Gravitational Wave Signatures of Highly Magnetized Neutron Stars

Motivated by the recent detection of GW170817 by the LIGO-VIRGO observatories, we study the effects of highly magnetized stars on the Love number and dimensionless tidal polarizability. We also investigate the fundamental quasinormal mode of neutron stars under such high magnetic fields. To perform our calculations we use the chaotic field approximation and consider both nucleonic and hyperonic stars. We conclude that the role played by the constitution of the stars is far more relevant than the intensity of the magnetic field and if massive stars are considered, the ones constituted by nucleons only present frequencies somewhat lower than the ones with hyperonic cores, a feature that can be used to point out the real internal structure of neutron stars. Moreover, highly magnetized stars could only be discriminated by the fundamental mode frequency in the limiting case of magnetic field intensities of the order of 3$\times 10^{18}$ G.


I. INTRODUCTION
On August 17, 2017 the LIGO-VIRGO collaboration observed the gravitational wave event GW170817, which consisted in the detection of a binary neutron star merger and, as consequence stablished a new channel to study the high density equation of state (EOS) that describes neutron stars. In these kind of events, before the merging, the neutron star components begin to react to their mutual tidal fields, and this effect can be detected in the phase modification of the gravitational wave impinging on the detector. This tidal response depends strongly on the neutron star composition and therefore important information can be obtained about the EOS [1,2]. In addition another source of information can be inferred from neutron star oscillations. In principle, before the merging, tidal interactions can excite the fluid modes by resonance [3,4] and also during and after the neutron star fusion, the fundamental mode can be greatly excited, with a strong influence on the respective gravitational wave emission [5][6][7][8].
At present, a trustworthy determination of the EOS of strongly interacting matter at densities above few times the nuclear saturation density remains a challenge. The EOS can be reliably obtained up to the nuclear saturation density, but at densities typical of neutron star interiors, which can be up to 6 times higher, its determination depends crucially on the knowledge of strong interactions in a regime that cannot be reached on earthly experiments. For this reason, the validity of both non-relativistic (Skyrme-type) and relativistic (RMF) models has to be checked according to verifiable constraints. In the present work we restrict our comments and calculations to relativistic models.
Not too long ago, 263 RMF models were confronted with nuclear bulk properties inferred from experiments and only 35 of them satisfied all constraints [9]. These 35 models were then investigated with respect to stellar matter properties in [10] and only 12 parameterizations resulted in neutron stars with a maximum mass in the range of 1.93-2.05 M ⊙ , as the ones measured in the present decade [11,12]. Very recently 34 out of the 35 models were also confronted with constraints imposed by GW170817 [13] (we come back to these constraints later in the text) and 24 were shown to satisfy them. Nevertheless, only 5 RMF parametrizations can simultaneously describe massive stars and GW170817 constraints. If hyperons are included in the calculations, the situation becomes even more complicated because the EOS must be soft at subsaturation densities and hard at higher densities to predict massive stars, but hyperons soften the EOS. There are different ways to circumvent this problem and one of them is by introducing the magnetic field in the Lagrangian density that describes the model, a field which is capable of stiffening the EOS.
However, the main reason to consider magnetic field effects on the EOS is the existence of magnetars, which are a special class of neutron stars bearing surface magnetic fields that are three orders of magnitude stronger than the ones present in their non-magnetized counterparts [14] (10 12 G). So far, only 30 of them have been clearly identified [15], but the launching of NICER [16] in 2017 and ATHENA [17], expected to take place in 2030, will certainly provide more information on these compact objects. Moreover, the known magnetars detected so far as either transient X-ray sources, known as soft-gamma re-peaters or persistent anomalous x-ray pulsars are isolated objects. Hence, all the analyses done in the present work in relation with possible GW170817 constraints, coming from a binary system, have to be taken with this fact in mind.
At this point, it is important to mention that there is some controversy on how strong magnetic fields can be incorporated to the Lagrangian density and the stress tensor. While some authors advocate that the EOS should be isotropic and no magnetization could appear in the EOS [18,19], others claim that the anisotropy is indeed present and magnetization effects should be considered [20][21][22]. For a discussion on this subject, the reader can refer to [23], where the different results obtained with the two formalisms can be seen. There is no doubt that the ideal situation is to use the LORENE code [24], which performs a numerical computation of the neutron star by taking into account Einstein-Maxwell equations and equilibrium solutions self consistently with a density dependent magnetic field. Unfortunately, this calculation is not always feasible for all purposes and it gives some estimates that may not be correct, as the case of the neutron star crust thickness discussed in [25]. However, the latter work shows clearly that although not too strong magnetic fields have negligible effect on the EOS itself, it strongly affects properties related to the crust, as the cases discussed next.
