFL}},$ (2.7)

and the energy density is obtained from

$\epsilon_{Q}=\sum_{i=u,d,s}\mu_{i}n_{i}+\Omega_{\rm{CFL}}=3\mu_{b}n_{b}-p_{Q}.$
(2.8)

Mirror dark matter (MDM) is a stable and self-interacting dark matter
candidate that emerges from the parity symmetric extension of the Standard
Model of particles [121, 122, 123, 124, 125, 126]. For reviews about MDM, see
refs. [127, 128, 129, 130, 131, 132].

In the minimal parity-symmetric extension of the standard model [126, 133],
the group structure is $G\otimes G$, where $G$ is the gauge group of the
standard model. In this model the two sectors are described by the same
lagrangians, the only difference between them is that ordinary particles have
left-handed interactions, while mirror particles have right-handed
interactions. Thus, the microphysics of MDM is the same as that of ordinary
matter and we could use the same EOS of the CFL SQM and MDM.

In this study, the CFL SQM is supposed to be the true ground state of baryonic
matter. Correspondingly, MDM (more exactly, the CFL mirror SQM consisting of
mirror up ($u^{\prime}$), mirror down ($d^{\prime}$) and mirror strange
($s^{\prime}$) quarks) is the ground state of the mirror partner of baryonic
matter.

The CFL SQM interacts with MDM by gravity. The interactions between quarks and
mirror quarks besides gravity have not been studied so far (the neutron–mirror
neutron mixing has been studied [134, 135, 136, 137, 138]). However, if such
interactions exist, it is reasonable to suppose that they are weak and can be
ignored in our study [105, 106].

##  3 Results and discussions

For a given EOS of the CFL SQM, the structure of SSs with a MDM core and their
tidal deformabilities are calculated using the two-fluid formalism. The
details can be found in our earlier paper [106]. The two-fluid formalism is
widely used in the study of the structure and tidal deformability of compact
stars with a dark-matter core [e.g., 104, 106, 63, 69, 102, 103]. In that
formalism, the CFL SQM and MDM do not interact directly, and these two sectors
interact only through the gravitational interaction. Following ref. [106], we
define the mass fraction of MDM by $f_{D}\equiv M_{D}/M$, where $M$ is the
total mass of the star and $M_{D}$ is the mass of the MDM core.

The mass-radius relation of the CFL SSs without a MDM core ($f_{D}=0$) is
shown in figure 1 for $a_{4}=0.6$. All the parameter sets of
[$B^{1/4}_{\rm{eff}}$,$\Delta$] are chosen to satisfy both the "2-flavor line"
and "3-flavor line" constraints, which will be shown later in figure 4. One
can see from figure 1 that for a fixed value of $B_{\rm{eff}}^{1/4}$, both the
maximum mass and the radius of the $1.4\,M_{\odot}$ star increase as the value
of $\Delta$ increases. While, for a fixed value of $\Delta$, both the maximum
mass and the radius of the $1.4\,M_{\odot}$ star increase as the value of
$B_{\rm{eff}}^{1/4}$ decreases.

Figure 1:  The mass-radius relation of the CFL SSs without a MDM core (i.e.,
the mass fraction of MDM $f_{D}=0$) for $a_{4}=0.6$. The black curves are for
$B_{\rm{eff}}^{1/4}=140$ MeV, and the blue curves are for
$B_{\rm{eff}}^{1/4}=170$ MeV. The dotted, solid, dashed lines are for
$\Delta=50$, 130, and 220 MeV, respectively. The orange region shows the mass
and radius estimates for the central compact object within the supernova
remnant HESS J1731-347 [16]. The grey region shows the mass of GW190814’s
secondary component [27]. The red data is the radius of a $1.4\,M_{\odot}$
compact star constrained by the observations of GW170817 [139]. The blue and
green regions show the mass and radius estimates of PSR J0030+0451 derived
from NICER data by Riley et al. [140] and Miller et al. [141], respectively.
The cyan and pink regions show the mass of PSR J0740+6620 [142, 143], and the
radius of it derived from NICER and XMM-Newton data by Riley et al. [144] and
Miller et al. [145], respectively. Figure 2:  The mass-radius relation of the
CFL SSs for $a_{4}=0.6$ and $\Delta=100$ MeV. The black lines and the blue
lines are for $B_{\rm{eff}}^{1/4}=129.4$ MeV and $B_{\rm{eff}}^{1/4}=141.5$
MeV, respectively. The solid, dashed, dotted, dash-dotted lines are for the
mass fraction of MDM $f_{D}=0$, 10%, 20%, and 30%, respectively. The data and
the shaded regions are the same as these in figure 1. Figure 3:  Relation
between the dimensionless tidal deformability of a $1.4\,M_{\odot}$ star
[$\Lambda(1.4)$] and the mass fraction of MDM ($f_{D}$) for $a_{4}=0.6$ and
$\Delta=100$ MeV. The shaded region corresponds to $70<\Lambda(1.4)<580$,
which is given by the observation of GW170817.

