Eur. Phys. J. C (2017) 77:440 DOI 10.1140/epjc/s10052-017-5006-3

Regular Article - Theoretical Physics

Compact bifluid hybrid : hadronic matter mixed with self-interacting fermionic asymmetric

Somnath Mukhopadhyay1,a, Debasis Atta1,2,b, Kouser Imam1,3,c,D.N.Basu1,d, C. Samanta4,e 1 Variable Energy Cyclotron Centre, HBNI, 1/AF Bidhan Nagar, Kolkata 700 064, India 2 Government General Degree College, Kharagpur II, West Bengal 721 149, India 3 Department of Physics, Aliah University, IIA/27, New Town, Kolkata 700156, India 4 Department of Physics and Astronomy, Virginia Military Institute, Lexington, VA 24450, USA

Received: 1 February 2017 / Accepted: 20 June 2017 / Published online: 3 July 2017 © The Author(s) 2017, corrected publication August 2017. This article is an open access publication

Abstract The masses and radii of non-rotating and rotat- Various theoretical models of dark matter are widespread, ing configurations of pure hadronic stars mixed with self- ranging from to to hot interacting fermionic asymmetric dark matter are calculated dark matter and from symmetric to asymmetric dark matter within the two-fluid formalism of equations [2Ð6]. Recent advances in cosmological precision tests fur- in general relativity. The Equation of State (EoS) of nuclear ther consolidate the minimal cosmological standard model, matter is obtained from the density dependent M3Y effec- indicating that the universe contains 4.9% ordinary matter, tive nucleonÐnucleon interaction. We consider the dark mat- 26.8% dark matter and 68.3% . Although being ter particle mass of 1 GeV. The EoS of self-interacting dark five times more abundant than ordinary matter, the basic matter is taken from two-body repulsive interactions of the properties of dark matter, such as particle mass and inter- scale of strong interactions. We explore the conditions of actions are unsolved. equal and different rotational frequencies of nuclear matter A dark composed mostly of normal matter and dark and dark matter and find that the maximum mass of differ- matter may have existed early in the universe before con- entially rotating stars with self-interacting dark matter to be ventional stars were able to form. Those stars generate heat ∼1.94 M with radius ∼10.4km. via annihilation reactions between the dark-matter particles. This heat prevents such stars from collapsing into the rela- tively compact sizes of modern stars and therefore prevent 1 Introduction among the normal matter atoms from being initiated [7]. In the universe there are large empty regions and dense One theory is that dark matter could be made of particles regions where the are distributed. This distribution called . Unlike protons, neutrons and electrons that is called the cosmic web that is speculated to be governed make up ordinary matter, axions can share the same quan- by the action of on the invisible mysterious “dark tum energy state. They also attract each other gravitationally, matter”. Recently, a research group led by Hiroshima Uni- so they clump together. Dark matter is hard to study because versity has suggested that the Cancer has nine it does not interact much with ordinary matter but dark such large concentrations of dark matter, each the mass of a matter could theoretically be observed in the form of Bose cluster [1]. stars [8]. The BoseÐEinstein condensation may come from the bosonic features of dark matter models. A phase transi- tion to condensation can occur either when the temperature The original version of this article was revised: on page 3 first decreases to below the critical value or when the density paragraph av was not correctly displayed. exceeds the critical value [9]. a e-mail: [email protected] The neutron stars could capture