In the present paper, we use the chaotic field approxi-mation introduced by Zel'dovich [26] and applied in [27] to account for strong magnetic fields in NS. The EOS is calculated for nuclear matter and hyperonic matter and both situations are investigated in the context of nonradial oscillations and tidal polarizabilities. The paper is organized as follows: in Sect. II we review some basic aspects of the EOS. In Sect. III we present the equilibrium configuration and the non-radial oscillation equations of the neutron star. In Sect. IV we present our results and some conclusions and in Sec. V we make our final remarks.

II. MAGNETIZED EQUATION OF STATE
If we believe that the standard model is correct, the physics of strong interacting matter is described by quantum chromodynamics (QCD). However, QCD provides no meaningful results in the region of the neutron star interior,i.e., high density and low temperature. To overcome this issue, we use an effective model, the quantum hadrodynamics (QHD). Originally developed in the early 70s [28], QHD considers the baryons, not the quarks, as the fundamental degrees of freedom. Also, the strong interaction is simulated by the exchange of massive mesons. In this work we use an extended version of the QHD whose lagrangian density reads [29]: in natural units. The sum in b stands just for the nucleons or for all the baryon octet, depending on our choice for the star constituents, ψ b are the Dirac fields of the baryons, σ, ω µ and ρ µ are the mesonic fields, and A µ is the electromagnetic four-potential. The g ′ s are the Yukawa coupling constants that simulate the strong interaction, m b and e b are the mass and the electric charge of the baryon b; m s , m v , and m ρ are the masses of the σ, ω, and ρ mesons respectively. The antisymmetric field tensors are given by their usual expressions as presented in [30]. The U (σ) is the self-interaction term introduced in ref. [31] to fix some of the saturation properties of the nuclear matter. We also define M * b as the effective mass of the baryon b: In the presence of a magnetic field B in the z direction, the energy eigenvalue E b , and the number density n b of charged baryons are quantized: where the discrete parameter ν is called Landau level (LL). The uncharged baryon energies are not modified by the magnetic interaction and keep their usual expressions [30]. The mesonic fields are obtained by mean field approximation [29,30,32] and the equation of state (EOS) by thermodynamic relations [33]. To construct a β stable matter, we also include leptons as a free Fermi gas and impose zero net charge and chemical equilibrium.
To describe the properties of nuclear matter, we use a slightly modified version of the well-known GM1 parametrization [34], which is a widely accepted parametrization [35,36] that is able to reasonably describe both, nuclear matter and stellar structure, consistent with experimental and astrophysical observations [37]. Here, we just reduce the strength of the ρ coupling, reducing the symmetry energy slope L from the original 94 MeV to 87.9 MeV [38], a value closer to what is inferred in recent observations [39,40]. The aim of the present work is to investigate the effects of the magnetic fields on certain observables and any other choice of parametrization would result on the same qualitative behavior.
In Table I we show the parameters of the model and its previsions for five nuclear matter properties at saturation density: saturation density point (n 0 ), incompressibility (K), binding energy per baryon (B/A), symmetry energy (S 0 ) and its slope (L).

Parameters
Previsions at n0 (gNω/mv) 2  Now, we discuss the presence (or the absence) of hyperons in the neutron stars core. The possibility of the hyperon onset in neutron stars is an old [41] but yet, very active subjetc of study [42]. This is the so called hyperon puzzle. The main problem is that we have very little knowledge of how the hyperons interact with nucleons and with each other, or in QHD words, what are the hyperon-meson coupling constants? Since we have six hyperons and three mesons, we have, in principle eighteen free parameteres besides those presented in table I. To overcome this difficulties we use symmetry group techniques. Following ref. [37], we use the hybrid symmetry group SU(6) to fix all hyperon-vector meson coupling constants and a nearly SU(6) symmetry to fix the hyperon-scalar meson coupling constants. This reduces the eighteen free parameters to just one, which in turn, is fixed using the Λ hyperon potential depth: U Λ = −28 MeV. The values are displayed in Eq. 4.
