{'fraction_non_alphanumeric': 0.031385803959439885, 'fraction_numerical': 0.010743602124577499, 'mean_word_length': 4.488736749116608, 'pattern_counts': {'":': 0, '<': 2, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 3, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Ferromagnetic Neutron Stars in Scalar-Tensor Theories of Gravity

Ferromagnetic spin ordering can take place in neutron stars. This phase transition alters the neutron star equation of state. Here, applying the scalar-tensor theories of gravity, we investigate the structure of neutron stars which are in the ferromagnetic phase. Considering the equation of state of ferromagnetic neutron matter with Skyrme-type interactions at zero temperature and using the scalar-tensor theories of gravity with sufficiently negative coupling constant, we explore the spontaneous scalarization in ferromagnetic neutron stars. In this regard, the mass versus the central density, the profiles of scalar field and mass and density, the central scalar field for different central densities, and the mass-radius relation of ferromagnetic neutron stars are presented. Moreover, we investigate the influences of the coupling constant on the scalarization of ferromagnetic neutron stars. The effects of the coupling constant on the critical densities of scalarization and the scalar charge of ferromagnetic neutron stars are also calculated. In addition, we study the maximum value of the coupling constant at which the spontaneous scalarization takes place presenting the influence of the neutron matter equation of state on this critical value.


I. INTRODUCTION
Ferromagnetic spin state in neutron stars is one of the most interesting subjects for both nuclear and particle astrophysicists. Different investigations verify that this ordering takes place in neutron stars [1][2][3][4][5][6][7][8][9][10][11][12][13][14]. The conditions for a ferromagnetic transition in neutron stars in the framework of exact general-relativistic equilibrium hydrodynamics and applying the relativistic equation of state (EOS) for interacting hard-core nucleons have been presented [1]. They showed that a ferromagnetic transition in these stars is not inconsistent with stellar equilibrium in possible superdense stars. The ferromagnetic transition has been explored in a dense neutron matter system employing the Landau's Fermi liquid function for a hard-sphere gas of neutrons [2]. Applying the relativistic Hartree-Fock approach to a pure neutron system with the densities appropriate for neutron stars shows that the ferromagnetic transition is specified by the Fock terms [3]. Within a relativistic Hartree-Fock approach and using different combinations of mesons and couplings, it has been clarified that a πmeson pseudoscalar coupling prefers the ferromagnetic phase in the pure neutron system [4]. Studying the neutron matter with Skyrme forces verifies that with the parameters which provide a very good parameterization of realistic neutron matter calculations, the ferromagnetic spin ordering occurs in the neutron matter [5]. The Landau Fermi liquid theory confirms that the spontaneous magnetization takes place in dense degenerate neutron system and this phase transition determines the neutron star density and magnetic field [6].
Studying the highly dense nuclear matter with the relativistic mean-field approach confirms that when the axial-vector interaction is negative enough, the system holds ferromagnetism [7]. Employing the Hartree-Fock theory for a system of nucleons interacting through a central spin-isospin schematic force and using the technique of the anomalous propagator, the critical values of the interaction that leads to a transition to a ferromagnetic phase have been obtained [8]. The axial anomaly acting on the parallel layers of neutral pion domain walls which spontaneously formed at high density in chiral nonlinear sigma model results in a ferromagnetic phase at high density for degenerate neutron matter [9]. Noting the chiral effective model incorporating magnetic fields and the chiral anomaly, it has been shown that with an axial vector meson condensation, there will be a possible realization of a QCD ferromagnetic phase and ferromagnetic magnetars [10]. The calculation in the framework of a relativistic σ +ω +π +ρ Hartree-Fock approach verifies that the interaction of protons with neutrons leads to the lower densities of spontaneous magnetization compared to the pure neutron matter [11]. Considering the presence of the protons with neutrons in a Hartree-Fock methed with Skyrme forces also leads to the reduction of the density of ferromagnetism [12]. For asymmetric nuclear matter in the framework of a Fermi liquid theory with the Skyrme effective interaction, the system experiences a phase transition to the spin polarized state with the oppositely directed spins of neutrons and protons, i.e. antiferromagnetic spin state [12]. However, considering the phase transition of symmetric nuclear matter in Fermi liquid theory with the Skyrme effective interaction confirms that the ferromagnetic spin state is preferable over the antiferromagnetic one [13]. The critical density of ferromagnetic transition in neutron matter with Skyrme-type interactions decreases with temperature [14].