Figure 2 shows the mass-radius relation of the CFL SSs for $a_{4}=0.6$ and
$\Delta=100$ MeV with different values of the mass fraction of MDM ($f_{D}$).
The value of $B_{\rm{eff}}^{1/4}$ is choosen to be $B_{\rm{eff}}^{1/4}=129.4$
MeV and $B_{\rm{eff}}^{1/4}=141.5$ MeV so that for the case of SSs without a
MDM core ($f_{D}=0$), all the constraints we focus later can be satisfied
except the one from the tidal deformability observation of GW170817 (see the
black dots in figure 4). One can see that the maximum mass and the radius of
the $1.4\,M_{\odot}$ star decrease with the increasing of $f_{D}$.

Figure 3 shows the relation between the dimensionless tidal deformability of a
$1.4\,M_{\odot}$ star [$\Lambda(1.4)$] and the mass fraction of MDM ($f_{D}$)
for $a_{4}=0.6$ and $\Delta=100$ MeV. One can see that $\Lambda(1.4)$
decreases with the increasing of $B^{1/4}_{\rm{eff}}$ for a given value of
$f_{D}$, and it also decreases with the increasing of $f_{D}$ for a fixed
value of $B^{1/4}_{\rm{eff}}$. For the case of the CFL SSs without a MDM core
(i.e., $f_{D}=0$), the tidal deformability of GW170817 cannot be satisfied for
both $B_{\rm{eff}}^{1/4}=129.4$ MeV and $B_{\rm{eff}}^{1/4}=141.5$ MeV. For
$B_{\rm{eff}}^{1/4}=141.5$ MeV, the tidal deformability of GW170817 can be
satisfied if the value of $f_{D}$ is larger than 15.4%. However, for
$B_{\rm{eff}}^{1/4}=129.4$ MeV, the tidal deformability of GW170817 cannot be
satisfied even if $f_{D}$ is as large as 50%.

We investigate the allowed parameter space of the CFL SQM model according to
the following five constraints [e.g., 146, 112, 147, 114, 148, 149, 106, 150,
20, 151].

Figure 4:  Constraints on $B_{\rm{eff}}^{1/4}$ and $\Delta$ for the CFL SSs
without a MDM core ($f_{D}=0$) for $a_{4}=0.6$. The blue-shadowed area shows
the parameter space which satisfies $M_{\rm max}\geq 2.6\,M_{\odot}$, the red-
shadowed area can satisfy $\Lambda(1.4)\leq 580$, and the purple-shadowed area
can satisfy both of these constraints. Note that all these shadowed areas can
fufill the 2-flavor constaint, the 3-flavor constraint and the HESS J1731-347
constraint. The black dots mark the two parameter sets of
[$B^{1/4}_{\rm{eff}}$(MeV),$\Delta$(MeV)] [namely, (129.4,100) and (141.5,
100)], which are employed for our discussions in figure 2 and figure 3.

First, the existence of the CFL SSs is based on the idea that the energy per
baryon of the CFL SQM is below the energy of the most stable atomic nucleus,
56Fe ($E/A\sim 930$ MeV) [9, 114]. The parameter region that satisfies this
constraint is below the 3-flavor line in figure 4.

The second constraint is given by assuming that non-strange quark matter
(i.e., two-flavor quark matter made of only $u$ and $d$ quarks) in bulk has an
energy per baryon higher than the one of 56Fe, plus a 4 MeV correction coming
from surface effects [1, 5, 114, 106, 20]. By imposing $E/A\geq 934$ MeV on
non-strange quark matter, one ensures that atomic nuclei do not dissolve into
their constituent quarks. The parameter region that satisfies this constraint
is above the 2-flavor line in figure 4.

The above two constraints from nuclear structure must be fulfilled when we
discuss the constraints from astrophysical observations. In other words, we
are only interested in the region between the 3-flavor line and the 2-flavor
line in the following.

The third constraint is from the mass and radius measurement of the HESS
J1731-347 compact object ($M=0.77_{-0.17}^{+0.20}\,M_{\odot}$ and
$R=10.4_{-0.78}^{+0.86}$ km [16]). These data are translated into the
$B^{1/4}_{\rm{eff}}$–$\Delta$ space in Fig. 4, and the region between the two
green lines (the upper one is for [$M\,(M_{\odot}$), $R$ (km)] set [0.97,
9.62] and the lower one is for [0.6, 11.26]) can fulfill this constraint.

