Constraints from compact star observations on non-Newtonian gravity in strange stars based on a density dependent quark mass model
aa r X i v : . [ a s t r o - ph . H E ] J a n Constraints from compact star observations on non-Newtonian gravity in strange stars based on adensity dependent quark mass model
Shu-Hua Yang , ∗ Chun-Mei PI , Xiao-Ping Zheng , , and Fridolin Weber , Institute of Astrophysics, Central China Normal University, Wuhan 430079, China School of Physics and Mechanical & Electrical Engineering,Hubei University of Education, Wuhan 430205, China Department of Astronomy, School of physics, Huazhong University of Science and Technology, Wuhan 430074, China Department of Physics, San Diego State University, San Diego, CA 92182, USA Center for Astrophysics and Space Sciences, University of California at San Diego, La Jolla, CA 92093, USA (Dated: November 2020)Using a density dependent quark mass (QMDD) model for strange quark matter, we investigate the e ff ects ofnon-Newtonian gravity on the properties of strange stars and constrain the parameters of the QMDD model byemploying the mass of PSR J0740 + ff ects are ignored. For the current quark masses of m u = .
16 MeV, m d = .
67 MeV, and m s =
93 MeV, we find that a strange star can exist for values of the non-Newtonian gravity parameter g /µ in the range of 4.58 GeV − ≤ g /µ ≤ − , and that the parameters D and C of the QMDD model arerestricted to 158.3 MeV ≤ D / ≤ − . ≤ C ≤ − .
12. It is found that the largest possiblemaximum mass of a strange star obtained with the QMDD model is 2 . M ⊙ , and that the secondary componentof GW190814 with a mass of 2 . + . − . M ⊙ could not be a static strange star. We also find that for the massand radius of PSR J0030 + + + I. INTRODUCTION
As hypothesized by Itoh [1], Bodmer [2], Witten [3], andTerazawa [4], strange quark matter (SQM) consisting of up( u ), down ( d ) and strange ( s ) quarks and electrons may bethe true ground state of baryonic matter. According to thishypothesis, compact stars made entirely of SQM, referred toas strange stars (SSs), ought to exist in the universe [5–10].E ff ects of non-Newtonian gravity on the properties of neu-tron stars and SSs have been studied extensively [e.g., 11–19].The conventional inverse-square-law of gravity is expectedto be violated in the e ff orts of trying to unify gravity withthe other three fundamental forces, namely, the electromag-netic, weak and strong interactions [20–22]. Non-Newtoniangravity arise due to either the geometrical e ff ect of the extraspace-time dimensions predicted by string theory and / or theexchange of weakly interacting bosons, such as a neutral veryweakly coupled spin-1 gauge U-boson proposed in the super-symmetric extension of the standard model [23, 24]. Althoughthe existence of non-Newtonian gravity is not confirmed yet,constraints on the upper limits of the deviations from New-ton’s gravity have been set experimentally (see [25] and refer-ences therein).For the standard MIT bag model, Yang et al. [19] foundthat if non-Newtonian gravity e ff ects are ignored, the exis-tence of SSs is ruled out by the mass of PSR J0740 + . + . − . M ⊙ for a 68.3% credibility interval; 2 . + . − . M ⊙ for a 95.4% credibility interval) [26] and the dimensionless ∗ Electronic address: [email protected] tidal deformability of a 1 . M ⊙ star of GW170817 ( Λ (1 . = + − ) [27, 28]. However, if non-Newtonian gravity e ff ectsare considered, Yang et al. [19] found that SSs can exist forcertain ranges of the values of the non-Newtonian gravity pa-rameter g /µ , and the bag constant B and the strong interac-tion coupling constant α S of the SQM model. For example,for a strange quark mass of m s =
95 MeV, SSs can exist for1.37 GeV − ≤ g /µ ≤ − , and limits on parametersof the SQM model are 141.3 MeV ≤ B / ≤ α S ≤ . ff ects. Similar to the results given by Yang et al. [19], theyfound that the observations of GW170817 and the mass ofPSR J0740 + . M ⊙ star from GW170817 directly. Instead, they employed the ra-dius of a 1 . M ⊙ star, which is R . = . + . − . km, derivedfrom the observations of GW170817 by Capano et al. [30].In this paper, we will investigate the e ff ects of non-Newtonian gravity on the properties of SSs and constrainthe parameter space of the QMDD model using the tidal de-formability of GW170817 and the mass of PSR J0740 + + ff ects. In Sec.III, numerical results and discussions are presented. Finally, abrief summary of our results is given in Sec. IV. II. EOS OF SQM INCLUDING THE NON-NEWTONIANGRAVITY EFFECTS
Before discussing the e ff ects of non-Newtonian gravity onthe EOS of SQM, we briefly review the phenomenologicalmodel for the EOS employed in this paper, namely the QMDDmodel.The key feature of the QMDD model is the use of den-sity dependent quark masses to express non-perturbative in-teraction e ff ects [33, 34]. The first few QMDD studies of theEOS of SQM were thermodynamicall inconsistent [e.g., 39–41]. Furthermore, while the original quark mass scaling for-malism barely accounted for the confinement interaction [e.g.,33, 42], an improved quark mass scaling taking into accountboth the linear confinement and leading order interactions hasbeen introduced by Xia et al. [41].Taking into account both the linear confinement and leadingorder interactions, the quark mass scaling is given by [41] m i = m i + m I ≡ m i + Dn / b + Cn / b . (1)Here m I is a density dependent term that includes the quarkinteraction e ff ects introduced through the adjustable parame-ters C and D , m i is the current mass of quark flavor i with m u = .