To finish this section, we discuss the influence of the magnetic field itself on the EOS. As pointed out earlier, the ideal situation is to use the LORENE code [24], but here we use an alternative, the so called chaotic magnetic field approximation. As presented by Zel'dovich [26],we can only use the concept of pressure, when we are dealing with a small-scale chaotic field. In this case, the stress tensor reads diag{B 2 /6, B 2 /6, B 2 /6}, restoring the thermodynamic consistency of the model, once, as expected, the pressure is scalar [18]. In this case, the EOS reads: where the subscript M stands for the matter contribution to the EoS. As mentioned in the Introduction, magnetars bear a magnetic field of the order of 10 15 G at the surface but according to the Virial theorem, stronger fields can be expected in their interior. To account for the growing of the magnetic field strength towards the neutron star core, we follow ref. [23,27,43] and use an energy densitydependent magnetic field: where ǫ c is the energy density at the center of the maximum mass neutron star with zero magnetic field and γ is any positive number, reducing the number of free parameters from two to only one. Moreover, as explained in detail in ref. [27], if we take γ > 2, we have a model without free parameters. B 0 is the fixed value of magnetic field, in our case can be 1.0 × 10 18 G, 3.0 × 10 18 G, or zero. Also, in this way the magnetic field is no longer fixed for all neutron star configuration. Each EOS produces a different value for ǫ c that enters into Eq.(5). For our particular case, ǫ c = 4.98f m −4 for neutron stars with hyperons and ǫ c = 5.65f m −4 for neutron stars without hyperons in the core, ensuring that the magnetic field does not exceed B 0 . In this work we use γ = 3.
Before we proceed, we show in Fig. 1 all the EOS that we use in the present work. As it is always the case, hyperons soften the EOS and magnetic fields within the chaotic approximation change the EOS only slightly, generally making them a bit stiffer. We can see two groups of curves, each one with three EOS, the softer ones describing matter with hyperons and the stiffer one nucleonic matter.

III. TIDAL DEFORMABILITY AND GW170817 CONSTRAINTS
To study the tidal deformability, at first we have to solve the equilibrium configuration for the neutron star, which is represented by the Tolman-Oppenheimer-Volkoff equations given by where m and ν is the gravitational mass and the metric potential respectively. The pressure p and the massenergy density ǫ are related by the equations of state  given in section II. The initial conditions at the centre are m(0) = 0, p(0) = p centre and ν(0) = ν centre . We stop the integration when the pressure becomes zero, and at this point we define the surface of the star whose radius is R. We also apply the junction condition to the metric ν(R) = ln(1 − 2M /R). We next use the EOS described in the last section as input to the TOV equations [44] to obtain the macroscopic properties of the neutron stars. The mass-radius relations are plotted in fig. 2 and the main properties are displayed in Table II. We see that the effect of the chaotic magnetic field is to increase a little the maximum mass of stars with hyperonic core, but causes no significant variation in the neutron stars without hyperons. This small increase on the maximum mass is in agreement with results obtained with the LORENE  code [24]. We can also see from fig. 2 that the low mass neutron stars with magnetic field bear a slightly smaller radii when compared with the stars without magnetic field. For a strong magnetic field ( 3 × 10 18 G) the radii of the canonical stars (the ones with 1.4 M ⊙ ) are always smaller. We can also see that with the GM1 model, even with hyperons, the maximum masses are very close to 2.0 solar masses, a necessary constraint since the observations of massive NS [11,12]. After the computation of the equilibrium configuration we proceed to the study of tidal deformations,which depend on the internal structure of neutron stars, and their main effect is to correct the gravitational wave phase of inspiraling binary systems. This phase shift has been observed by the existing gravitational wave detectors, and our purpose is to use it to constraint the equation of state of compact stars. For this objective we present, in the next lines, some comments about the theory of tidal deformabilities, the main equations and relationships useful for our calculations.
The relativistic theory of tidal effects was deduced by Damour and Nagar, Binnington and Poisson [45,46]. They concluded that the tidal deformation of a neutron star is characterized by the gravito-electric K el 2 and gravito-magnetic K mag 2 Love numbers, where the former is related to the mass quadrupole and the second to the current quadrupole induced by the companion star. Further researches by Flanagan and Hindeler concluded that only a single detection should be sufficient to impose upper limits on K el 2 at 90% confidence level [9]. Since then intense research has been invested on the computing of Love numbers of neutron stars [47][48][49][50][51][52].
