Contraction of cold neutron star due to in the presence a quark core

Motivated by importance of the existence of quark matter on structure of neutron star. For this purpose, we use a suitable equation of state (EoS) which include three different parts: i) a layer of hadronic matter, ii) a mixed phase of quarks and hadrons, and, iii) a strange quark matter in the core. For this system, in order to do more investigation of the EoS, we evaluate energy, Le Chatelier's principle and stability conditions. Our results show that the EoS satisfies these conditions. Considering this EoS, we study the effect of quark matter on the structure of neutron stars such as maximum mass and the corresponding radius, average density, compactness, Kretschmann scalar, Schwarzschild radius, gravitational redshift and dynamical stability. Also we find an upper limit for the maximum mass of these stars. Indeed, our results show that there is no any massive neutron star with mass more than $2M_{\,\odot }$ ( $M_{\,\odot }$ is solar mass), when we consider a quark core for this object. This means that in the center of massive neutron stars with mass more than $2M_{\,\odot }$ (for example $4U~1700-377$ and $J1748-2021B$ with masses about $2.4M_{\,\odot }$ and $2.7M_{\,\odot }$, respectively), there is no any quark core. In addition, our analyze indicates that neutron stars are under a contraction due to the existence of quark core.


I. INTRODUCTION
Neutron stars which are born in the aftermath of core-collapsing supernova (SN) explosions, are a cosmic laboratory and the best environment for the studying dense matter. It is notable that, in the center of neutron star because of high densities, the matter is envisaged to have a transition from hadronic matter to strange quark matter, see refs. [1][2][3][4], for more details. Also, Glendenning in ref. [5], showed that proper construction of phase transition between the hadron and quark, inside the neutron stars implies the coexistence of nucleonic matter and quark matter over a finite range of the pressure. Accordingly, a mixed hadron-quark phase exists in the neutron star, so that its energy is lower than that of the quark matter and nucleonic matter. The effects of quark-hadron matter in the center of neutron stars have been studied in ref. [6,7], and the obtained results have shown that for P SR J1614 − 2230, and P SR J0348 + 0432 with masses about 2M ⊙ , may contain a region of quark-hybrid matter in their center. S. Plumari et al. have investigated the effects of a quark core inside neutron star by considering the quark-gluon EoS in the framework of field correlator model [8]. They found an upper limit for the mass of neutron stars by adjusting some parameters. This limit was in the range M max ≃ 2M ⊙ . In addition, H. Chen et al. have studied cold dense quark matter and hybrid neutron stars with a Dyson-Schwinger quark model and various choices of the quark-gluon vertex [9]. They showed that hadron states have the maximum mass lower than the pure nucleonic neutron stars, but higher than two solar masses. Their results depended on parameters of EoS. Also, R. Lastowiecki et al. have found that compact stars masses of about 2M ⊙ such as P SR J1614 − 2230 and P SR J3048 + 0432 were compatible with the possible existence of deconfined quark matter in their core [10]. Neutrino emissivity in the quark-hadron mixed phase of neutron stars have been investigated in ref. [11]. According to importance of existence of quark matter in the neutron stars , we consider a neutron star to be composed of a hadronic matter layer, a mixed phase of quarks and hadrons, and in the core of star, a quark matter. One of our aims in this work is determining structure of neutron star with a quark core and comparing it with observation data.
On the other hand, in order to study the structure of stars and their phenomenological properties, we must use the hydrostatic equilibrium equation (HEE). Indeed, this equation is based on the fact that a typical star will be in equilibrium when there is a balance between the gravitational force and the internal pressure. The first HEE equation was introduced by Tolman, Oppenheimer and Volkoff (TOV) [39][40][41] in the Einstein gravity which is known as TOV equation. Considering TOV equation, the structure of compact stars have been evaluated by many authors in refs. [42][43][44][45][46][47][48][49][50].
According to this fact that, there are some massive neutron stars with mass more than two times of solar mass, M ≥ 2M ⊙ , for example, 4U 1700 − 377 with M = 2.4M ⊙ [51], and J1748 − 2021B with M = 2.7M ⊙ [52], in this work we intend to answer this question: is there a quark core inside neutron stars with mass more than 2M ⊙ (M ≥ 2M ⊙ )? For this goal, we use a suitable EoS which obtained by combining three different parts, a layer of hadronic matter, a mixed phase of quarks and hadrons, and a strange quark matter in core.
The outline of the paper will be as follows; First, in order to investigate the structure of neutron stars, we evaluate a suitable EoS which includes three different layers. Then, we compare the structure of neutron stars by considering the EoS with and without a quark core. Indeed, we study the effects of EoS by applying a quark core on the structure of neutron stars. Next, we compare our obtained results with those of observational data. The last section is devoted to closing remarks.