Scalar-tensor theories of gravity (STTs) which are among the extensions of general relativity (GR) have been studied and developed by Fierz [15], Jordan [16], Brans and Dicke [17], Bergman [18], Nordtvedt [19], Wagoner [20], Damour and Esposito-Farese [21]. These theories which include one or more scalar fields coupled to matter, predict scalar gravitational wave that can be detected by the laser interferometer [22,23]. These theories of gravity have been extensively applied to investigate the relativistic compact objects [24][25][26][27][28][29][30][31][32]. It has been confirmed that a wide class of STTs passes the weak-field gravitational tests indicating nonperturbative strong-field deviations away from GR in neutron stars [24]. The instability in spherically symmetric stars in STTs is induced by the scalar field for some ranges of the value of the first derivative of the coupling function [25]. In the gravitational collapse of neutron star toward a black hole, the amplitude of the gravitational wave increases with the value of the parameter of the coupling function [26]. Using a turning point method, the secular stability of neutron stars in STTs against spherically symmetric perturbations has been studied [27]. Considering spherical neutron star models in STTs, it has been shown that with some conditions on the second derivative of the coupling function and on star's mass, there exist two strong-scalar-field solutions as well as the usual weak-field one [28]. For lower sound velocity in the core of neutron stars, the maximum mass limit of the scalarized neutron stars is larger than that of in GR, while for the stiff EOSs with high sound velocity, the maximum mass limit in GR is larger than the one in STT [29]. The coupling to additional degrees of freedom in massless STTs leads to new families of modes in the spectrum of pulsating neutron stars, with no counterpart in GR [30]. Relativistic stars in degenerate higher-order STTs which evade the constraint on the speed of gravitational waves imposed by GW170817 have been investigated [31]. Their results indicate that for high density stars, the mass-radius relation changes from the one in GR. For the coupling constant confined to values provided by the astronomical observations, the maximum compactness of neutron stars in GR is higher than the one in STT [32]. Regarding the above discussions, the EOS of ferromagnetic neutron matter can have significant effects on the scalarized neutron stars.
The main goal of this work is investigating the effects of the ferromagnetic neutron matter on the structure of neutron stars in STTs. Considering the ferromagnetic neutron matter with Skyrme-type interactions within the STT, we investigate the structure as well as the spontaneous scalarization of ferromagnetic neutron stars.
Moreover, t 0 , t 1 , t 2 , t 3 , x 0 , x 1 , x 2 , x 3 , γ, and W 0 are the Skyrme-force parameters. The total energy of neutron matter per particle is calculated by E/N = ψ|H|ψ /N through the Hartree-Fock method in which ψ and H denote the wave function and the Hamiltonian of system, respectively. For the non-ferromagnetic neutron matter (NFM), the energy per particle is as follows [33], with Θ = t 1 (1 − x 1 ) + 3t 2 (1 + x 2 ). We also consider a system of polarized neutron matter with the spin-up number density n ↑ , spin-down number density n ↓ , and spin polarization parameter ∆ = (n ↑ − n ↓ )/n. The energy per particle for ferromagnetic (FM) neutron matter is given by [14], with in which + and − denote the up and down spin projections. It is possible to calculate the equilibrium value of the spin polarization parameter, i.e. ∆ min , at each number density n from Eq. (3). By the equilibrium value, we mean the value at which the energy is minimum. Besides, the energy at each ∆ min can be calculated using Eq. (3). Here, we employ the SLy230a model to describe the neutron matter EOS [33]. At lower densities, ∆ min is equal to zero and neutron matter is fully unpolarized. At a density about 5.40ρ 0 in which ρ 0 = 1.66×10 17 kg/m 3 , this parameter spontaneously increases and it reaches to 1 at a density about 10.41ρ 0 . Neutron matter with the densities between 5.40ρ 0 < ρ < 10.41ρ 0 is partially polarized. Besides, with the densities higher than 10.41ρ 0 , the neutron matter becomes fully polarized and it is in ferromagnetic state. Applying the first law of thermodynamics, i.e. P = n 2 ( ∂(E/N ) ∂n ), the EOSs for the NFM and FM neutron matter are calculated. Fig. 1 shows the EOSs of non-ferromagnetic and ferromagnetic neutron matter in Sly230a model.