weakly interacting dark b e-mail: [email protected] matter particles (WIMPs) because of their strong gravita- tional field, high density and finite, but very small, WIMP- c e-mail: [email protected] to-nucleon cross section. In fact, if there is no baryon-dark d e-mail: [email protected] matter interaction, a purely baryonic would not e e-mail: [email protected] 123 440 Page 2 of 9 Eur. Phys. J. C (2017) 77 :440 capture dark matter at all. A dark star of comparable mass Although dark matter particles can have only very weak may as well accrete neutron star matter to form a dark matter interactions with standard model states, it is an intriguing pos- dominated neutron star. In 1978, Steigman et al. [10] sug- sibility that they experience much stronger self-interactions gested that capture of WIMPs by individual stellar objects and thereby alter the behavior of dark matter on astrophysi- could affect the stellar structure and evolution. The effects of cal and cosmological scales in striking ways. Recent stud- self-annihilating dark matter on first-generation stars and on ies [30Ð35] have provided constraints on the dark matter the evolution path of stars have been studied self-interaction cross section. The constraints are based on extensively [11,12]. For non self-annihilating dark matter, its the cusp-core problem and the “Too big to fail” problem of impact on main sequence stars [13] and neutron stars [14,15] galaxies. According to them the dark matter self-interaction has been studied in different dark matter models. Gravita- cross section per unit mass is about 0.1Ð100 cm2/g ∼0.1Ð1 tional effects of non self-annihilating condensate dark mat- barn/GeV, typical of the scale of strong interactions. ter on compact stellar objects has been studied [16] assuming In this work we consider fermionic asymmetric dark mat- dark matter as ideal Fermi gas and considering the ter (ADM) particles of a mass of 1 GeV and the self- process through dark matter self-interaction from the sur- interaction mediator mass of 100 MeV (low mass imply- rounding halo. Non-annihilating heavy dark matter of mass ing strong interaction), mixed with rotating and non-rotating greater than 1 GeV is predicted to accumulate at the center of neutron stars. ADM, like ordinary baryonic matter, is charge a neutron star leading it to a possible collapse [17]. The effect asymmetric with only the dark baryon (or generally only of this accumulation is observable only in cases where the the particle) excess remaining after the annihilation of most annihilation cross section is extremely small [18,19]. The antiparticles after the Big Bang. Hence these ADM particles capture is fully efficient even for WIMP-to-nucleon cross are non self-annihilating and behave like ordinary free parti- sections (elastic or inelastic) as low as 10−18 mb. Moreover, cles. The gravitational stability and massÐradius relations of a dark star of comparable mass may as well accrete neutron static, rigid and differentially rotating neutron stars mixed star matter to form a dark matter dominated neutron star. In with fermionic ADM are calculated using the LORENE addition to axions and WIMPs, a general class of dark matter code [36]. It is important to note that we do not allow any candidates, called Macros, have been suggested that would phase transition of the nuclear matter and that the interac- have macroscopic size and mass [20]. tion between nuclear matter and dark matter is only through Since dark matter interacts with normal baryonic matter gravity. through gravity, it is quite possible for white dwarfs and neu- tron stars to accrete dark matter and evolve to a dark matter admixed [12,15,17,21Ð26]. The large baryonic density in compact stars increases the