II. EQUATION OF STATE
A neutron star with a quark core composed of a hadronic matter layer, a mixed part of quarks and hadrons and a quark matter in core. Thus we calculate the EoS of different parts of this star in the following subsections.

A. Hadron Phase
We use the lowest order constrained variational (LOCV) many-body method to determine the EoS of nucleonic matter [53][54][55][56]. We consider a cluster expansion of the energy functional up to the two-body term, in which H is the Hamiltonian of the system. Also, ψ is the total wave function in which we consider a trail manybody wave function as ψ = F φ. Here φ is a uncorrelated ground-state wave function of N independent nucleons, and F is a proper N -body correlation function which is taken according to the Jastrow ansatz, F = S i>j f (ij), in which S and f (ij) are a symmetrizing operator and the two-body correlation function, respectively. Using the Jastrow ansatz and after some algebra, the energy is calculated (see [57], for more details). The one body term is ρi ρ , for an asymmetrical nucleonic matter, that ρ i are the nucleonic densities associated with the protons and neutrons (ρ = ρ p + ρ n ) and k i = (6π 2 ρ i ) 1/3 is the Fermi momentum of particle i. The two-body energy is E 2 = 1 2N ij < ij|ν (12)|ij − ji >. The operator ν(12) is the effective nuclear potential. There is a complete calculation for nuclear matter in ref. [54].

B. Quark Phase
The total energy of strange quark matter with deconfined up (u), down (d) and strange (s) quarks within MIT bag model [58,59] is given by The quark confinement in MIT bag model is satisfied by a density dependent bag constant B, that is interpreted as the difference between energy densities of non interacting and interacting quarks. We use a density dependent with the Gaussian form [60,61], In the equation ( [60] for more details. Now, by using the energy density from Eq. (2), we can obtain the EoS of quark matter in the MIT bag model,

C. Mixed phase
The hadron-quark phase happens in the high range of baryon density values. The occupied fraction of space by quark matter smoothly increases from zero where there is no quark to unity when the last nucleons dissolve into the quarks. In this phase, we have a mixture of hadrons, quarks and electrons. According to the Gibss equilibrium condition, the pressures, the temperatures and chemical potentials of both hadron and quark phases are equal [5], are neutrons (protons) chemical potential in the hadron phase and the quark phase, respectively. It is notable that µ n and µ p are given by As the chemical potentials determine the charge densities, the volume fraction occupied by quark matter, χ, can be obtained by the requirement of global charge neutrality. Then the total energy density and baryon density of mixed phase could be determined, Calculations regarding the EoS of mixed phase has been fully discussed in Ref. [60]. At this stage we can determine EoS of neutron star with quark core using the results of proceeding sections. Also we investigate the energy and stability conditions for our results. For this purpose at first, we extract a mathematical form for the EoS presented as a polynomial function in the following form where a i are In order to do more investigation of the obtained EoS, we evaluate energy, Le Chatelier's principle and stability conditions as follows.

D. Energy conditions
The null energy condition (NEC), weak energy condition (WEC), strong energy condition (SEC) and dominant energy condition (DEC) in the center of neutron star are given as follows where E c and P c are the density and pressure in the center of neutron star (r = 0), respectively. The results of above conditions for our EoS are given in Table I. We see that our EoS satisfied all mentioned energy conditions.

E. Stability
According to the stability condition for the EoS of neutron star matter in a physically acceptable model, the corresponding extracted velocity of sound (v) must be less than the light velocity (c) [62,63]. Thus the stability condition is given 0 Using Eq. 9 we compute v 2 c 2 versus density which is presented in in Fig. 1. It is evident that stability condition is satisfied by the EoS of neutron star with quark matter.

