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Constraints on interquark interaction parameters with GW170817 in a binary

strange star scenario

Article · November 2017

DOI: 10.1103/PhysRevD.97.083015

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Constraints on interquark interaction parameters with GW170817 in a binary strange star scenario

En-Ping Zhou,1, 2 Xia Zhou,3 and Ang Li4, ∗

1
State Key Laboratory of Nuclear Science and Technology and School of Physics, Peking University, Beijing 100871, China
2
Institute for Theoretical Physics, Frankfurt am Main 60438, Germany
3
Xinjiang Astronomical Observatory, Chinese Academy of Sciences, Urumqi, Xinjiang 830011, China
4
Department of Astronomy, Xiamen University, Xiamen, Fujian 361005, China
(Dated: November 28, 2017)
The LIGO/VIRGO detection of the gravitational waves from a binary merger system, GW170817, has put
a clean and strong constraint on the tidal deformability of the merging objects. From this constraint, deep in-
sights can be obtained in compact star equation of states, which has been one of the most puzzling problems
for nuclear physicists and astrophysicists. Employing one of the most widely-used quark star EOS model, we
characterize the star properties by the strange quark mass (ms ), an effective bag constant (Beff ), the perturba-
tive QCD correction (a4 ), as well as the gap parameter (∆) when considering quark pairing, and investigate
the dependences of the tidal deformablity on them. We find that the tidal deformability is dominated by Beff ,
and insensitive to ms , a4 . We discuss the correlation between the tidal deformability and the maximum mass
(MTOV ) of a static quark star, which allows the model possibility to rule out the existence of quark stars with
future gravitational wave observations and mass measurements. The current tidal deformability measurement
implies MTOV ≤ 2.18 M (2.32 M when pairing is considered) for quark stars. Combining with the con-
straint of two-solar-mass pulsar observations, we also find that the poorly known gap parameter ∆ could have a
new lower limit of 40 MeV for color-flavor-locked quark matter.

I. INTRODUCTION to the estimated ejecta mass [19] and nucleosynthesis process

associated with a BQS merger [21], it’s possible to explain the
The direct detection of the gravitational wave (GW) orig- kilonova observation (AT 2017gfo) by a low opacity ejecta to-
inating from a binary system by LIGO and VIRGO network gether with spin down power injection as suggested by [22].
[1], as well as its electromagnetic (EM) counterparts detected QSs are quite different from NSs in many aspects, due to
by ∼ 70 astronomical observatories [2], has announced the the self-bound nature. QSs show a different mass-radius re-
birth of the long-anticipated multi-messenger astronomy era. lation compared with NSs. When supported by uniform rota-
In addition to enriching our comprehension on the central en- tion, QSs can have more enhanced mass shedding limits from
gine of short gamma ray bursts [3, 4] and the abundance of their MTOV values than NSs [23, 24]: 40% vs. 20% [18].
heavy elements in the Universe [5, 6], it also contains effec- The finite surface density of QSs also requires a correction
tive information of the equation of state (EOS) of the merg- on the surface when calculating tidal deformability [25, 26].
ing objects. Following the observation of GW170817, various Therefore, those constraints on NSs according to the observa-
works have been done on constraining the EOSs. According tion of GW170817 cannot be simply applied in the scenario of
to the EM counterpart, a similar upper limit of the maximum BQS merger. Therefore, under the intriguing possibility that
mass of a static spherically symmetric star (MTOV ∼ 2.2M ) GW170817 may be from a BQS system, it will be interest-
has been suggested by different groups within the neutron star ing and important to study what this GW observation of tidal
(NS) model [7–10]. The tidal deformability measurement has deformability means for QS EOS or SQM properties.
also been invoked to constrain the stiffness of a generic group The paper is organized as follows: In Section II, we in-
of NS EOSs [11]. troduce the calculations of EOS and tidal deformability of
Nevertheless, the EOS of compact stars is still in lively QSs; The main results are presented in Section III, where var-
debate, as it originates from complicated problems in non- ious EOS models are extensively investigated with different
perturbative quantum choromodynamics (QCD). Besides the choices of parameters for the strange quark mass (ms ), an ef-
conventional NS model, strange quark stars (QS) are also sug- fective bag constant (Beff ), the perturbative QCD correction
gested as a possible nature of compact stars [12, 13], after it parameter (a4 ), and the pairing energy gap (∆). They are
was conjectured that strange quark matter (SQM) consists of all confronted with the tidal deformability (Λ) measurement
up, down and strange quark could be the true ground state of from GW170817 for systematic constraints on those parame-
strong interaction [14, 15]. Moreover, due to the lack of infor- ters. Finally, we summary our work and conclude in Section
mation on the post-merger remnant [16, 17], the GW obser- IV.
vation itself cannot exclude the possibility of a binary quark
star (BQS) merger as the origin of GW170817. Additionally,
attempts in understanding the EM counterparts of BQS merg- II. EOS MODELS AND TIDAL DEFORMABILITY
ers or remnant QSs have also been made [18–20]. According
Making use of the simple but widely-used MIT model
[12, 27], we describe the unpaired SQM as a mixture of quarks
(u, d, s) and electrons (e), allowing for the transformation due
∗ liang@xmu.edu.cn to weak interaction between quarks and leptons. The expres-
 2

