Can the maximum mass of neutron stars rule out any equation of state of dense stellar matter before gravity is well understood?

0.28 MB·English

by  De-Hua Wen

Draftversion January10,2011
PreprinttypesetusingLATEXstyleemulateapjv.11/10/09
CAN THE MAXIMUM MASS OF NEUTRON STARS RULE OUT ANY EQUATION OF STATE OF DENSE
STELLAR MATTER BEFORE GRAVITY IS WELL UNDERSTOOD?
De-Hua Wen1,2, Bao-An Li 2,3, Lie-Wen Chen4
1DepartmentofPhysics,SouthChinaUniversityofTechnology, Guangzhou 510641, P.R.China
2 DepartmentofPhysicsandAstronomy,Texas A&MUniversity-Commerce,Commerce,Texas 75429-3011, USA
3 DepartmentofAppliedPhysics,Xi’anJiaoTongUniversity,Xi’an710049, P.R.Chinaand
4 DepartmentofPhysics,Shanghai JiaoTongUniversity,Shanghai 200240, P.R.China
Draft version January 10, 2011
1
ABSTRACT
1
0 ProbablyNo! Asanexample,usingsoftEOSsconsistentwithexistingterrestrialnuclearlaboratory
2 experiments for hybrid neutron stars containing a quark core described with MIT bag model using
reasonable parameters, we show that the recently discovered new holder of neutron star maximum
n
a mass PSR J1614-2230 of 1.97±0.04M⊙ can be well described by incorporating a Yukawa gravita-
J tional correction that is consistent with existing constraints from neutron-proton and neutron-lead
scatterings as well as the spectroscopy of antiproton atoms.
7
Subject headings: hybrid star, hyperon, quark, gravity

]
R
S 1. INTRODUCTION forces. Moreover, neutron stars are among the dens-
est objects with the strongest gravity in the Universe,
. What is gravity? Are there additional spacetime di-
h making them ideal places to test strong-fieldpredictions
mensions? These are among the Eleven Science Ques-
p of General Relativity (GR) (Psaltis 2008). The masses
tions for the New Century identified by the Commit-
- and radii of neutron stars are solely determined by both
o tee on the Physics of the Universe, US National Re-
the strong-field behavior of gravity and the Equation of
r search Council (CPUNRC 2003). Interestingly, despite
t State(EOS)ofdensestellarmatter. However,thereisno
s the fact that gravity is the first force discovered in
a nature, the quest to unify it with other fundamental fundamental reason to choose Einstein’s equations over
[ other alternatives and it is known that the GR theory
forces remains elusive because of its apparent weakness

1 at short-distance, see, e.g., refs. (Arkani-Hamed et al. itselfmaybreakdownatthelimitofverystronggravita-
tionalfields,see,e.g.,ref. (Psaltis2008)foracomprehen-
v 1998;Pea2001;Hoyle2003;Long et al.2003;Jean et al.
sive review. In fact, effects of modified gravity on prop-
4 2003; Boehm et al. 2004a,b; Decca et al. 2005). In de-
erties of neutron stars have been under intense investi-
0 veloping grand unification theories, the conventional
gation. Asexpected, resultsofthesestudies arestrongly
5 inverse-square-law(ISL)ofNewtoniangravitationalforce
model dependent, see, e.g., refs. (Germani et al. 2001;
1 has to be modified due to either the geometrical ef-
Wiseman 2002; Azam et al. 2008; Krivoruchenkoet al.,
. fect of the extra spacetime dimensions predicted by
1 2009; Wen et al. 2009). Nevertheless, it is very interest-
string theories and/or the exchange of weakly interact-
0 ing to note that alternative gravity theories that have
ing bosons, such as the neutral spin-1 vector U-boson
1 all passed low-field tests but diverge from GR in the
(Fayet2009),proposedinthesuper-symmetricextension
1
strong-field regime predict neutron stars with signifi-
: of the Standard Model, see, e.g., refs. (Adelberger et al.