probability of dark 2 Equation of state of β-equilibrated nuclear matter matter capture within the star and eventually results in grav- itational trapping. It may also be possible for dark matter The nuclear matter EoS is calculated using the isoscalar and alone to form gravitationally bound compact objects and thus the isovector [37,38] components of M3Y interaction along mimic black holes [27]. with density dependence. The density dependence of this The hydrostatic equilibrium configuration of an admixture DDM3Y effective interaction is completely determined from of degenerate dark matter and normal nuclear matter was nuclear matter calculations. The equilibrium density of the studied by using a general relativistic two-fluid formalism nuclear matter is determined by minimizing the energy per taking non self-annihilating dark matter particles of a mass nucleon. The energy variation of the zero range potential is of 1 GeV. A new class of compact stars was predicted that treated accurately by allowing it to vary freely with the kinetic consisted of a small normal matter core with radius of a few energy part kin of the energy per nucleon  over the entire kilometers embedded in a 10 km-sized [15]. range of . This is not only more plausible but also yields Compact objects formed by non self-annihilating dark excellent results for the incompressibility K∞ of the SNM matter admixed with ordinary matter has been predicted with which does not suffer from the superluminosity problem [39]. Earth-like masses and radii from few km to few hundred km In a Fermi gas model of interacting neutrons and protons ρ −ρ for weakly interacting dark matter. For the strongly inter- n p with isospin asymmetry X = ρ +ρ ,ρ = ρn + ρp, where acting dark matter case, dark compact are suggested n p ρn, ρp and ρ are the neutron, proton and nucleonic densities, to form with Jupiter-like masses and radii of few hundred respectively, the energy per nucleon for isospin asymmetric km [28]. Possible implications of asymmetric fermionic dark nuclear matter can be derived as [39] matter for neutron stars have been studied that apply to vari- ous dark fermion models such as models and to     2 2 other models where the dark fermions have self-interactions 3h¯ k ρ JvC (ρ, X) = F F(X) + (1 − βρn) (1) [29]. 10m 2 123 Eur. Phys. J. C (2017) 77 :440 Page 3 of 9 440  2 1 2 2 where m is the nucleonic mass, kF = (1.5π ρ)3 which 2h¯ k F0 equals the Fermi momentum in the case of SNM, the kinetic C =− , (5) qh¯ 2k2 (1−βρn) 3h¯ 2k2 0 n F0 0 kin =[ F ] ( ) ( ) = 5mJv ρ0 1 − (n + 1)βρ −  energy per nucleon 10m F X with F X 00 0 10m 0 / / [ (1+X)5 3+(1−X)5 3 ] = + 2 2 and Jv Jv00 X Jv01, Jv00 and Jv01 represent the volume integrals of the isoscalar and the isovec- respectively. It is quite obvious that the constants of the den- tor parts of the M3Y interaction. The isoscalar t M3Y and the 00 sity dependence C and β obtained by this method depend isovector t M3Y components of M3Y interaction potential are 01 on the saturation energy per nucleon 0, the saturation den- given by sity ρ0, the index n of the density dependent part and on exp(−4s) exp(−2.5s) the strengths of the M3Y interactions through the volume t M3Y (s,) =+ − 00 7999 2134 . 