sions for the grand canonical potential per unit volume is writ- 3000
ten as:
3  µ 4
X b 2500
Ωfree = Ω0i + 2 (1 − a4 ) + Beff . (1)
i
4π 3
2000
In Eq.1, Ω0i in the first term at the right hand side is the grand
canonical potential for each species of the particles described

Λ2
1500
as ideal Fermi gases (i = u, d, s, e). The second term accounts
for the perturbative QCD corrections due to gluon mediated

90
1000

%
quark interactions to O(αs2 ) [28–31]. The perturbative QCD
ms = 0
correction parameter a4 characterizes the degree of the quark
interaction correction, with a4 = 1 corresponding to no QCD 500 ms = 90

50
%
corrections (Fermi gas approximation). µb = µu + µd + µs ms = 100
is the baryon chemical potential, with the total baryon num- 0
ber density n = (nu + nd + ns )/3. The effective bag con- 0 500 1000 1500 2000 2500 3000
stant (Beff ) also includes a phenomenological representation Λ1
of nonperturbative QCD effects.
In the calculation we take mu = md = me = 0, and
ms = 0, 90 MeV, 100 MeV [32]. For a4 and Beff , we treat FIG. 1. Tidal deformability (Λ1 , Λ2 ) for ms = 0, 90 MeV,
them as free parameters chosen from the stability window 1/4
100 MeV, with fixed a4 = 0.61 and Beff = 138 MeV. Λ1 and
bounded by the “2 flavor” line and the “3 flavor” line [33], Λ2 are calculated by employing the 90% most probable fraction of
and explore possible further limits on then with GW170817. component masses M1 and M2 for GW170817. They are compared
With the constraint of “2 flavor” line we ensure that normal with the observation [1]: 2 dashed lines enclose 50% and 90% of the
atomic nuclei shouldn’t decay into non-strange quark matter. of the probability density, respectively, in the low spin prior case.
With the constraint of “3 flavor” line we ensure that strange
quark matter would be more stable than normal nuclear mat-
ms [MeV] nc [fm−3 ] R [km] M/R k2 Λ
ter, namely the Bodmer-Witten’s conjecture [14, 15].
For a set of parameters (ms , a4 , Beff ), from Ωfree in 0 0.327 11.814 0.17499 0.19510 792.8
Eq.1 one can deduce the EOS of unpaired SQM, i.e., the 90 0.355 11.478 0.18016 0.18357 644.9
pressure p as a function of the energy density  (the num- 100 0.361 11.415 0.18115 0.18133 619.7
ber density n). Then by solving the TOV equation using
p() as input, one can obtain the star’s mass-radius relation TABLE I. Properties of a 1.4 M QS, including the central number
density nc , the radius R, the compactness M/R, the Love number k2
M (R). The obtained static maximum mass MTOV should
and the tidal deformability Λ, for different ms with fixed a4 = 0.61
necessarily reach the present maximum mass measurement of 1/4
and Beff = 138 MeV.
2.01 ± 0.04M [34]. The GW170817 observation puts in-
dependent constraints on EOS through the tidal deformabil-
ity Λ = (2/3)k2 /(GM/c2 R)5 , where k2 is the second Love