v 2003, 2009; Fischbach 1999; Newman 2009; Uzan 2003; cantly different properties than their GR counterparts
Xi Reynaud 2005) for recent reviews. The modified (DeDeo et al. 2003). Moreover, the deviations for neu-
tron star properties from the GR predictions for these
gravity has also been proposed as an explanation for
r theoriesarelargerthantheuncertaintyduetothepoorly
a the present period of cosmological acceleration, see,
knownEOSofdensematterinneutronstars. Itwasalso
e.g., ref. (DeDeo et al. 2008). The search for evi-
clearlyshownthattheneutronstarmaximummassalone
dence of modified gravity is at the forefront of re-
cannot distinguishgravitytheories (DeDeo et al.2003).
search in several sub-fields of natural sciences includ-
Furthermore, in the endeavor of testing GR theory of
ing geophysics, nuclear and particle physics, as well
gravity using properties of neutron stars, it is known
as astrophysics and cosmology, see, e.g., refs. (Fujii
that there is a degeneracy between the matter content
1971; Pea 2001; Hoyle 2003; Arkani-Hamed et al. 1998;
and gravity. This degeneracy is tied to the fundamental
Long et al. 2003; Adelberger et al. 2009; Kapner et al.
Strong Equivalence Principle and can only be brokenby
2007; Nesvizhevsky et al. 2008; Kamyshkov et al. 2008;
using at least two independent observables (Yunes et al.
Azam et al. 2008; Geraci et al. 2010; Lucchesi et al.
2010).
2010). Various upper limits on the deviation from the
Recently, using the general relativistic Shapiro delay
ISL has been put forward down to femtometer range.
the mass of PSR J1614-2230 was precisely measured to
Since the composition of neutron stars are determined
mainly by the weak and electromagnetic forces through be 1.97±0.04M⊙(Demorest et al. 2010), making it the
newholderofthemaximummassofneutronstars. Com-
the β equilibriumandchargeneutralityconditions while
paringwithmass-radiusrelationspredictedfromsolving
theirstabilityismaintainedbythebalanceofstrongand
theTOVequationusingvariousEOSswithinGRtheory
gravitationalforces,neutronstarsarethusanaturaltest-
of gravity, it was shown that the mass of PSR J1614-
ing ground of grand unification theories of fundamental
2
2230 can rule out almost all soft EOSs especially those
associated with hyperon or boson condensation. While
conventional quark stars with soft EOSs are also ruled
out by this observation, neutron stars with strongly in-
teracting quark cores are allowed (Demorest et al. 2010;
O¨zel et al. 2010; Lai et al. 2010). It was further shown 100
that a transition to quark matter in neutron star cores
)
can occur at densities comparable to the nuclear satu- -3 m
ration density ρ only if the quarks are strongly inter-
0 f
acting and are color superconducting (O¨zel et al. 2010). eV 10 µ
The massofPSRJ1614-2230wasthenusedtoconstrain M npe matter
the interacting parameters of quarks. It was also shown P( B1/4=170MeV
thatneutronstarswithinteractingquarkclustersintheir B1/4=180MeV
coresorsolidquarkstarscanbe verymassive. Usingthe 1/4 2 2 -2
B =170MeV&g/ =50GeV
Lennard-Jones potential for interactions between quark 1 B1/4=180MeV&g2/ 2=40GeV-2
clusters, the mass of the PSR J1614-2230 was used to
constrain the number of quarks inside individual quark 0 1 2 ρ/ρ 3 4 5
clusters(Lai et al.2010). Inthiswork,usingsoftnuclear 0
EOSs for hybrid stars containing a quark core described
Fig.1.— (Color online) Model EOSs for hybrid stars with
by the MIT bag model with reasonable parameters, we and without the Yukawa contribution using MIT bag constant
show that the mass of PSR J1614-2230 is readily ob- B1/4=170MeVandB1/4=180MeV,respectively.