16 MeV, m d = .
67 MeV, and m s =
93 MeV [43],and n b is the baryonic density n b = X i n i , (2)where the number density of each quark species n i is given byEq. (6).The EOS of SQM with the above density dependent quarkmasses is to be determined subject to the following fullyconsistent thermodynamic conditions [41]. At zero temper-ature, the thermodynamic potential of free unpaired particlesis given by Ω = − X i g π " µ ∗ i ν i ν i − m i ! + m i ln µ ∗ i + ν i m i , (3)where g = µ ∗ i is the e ff ectivechemical potential of quark flavor i , and it is related to thechemical potential µ i through the following equation, µ i = µ ∗ i + ∂ m I ∂ n b ∂ Ω ∂ m I . (4)The quantity ν i denotes the Fermi momentum of a quark oftype i , ν i = q µ ∗ i − m i (5)and the corresponding particle number densities are given by n i = g π ( µ ∗ i − m i ) / = g ν i π . (6) The energy density without the e ff ects of the non-Newtoniangravity is given by ǫ Q = Ω − X i µ ∗ i ∂ Ω ∂µ ∗ i , (7)and the pressure is obtained from p Q = − Ω + X i , j ∂ Ω ∂ m j n i ∂ m j ∂ n i , (8)which can be written in the more convenient form p Q = − Ω + n b ∂ m I ∂ n b ∂ Ω ∂ m I . (9)In addition, chemical equilibrium is maintained by theweak-interaction of SQM, which leads for the chemical po-tentials to the following conditions, µ d = µ s , (10) µ s = µ u + µ e . (11)The electric charge neutrality condition is given by23 n u − n d − n s − n e = . (12)Non-Newtonian gravity is often characterized e ff ectively byadding a Yukawa term to the normal gravitational potential[44]. The Yukawa-type non-Newtonian gravity between thetwo objects with masses m and m is [20–22] V ( r ) = − G ∞ m m r (cid:16) + α e − r /λ (cid:17) = V N ( r ) + V Y ( r ) , (13)where V Y ( r ) is the Yukawa correction to the Newtonian poten-tial V N ( r ). The quantity G ∞ = . × − N m / kg is theuniversal gravitational constant, α is the dimensionless cou-pling constant of the Yukawa force, and λ is the range of theYukawa force mediated by the exchange of bosons of mass µ (given in natural units) among m and m , λ = µ . (14)In this picture, the Yukawa term is the static limit of an inter-action mediated by virtual bosons. The strength parameter inEq. (13) is given by α = ± g π G ∞ m b , (15)where the ± sign refers to scalar (upper sign) or vector (lowersign) bosons, g is the boson-baryon coupling constant, and m b is the baryon mass. An extra Yukawa term also naturally arises in the weak-field limit of somemodified theories of gravity, e.g., f(R) gravity, the nonsymmetric gravita-tional theory, and Modified Gravity. See [45], and references therein.