The fourth constraint is that the maximum mass of the CFL SSs must be greater
than the mass of GW190814’s secondary component (we take $M_{\rm max}\geq
2.6\,M_{\odot}$ here). By employing this constraint, the allowed parameter
space is limited to the region below the blue line. The blue-shadowed area in
figure 4 shows the parameter space which satisfies $M_{\rm max}\geq
2.6\,M_{\odot}$ and the first three constraints mentioned above, namely, the
2-flavor constaint, the 3-flavor constraint and the HESS J1731-347 constraint.

The last constraint is from the tidal deformability observation of GW170817,
$\Lambda(1.4)=190_{-120}^{+390}$, where $\Lambda(1.4)$ is the dimensionless
tidal deformability of a $1.4\,M_{\odot}$ star. The parameter region that
satisfies this constraint is limited to the area above the red line in figure
4. The red-shadowed area in figure 4 shows the parameter space which satisfies
$\Lambda(1.4)\leq 580$ and the first three constraints mentioned above (the
fourth constraint $M_{\rm max}\geq 2.6\,M_{\odot}$ cannot be satisfied).

As can be seen from figure 4, the blue-shadowed area overlaps with the red-
shadowed area in the purple-shadowed region, which satisfies all the five
constraints. We reach to the conclusion that to satisfy all the five
constraints, $B^{1/4}_{\rm{eff}}$ must be larger than 168.8 MeV and $\Delta$
must be larger than 212.2 MeV, since the $M_{\rm max}=2.6M_{\odot}$ line cuts
across the $\Lambda(1.4)=580$ line at the point (212.2, 168.8).

Figure 5:  Constraints on $B_{\rm{eff}}^{1/4}$ and $\Delta$ of the CFL SQM for
$a_{4}=0.6$. The only difference from figure 4 is that the dashed, dotted and
dash-dotted red lines are added, which are for $\Lambda(1.4)=580$ with
$f_{D}=$ 10%, 20% and 21.6% , respectively.  Figure 6:  Similar to figure 5,
but for $a_{4}=0.8$. The dashed, dotted and dash-dotted red lines are for
$\Lambda(1.4)=580$ with $f_{D}=$ 10%, 20% and 17.2% , respectively.  Figure 7:
Similar to figure 5, but for $a_{4}=0.55$. The dashed, dotted and dash-dotted
red lines are for $\Lambda(1.4)=580$ with $f_{D}=$ 10%, 20% and 22.8% ,
respectively.

Figure 5 is almost the same as figure 4 except that the curves of
$\Lambda(1.4)=580$ for the case of $f_{D}=$ 10%, 20% and 21.6% are presented.
We stress that the blue-shadowed area in figure 5 is for the case of the CFL
SSs without a MDM core ($f_{D}=0$) and satisfies all the five constraints
mentioned above except $\Lambda(1.4)\leq 580$. One can see from figure 5 that
as the value of $f_{D}$ increases, the parameter space region which satisfies
$\Lambda(1.4)<580$ (the region above the $\Lambda(1.4)=580$ line) shifts
downward. Therefore, assuming GW190814’s secondary component and the HESS
J1731-347 compact object are CFL SSs without a MDM core, the parameter space
which satisfies all the five constraints becomes larger if one assumes the
compact stars in GW170817 have a larger MDM core. It turns out that for the
case of $f_{D}<$ 21.6%, the minimum values of $B^{1/4}_{\rm{eff}}$ and
$\Delta$ needed to satisfy all the five constraints decrease with the
increasing of the value of $f_{D}$. For examples, $B^{1/4}_{\rm{eff}}\geq
168.8$ MeV and $\Delta\geq$ 212.2 MeV are needed for $f_{D}=$ 0;
$B^{1/4}_{\rm{eff}}\geq 151.5$ MeV and $\Delta\geq$ 136.4 MeV are needed for
$f_{D}=$ 10%; while $B^{1/4}_{\rm{eff}}\geq$ 131.9 MeV and $\Delta\geq$ 63.6
MeV are needed for $f_{D}=$ 20%. However, the $f_{D}=$ 21.6% line, the $M_{\rm
max}=2.6M_{\odot}$ line and the 2-flavor line meet at the point (49.4, 128.6),
which means that for the case of $f_{D}\geq$ 21.6%, the minimum allowed values
of $B^{1/4}_{\rm{eff}}$ and $\Delta$ are fixed to 128.6 MeV and 49.4 MeV,
respectively. Of course, GW190814’s secondary component and the HESS J1731-347
compact object could also be CFL SSs with a MDM core. In that case, the blue
line and the green lines will shift downword, and the minimum allowed value of
$\Delta$ will be a larger one.