tained by incorporating a Yukawa gravitational correc-
tionthatisconsistentwithexistingconstraintsfromter-
vector boson contributes to the EOS only through the
restrial nuclear laboratory experiments.
combination g2/µ2, while both the coupling constant
g and the mass µ of the light and weakly interacting
2. NON-NEWTONIANGRAVITYANDMODELEOSOF
bosons are small, the value of g2/µ2 can be large. On
HYBRIDSTARS
the other hand, by comparing with the g2/µ2 value of
Fujii (Fujii 1971) first proposed that the non-
the ordinary vector boson ω, Krivoruchenko et al. have
Newtoniangravitycanbe describedbyaddingaYukawa
pointed out that as long as the g2/µ2 value of the U
termto theconventionalgravitationalpotentialbetween
boson is less than about 200 GeV−2 the internal struc-
two objects of mass m and m , i.e.,
1 2 tures of both finite nuclei and neutron stars will not
Gm m change (Krivoruchenkoet al., 2009). However, global
V(r)=− 1 2(1+αe−r/λ), (1)
r properties of neutron stars can be significantly modified
(Krivoruchenkoet al.,2009;Wen et al.2009)One ofthe
where α is a dimensionless strength parameter, λ is the keycharacteristicsoftheYukawacorrectionisitscompo-
length scale and G is the gravitational constant. In sition dependence, unlike Einstein’s gravity. Thus, ide-
the boson exchange picture, α = ±g2/(4πGm2b) where allyoneneedstousedifferentcouplingconstantsforvar-
± stands for scalar/vector bosons and λ = 1/µ (in
ious baryons existing in neutron stars. Moreover,to our
natural units). The g2 and µ are the boson-baryon best knowledge, it is unknown if there is any and what
coupling constant and the boson mass, respectively. might be the form and strength of the Yukawa term in
The light and weakly interacting U-boson is a favorite thehadron-quarkmixedphaseandthepurequarkphase.
candidate mediating the extra interaction (Fayet 2009; Nevertheless,insteadofintroducingmoreparameters,for
Krivoruchenkoet al., 2009; Zhu 2007). Similar to the thepurposeofthisexploratorystudy,weassumethatthe
degeneracy between matter content and gravity, there P term is an effective Yukawa correction existing in
UB
appears to be a duality of incorporating effects of the allphases with the g consideredas anaveragedcoupling
Yukawa term in either the TOV equation or the input constant.
EOS. Nevertheless, according to Fujii (Fujii 1988), the Including the Yukawa term the EOS becomes P =
Yukawatermissimplypartofthe mattersystemingen- P +P where P is the conventional pressure inside
0 UB 0
eral relativity. Therefore, only the EOS is modified and neutronstars. Forthelatter,weusetypicalmodelEOSs
the TOV equation remains the same. Within the mean- for hybrid stars containing a quark core covered by hy-
field approximation, the extra energy density due to the peronsandleptons. Thequarkmatterisdescribedbythe
Yukawa term is (Long et al. 2003; Krivoruchenko et al., MIT bag model with reasonableparameters widely used
2009) in the literature (Chodos et al. 1974; Heinz et al. 1986).
