Dark matter is hypothesized to interact with ordinary matter solely through gravity and may be present in compact objects such as strange quark stars. We treat strange quark stars admixed with dark matter as twofluid systems to investigate the potential effects of dark matter on strange quark stars. Quark matter is described by the quasiparticle model and the extended MIT bag model for comparison. Dark matter is treated as asymmetric selfinteracting and composed of massive fermionic particles. The twofluid TolmanOppenheimerVolkoff TOV equations are employed to solve for specific stellar properties. Our analysis yields relations between central energy density and mass radius and mass as well as tidal deformability and mass. The calculated curves generally align with observational data. In particular we find that the pattern in which fermionic asymmetric dark matter affects the properties of strange quark stars may not be influenced by the equation of state EOS of strange quark matter.
['Astrophysics - High Energy Astrophysical Phenomena']
Impact of dark matter on strange quark stars described by different quark models
{'Impact of dark matter on strange quark stars described by different quark models': 'Yida Yang, 1 Chen Wu, 2 and Ji-Feng Yang 1  \n1 School of Physics and Electronic Science, East China Normal University, Shanghai 200241, China 2 Xingzhi College, Zhejiang Normal University, Jinhua, 321004, China. (Dated: September 17, 2024)  \nDark matter is hypothesized to interact with ordinary matter solely through gravity and may be present in compact objects such as strange quark stars. We treat strange quark stars admixed with dark matter as two-fluid systems to investigate the potential effects of dark matter on strange quark stars. Quark matter is described by the quasiparticle model and the extended MIT bag model for comparison. Dark matter is treated as asymmetric, self-interacting, and composed of massive fermionic particles. The two-fluid Tolman-Oppenheimer-Volkoff (TOV) equations are employed to solve for specific stellar properties. Our analysis yields relations between central energy density and mass, radius and mass, as well as tidal deformability and mass. The calculated curves generally align with observational data. In particular, we find that the pattern in which fermionic asymmetric dark matter affects the properties of strange quark stars may not be influenced by the equation of state (EOS) of strange quark matter.', 'I. INTRODUCTION': 'If strange quark matter is the true ground state of strongly interacting matter [1-4], then there could exist a self-bound compact object composed of deconfined u, d, and s quarks [5, 6]. Strange quark stars are such hypothetical compact objects. They can be formed by the conversion of neutron stars [7-9], but their macroscopic properties would differ from those of traditional neutron stars. At the same mass, strange quark stars tend to have smaller radii, and both their mass and radius can be arbitrarily small [10]. Due to these characteristics, strange quark stars have become an attractive research subject in astrophysics and particle physics.  \nIn addition to ordinary matter, there is evidence suggesting the existence of dark matter in the universe [1118]. It does not interact directly with ordinary matter, causing detection difficulties [19, 20]. However, it produces significant gravitational effects, such as on compact objects like neutron stars. Dark matter may appear inside compact stars through capture processes [21-25] or accumulation during stellar formation [26, 27]. When dark matter admixes with ordinary matter in sufficient amounts, it will influence some observable properties of compact stars (such as mass-radius relation and tidal deformability) [19, 21]. Since strange quark stars are also a type of compact object, incorporating dark matter can expand our research on strange quark stars and in turn explore potential properties of dark matter [28].  \nIn this work, we study strange quark stars admixed with dark matter as a two-fluid system [28-30]. This requires solving the two-fluid Tolman-Oppenheimer-Volkov (TOV) equations [31] by incorporating the equations of state for quarks and dark matter. For strange quark matter, there are currently no reliable first-principle quantum chromodynamics (QCD) calculations [32, 33] of equation of state for strange quark matter at high densities. To this end, many phenomenological models have been developed, such as the MIT bag model [30, 34],  \nNambu-Jona-Lasinio (NJL) model [6, 35-37], quasiparticle model [38-43], and confined-density-dependent-mass (CDDM) model [44]. For comparison, we use the quasiparticle model [39, 45, 46] and the extended MIT bag model [47-49] to describe strange quark matter.  \nDue to the unclear origin and properties of dark matter, a rich variety of models have been proposed in literature. Examples include weakly interacting massive particles (WIMPs), sterile neutrinos, axions, mirror particles, supersymmetry particles, fermionic/bosonic asymmetric dark matter, etc (for a review, see [17, 50, 51]). The development of these dark matter models are motivated by various reasons, such as to solve the gauge hierarchy problem or the strong CP problem. In this work, we consider the possibility of zero-temperature fermionic asymmetric dark matter with self-interactions [30, 52]. Here, the asymmetric nature of dark matter prevents its self-annihilation [21].  \nThis paper is organized as follows. In Sec. II, the models to describe strange quark matter are presented. In Sec. III, we show the model of dark matter considered in this work. The TOV equations for two-fluid systems and the corresponding equations for solving tidal deformability are introduced in Sec. IV. Our numerical calculation results regarding the properties of strange quark stars admixed with dark matter are presented in Sec. V. Included in Sec. VI our the summaries and discussions.', 'II. STRANGE QUARK MATTER': 'Considering the electroweak reactions in quark stars, strange quark matter needs to fulfill the following chemical equilibrium conditions and electric charge neutrality condition:  \nµ d = µ u + µ e , (1)  \nµ s = µ u + µ e , (2)  \n2 3 ρ u = 1 3 ρ s + 1 3 ρ d + ρ e , (3)  \nwhich are necessary for calculating the energy density and pressure of strange quark matter. In this work, two different models are used to describe strange quark matter.', 'A. Quasiparticle model': 'As mentioned earlier, the current strange quark mass m s 0 in the quasiparticle model is treated as a free parameter. Here we compare the numerical results with m s 0 set to 120 MeV and 90 MeV respectively.  \nFirst, we study the relationship between the mass of admixed quark stars and their central energy density. The central energy density is divided into that of quark matter ( ϵ qm 0 ) and dark matter ( ϵ dm 0 ). And contour lines of stellar mass can be drawn on the ϵ dm 0 -ϵ qm 0 plane.  \nThen the dependence of the total stellar mass (in solar mass M ⊙ ) on the central energy densities of the two fluids is presented in Fig. 4. The region to the right of the white scattered line represents unstable stars and is not considered. It is evident that when ϵ qm 0 is fixed, increasing ϵ dm 0 leads to a decrease in the total stellar mass. Conversely, when ϵ dm 0 is fixed, changing ϵ qm 0 produces the opposite effect. It is also noted that the impact on mass due to changes in ϵ dm 0 is far less significant than that caused by changes in ϵ qm 0 .  \nNext, we present the relation between mass and radius in Fig. 5. Multi-messenger astronomical observation data are presented in the figure as regions/bands of different colors. The first and most obvious point is that the M -R curves of admixed quark stars are consistent with most of the observational data shown in Fig. 5 . However, some of the maximum masses cannot reach the data of PSR J0348+0432. This is partly due to the choice of parameters in the quark model, where a larger m s 0 would reduce the maximum mass of the star. On the other hand, the addition of dark matter also reduces the maximum mass of admixed quark stars. And the M -R curve of quark stars without dark matter almost envelops that of those with dark matter.  \n)  \n3  \nm  \nf  \n/  \nV  \ne  \nM  \n(  \nε  \nFIG. 5: Mass-radius ( M -R ) relation of strange quark star admixed with dark matter. The current strange quark mass m s 0 is 120 MeV for panel (a) and 90 MeV for panel (b). The color of the solid line represents the strange quark star admixed with (orange) or not admixed with (blue) dark matter. The light green region represents the observational data of PSR J0030+0451 after Bayesian analysis [77], while the light blue region is from the observational data of the central compact object within the supernova remnant HESS J1731-347 [73]. The area enclosed by the black dashed line comes from the gravitational wave data analysis of GW170817 [74, 75] and the yellow band corresponds to the mass measurement of PSR J0348+0432 [76].  \n<!-- image -->  \n<!-- image -->  \n)  \n3  \nm  \nf  \n/  \nV  \ne  \nM  \n(  \nε  \nFIG. 4: Contour plot with ϵ qm 0 , ϵ dm 0 and M/M ⊙ as the x, y, and z axes, respectively. The current strange quark mass m s 0 is 120 MeV for panel (a) and 90 MeV for panel (b). The white line with scattering points represents the maximum mass of quark stars admixed with dark matter. Such stars in the region to the right of this line are unstable. The region where the maximum mass exceeds 2 M ⊙ is delineated by a white curve.  \n<!-- image -->  \nR  \n(b)  \nTo further investigate the gravitational effects of adding dark matter to strange quark stars, the relationship between tidal deformability (Λ) and mass is presented in Fig. 6. It can be seen that for 1.4 M ⊙ admixed quark stars, increased dark matter and smaller m s 0 reduce tidal deformability, aligning partially with GW170817 data. And corresponding to the position of 1.4 M ⊙ on the M -R curve, it can also be observed that the decrease in tidal deformability is accompanied by a  \nreduction in radius.', 'B. Extended MIT bag model': 'When describing quark matter using this model, we employ two sets of parameters: g = 2 , B 1 / 4 = 141 MeV and g = 3 , B 1 / 4 = 136 MeV. Similar to the previous subsection, here we also present three figures illustrating  \n2.0  \n1.5  \n1.0  \n0.5  \nPSR J0348  \n+  \n0432  \nPSR J0030  \n+  \n0451  \nWith dark matter  \nWithout dark matter  \n(a)  \nM/M  \n/s9737  \n⊙  \nM  \n/  \nM  \n(b)  \n8  \n10  \n12  \n14  \n16  \n18  \nGW170817  \nHESS J1731  \n-  \n347  \n(  \nkm  \n)  \nM/M  \n/s9737  \nFIG. 6: Relation between tidal deformability and the mass (Λ -M ) of admixed quark star. The current strange quark mass m s 0 is 120 MeV for panel (a) and 90 MeV for panel (b). The black error bar represents the range of tidal deformability for GW170817 at 1.4 M ⊙ [74]. The color of the solid line represents the strange quark star admixed with (orange) or not admixed with (blue) dark matter.  \n<!-- image -->  \nthe following relationships: ϵ dm 0 -ϵ qm 0 -M (Fig. 7), M -R (Fig. 8) and Λ -M (Fig. 9).  \nIn Fig. 7, we can observe that the trend of maximum mass variation with central energy density ( ϵ qm 0 and ϵ dm 0 ) is consistent with that of the quasiparticle model. Subsequent analysis focuses on the M -R plot (Fig. 8) and Λ -M plot (Fig. 9). The pattern of how dark matter content affects the two curves is similar to that in the quasi-particle model. Both the maximum mass and Λ 1 . 4 will slightly decrease as the amount of dark matter increases. However, comparatively, changes in the parameter set have a much greater impact on stellar properties than changes in the amount of dark matter. It can be found in Fig. 9 that once Λ 1 . 4 (without dark matter) exceeds a certain threshold, increasing dark matter content fails to reduce Λ 1 . 4 to the observational data range.', 'III. DARK MATTER': 'As mentioned in the introduction, the dark matter considered in this work is asymmetric. Asymmetric dark  \nFIG. 2: Equations of state for quark matter described by the extended MIT bag model. The color of the lines indicates different parameter choices, with g = 2 , B 1 / 4 = 141 MeV shown in dark orange and g = 3 , B 1 / 4 = 136 MeV in blue.  \n<!