Figure 6 is similar to figure 5, but for $a_{4}=0.8$. From this figure, we
find that for the scenario that all the stars do not have a MDM core, the
minimum allowed value of $B^{1/4}_{\rm{eff}}$ and $\Delta$ are 168.9 MeV and
227.6 MeV, respectively. However, if the compact stars in GW170817 have a MDM
core with $f_{D}\geq$ 17.2% (asuming GW190814’s secondary component and the
HESS J1731-347 compact object are CFL SSs without a MDM core), the minimum
allowed values of $B^{1/4}_{\rm{eff}}$ and $\Delta$ are 137.7 MeV and 90.7
MeV, respectively. On the other hand, if the compact stars in GW170817 have a
MDM core with $f_{D}<$ 17.2%, the minimum allowed values of
$B^{1/4}_{\rm{eff}}$ and $\Delta$ will be larger.

We also find that assuming the compact stars in GW170817 have a large enough
MDM core, the minimum allowed values of $B^{1/4}_{\rm{eff}}$ and $\Delta$
depend on the location of the intersection point of the $M_{\rm
max}=2.6M_{\odot}$ line and the 2-flavor line. Comparing figure 5 with figure
6, we find that these values decrease with the decreasing of $a_{4}$. The
reduction of $\Delta$ has significant consequences on the conformality of the
CFL SQM. Oikonomou and Moustakidis found that whether the conformal bound
($v_{s}\leq c/\sqrt{3}$, where $v_{s}$ is the speed of sound and $c$ is the
speed of light) or the recently proposed positive trace anomally bound
($\left\langle\Theta\right\rangle_{\mu_{B}}\geq 0$ [36], where
$\left\langle\Theta\right\rangle_{\mu_{B}}$ is the trace anomaly at high
baryon density) is violated or not depends on the sign of
$m^{2}_{s}-4\Delta^{2}$ [20]. These bounds will not be violated unless
$\Delta<m_{s}/2$ (i.e., $\Delta<46.5$ MeV, since we take $\Delta=93$ MeV). By
throughly investigating different cases with different values of $a_{4}$, we
find that when $a_{4}<0.589$, the minimum allowed value of $\Delta$ is less
than 46.5 MeV (the minimum allowed value of $B^{1/4}_{\rm{eff}}$ is less than
128.0 MeV correspondingly). For example, it is shown in figure 7 that for the
case of $a_{4}=$ 0.55, if $f_{D}\geq$ 22.8%, the minimum allowed value of
$\Delta$ is 35.4 MeV (the minimum allowed value of $B^{1/4}_{\rm{eff}}$ is
125.9 MeV correspondingly). Thus, for $a_{4}<0.589$, if the MDM core in the
compact stars of GW170817 is large enough, all of the five constraints could
be satisfied without violating the conformal bound or the positive trace
anomally bound.

##  4 Summary

In this paper, we study the structure and the tidal deformability of the CFL
SSs with a MDM core and explain the observations of the central compact object
within the supernova remnant HESS J1731-347 and the compact objects in the
GW190814 and GW170817 events simultaneously. Our explanation is based on the
fact that the mass fraction of MDM ($f_{D}$) of each CFL SS could be different
and it depends on the individual history, which was first realized by
Ciarcelluti and Sandin [103]. We find that if the CFL SSs in GW170817 have a
MDM core (assuming GW190814’s secondary component and the compact object in
HESS J1731-347 are CFL SSs without a MDM core. Our results will not change
qualitatively if these stars also have a MDM core), the observations of these
compact objects could be explained simultaneously without the use of the
extremely large pairing gap ($\Delta>$ 200 MeV). Moreover, we find that for
the case of $a_{4}<0.589$, if the MDM core in the compact stars of GW170817 is
large enough, the minimum allowed value of $\Delta$ could be less than 46.5
MeV ($m_{s}/2$), which leads to the result that all of these observations
could be satisfied without violating the conformal bound or the positive trace
anomally bound.

To avoid the violating the conformal bound or the positive trace anomally
bound, the CFL SSs in GW170817 should have a large enough MDM core (e.g.,
$f_{D}>$ 22.8% for $a_{4}=$ 0.55). Such high MDM fraction cannot be easily
obtained from normal accretion processes [63]. However, it is reachable
through other mechanisms, e.g., a CFL SS could acquire such a large MDM core
if it mergers with a compact star made of MDM [103] (note that mirror NSs have
been studied in ref. [152]. It’s reasonable to suppose the existence of mirror
SSs made of mirror CFL SQM). Other scenarios of the formation of the high dark
matter fractions inside compact stars are presented in refs. [63], which apply
equally to our case of MDM inside the CFL SSs.

Finally, we want to stress that although our results are based on the
calculation of MDM, they are qualitatively valid for other kinds of dark
matter that could form stable cores inside the CFL SSs [103, 106].

## Acknowledgments

This work is supported by the National Key R&D Program of China (Grant No.
2021YFA0718504) and the Scientific Research Program of the National Natural
Science Foundation of China (NSFC, Grant No. 12033001).

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