The hyperonic EOS is modelled by using an extended
1 g2 e−µr 1g2
ε = ρ(~x ) ρ(~x )d~x d~x = ρ2, (2) isospin- and momentum-dependent effective interaction
UB 2V Z 1 4π r 2 1 2 2µ2 (MDI (Das et al. 2003)) for the baryon octet with pa-
rametersconstrainedbyempiricalpropertiesofsymmet-
where V is the normalization volume, ρ is the baryon
ricnuclearmatter,hyper-atoms,andheavy-ionreactions
number density and r = |~x − ~x |. Assuming a con-
1 2 (J. Xu et al. 2010). In particular, the underlying EOS
stantbosonmassindependentofthedensity,oneobtains
of symmetric nuclear matter is constrained by compar-
the corresponding addition to the EOS PUB = 12µg22ρ2, ing transport model predictions with data on collective
which is just equal to the additional energy density. As flow and kaon production in relativistic heavy-ion colli-
it was emphasized by Fujii (Fujii 1988), since the new sions(Danielewicz et al.2000;Li et al.2008). Moreover,
3

causality
2.0 ) 2.0
J1614-2230 60
M
50 s ( J1614-2230
M)1.5 30 40 Mas 1.8
M( um
mi
1.0 5 ax
M 1.6
1/4 0 1/4
B =170MeV B =170MeV
1/4
0.5 rotation B =180MeV
1.4
0 10 20 30 40 50 60
10 12 14 16 18
2 2 -2
R(km) g/ (GeV )
Fig.2.— (Color online) The mass-radius relation of static neu- Fig. 3.— (Color online) The maximum mass of neutron stars
tronstarswithB1/4=170MeVandvariousvaluesofg2/µ2inunits as afunctionof g2/µ2 withB1/4=170 MeV andB1/4=180MeV,
ofGeV−2 indictedusingnumbersabovethelines. respectively.
the nuclear symmetry energy E (ρ) with this inter-
sym
action is chosen to increase approximately linearly with
density (i.e., the MDI interaction with a symmetry en- µ
log( (MeV))
ergy parameter x=0 (J. Xu et al. 2010)) in agreement
with available constraints around and below the satura- 2 1 0 -1 -2
40
tion density (C. Xu et al. 2010). It is worth noting that
this E (ρ) is verysimilar to the well-knownAPR pre- 1 0
sym
diction up to about 5ρ0 (Akmal et al. 1998; Xiao et al. 35 2
2009). The Gibbs construction was adopted to describe
the hadron-quark phase transition (Glendenning 2001). α|) g2/µ2≈ 40−50GeV-2 3 -5 log
Sbirmidilsatrartoistdhievipdreedviionutos twhoerkliqiunidthceorleit,eirnantuerrec,ruthsteahnyd- og(| 30 (g) 2
l
outer crust from the center to surface. For the inner -10
crust, a parameterizedEOS of P =a+bǫ4/3 is adopted 1, 2. Kamyshkov (2008): np scattering
0 25
as in refs. (J. Xu et al. 2009). For the outer crust, the 3. Pokotilovshi (2006): spectroscopy of antiproton atoms
BPS EOS (Baym et al. 1971) is adopted. As an exam- -15
ple, shown in Fig. 1 are the EOSs for hybrid stars with
20
MIT bag constant B1/4 = 170 MeV and B1/4 = 180 -15 -14 -13 λ -12 -11 -10
MeV, respectively, with and without the Yukawa contri- log( (m))
Fig.4.— (Color online) Constraints on the strength and range
bution. The corresponding hadron-quark mixed phase
of the Yukawa term from terrestrial nuclear experiments in com-
above/below the pure hadron/quark phase covers the parisonwithg2/µ2≈40−50GeV−2.
density range of ρ/ρ =1.31 to 6.56 and ρ/ρ = 2.19 to
0 0
8.63, respectively. Including the Yukawa term the EOS bagconstantBandtheYukawaterm,showninFig.3are
stiffensasthevalueofg2/µ2 increases. Itisseenthatthe the maximum stellar masses as a function of g2/µ2 with
two sets of parameters, B1/4 = 170 MeV and g2/µ2 = B1/4 = 170 MeV and B1/4 = 180 MeV, respectively.
50 GeV−2 or B1/4 = 180 MeV and g2/µ2 = 40 GeV−2, As expected, with B1/4 = 180 MeV a smaller value of
lead to approximately the same total pressure. As a ref- g2/µ2 =40GeV−2 isneededtoobtainamaximummass
erence,the MDI EOSforthe npeµ matter is alsoshown. consistent with the observed mass of PSR J1614-2230.