0 4s 2 5s integral Jv00. +J00(1 − α)δ(s) The calculations are performed using the values of the (− ) (− . ) ρ = . −3 M3Y ( ,) =− exp 4s + exp 2 5s saturation density 0 0 1533 fm [42] and the satura- t01 s 4886 1176 4s 2.5s tion energy per nucleon 0 =−15.26 MeV [43]forthe +J01(1 − α)δ(s) (2) SNM obtained from the coefficient of the volume term of the BetheÐWeizsäcker mass formula which is evaluated by where s represents the relative distance between two interact- 3 3 fitting the recent experimental and estimated atomic mass ing nucleons, J00 =−276 MeV.fm , J01 =+228 MeV fm − excesses from the AudiÐWapstraÐThibault atomic mass table and the energy dependence parameter α = 0.005 MeV 1. [44] by minimizing the mean square deviation incorporating The Yukawa strengths were extracted by fitting the matrix correction for the electronic binding energy [45]. In a similar elements in an oscillator basis to those elements of a G-matrix recent work, addressing the surface symmetry energy term, obtained with the ReidÐElliott soft core NN interaction. The the Wigner term, the shell correction and the proton form ranges were selected to ensure OPEP tails in the relevant factor correction to the Coulomb energy, the av turns out to channels as well as a short-range part which simulates the σ - be 15.4496 MeV and when the A0 and A1/3 terms are also exchange process [40]. The density dependence is employed included it becomes 14.8497 MeV [46]. Using the usual val- to account for the Pauli blocking effects and the higher order ues of α = 0.005 MeV−1 for the parameter of the energy exchange effects [41]. Thus the DDM3Y effective NN inter- M3Y dependence of the zero range potential and n = 2/3, the val- action is given by v i (s,ρ,) = t (s,)g(ρ) where the 0 0i ues obtained for the constants of density dependence C and density dependence g(ρ) = C(1 − βρn) [39] with C and β β and the SNM incompressibility K∞ are 2.2497, 1.5934 being the constants of the density dependence. fm2 and 274.7 MeV, respectively. The saturation energy per Equation (1) can be differentiated with respect to ρ to nucleon is the volume energy coefficient and the value of yield an equation for X = 0:   −15.26 ± 0.52 MeV covers, more or less, the entire range 2 2 ∂ h¯ k Jv C   of values obtained for av for which now we have the values = F + 00 1 − (n + 1)βρn ∂ρ 5mρ 2 of C = 2.2497 ± 0.0420, β = 1.5934 ± 0.0085 fm2 and the   SNM incompressibility K∞ = 274.7 ± 7.4MeV.   ¯ 2 2 n h kF The symmetric nuclear matter incompressibility K∞,the − α J00C 1 − βρ . (3) 10m nuclear symmetry energy at saturation density Esym(ρ0),the slope L and the isospin dependent part Kτ of the isobaric The equilibrium density of the cold SNM is determined from incompressibility are also tabulated in Table 1, since these the saturation condition. Then Eqs. (1) and (3) with the sat- ∂ are all in excellent agreement with the recently extracted uration condition = 0atρ = ρ0,  = 0 can be solved ∂ρ constraints from the measured isotopic dependence of the simultaneously for fixed values of the saturation energy per giant monopole resonances in even-A Sn isotopes [47], from nucleon 0 and the saturation density ρ0 of the cold SNM to the neutron skin thickness of nuclei and from analyses of obtain the values of β and C. The constants of the density experimental data on isospin diffusion and isotopic scaling dependence β and C, thus obtained, are given by in intermediate energy heavy-ion collisions.  ( − ) + − 3q ρ−n The calculations for masses and radii are performed using 1 p q p 0 β =  (4) the EoS covering the crustal region of a compact star which ( + ) − ( + ) + − 3q are the FeynmanÐMetropolisÐTeller (FMT) [48], BaymÐ 3n 1 n 1 p q p PethickÐSutherland (BPS) [49] and BaymÐBetheÐPethick − [  ] α 3 = 10m 0 = 2 0 J00 0 = (kin) (BBP) [50] cases up to number density of 0.0582 fm and where p 2 2 , q 0 , Jv Jv00 implying [h¯ k ] Jv 00 0 β F0 00 -equilibrated neutron star matter beyond this value. Figures kin = kin Jv00 at 0 , the kinetic energy part of the saturation 1 and 2 represent the massÐcentral density and massÐradius =[ . π 2ρ ]1/3 energy per nucleon of SNM, kF0 1 5 0 and plots, respectively, for slowly rotating pure neutron stars with 123 440 Page 4 of 9 Eur. Phys. J. C (2017) 77 :440