number. Λ describes the amount of induced mass quadrupole perturbation introduced into the metric and satisfies a second
moment when reacting to a certain external tidal field [35, 36]. order differential equation with metric and fluid variables. s
If a low spin prior is assumed for both stars in the binary, is the surface energy density.
which is reasonable considering the magnetic braking during
the binary evolution, the tidal deformability for a 1.4 M star
(denoted as Λ(1.4) in below) was concluded to be smaller than III. RESULTS
800 [1].
The Love number k2 measures how easily the bulk of the A. Dependence of Λ on ms
matter in a star is deformed by an external tidal field. Follow-
ing the instructions of previous works [25, 37], we introduce
a static l = 2 perturbation to the TOV solution and solve the ms has been well-constrained, for a recent result of 95 ±
perturbed Einstein equation both inside and outside the star. 5 MeV [32]. The tiny uncertainty of ms allow us to verify
In particular for QSs with a finite surface density, a special easily whether there is strong dependence of Λ on it. For this
boundary treatment on the stellar surface has to be done to purpose, we have applied three models with same usual val-
1/4
join the interior solution with the exterior [25, 26]: ues of a4 (0.61) and Beff (138 MeV) but different ms (0,
90 MeV and 100 MeV). The results are shown in Fig.1 and
ext int s /c2 Table.I. Three QS EOSs are all consistent with the observa-
yR = yR − . (2) tion of GW170817, justifying the possibility of BQS merger
M/4πR3
despite those differences between QSs and NSs. The mass-
In Eq.2, y = rH 0 /H is a variable relating to k2 with a com- less case gives Λ(1.4) = 791.4, still managing to be inside
plicated algebraic expression and H (as a function of r) is the the boundary of GW170817 constraint.
 3

3.1
144 a4 = 0.57
142
3.0 a4 = 0.585
2.01 M a4 = 0.61
140 2.9 a4 = 0.72
Λ(1.4) = 600
Beff [MeV]

line

logΛ(1.4)
138 a4 = 0.83

e
or
2.8

lin
ms = 90
av

r
1/4

vo
3 fl

fla
136
Λ(1.4) = 800

2
2.7
134 2.2 M
2.6
132

130 2.5
0.5 0.6 0.7 0.8 0.9 1.0 0.28 0.30 0.32 0.34 0.36
a4 log(MTOV /M)

FIG. 2. Constraints on a4 and Beff with ms = 100 MeV. The FIG. 3. Correlation between Λ(1.4) and MTOV for various a4 and
grey shaded region is the allowed parameter space jointly constrained Beff , for ms = 100 MeV (color symbols) or ms = 90 MeV (black
by measurement of Λ according to GW170817 (bottom red line), symbols). The solid line indicates our linear fit of Eq.3. Symbols
mass measurement of pulsars (top black line) and stability condition with different colors and shapes lying in the same line demonstrate
(green lines). The Λ(1.4) = 600 line and the MTOV = 2.2 M line that the fitting formula is independent of a4 and ms .
indicate possible future observational constraints.