3. MAXIMUMMASSOFHYBRIDSTARSWITH 4. COMPARISONWITHTERRESTRIALCONSTRAINTSON
NON-NEWTONIANGRAVITY NON-NEWTONIANGRAVITYATSHORTDISTANCE
As an example, shown in Fig. 2 is the mass-radius As mentioned earlier, significant efforts have been de-
relationof hybridstars with the bag constantB1/4=170 voted to constrain the possible non-Newtonian grav-
MeV and varying values of g2/µ2. First of all, without ity using terrestrial experiments. These experiments
the Yukawacontribution(blacksolidline)the maximum have established a clear trend of increased strength α
stellar mass supported is only about 1.46M⊙. Including at shorter length λ. In the short range down to λ ≈
the Yukawa term, as the EOSs are increasingly stiffened 10−14−10−8 m, neutron-proton and neutron-lead scat-
with larger values of g2/µ2, the maximum stellar mass tering data as well as the spectroscopy of antiproton
increases. With g2/µ2 = 50 GeV−2 the maximum mass atomshavebeen usedto setupper limits onthe value of
of 1.97M⊙ is just in the middle of the measured mass g2/µ2 (orequivalentlythe|α|vsλ). Itisthusinteresting
band of PSR J1614-2230. The corresponding radius is to compare the values of g2/µ2 necessary to support the
about12.4km. Toseemoreclearlyrelativeeffectsofthe PSR J1614-2230within the model presented above with
4
the constraints extracted from terrestrial experiments. out any EOS. As an example, using soft nuclear EOSs
While the range parameter λ is expected to be much consistent with existing terrestrial experiments for hy-
larger (smaller) than the radii of finite nuclei (neutron brid starscontaining a quark core described by the MIT
stars), the maximum mass of neutron stars alone is not bag model with reasonable parameters, the maximum
sufficient to set separate constraints on the values of α mass of PSR J1614-2230is readily obtained by incorpo-
and λ. Shown in Fig. 4 is a comparison with the terres- rating the Yukawa gravitational correction that is con-
trial constraints (Kamyshkov et al. 2008; Barbieri et al. sistent with existing constraints from terrestrial nuclear
1975;Pokotilovski2006;Nesvizhevsky et al.2008)inthe laboratory experiments.
|α| vs λ plane. The straight line is for g2/µ2 ≈ 40− We thank W.Z. Jiang, W. G. Newton, A.W. Steiner
50 GeV−2. It is seen that the values of g2/µ2 neces- and Y. Zhang for useful discussions. D.H. Wen is sup-
sary to describe the maximum mass of PSR J1614-2230 ported in part by the National Natural Science Founda-
are consistent with the upper limits from the terrestrial tion of China under Grant No.10947023 and the Fun-
experiments. damental Research Funds for the Central University,
China under Grant No.2009ZM0193. B.A. Li is sup-
ported in part by the US National Science Foundation
undergrantPHY-0757839,theNationalAeronauticsand
5. CONCLUSIONS SpaceAdministrationundergrantNNX11AC41Gissued
Among all fundamental forces, gravity remains the through the Science Mission Directorate and the Texas
most uncertain one despite being the first discovered in Coordinating Board of Higher Education under grant
nature. Neutron stars are natural testing grounds of No. 003565-0004-2007. L.W. Chen is supported in part
grandunificationtheories of fundamental forces. In par- by the National Natural Science Foundation of China
ticular,theyareidealplacestotestGRpredictionsatthe under Grant Nos. 10675082 and 10975097, MOE of
strong-field limit. Interpretations of observed properties China under project NCET-05-0392, Shanghai Rising-
of neutron stars require a comprehensive understanding Star Program under Grant No. 06QA14024, the SRF
ofbothgravityandtheEOSofdensestellarmatter. Be- for ROCS, SEM of China, the National Basic Research
fore strong-field gravity is well understood, it is unlikely Program of China (973 Program) under Contract No.
that the maximum mass of neutron stars alone can rule 2007CB815004.
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