F. Le Chatelier's principle
Le Chatelier's principle is defined as: the matter of star satisfies dP/dE ≥ 0 which is a essential condition of a stable body both as a whole and also with respect to the non-equilibrium elementary regions with spontaneous expansion or contraction [64]. According the Fig. 1, the Le Chatelier's principle is established.
The above results show that we encounter with a suitable EoS for studying the neutron stars with a quark matter in the core. Therefore, we consider our EoS and investigate the structure of neutron star with three different layers.

III. STRUCTURE OF THE NEUTRON STAR WITH AND WITHOUT QUARK MATTER
In this section, we study the effect of quark matter on the structure of neutron stars, and then compare them with the structure of neutron stars without quark matter. Employing the obtained EoS for neutron star with the quark matter and using TOV equation, we have gotten interesting results which are given in Table II. As one can see, considering the quark matter in the core of neutron stars changes their structure properties. Indeed, there are some interesting results when we consider the quark matter in the calculation of structure of neutron stars. For example, by considering the quark matter, the maximum mass decreases (see Table II and Fig. 2, for more details). In order to do more investigation of the effect of quark matter, we calculate another properties of the neutron star such as the average density, compactness, Kretschmann scalar, gravitational redshift and dynamical stability.

A. Average density
The average density of a object has the following form, for our system, M and R are the gravitational mass and radius of a neutron star. The results presented in Table II show that by applying the quark matter to neutron star matter calculations, the average density from the perspective of a distant observer (or a observer outside the neutron star), decreases. Indeed, by adding the quark matter to neutron star, the total average density (this average density is not related to density of center of neutron star) of this system decreases.

B. Compactness
The compactness of a spherical object is usually defined as the ratio of Schwarzschild radius to the radius of object (σ = R Sch R ), which may be indicated as the strength of gravity of compact objects. Our results for the compactness are presented in Table II. Similar to average density, by applying the quark matter in the structure of neutron star, the compactness from the perspective of a distant observer (or a observer outside the neutron star) decreases.

C. Kretschmann scalar
Another quantity that gives us information of the strength of gravity is related to the spacetime curvature. According to this fact that, in the Schwarzschild spacetime, the components of the Ricci tensor (R µν ) and the Ricci scalar (R µν R µν ) are zero outside the star, and therefore these quantities do not give us any information about the spacetime curvature. So, we use another quantity in order to evaluate the curvature of spacetime. The quantity that can help us to find out the curvature of spacetime is related to the Riemann tensor (R µνγδ ). The Riemann tensor may have more components, and also for simplicity, we can evaluate the Kretschmann scalar for measurement of the curvature in a vacuum. So, the curvature at the surface of a neutron star is given as [65][66][67] Our results confirm that by adding the quark matter to neutron star matter, the strength of gravity decreases (see Table II).

D. Gravitational redshift
The gravitational redshift is given as follows, The results related to the gravitational redshift are given in Table II. This result shows that by considering a quark core inside a neutron star, the gravitational redshift decreases.

E. Dynamical Stability
Chandrasekhar in 1964 [68], introduced the dynamical stability of stellar model against the infinitesimal radial adiabatic perturbation. Then, some authors developed this stability condition and applied it to astrophysical cases in refs. [69][70][71][72][73]. For investigating the dynamical stability we use of adiabatic index (γ) which is defined in the following form, In order to have the dynamical stability, the adiabatic index must be more than 4/3 (γ > 4/3 = 1.33) everywhere within the obtained neutron stars with the quark matter. So, we plot diagram related to γ versus radius for neutron stars with and without quark matter in Fig. 3. We find an interesting result about the dynamical stability of neutron stars with quark cores. In other words, our results show that, the centers of neutron stars with quark cores are unstable against the radial adiabatic infinitesimal perturbations, whereas the neutron stars without quark matter are stable.
Another interesting results is related to contraction of neutron stars due to the quark core (see Tables III and IV, for more details). In the other words, the obtained results of Tables III and IV and Fig. 2 show that the radius of neutron stars without the quark matter (NS) and with the gravitational mass equal to 1.4M ⊙ (or 1.8M ⊙ ) are greater  than the radius of neutron stars with the quark matter (NS+Q). Indeed, for the same gravitational masses of NS and NS+Q, the compactness, the Kretschmann scalar and the gravitational redshift increase due to the reduced radius. In addition, this difference appears for neutron stars with the gravitational mass higher than the solar mass (M ≥ M ⊙ ) (see Fig. 2). Briefly, the existence of quark matter inside the neutron stars lead to decreasing for the maximum mass and so it contracts them. These results indicate that the cores of heavy neutron stars do not have any quark core. Indeed, massive neutron stars (with mass more than 2M ⊙ ) can not have the quark cores, because by adding the quark matter to the structure of these stars, the maximum masses decreases.