1/4
a4 Beff [MeV] R [km] M/R k2 Λ and less likely to be tidally deformed [26, 38].
0.61 133 12.046 0.17166 0.19973 893.4 Both MTOV and Λ(1.4) increases with increasing a4 .
0.61 136 11.662 0.17731 0.18865 717.7 Namely, the perturbative QCD corrections soften the EOS as
0.61 138 11.415 0.18115 0.18133 619.7 well as the finite strange quark mass, although the softening
effect is quite modest and the constant MTOV lines in Fig.2
0.72 138 11.453 0.18055 0.18262 634.5
is close to horizontal ones. Unlike the weak dependence of
0.83 138 11.482 0.18008 0.18367 646.6
Λ(1.4) on ms and a4 , the effective bag constant is dominating
TABLE II. Properties of a 1.4 M QS, including the radius R, the for determining the tidal deformability and maximum mass.
compactness M/R, the Love number k2 and the tidal deformability Consequently, a rather proper constraint can be set on Beff
Λ, for various choices of a4 and Beff with fixed ms = 100 MeV. with GW170817.
Combining the GW170817 constraint on Λ(1.4), the two-
solar-mass constraint on MTOV and the stability window
The effect of bringing in finite strange quark mass is to for quark matter, we find that the QS model parameters to
soften the EOS. Consequently, the TOV maximum mass de- 1/4
be compatible with Beff ∈ (134.1, 141.4) MeV and a4 ∈
creases from 2.217 M in the massless case to 2.101 M (0.56, 0.91). Further limits can be added once more obser-
(2.079 M ) for ms = 90 MeV (ms = 100 MeV). Also vations are made in the future, as indicated Fig.2 with the
for a fixed gravitational mass of 1.4 M , with increasing ms , Λ(1.4) = 600 line and the MTOV = 2.2 M line.
the central density increases, the star radius decreases, result-
ing an increasing compactness and a decreasing Love number.
Those lead to a decreasing tidal deformability. However, the
differences between Λ(1.4) for the two finite ms models are C. Correlation between Λ and MTOV
negligible. That is, Λ(1.4) only weakly depends on ms .
Since both MTOV and Λ(1.4) indicate the stiffness of QS
EOSs, particularly, their dependences on both Beff and a4 are
B. Dependence of Λ on a4 and Beff almost identical, it will be useful to present a straightforward
relation between them. As can be seen from Fig.3, a strong
Since the dependence of Λ on ms is very weak, we fix linear dependence is found between MTOV and Λ(1.4) in log-
strange quark mass as 100 MeV in the following. Selected arithm scale, which is independent of a4 . We fit the data in
solutions with several sets of (a4 , Beff ) are presented in Ta- the following form:
ble.II and interpolations have been done to produce a series of  5.457
constant MTOV lines and Λ(1.4) lines in Fig.2. It is clearly MTOV
Λ(1.4) = 510.058 × , (3)
shown that softer EOSs are more compact for a given mass, 2.01 M
 4

with coefficient of determination R2 = 0.9996. We also 160

li ne
check that the ms = 90 MeV case can also satisfy the same vor 1) a4 = 1
fitted formula. 3 fla
A direct application of the strong MTOV -Λ(1.4) correlation 155
is to set a constraint on QS EOS directly from a tidal obser-
vation. For example, the corresponding MTOV for Λ(1.4) =

Beff [MeV]
150
800 is 2.18 M . Although the physical picture is different in 600
many aspects, we coincidentally end up with a similar upper 1M

.4 )=
2. 0 Λ(1

1/4
limit of MTOV for QSs as NSs [7–10] from the observation of 00
145 ) =8
GW170817. 2 flavor line (1.4
M  Λ
On the other hand, it is also possible to find the minimum 2. 2
possible Λ(1.4) value with the current two-solar-mass con- 140
straint, which is 510.1. Once future GW observations find
a smaller upper limit, it will be impossible for a QS EOS to
accommodate in the present model, unless other quark-matter 135
10 20 30 40 50 60 70 80 90 100
phases is included such as quark pairing. We mention here ∆ [MeV]
that Λ(1.4) for the NS EOS of APR4 (consists of n, p, e, and
µ [39]) is 255.8.
150
2) a4 = 0.61
ine
145 rl
D. Dependence of Λ on ∆
avo
fl
3