Krivoruchenko et al. [11] suggested that a neutral veryweakly coupled spin-1 gauge U-boson proposed in the super-symmetric extension of the standard model is a favorite candi-date for the exchanged boson [23, 24]. This light and weaklyinteracting U-boson has been used to explain the 511 keV γ -ray observation from the galatic bulge [46–48], and variousexperiments in terrestrial laboratories have been proposed tosearch for this boson [49]. Since the new bosons contributeto the EOS of dense matter in terms of g /µ [50], which canbe large even when both the coupling constant g and the mass µ of the light and weakly interacting bosons are small, thestructure of compact stars may be greatly influenced by thenon-Newtonian gravity e ff ects.It has been shown by Krivoruchenko et al. [11] that an in-crease of g (a decrease of µ ) of scalar bosons has a negativecontribution to pressure, which makes the EOS of dense mat-ter softer and reduces the maximum mass of a compact star.By contrast, an increase of g (a decrease of µ ) of vector bosonsmakes the EOS of dense matter sti ff er and increases the max-imum mass of a compact star. In the following, we will onlystudy the case of vector bosons since a sti ff EOS of SQM isneeded to accomodate the tidal deformability of GW170817and the mass of PSR J0740 + V Y ( r ) of Eq. (13)to the energy density of SQM is obtained by integrating overthe quark densities n b ( ~ x ) and n b ( ~ x ) inside a given volume V [11, 12, 17, 51], ǫ Y = V Z n b ( ~ x ) g π e − µ r r n b ( ~ x ) d ~ x d ~ x , (16)where r = | ~ x − ~ x | . The prefactors of 3 in front of the quarkdensities are required since the baryon number of quarks is1 /
3. Equation (16) can be evaluated further since the quarkdensities n b ( ~ x ) = n b ( ~ x ) ≡ n b are essentially independent ofposition [7–10]. Moving n b outside of the integral then leadsfor the energy density of SQM inside of V = π R / ) to [17, 19] ǫ Y = g n b Z R re − µ r dr . (17)Upon carrying out the integration over the spherical volumeone arrives at ǫ Y = g n b µ h − (1 + µ R ) e − µ R i . (18)Because the system we are considering is in principle verylarge, we may take R → ∞ in Eq. (18) to arrive at ǫ Y = g µ n b . (19) The actual geometry of the volume is unimportant since we are only in-terested in the local modification of the energy (Eq. (19)) caused by theYukawa term.
This analysis shows that the additional contribution to the en-ergy density from the Yukawa correction, V Y , is simply deter-mined (aside from some constants) by the number of quarksper volume. The total energy density of SQM is obtained byadding ǫ Y to the standard expression for the energy density ofSQM given by Eq. (7), leading to ǫ = ǫ Q + ǫ Y . (20)Correspondingly, the extra pressure due to the Yukawa correc-tion is p Y = n b ddn b (cid:18) ǫ Y n b (cid:19) = g n b µ (cid:18) − n b µ ∂µ∂ n b (cid:19) . (21)Assuming a constant boson mass (independent of the density)[11, 12, 17], one obtains p Y = ǫ Y = g µ n b . (22)The total pressure including the non-Newtonian gravity(Yukawa) term then reads p = p Q + p Y , (23)where p Q is given by Eq. (9). III. RESULTS AND DISCUSSIONS
For a given SQM EOS, the structure of strange starsand their tidal deformability is calculated from the Tolman-Oppenheimer-Volko ff equation, as described in Refs. [52–57].The mass-radius relations of SSs for di ff erent non-Newtonian gravity parameters are shown in Fig. 1. We choose D / = . C = − .
23 because for this set of param-eter, the observations of PSR J0740 + + g /µ = .
77, as will beshown in Fig. 3. The dash-dotted line for g /µ = .
32 sat-isfies the constraints on PSR J0030 + D / = . C = − .
23, and g /µ = .
32 is ruled out by the constraints employed by thispaper later, which can be seen in Fig. 2(e).We investigate the allowed parameter space of QMDDmodel according to the following five constraints [e.g., 19, 29,58–62]:First, as pointed out by Backes et al. [29], the quark massescould become negative at high densities and a negative masshas no physical meaning, resulting in a regime where themodel is not valid. Following Backes et al. [29], we presentthe invalid D / – C parameter regions in Fig. 2 (namely, theyellow-shaded regions), which are separated with other areasby requiring m u = n = . − (around ten times thenuclear saturation density). These yellow-shaded regions areruled out because for the parameters located in these regions, Miller et al. (2019) M ( M ) R(km)
FIG. 1: (Color online) The mass-radius relation of SSs with D / = . C = − .