For both NFM and FM neutron matter, the pressure grows by increasing the density. At most values of ρ, the rate at which the pressure increases is higher for NFM neutron matter.
Therefore, the EOS of FM neutron matter is softer than NFM one. As wee see in the following, the softening of the EOS affects the properties of the scalarized neutron stars, i.e. mass, radius, and spontaneous scalarization. The main goal of this work is investigating the effects of neutron matter ferromagnetism on the properties of neutron stars in STTs.
To describe the neutron stars in STTs, we start with the spacetime line element in the Einstein frame for a spherical symmetry static neutron star as follows [34], stress-energy tensor as follows,T µν =ǫũ µũν +p(g µν +ũ µũν ), with the fluid's total energy densityǫ, pressurep, and 4-velocityũ. It should be noted that tilde presents the quantities in Jordan frame. Besides, we denote the physical metric or Jordan metric byg µν := a(φ) 2 g µν in which a(φ) and φ are the coupling function and scalar field, respectively. For the coupling function a(φ), we consider two models as follows [24,34], in which β is the coupling constant and we assume φ 0 = 0. In addition, α(φ) := d ln a(φ) dφ . It has been verified that a nonperturbative amplification mechanism of the coupling strength of the scalar field exists when the logarithm of the coupling function has a sufficiently negative curvature around φ 0 [24]. They showed the existence of strong-field deviations from GR in STTs with β −4. Besides, the negative values of β result in the scalar field nonlinearities and these reinforce the naturally attractive character of scalar interactions [37]. Moreover, according to the results of Ref. [27], the spontaneous scalarization occurs for β ≤ −4.35. Therefore, in this work, we consider the negative values of the coupling constant, i.e. β ≤ −4.35. By applying some calculations, the field equations in STTs lead to the following equations which describe the structure of neutron stars in STTs [34], in which m b denotes the baryonic mass. These are the generalized Tolman-Oppenheimer- Volkoff equations in STTs. The boundary conditions to solve these equations together with the neutron matter EOS are as follows, where R s is the radius of the star. Using a fourth-order Runge-Kutta algorithm [35], we integrate the above equations. The integration is done with the boundary conditions at r = 0. In addition, with a guess for the scalar field at the center, i.e. φ(0) = φ c , we do the iteration on φ c such that the following condition satisfies [24,34], In the above equation, the subscript s denotes the quantities on the surface of star. Besides, Moreover, the ADM mass, M ADM , and scalar charge, ω, are given by [24,34], It should be mentioned that the scalar charge is introduced through the asymptotic behavior of the scalar field with r → ∞ as follows [21],  for higher values of the coupling constant, the neutron star mass increases by increasing the density to a special value of the central density. For the densities higher than that special value, the neutron star mass is constant. This special value of the central density is greater for NFM neutron stars. At lower densities, the mass of FM neutron stars is greater than that of NFM ones. But at higher densities, the mass of NFM neutron stars is higher than that of FM stars. This is due to this fact that the EOS of NFM neutron matter is stiffer than the EOS of FM neutron matter. In addition, in M1 for different values of β and in M2 for higher values of β, for both NFM and FM neutron stars, the results of GR and STT are nearly equal. This is while the results of STT and GR are different for lower values of β.
For the cases that the results of GR and STT are different, the neutron stars are scalarized.
In M2 with lower values of β, the mass of the stars with lower central densities in STT is smaller than GR. However, for some stars with higher densities, the mass in STT is greater than GR. As we see in the following, this fact that the result of STT how is different from GR is related to the variation of the central scalar field respect to the central density. For NFM neutron stars with high densities, the results of STT and GR are the same. Therefore, there is no scalarized NFM neutron stars with high central density. However, for the FM neutron stars with densities greater than a special value, the results of GR and STT are not the same. This difference continues to highest values of the density considered in this work, i.e. ρ c = 14ρ 0 . Therefore, even the FM neutron stars with high densities are scalarized unlike NFM stars. In fact, this phenomenon (scalarization of FM neutron stars with high densities) is the main distinction of NFM and FM neutron stars. For the lower values of β, the deviation of STT from GR is more considerable. Besides, the difference of the results in STT and GR is more significant in M2 compared to M1.