2 Table 1 Results of present calculations for n= 3 of symmetric nuclear matter incompressibility K∞, nuclear symmetry energy at saturation density Esym(ρ0), the slope L and the isospin dependent part Kτ of the isobaric incompressibility (all in MeV) [54,55] are tabulated

K∞ Esym(ρ0) LKτ

274.7 ± 7.430.71 ± 0.26 45.11 ± 0.02 −408.97 ± 3.01

Fig. 2 MassÐequatorial radius plot of slowly rotating neutron stars for the DDM3Y EoS

  2 2 1 3 1 p c 3 Pχ = pvn pd p =  n pd p (6) 3 3 2 2 2 4 (p c + mχ c )

where mχ is the rest mass of dark particles, v is the velocity  3 Fig. 1 Mass vs. central baryonic density plot of slowly rotating neutron of the particles with momentum p and n pd p is the num- stars for the DDM3Y EoS ber of particles per unit volume having momenta between   +  1 p and p d p. The factor 3 accounts for the fact that, on 1 3 average, only of total particles n pd p are moving in a par- DDM3Y EoS. The maximum mass goes to 1.9227 M with 3 ticular direction. For fermions having spin 1 , degeneracy = 2, a radius of 9.7559 km [51Ð53]. 2 8πp2dp n d3 p = nχ p h3 and hence the number density is given by 3 Equation of state of non-interacting fermionic  pF π 3 3 3 8 pF xF asymmetric dark matter nχ = n d p = = p 3 2 3 (7) 0 3h 3π λχ We consider the non-interacting fermionic ADM to be a com- pletely degenerate free Fermi gas of particle mass mχ at zero where pF is the Fermi momentum, which is the maximum = pF temperature. By the Pauli exclusion principle no quantum momentum possible at zero temperature, xF mχ c is a h¯ state can be occupied by more than one fermion with an iden- λχ = dimensionless quantity and mχ c is the Compton wave- tical set of quantum numbers. Thus a non-interacting Fermi length. The energy density εχ is given by gas, unlike a Bose gas, is prohibited from condensing into a BoseÐEinstein condensate. The total energy of the Fermi gas    pF pF π 2 3 8 p dp at absolute zero is larger than the sum of the single-particle εχ = En d p = (p2c2 + m2 c4) , p χ 3 ground states because the Pauli principle implies a degener- 0 0 h acy pressure that keeps fermions separated and moving. (8) The non-interacting assembly of fermions at zero temper- ature exerts pressure because of kinetic energy from different which, along with Eq. (7), turns out upon integration to be states filled up to Fermi level. Since the pressure is the force per unit area, which means the rate of momentum transfer mχ c2 mχ c2 εχ = χ(x ); Pχ = φ(x ), 3 F 3 F (9) per unit area, it is given by λχ λχ 123 Eur. Phys. J. C (2017) 77 :440 Page 5 of 9 440 where To proceed to the EoS, we calculate the total energy of a system of particles classically by summing over the inter-    1 actions of all pairs of particles. To facilitate the calcula- χ(x) = x 1 + x2(1 + 2x2) − ln(x + 1 + x2) 8π 2 tion, we assume that the macroscopic assembly is uniformly (10) distributed, thereby neglecting the influence of the interac- tion on the mean interparticle separation. In other words, and we ignore any correlations between particle positions due to their mutual interaction. Finally, we assume that the number     1 2x2 of particles is sufficiently large so that we can replace sums φ(x) = x 1 + x2 − 1) + ln(x + 1 + x2 . 8π 2 3 by integrals and the characteristic size of the assembly R  / (11) satisfies R 1 m I [56]. The total Yukawa potential energy of a system of N par- ticles in volume  is  4 Equation of state of strongly self-interacting  −m I rij 1 1 2 2 e fermionic asymmetric dark matter E = Vij = n g di d j , (18) 2 2 4πr i= j ij In order to calculate the EoS of a strongly interacting fermionic ADM we turn to massive vector field theory sim- where n is the number density. ilar to the meson exchange of the nuclear interaction. The Choosing one particle at the origin and integrating to infin- Lagrangian density (in natural units) of a massive vector field ity (ignoring surface terms) we find is given by 1 2 2 E = n g , (19) 2m2 1 μν 1 2 μ μ I L =− Fμν F + m Aμ A − jμ A (12) 4 2 I so that the interaction energy density can be written as μ μ where Fμν = ∂μ Aν − ∂ν Aμ , A is the four-vector field, j E 1 is the four-current and m is the mass of the field quanta. The ε = = n2g2. (20) I int  2 equation of motion is given by 2m I 2/ = = / (∂ ∂ν + 2) μ = μ. Now putting g 2 1 for convenience, x f k f mχ , where ν m I A j (13) mχ is the rest mass of the dark matter particle and using the relation k = (3π 2n)1/3 we get putting back h¯ and c Now considering a charge of g at rest at the origin f   we have 2 6 6 1 x f mχ εint = , (21) 0 3  π 2 (¯ )3 2 j = gδ (x) j = 0. (14) 3 hc m I