23. The solid, dashed, dotted, dash-dotted linesare for g /µ = .
0, 3.0, 5.77, and 9.32 GeV − , respectively. Thered data is R . = . + . − . km, which is the radius of 1 . M ⊙ con-strained by the observations of GW170817 [30]. The blue and greenregions show the mass and radius estimates of PSR J0030 + R = . + . − . km, M = . + . − . M ⊙ ) and Miller et al. [32] ( R = . + . − . km, M = . + . − . M ⊙ ). m u becomes negative on densities lower than n = . − ,which may happen in the cores of the massive SSs.Second, the existence of SSs is based on the idea that thepresence of strange quarks lowers the energy per baryon ofa mixture of u , d and s quarks in beta equilibrium below theenergy of the most stable atomic nucleus, Fe ( E / A ∼ . This constraint results in the 3-flavor lines (thedash-dotted lines) shown in Fig. 2.Here we want to stress that atomic nuclei do not transitionto (lumps of) SQM, and neither the EOS of ordinary nuclearmatter nor NN scattering data are impacted by the possibleabsolute stability of SQM. The reason is that the creation ofSQM requires a significant fraction of strange quarks to bepresent. Conversion of an Fe nucleus, for instance, intoSQM requires a very high-order weak interaction to simul-taneously change dozens of u and d quarks into s quarks.The probability of this happening is astronomically small. Forlower baryon numbers, the conversion requires a lower-orderweak interaction, but finite-size e ff ects and the positive elec- It is common practice to compare the energy of SQM to Fe. The energyper baryon of Fe, however, is only the third lowest after Ni and Fe. trostatic potential of SQM destabilize small junks of SQM sothat they become unstable even if SQM is stable in bulk.The third constraint is given by assuming that non-strangequark matter (i.e., two-flavor quark matter made of only u and d quarks) in bulk has an energy per baryon higher than theone of Fe, plus a 4 MeV correction coming from surfacee ff ects [5, 9, 59, 62]. By imposing E / A ≥
934 MeV on non-strange quark matter, one ensures that atomic nuclei do notdissolve into their constituent quarks. This leads to the 2-flavor lines (dotted lines) in Fig. 2. The cyan-shaded areasbetween the 3-flavor lines (the dash-dotted lines) and the 2-flavor lines (dotted lines) in Fig. 2 show the allowed D / – C parameter regions where the second and the third constraintsdescribed just above are fulfilled.The fourth constraint is that the maximum mass of SSsmust be greater than the mass of PSR J0740 + M max ≥ . M ⊙ . By employing this constraint, the allowed parameterspace is limited to the region below the solid lines in Fig. 2.The last constraint follows from Λ (1 . ≤ Λ (1 .
4) is the dimensionless tidal deformability of a 1 . M ⊙ star. The parameter space satisfies this constraint correspondsto the region above the dashed lines in Fig. 2. The magenta-shaded areas between the solid lines and the dashed lines inFig. 2 show the allowed D / – C parameter regions where bothconstraints from the mass PSR J0740 + D / – C parameter space of QMDD model is restrictedto the red-shadowed regions shown in Fig. 2(c) and 2(d),which are obtained for non-Newtonian gravity parameter val-ues of g /µ = .
89 GeV − , and g /µ = .
77 GeV − , respec-tively. An overlapping region where all the five constraintsare simultaneously satisfied does not exist for all other casesshown in Fig. 2, panels (a), (b), (e), which correspond to g /µ = g /µ = .
58 GeV − , and g /µ = .
32 GeV − ,respectively.From Fig. 2(a), one sees that for the case of g /µ = g /µ becomes bigger until it is as large as 4.58 GeV − , in whichcase the M max = . M ⊙ line, the Λ (1 . =
580 line and the2-flavor line intersect at the point (158.3, -0.15)(see Fig. 2(b)).The allowed parameter space vanished entirely for g /µ > .
32 GeV − , as shown in Fig. 2(e).Let us focus on Fig. 2, panels (b), (c) and (d) once again. InFig. 2(b), the M max = . M ⊙ line, the Λ (1 . =
580 line andthe 2-flavor line intersect at the point (158.3, − . D / is 158.3 MeV. In Fig. 2(c),the M max = . M ⊙ line, the Λ (1 . =
580 line and the 3-flavor line intersect at the point (158.5, − . C is − .
12. Whereas, in Fig. 2(d) ,the M max = . M ⊙ line, the 3-flavor line and the m u = − . D / is 181.2 MeV and the lower limit of C is − . -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4130140150160170180190200 (a) g / =0 D / ( M e V ) C max = (1.4)=580 m u0 =0 D / ( M e V ) C -0.6 -0.4 -0.2 0.0 0.2 0.4 (b) g / =4.58GeV -2 -0.6 -0.4 -0.2 0.0 0.2 0.4 (e) g / =9.32GeV -2 (d) g / =5.77GeV -2 (c) g / =4.89GeV -2 C -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4130140150160170180190200 C -0.6 -0.4 -0.2 0.0 0.2 0.4 C FIG. 2: (Color online) Constraints on D / and C for g /µ = g /µ = − (b), g /µ = − (c), g /µ = − (d),and g /µ = − (e), respectively. The red-shadowed regions in panels (c) and (d) indicate the allowed parameter spaces. (See text fordetails.) D / = . C = − .