-- image -->  \nmatter may be composed of weakly-interacting massive particle [63]. We assume these weakly-interacting massive particles are fermions with a mass of 100 GeV (as in Ref. [30]). First we take the dark matter as a free Fermi gas, its energy density, pressure and number density can  \nbe expressed as,  \nϵ = 1 π 2 ∫ k F 0 k 2 √ m 2 f + k 2 dk = m 4 f 8 π 2 [ (2 z 3 + z ) √ 1 + z 2 -sinh -1 ( z ) ] (17) P = 1 3 π 2 ∫ k F 0 k 4 √ m 2 f + k 2 dk = m 4 f 24 π 2 [ (2 z 3 -3 z ) √ 1 + z 2 +3sinh -1 ( z ) ] (18) (19)  \nn = k 3 F 3 π 2 ≡ m 3 f z 3 3 π 2 ,  \nwhere z = k F /m f is the dimensionless Fermi momentum, k F and m f are the Fermi momentum and masses of dark fermions. Next we include interactions between dark fermions. To proceed, we consider the simplest twobody repulsive interactions [52]. In the lowest order approximation, the energy density of self-interaction is proportional to n 2 [52, 64]. The resulting equation of state is given by [30]  \nϵ = m 4 f 8 π 2 [ (2 z 3 + z ) √ 1 + z 2 -sinh -1 ( z ) ] +   ( m 3 f z 3 3 π 2 ) 2 η 2 m 2 f   (20)  \nP = m 4 f 24 π 2 [ (2 z 3 -3 z ) √ 1 + z 2 +3sinh -1 ( z ) ] +   ( m 3 f z 3 3 π 2 ) 2 η 2 m 2 f   , (21)  \nwhere η = m f /m I is the dimensionless interaction strength parameter and m I is the interaction mass scale [30, 52, 65, 66].  \nThe typical interaction mass scale of strong interaction is m I ∼ 100 MeV. In this case, dark matter fermions with a mass of 100 GeV can have strong self-interactions with a strength of η ∼ 10 3 . Thus, here we focus on the situation of strongly self-interacting dark matter with m I = 100 MeV and η = 10 3 . The resulting EOS in this case is presented in Fig .3.', 'IV. TWO-FLUID TOV EQUSTIONS AND TIDAL DEFORMABILITY': 'In order to obtain the properties of a strange quark star admixed with dark matter, the two-fluids TOV equations are needed. In a two-fluid system, each independent fluid has its own hydrodynamic equilibrium condition, and there are only gravitational interactions between the fluids. The two-fluid TOV equations that we use here are  \nFIG. 3: Equation of state for fermionic dark matter with strong self-interaction.  \n<!-- image -->  \n[28-31, 65, 67]  \ndp 1 dr = -G r 2 [ M ( r ) + 4 πr 3 ( p 1 ( r ) + p 2 ( r )) ] × [ ϵ 1 ( r ) + p 1 ( r )] [ 1 -2 G M ( r ) r ] -1 (22)  \ndp 2 dr = -G r 2 [ M ( r ) + 4 πr 3 ( p 1 ( r ) + p 2 ( r )) ] × [ ϵ 2 ( r ) + p 2 ( r )] [ 1 -2 G M ( r ) r ] -1 (23)  \ndM 1 dr = 4 πr 2 ϵ 1 ( r ) (24)  \ndM 2 dr = 4 πr 2 ϵ 2 ( r ) (25)  \nM ( r ) = M 1 ( r ) + M 2 ( r ) . (26)  \nTo solve these equations, we need the boundary conditions M 1 (0) = 0 and M 2 (0) = 0, as well as the values of central pressures p 1 (0) and p 2 (0) . Then the functions of p 1 ( r ) , p 2 ( r ) and M ( r ) can be solved. The pressures of both fluids drop to zero at radial coordinate points R 1 and R 2 respectively, i.e., p 1 ( R 1 ) = p 2 ( R 2 ) = 0. The larger value among R 1 and R 2 is defined as the overall admixed star radius R out . Finally, the total mass of the admixed star can be obtained by substituting R out into M ( r ).  \nIn addition to mass and radius, tidal deformability is also a property we are interested in. In order to obtain the tidal deformability of the star, we need to solve the following equation along with the TOV equation simul-  \ntaneously [68, 69]:  \nr dy ( r ) dr + y 2 ( r ) + y ( r ) F ( r ) + r 2 Q ( r ) = 0 , (27)  \nF ( r ) = r -4 πGr 3 ∑ i ( ϵ i -p i ) r -2 GM (28)  \n(29)  \nQ ( r ) = 4 πGr r -2 GM × { ∑ i [ 5 ϵ i +9 p i + ∂ϵ i ∂p i ( ϵ i + p i ) ] -6 4 πGr 2 } -[ 8 πGr 3 ∑ i p i +2 GM r ( r -2 GM ) ] 2 .  \nThe sum over i in functions F ( r ) and Q ( r ) implies considering multiple fluids. Here y ( r ) is the logarithmic derivative for metric perturbation [70, 71].  \nThere is the squared adiabatic speed of sound dϵ dp in Eq. (27) for y ( r ). It should be noted that according to models of strange quark stars, there is a finite energy density at their surface. Such situation similar to a first-order phase transition at constant pressure leads to c 2 s ≡ dp dϵ , while dϵ dp is discontinuous. To address this, the expression needs to be modified as follow [70, 72]  \n̸  \ndϵ dp = 1 c 2 s = dϵ dp ∣ ∣ ∣ ∣ ∣ p = p d +∆ ϵδ ( p -p d ) . (30)  \nThe ∆ ϵδ ( p -p d ) term results in an extra term in the solution of y ( r ), which is written as [70]:  \ny ( r + d ) -y ( r -d ) = -4 πr 3 d ∆ ϵ m ( r d ) + 4 πr 3 d p ( r d ) . (31)  \nHere r d is the position where dϵ dp is discontinuous, and r ± d represents two points infinitesimally distant from r d in opposite directions. For a admixed quark star, r d here represents the radius of the portion containing strange quark matter ( R qm ). And ∆ ϵ = ϵ ( R -qm ) is the energy density just inside this strange quark matter region.  \nAfter obtaining the solution at surface of the star  \ny R = y ( R out ) + extra term,  \nand substitute it into the expression for the l = 2 tidal Love number,  \nk 2 = 8 5 β 5 (1 -2 β ) 2 [2 -y R +2 β ( y R -1)] ×{ 2 β [6 -3 y R +3 β (5 y R -8)] +4 β 2 [13 -11 y R + β (3 y R -2) + 2 β 2 (1 + y R )] +3(1 -2 β ) 2 [2 -y R +2 β ( y R -1)] ln(1 -2 β ) } -1 , (32)  \nwhere β = GM ( R out ) /R out . Finally, the tidal deformability of the star can be calculated through the formula, which reads:  \nΛ = 2 3 k 2 β -5 . (33)', 'V. PROPERTIES OF STRANGE QUARK STARS WITH DARK MATTER': 'In this section, we use the two quark models introduced above to investigate the properties of strange quark stars with fermionic dark matter inside.  \nTo study the influence of dark matter on the properties of strange quark stars, observational data is essential for comparison. A compact object has recently been discovered at the center of the supernova remnant HESS J1731-347 [73]. It is speculated to be either the lightest known neutron star or a strange quark star. The gravitational wave event GW170817 [74, 75] is also an important data source. Existing analyses based on this event have yielded a range of masses and radii, as well as an interval for tidal deformability of a 1 . 4 M ⊙ compact star Λ 1 . 4 = 190 +390 -120 . In addition to the aforementioned data, in this work we also employ data concerning PSR J0348+0432 (2 . 01 ± 0 . 04 solar mass M ⊙ ) [76] and PSR J0030+0451 [77, 78]. See the figures drawn later in this section for details.', 'VI. SUMMARY AND DISCUSSION': 'In this work, we have investigated the effects of dark matter on strange quark stars. The strange quark matter is described by the quasiparticle model and the extended MIT bag model, respectively. Dark matter is assumed to be self-interacting Fermi gas, and interacts with quark matter only through gravitational interactions. The twofluid TOV equations are solved to study the properties of strange quark stars admixed with dark matter.  \nBoth the quasiparticle model and the extended MIT bag model employ the quasiparticle concept and account for medium effects. The impact of fermionic asymmetric dark matter on strange quark stars described by these models may follow a similar pattern. For a fixed ϵ qm 0 , the mass of stable admixed quark star decreases as the ϵ dm 0 increases. To maintain a constant mass while increasing the ϵ dm 0 , a higher ϵ qm 0 is required. Similarly, the increase in ϵ dm 0 drives stable admixed quark stars towards greater ϵ qm 0 to attain maximum mass. Concurrently, the maximum mass gradually decreases with increasing central energy density of dark matter. Besides the maximum mass, Λ 1 . 4 also exhibits this pattern, i.e., The increase in the central energy density of dark matter causes Λ 1 . 4 to decrease.  \nEvidently, most of our numerical results for M -R and Λ -M relations are consistent with observational data. It had been expected that the addition and increase of fermionic asymmetric dark matter would significantly alter the calculated M -R curve and Λ -M curve. However, our calculations in this work show that its impact appears to be less significant. Changes in certain parameters of the quark model used in this work appear to have an even greater impact on these curves. Consequently, the effects of different types of dark matter on strange quark stars described by various quark models still have significant room for future investigation. It is important for the study of properties of dark matter and strange quark stars.  \n)  \n3  \nm  \nf  \n/  \nV  \ne  \nM  \n(  \nε  \nFIG. 8: Mass-radius ( M -R ) relation of strange quark star admixed with dark matter. The parameter set is g = 2 , B 1 / 4 = 141 MeV for panel (a) and g = 3 , B 1 / 4 = 136 MeV for panel (b). The color of the solid line represents the strange quark star admixed with (orange) or not admixed with (blue) dark matter. The light green region represents the observational data of PSR J0030+0451 after Bayesian analysis [77], while the light blue region is from the observational data of the central compact object within the supernova remnant HESS J1731-347 [73]. 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