where mχ and m are expressed in MeV. Substituting the above in the right side of Eq. (14) and also I The pressure due to the interacting energy density can noting that A0 = V and A = 0 we finally get be computed with the help of the thermodynamic relation ε = 2 d int (∇2 − 2) =− δ3() Pint n , which yields m I V g x (15) dn n   2 6 6 whose solution is the Yukawa potential: 1 x f mχ Pint = . (22) 3π 2 (hc¯ )3m2 − I e m I r V (r) = g . (16) 4πr Hence the total energy density and pressure of self- interacting dark matter particles are given by Hence the potential energy of two like charges of magnitude   g is 2 6 6 mχ 1 x f mχ εχ = εχ + ε = χ(x ) + , (23) int int 3 F 2 3 2 − λχ 3π (hc¯ ) m e m I r I V (r) = g2 (17) 12 π   4 r 2 6 6 mχ 1 x f mχ Pχ = Pχ + P = φ(x ) + . (24) and is repulsive in nature. int int λ3 F π 2 (¯ )3 2 χ 3 hc m I 123 440 Page 6 of 9 Eur. Phys. J. C (2017) 77 :440

5 Two-fluid TOV equation

We consider two ideal fluids—the nuclear matter and fermionic dark matter with the above two EoSs coupled grav- itationally to form the structure of the mixed neutron star. The energy-momentum tensor of the mixed fluid can be written as [29,57] μν = μν + μν = (ε + ) μ ν − μν T Tnuc Tdark nuc Pnuc u1 u1 Pnucg + (ε + ) μ ν − μν dark Pdark u2 u2 Pdarkg (25) μ ε where u1 , nuc and Pnuc are the 4-velocity, energy density and pressure of nuclear matter, respectively, while the corre- sponding quantities in the second term are for dark matter. For a non-rotating case the metric is spherically symmetric Fig. 3 Plots of mass vs. central density for static and rotating fermionic and the hydrostatic equations of the two fluids can be written asymmetric dark matter stars as coupled two-fluid TolmanÐOppenheimerÐVolkoff (TOV) equations,   dPnuc(r) GM(r)ρnuc(r) Pnuc =− 1 + dr r2 εnuc     3 −1 4πr (Pnuc + P ) 2GM(r) × 1 + dark 1 − M(r)c2 rc2   dP (r) GM(r)ρ (r) P dark =− dark 1 + dark 2 dr r εdark     3 −1 4πr (Pnuc + P ) 2GM(r) × 1 + dark 1 − M(r)c2 rc2 ( ) dMnuc r 2 = 4πr ρnuc(r), dr dM (r) dark = 4πr2ρ (r), dr dark M(r) = Mnuc(r) + Mdark(r), (26)

2 where ρnuc = εnuc/c , Mnuc is the mass density and the total Fig. 4 MassÐequatorial radius plots for static and rotating fermionic mass of nuclear matter, while the corresponding quantities asymmetric dark matter stars in the second equation are for dark matter. M(r) is the total mass of nuclear and dark matter.