In Table I   equal to one in GR. In both gravities, the maximum mass of FM stars is lower than the NFM ones. This is due to the fact that the EOS of FM neutron matter is softer than the NFM one.  significant compared to M1. The rate at which the scalar field reduces is higher in M2.
Our calculations verify that by increasing β, the scalar field in neutron star decreases. This is due to the fact that for lower values of β, the coupling of scalar field to metric is more significant. In addition, the scalar field for FM neutron star is smaller than the NFM one. Therefore, with the softer EOS, the magnitude of the scalar field is smaller than the one with the stiffer EOS. The difference between the profile of scalar field in two models is more considerable for NFM neutron stars. Moreover, with different values of β, for both NFM and FM neutron stars, the difference of M1 and M2 is more important in the center of star compare to its surface. and FM neutron stars. This difference is more significant for NFM stars. With higher values of β, the profiles approach to each other. The mass profile of FM stars in two models is lower than the NFM one. This is due to the fact that the FM EOS is softer than NFM one. For the profile of density, the difference between two models is more considerable at lower coupling constants. This difference is nearly negligible for FM stars.  Therefore, the high density FM neutron stars are also scalarized. This is the main difference of NFM and FM stars. This phenomenon is related to the one explained in Fig. 2. In fact, the difference of the GR and STT solutions up to high densities for the mass verses the density in FM stars is a result of the nonzero scalar field in these stars. Therefore, the EOS of star affects its scalarization. It can be seen from Fig. 6 that with lower values of β, the first critical densities of scalarization for NFM and FM stars are more close to each other.
Besides, for lower values of β, the ranges of density at which the NFM and FM stars are scalarized have more overlap with each other. For NFM stars in two models, the value of ρ cr1 increases as the coupling constant grows, in agreement with the result of Ref. [32]. In addition, for these stars, ρ cr2 decreases by increasing β. It can be concluded that the range at which the NFM stars are scalarized is larger for lower values of β. With lower values of the coupling constant, ρ cr1 in FM stars is smaller than NFM ones. Therefore, with lower values of β, lower density FM stars can be also scalarized. We can found that the range of scalarization in FM neutron stars is greater than NFM one. In fact, softening of EOS leads to a larger range of scalarization. This effect agrees with the result reported in Ref. [36].
Another result of Fig. 6 is that by increasing the coupling constant, the maximum value of the central scalar field decreases. For that reason, neutron stars are more scalarized with lower values of β. This result has also reported in Refs. [24,26,32]. Fig. 6 also verifies that M2 predicts higher values of the scalar field compared to M1. Therefore, the stars are more scalarized in M2.  and with high values of β in M2, for NFM and FM neutron stars, the smaller stars have larger masses. Besides, for that conditions, the bigger stars have lower masses. In fact, they are gravitationally bound stars. Fig. 7 shows that in two models for neutron stars, the deviation of STT from GR is more significant with lower values of β. According to Fig. 7, for massive NFM neutron stars, the results of STT and GR are equal. However, for massive FM neutron stars, these results are not the same. This is due to the fact that high density FM stars unlike the NFM ones are scalarized (see Fig. 6). In addition, this deviation in M2 is more considerable than M1. Because M2 predicts more scalarization for neutron stars compared to M1. For lower values of β, the deviation of STT from GR in the mass-radius relation takes place in a greater region. This is a result of the fact that with lower values of β, the range of scalarization is bigger (see Fig. 6). For scalarized NFM stars (specially with β = −6.0 in M2), the massive stars are bigger while the lower mass ones are smaller. This means that the slop of mass-radius relation for these stars is different from the one in GR.
In fact, in these cases, the stars are self bound. However, the scalarized FM neutron stars are still gravitationally bound. It is clear from Fig. 7 that the deviation of STT from GR is more significant for NFM stars compared to FM ones. Therefore, it can be concluded that neutron stars, the mass of scalarized stars is lower than the stars in GR. This is similar to the result of Ref. [34].