60 and g /µ = .
32 GeV − ,which is 2 . M ⊙ .Recently, the NICER observations of the isolated pulsarPSR J0030 + M = . + . − . M ⊙ and R eq = . + . − . km [31], and M = . + . − . M ⊙ and R eq = . + . − . km [32]. In Fig. 3, these data on the M – R plane is translated into the D / – C space (namely, the gray-shaded regions) for the case of g /µ = − . The graylines in Fig. 3(a) are for ( M ( M ⊙ ), R (km)) sets (1.49, 11.52),(1.34, 12.71), and (1.18, 13.85) from top to bottom, and theseparameter sets correspond to the data given by Riley et al.[31]. The gray lines in Fig. 3(b) are for (1.59, 11.96), (1.44,13.02), and (1.30, 14.26) from top to bottom, and these pa-rameter sets come from the data given by Miller et al. [32].We can see from Fig. 3(a) that the allowed parameter spaceconstrained by the mass and radius of PSR J0030 + + + g /µ = − in Fig. 3, we have checked some other cases between4.58 GeV − < g /µ < − and find that one alwaysarrives at the above conclusion. -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4130140150160170180190200 (a) Riley et al. (2019) max = (1.4)=580 m u0 =0 D / ( M e V ) C -0.6 -0.4 -0.2 0.0 0.2 0.4 (b) Miller et al. (2019) C FIG. 3: (Color online) Constraints on D / and C for g /µ = − . The gray-shaded regions in panels (a) and (b) indicate the parameterspaces restricted by the mass and radius of PSR J0030 + IV. SUMMARY
In this paper, we have investigated the e ff ects of non-Newtonian gravity on the properties of SSs and constraintthe parameter space of the QMDD model using astrophysi-cal observations related to PSR J0740 + ff ectsare not included. In other words, the existence of SSs is ruledout in this case.Considering the non-Newtonian gravity e ff ects, for the cur-rent quark mass m u = .
16 MeV, m d = .
67 MeV, and m s =
93 MeV [43], an allowed parameter space of D / and C exists only when 4.58 GeV − ≤ g /µ ≤ − , andthe parameters of the QMDD model are restricted to 158.3MeV ≤ D / ≤ − . ≤ C ≤ − .
12. Asshown in Fig. 4, theoretical bounds on g /µ of 4.58 GeV − ≤ g /µ ≤ − for which QSs are found to exist (indi-cated by the cyan-colored strip in the figure) is excluded bysome experiments (curves labeled 4, 6, 8, 9) but allowed byothers (curves labeled 1, 2, 5 and parts of curves 3 and 7).We also find that the largest allowed maximum mass ofSSs for the QMDD model is 2 . M ⊙ , corresponding to theparameter set D / = . C = − .
60 and g /µ = .
32 GeV − . Therefore, even considering the non-Newtoniane ff ect, the GW190814’s secondary component with mass2 . + . − . M ⊙ [69] could not be a static SS. However, it couldbe a rigid or di ff erentially rotating SS [70].Moreover, by translating the mass and radius of PSRJ0030 + D / – C space, wefind that for the analysis by Riley et al. [31], there exists avery tiny allowed parameter space for which SSs constructedwith the QMDD model agree with the observations related toPSR J0740 + + Acknowledgments
The authors are especially indebted to the anonymous ref-eree for his / her valuable comments. We thank J. Scha ff ner-Bielich for discussions on the stability of strange quark matter.This work is supported by National SKA Program of ChinaNo. 2020SKA0120300, and the Scientific Research Programof the National Natural Science Foundation of China (NSFC,grant Nos. 12033001, 11773011, and 11447012). F.W. is sup-ported through the U.S. National Science Foundation underGrants PHY-1714068 and PHY-2012152. [1] N. Itoh, Prog. Theor. Phys. 44, 291 (1970). [2] A.R. Bodmer, Phys. Rev. D 4,1601 (1971). -15 -14 -13 -12 -11 -10 -9 -8 -7 -610152025303540 2 0 -2 -4 -6 -25-20-15-10-50 g / =4.58 GeV -2 g / =9.32 GeV -2
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