IV. SUMMARY AND CONCLUSION
The paper at hand studied the structure of cold hybrid neutron stars which included three different parts: i) a layer of hadronic matter, ii) a mixed phase of quarks and hadrons, and iii) a quark matter in the core. For layer of hadronic matter we used the lowest-order constrained variational (LOCV) many-body method employing the U V 14 + T N I potential for the nucleon-nucleon interaction (for more details about neutron star with hadron matter and U V 14 + T N I potential see ref. [74]). In another layer (mixed phase of quarks and hadrons), we considered Gibss equilibrium condition. Indeed for this layer, the temperature, pressures and chemical potentials of both hadron and quark phases are equal (see refs. [5,60] for more details). Finally, we considered a quark matter in the core of neutron stars, and for this region, we used the total energy which included up, down and strange quarks within MIT bag model [58][59][60][61]. We applied TOV equation for obtaining the structure properties of these stars. Our results indicated that the maximum mass, the average density, compactness, gravitational redshift, and Kretschmann scalar of neutron stars decrease by adding a quark core to the neutron star matter. This results led to contraction of hybrid neutron stars with the quark core. On the other hand, we found an upper limit for the maximum mass of neutron stars with a quark core. In other words, the neutron stars with a quark core can not be more massive than two times of the solar mass (M max ≤ 2M ⊙ ). It is notable that, C. Hoyos et al. evaluated the properties of neutron stars by applying top-down holographic model for strongly interacting quark matter [75]. They obtained an EoS which was matched with state-of-the-art results for dense nuclear. Then solved TOV equation with their EoS, and found the maximal stellar masses in the excess of two solar masses. Their results showed that there are no any quark matter inside these massive neutron stars. In other words, their results confirm our conclusions about the existence an upper limit for neutron stars with a quark core, i.e, M max ≤ 2M ⊙ . However there were some difference between these results. For example, the obtained radius of neutron stars in our calculations were greater than their results (the radius of a neutron star with M ≃ 2M ⊙ was about 9.7 km [75], but our calculation indicated that the radius of a neutron star with the same mass was about 10 km). Finally, we investigated the dynamical stability of neutron stars with and without the quark core. Our calculations indicated that the centers of neutron stars with quark cores are unstable against the radial adiabatic infinitesimal perturbations, whereas the neutron stars without quark matter are stable. Briefly, we obtained the quite interesting results for the neutron stars with quark cores, such as the following: i) The EoS derived in this work satisfied the energy, Le Chatelier and stability conditions. ii) Consequently the presence of quark cores, the obtained maximum mass of cold neutron stars could not be more than 2M ⊙ (M max ≤ 2M ⊙ ). In other words, there are not cold massive neutron stars with quark matter in the mass range M max > 2M ⊙ . Therefore, our results showed that inside the neutron stars such as 4U 1700 − 377 [51] with the mass about 2.4M ⊙ , and J1748 − 2021B [52] with the mass about 2.7M ⊙ , there are no any quark matter.
iii) The maximum mass, the average density, compactness, the Kretschmann scalar, and gravitational redshift of neutron stars decrease owing to the existence of quark matter in the structure them.
iv) The neutron stars are contracted due to the presence of quark matter in their center. v) The obtained results indicated that the centers of neutron stars with quark cores are unstable against the radial adiabatic infinitesimal perturbations, whereas the neutron stars without quark matter are stable.
vi) For neutron stars with the gravitational mass more than one solar mass (M ≥ M ⊙ ), there is a difference between NS and NS+Q. In other words, there are no any difference between the properties of NS and NS+Q in the range M < M ⊙ .