Beff [MeV]
If quarks form Cooper pairs and the SQM is in the color- 140
flavor-locked (CFL) phase, an additional term corresponding

to the energy of the diquark condensate has to be added: 1M M
2. 0
1/4

135 2. 2
0
60
3 = 0
ΩCFL = Ωfree − 2 ∆2 µ2b . (4) . 4) 80
1 =
π Λ( .4)
1
130 Λ(
∆ in Eq.4 is the pairing energy gap for the phase, lack-
ing an accurate calculation within a typical range (0 − 100 2 flavor line
125
MeV [40–43], possibly up to 150 MeV [44]). Starting with 10 20 30 40 50 60 70 80 90 100
this grand canonical potential, we here explore the possi- ∆ [MeV]
bility of a new constraint for ∆ with GW170817. There
may be other quark-matter phases, such as two-flavor color-
superconducting (2SC) phase (e.g., [45]), but we leave a study
FIG. 4. Constraints on the pairing gap ∆ with ms = 100 MeV, for
of these to future work.
both a4 = 1 (upper panel) and a4 = 0.61 (lower panel). a4 = 1
The result is shown in Fig.4. The Λ(1.4) ≤ 800 constraint corresponds to no perturbative QCD correction, and a4 = 0.61 is
is found to be consistent with the previous upper value of ∆ ≥ close to the previous calculated result with different choice of the
100 MeV, except for when Beff choice is close to the “2 flavor renormalization scale [28]. Other notions applied are the same with
line” condition. This is true for both ideal a4 = 1 case and a Fig.2.
more realistic a4 = 0.61 case. In the latter case for example,
1/4
when setting Beff to its lower limit of 128 MeV, an upper
limit of ∼ 69 MeV is found for the pairing gap. The two- IV. CONCLUSION
solar-mass constraint bounds the lower limit of ∆ at the order
of ∼ 40 MeV in both cases, indicating 40 MeV as a possible The properties of QSs have been systematically studied us-
lower limit for ∆. ing one of the most widely-used QS EOS model, for the pur-
Different from the unpaired case, the constant Λ(1.4) lines pose of constraining the QS EOS and interquark interaction
here deviate from the constant MTOV lines. However, it’s parameters with the tidal deformability of GW170817. We
still possible to deduce the maximum possible MTOV in the find that a QS should have MTOV ≤ 2.18 M (2.32 M
allowed parameter space, which is roughly 2.32 M for both when considering CFL phase). In particular, a power law rela-
a4 = 1 and a4 = 0.61. The minimum Λ(1.4) is ∼390 when tion between MTOV and Λ(1.4) is newly found and fitted for
1/4 normal SQM.
choosing maximum possible Beff and ∆, which is smaller
than the unpaired case but larger than that of APR4. Over- We also demonstrate that finite ms play only minor role,
all, incorporating pairing in SQM brings more freedom when while it is Beff dominating the EOS stiffness, compared with
comparing QS EOS models with current observational con- a weak influence of a4 . Combining GW and pulsar observa-
1/4
straints. tions, Beff is limited in a range of (134.1,141.4) MeV, and a4
 5

between 0.56 to 0.91. Furthermore, a lifted lower limit of 40 QSs might be excluded as possible type of compact stars us-
MeV is found for the gap parameter ∆, and the Λ(1.4) ≤ 800 ing the present model.
constraint of GW170817 is found to be consistent with its pre-
vious upper value (≥ 100 MeV).
ACKNOWLEDGMENTS
In the present study it is shown that for both paired and
unpaired SQM, GW170817 has the possibility of originating We thank Dr. A. Tsokaros and C. Pan for useful discus-
from a BQS merger, and the GW observation of tidal deforma- sion and helps in the numerical calculation. E. Z. is grate-
bility can translate into constraints for high-density QS EOSs. ful for China Scholarship Council for supporting the joint
By combining with the constraints from massive (> 2.0M ) training in Frankfurt. The work was supported by the Na-
pulsars, the model parameters of QS EOSs are ready to be tional Natural Science Foundation of China (Nos. U1431107
more tightly defined by future observations. If future GW and and 11373006), and the West Light Foundation of Chinese
pulsar observations could not be reconciled with each other, Academy of Sciences (No. ZD201302).

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