The mass of the exchange boson determines the strength 6 Theoretical calculations and the range of the interaction, implying that the lower the mass is, the stronger the interaction. For non-interacting dark The massÐradius relationship of non-rotating, rigidly rotating matter m I is infinite and second terms in the above equations and differentially rotating neutron stars admixed with dark are absent. matter is calculated using the LORENE code. The nuclear Figures 3 and 4 depict the plots of mass vs. central dark matter and dark matter EoSs are fitted to a polytropic form matter density and mass vs. equatorial radius, respectively, P = Kργ where P is the pressure, ρ is the mass density, for static and rotating stars, using a self-interacting dark mat- K the polytropic constant and γ the polytropic index for the ter EoS. We see that the maximum mass for non-rotating stars corresponding fluid. For interacting nuclear matter γ = 2.03 goes to 3.0279 M with a radius of 16.2349 km and that the and K = 5.65283 × 1035 in C.G.S. units. For interacting maximum mass for rotating stars goes to 3.1460 M with dark matter γ = 1.97562 and K = 1.33404 × 1036 in equatorial radius of 19.2173 km. Now, if we take the dark C.G.S. units. We take the dark matter particle mass to be 1 matter particle mass mχ to be 0.5 GeV the maximum mass GeV and the exchange boson mass m I = 100 MeV, typical 2 goes to ∼12.6M using the relation mass ∝ 1/mχ [24], of the strong interaction. First, we assume the dark matter thus mimicking stellar mass black holes. central enthalpy to be 0.24c2 (fixed) and vary the nuclear 123 Eur. Phys. J. C (2017) 77 :440 Page 7 of 9 440 matter central enthalpy for static, rigidly rotating and differ- interacting dark matter goes to 1.3640 M with a corre- entially rotating configurations and next we reverse the roles sponding radius of 6.7523 km for differential rotation (fre- of nuclear and dark matter. quency of dark matter to be 300 Hz and that of nuclear mat- ter to be 700 Hz) as shown in Fig. 5.FromFig.6 we see that the corresponding central baryonic number density is − 7 Results and discussions 2.1060 fm 3. In this case the maximum gravitational mass is 1.3640 M, the corresponding matter mass is 1.5024 M, In Fig. 5 the plots of total mass vs. equatorial radius of static, which is constituted of nuclear matter 1.4719 M and dark rigidly and differentially rotating neutron stars mixed with matter 0.0305 M. fermionic self-interacting dark matter are shown for fixed In Fig. 7 the plots of total mass vs. equatorial radius of dark matter central enthalpy (0.24c2) and varying nuclear static, rigidly and differentially rotating neutron stars mixed matter central enthalpies. In Fig. 6 the corresponding plots with fermionic self-interacting dark matter are shown for of mass vs. central baryonic number density are shown. The fixed nuclear matter central enthalpy (0.24c2) and varying maximum mass of the neutron star mixed with strongly self- dark matter central enthalpies. In Fig. 8 the corresponding

Fig. 5 Plots of total mass vs. equatorial radius of static, rigidly Fig. 7 Plots of total mass vs. equatorial radius of static, rigidly rotat- rotating and differentially rotating neutron stars mixed with interact- ing and differentially rotating neutron stars mixed with interacting ing fermionic asymmetric dark matter with fixed dark matter central fermionic asymmetric dark matter with fixed nuclear matter central . 2 enthalpy (0 24c ) and varying nuclear matter central enthalpies enthalpy (0.24c2) and varying dark matter central enthalpies

Fig. 6 Plots of total mass vs. central baryonic density of static, rigidly rotating and differentially rotating neutron stars mixed with self- Fig. 8 Plots of total mass vs. central dark matter density of static, interacting fermionic asymmetric dark matter with fixed dark matter rigidly rotating and differentially rotating neutron stars mixed with self- central enthalpy (0.24c2) and varying nuclear matter central enthalpies interacting fermionic asymmetric dark matter with fixed nuclear matter central enthalpy (0.24c2) and varying dark matter central enthalpies