E. Critical Density of Scalarization
In this part, we are going to investigate the effects of the coupling constant on the critical density of scalarization for NFM and FM stars. Fig. 8 gives the first and second critical densities for different stars in M1 versus the coupling constant. For both NFM and FM stars, the first critical density, ρ cr1 , increases as the coupling constant grows. This means that for higher values of β, the stars become scalarized at higher densities. It should be noted that our results confirm that the critical density of scalarization does not depend on the coupling function model. It is clear from Fig. 8 that for the most values of β, the first critical density in FM stars is higher than the one in NFM stars. Moreover, the rate at which ρ cr1 grows with β is greater for FM stars. The effects of the spin polarization of neutron matter on ρ cr1 is more significant when the coupling constant is higher. Fig. 8 also shows the second critical density of scalarization for NFM stars versus β. It should be mentioned that since the FM neutron stars are scalarized up to high densities, the second critical density of scalarization is not defined for the FM stars. In NFM stars, the second critical density, ρ cr2 , decreases as β grows. Regarding the increase of ρ cr1 with β, it is possible to conclude that for NFM neutron stars, the range of density at which the scalarization takes place decreases as the coupling constant grows. Therefore, the solution of STT approaches to GR when β increases. According to our results, for NFM neutron stars with β = the result of Ref. [37]. For NFM neutron stars with high masses, the scalar charge again becomes zero. The range of mass with nonzero scalar charge is precisely the range at which the central scalar field is nonzero (see Fig. 6). In fact, the more scalarized the neutron star, the more amount of scalar charge for star. Fig. 9 shows that the mass at which the scalar charge again becomes zero reduces as β increases. In addition, the maximum value  In two models with all values of β for NFM and FM neutron stars with low compactness, the scalar charge is zero. In NFM neutron stars with the compactness higher than a special value, the scalar charge increases with the compactness. Besides, the scalar charge again becomes zero for NFM stars with high compactness. However, for FM stars, the scalar charge increases monotonically with the increase of the compactness. Fig. 10 confirms that the compactness at which the scalar charge becomes nonzero is higher for NFM neutron stars compared to FM ones. This means that the FM stars with lower compactness can also have scalar charge. For all stars in two models, the value of compactness at which the scalar charge becomes nonzero increases when the coupling constant grows. It is due to the fact that with lower values of |β| and approaching to GR, the more compactness is needed to have the scalar charge. In addition, for both NFM and FM neutron stars in two models, the range of compactness at which the scalar charge is nonzero decreases by increasing β. It means that with the lower values of |β| and approaching to GR, the chance for finding the stars with scalar charge is lower.

IV. SUMMARY AND CONCLUDING REMARKS
The structure of ferromagnetic (FM) neutron stars in scalar-tensor theories of gravity has bee studied. To describe the neutron star, we employ the equation of state of FM neutron matter with Skyrme-type interactions. We found that the soft EOS of FM neutron matter leads to the lower values of the neutron star mass compared to the non-ferromagnetic (NFM) one. Our results confirm that with the lower values of the coupling constant, the results of STT deviate significantly from the ones in GR. In the cases which the results of STT are different from GR, the neutron stars are scalarized. For high density NFM neutron stars, the results of STT and GR are the same and there is no scalarized NFM neutron stars with high central density. However, for the FM neutron stars up to high central density considered in this work, the results of GR and STT are not the same and these stars even with high densities are scalarized. We found that the densities at which the scalarization takes place are not equal for NFM and FM neutron stars. Our calculations show that for both NFM and FM neutron stars, the first critical density of scalarization increases as the coupling constant grows. In addition, in NFM stars, the second critical density decreases as the coupling constant increases. The range of scalarization in FM neutron stars is greater than NFM ones. For both NFM and FM stars, the maximum value of the central scalar field reduces as the coupling constant increases. We showed that the deviation of STT from GR in the mass-radius relation is more significant with lower values of the coupling constant.
Besides, this deviation in the mass-radius relation as well as the scalarization are seen in a greater region when the coupling constant takes lower values. We found that the deviation of STT from GR is more significant for NFM stars compared to FM ones. It means that the EOS of neutron matter affects the amount of deviation of STT from GR. For NFM neutron stars, the scalarized ones are larger in size compared to the GR solutions. Moreover, for FM neutron stars, the mass of scalarized stars is lower than the ones in GR. Our results verify that the compactness at which the scalar charge becomes nonzero is greater for NFM neutron stars compared to FM ones.