123 440 Page 8 of 9 Eur. Phys. J. C (2017) 77 :440 plots of mass vs. central dark baryonic number density are The massÐradius relations of pure hadronic stars mixed shown. In this case the maximum mass goes to 1.9355 M with self-interacting fermionic asymmetric dark matter have with a corresponding radius of 10.3717 km for differential been obtained using the LORENE code. In the case of pure rotation (frequency of dark matter to be 700 Hz and that of dark matter stars consisting of less massive dark particles we nuclear matter to be 300 Hz) as shown in Fig. 7.FromFig.8 see that the maximum masses are comparable to those of stel- we see that the corresponding central dark baryonic number lar mass black holes. In the case of hadronic stars mixed with density is 1.1605 fm−3. In this case the maximum gravita- dark matter we considered three different configurations— tional mass is 1.9355 M, the corresponding matter mass is static, rigid rotation and differential rotation of nuclear mat- 2.1105 M, which constitutes of nuclear matter 0.1179 M ter and dark matter fluids. From the results we conclude that and dark matter 1.9926 M. for the dark matter dominated configurations the masses are It is seen that the polytropic indices γ for nuclear and self- higher, viz. for the static case the maximum masses of these interacting dark matter EoSs are approximately equal. How- hybrid stars can reach up to ∼1.88 M with corresponding ever, the polytropic coefficient K for dark matter is about 2.5 radii ∼9.5 km, whereas in the rigid and differential rotational times larger than that of nuclear matter making dark matter cases the maximum masses of these hybrid stars can reach EoS stiffer. Consequently, configurations of stars with vary- up to ∼1.94 M with corresponding equatorial radii ∼10.4 ing dark matter central enthalpy with fixed nuclear matter km. central enthalpy are more massive than those obtained in the We also find that the dark matter dominated neutron star reverse case. behaves differently from the nuclear matter dominated one, From Fig. 7 we see that the dark matter dominated neu- which shows characteristics similar to low mass self-bound tron star behaves differently than the nuclear matter dom- strange stars. This is because of the very strong two-body inated one as shown in Fig. 5.InFig.7, the plots of low repulsive interactions of dark matter which is dominant in mass neutron stars admixed with dark matter typically show the low mass region where it counteracts gravity effectively the characteristics similar to low mass self-bound strange to make radius much smaller. Thus, while the nuclear matter stars. This is because of the very strong two-body repulsive dominance induces gravitational binding, dark matter dom- interactions of dark matter, which is dominant in the con- inant low mass neutron star becomes more compact. How- figuration of Fig. 7, which counteract the gravity effectively ever, if the dark matter particle mass is small compared to the for the low mass region and make the radius much smaller nucleon mass, the maximum mass may well be above 2 M, compared to the pure neutron star of a similar mass (vide Fig. provided no phase transition from nuclear to quark matter 2). Thus, while the nuclear matter dominance induces grav- occurs. itational binding, a dark matter dominant low mass neutron In the past, phase transition and the possible existence of star becomes gravitationally bound at a much smaller radius. a pion condensate or quark matter inside compact stars have The maximum mass for non-rotating dark matter stars been studied extensively [58Ð62]. The phase transition from goes to 3.0279 M with a radius of 16.2349 km for par- baryonic matter to quark matter is determined by the tran- ticle mass mχ = 1 GeV. For rotating stars the maximum sition density. It is observed that a deconfinement transition mass goes to 3.1460 M with a radius of 19.2173 km. How- inside a neutron star causes reduction in its mass. It would be ever, if one takes mχ to be 0.5 GeV, then the maximum mass interesting to find the effect of dark matter on the compact 2 goes to ∼12.6M using the relation mass ∝ 1/mχ [24], hybrid stars (baryonic matter with quark core), mixed with thus mimicking stellar mass black holes. self-interacting fermionic asymmetric dark matter. Such an effect cannot be predicted a priori without full calculations and we leave it for future investigation. 8 Summary and conclusions

Open Access This article is distributed under the terms of the Creative In this work we consider fermionic asymmetric dark mat- Commons Attribution 4.0 International License (http://creativecomm ter (ADM) particles of mass 1 GeV and the self-interaction ons.org/licenses/by/4.0/), which permits unrestricted use, distribution, mediator mass of 100 MeV (low mass implying strong inter- and reproduction in any medium, provided you give appropriate credit action), mixed with rotating and non-rotating neutron stars. to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. These ADM particles are non self-annihilating and behave Funded by SCOAP3. like ordinary free particles. We have shown that a massive exotic neutron star with a strong two-body self-interacting fermionic dark matter is gravitationally stable with equal or References unequal rotational frequencies of the two fluids. This pro- ∼ vides an alternative scenario for the existence of 2M 1. Y. Utsumi et al., Astrophys. J. 833, 156 (2016) neutron stars with ‘stiff’ equations of state. 2. G. Bertone, D. Hooper, J. Silk, Phys. Rep. 405, 279 (2005) 123 Eur. Phys. J. C (2017) 77 :440 Page 9 of 9 440

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