In a binary system the induced quadrupole moment Q ij in one neutron star due to the external tidal field E ij created by a companion compact object can be written as [51,52], where, λ is the tidal deformability parameter, which can be expressed in terms of dimensionless l = 2 quadrupole tidal Love number k 2 as To obtain k 2 we have to solve the following differential equation where the coefficients are given by and with E = 1 − 2m/r, ε and p are the energy density and pressure profiles inside the star. Therefore the Love number k 2 can be obtained as where y R = y(r = R) and C = M/R are the star compactness, M and R are the total mass and radius of the star respectively. Equation (11) has to be solved coupled to the TOV equations. The dimensionless tidal deformability Λ (i.e., the dimensionless version of λ) is connected with the compactness parameter C through In Fig. 3(a) and Fig.3(b) the Love number is plotted as a function of the compactness and stellar mass and in Fig.  3(c) the dimensionless tidal polarizability is shown as a function of the stellar mass. If we compare Fig. 3(a) with the ones produced with other models in the literature, we observe that the second Love number lies at about the same range as many of the results obtained with other RMF parametrizations [13] (notice the difference in the x scale), but below most of the results found with different versions of the quark-meson coupling model [53]. These results are clearly model dependent, but our point here is to confirm that most of the differences reside on the constitution of the star (containing hyperons or not) and the effects of the magnetic field are noticeable, but minor. This observed features are certainly expected because in our choice of modelling the magnetic field, its value at the crust is set to be B = 10 15 G and this low magnetic field hardly affects the EOS. In Fig. 3(c), Λ is plotted alongside recent results of the canonical Λ 1.4 = 190 +390 −120 obtained by the LIGO and Virgo Collaboration [54] and we see that our results lie outside its range. Again, it is worth pointing out that a similar result is obtained in [53] and it is due to the choice of parameters. It is also fair to mention that more experimental results should be obtained before any strong conclusion can be drawn.
In Fig. 3(d) we show the tidal deformabilities (Λ 1 , Λ 2 ) for the binary system (m 1 , m 2 ), with m 1 > m 2 . The plots are calculated using the equation for the chirp mass and the diagonal solid line corresponds to the case m 1 = m 2 . The lower and upper solid orange lines correspond to 50% and 90% confidence limits respectively, which are obtained from the GW170817 event. The shadow region represents recently published theoretical results [13], obtained with non-magnetized EOS. We see, once again, that our results lie on the border of the 90% confidence line and the effects of the magnetic field are very small. Therefore, we can see that all the results presented in Figures 3(a), 3(b) and 3(c) are sensitive to the presence of hyperons but are hardly affected by the intensity of the magnetic field. Nevertheless, the effect on the (Λ 1 , Λ 2 ) window shown in Fig. 3(d) is always negligible.
A more recent constraint concerns the radii of the canonical stars, the ones with M = 1.4M ⊙ . Although in the past, studies pointed that the radii of the canonical stars could be as larger as 17 km [55], nowadays this value is believed to be significant lower. More conservative results point towards a maximum radius of 13.9 km [56,57], while more radical studies point to 13 km as the maximum radius [58,59]. Recently, the LIGO and Virgo collaboration stated that the tidal polarizability of canonical stars should lie in the range 70 ≤ Λ 1.4 ≤ 580 [54] and this restriction imposed another constraint to the radii of the corresponding stars. According to [48], the values should lie in the region 11.82 km ≤ R 1.4M⊙ ≤ 13.72 km and according to [54], in the range 10.5 km ≤ R 1.4M⊙ ≤ 13.4 km. Whichever constraint we consider correct, we see that our results for the radii are very close to the inner (or outer) border.

IV. NEUTRON STAR OSCILLATIONS
The oscillations of neutron stars can be excited by the violent dynamics of the binary system. The theory to study the quasi-normal modes of compact stars is well established [60][61][62]. In this work we use the Lindblom and Detweiler method that is widely used to compute the fluid modes. In the rest of this section we briefly present the perturbative formalism to study neutron star oscillations.   The polar non-radial perturbations of a non-rotating star can be described through a set of equations presented in [62,63]. The perturbed metric tensor reads where the metric perturbations are given by and the polar perturbations in the fluid are given by the following Lagrangian displacements where Y ℓ m (θ, φ) are the spherical harmonics, and l is restricted to the l = 2 component, which dominates the emission of gravitational waves.
Non-radial oscillations are then described by the following set of first order linear differential equations [63]: where the prime denotes a derivative with respect to r and γ is the adiabatic index. The function X is given by and H 0 fulfills the algebraic relation b 4 = 1 2 (l + 2)(l − 1)r − ω 2 r 3 e −ν −r −1 e λ (M + 4πr 3 p)(3M − r + 4πr 3 p). (34) Outside the star, the perturbation functions that describe the motion of the fluid vanish and the system of differential equations reduces to the Zerilli equation: where Z(r * ) and dZ(r * )/dr * are related to the metric perturbations H 0 (r) and K(r) by transformations given in Refs. [62,63]. The "tortoise" coordinate is r * = r + 2M ln(r/(2M ) − 1), and the effective potential V Z (r * ) is given by (36) with n = (l − 1)(l + 2)/2.
The quasi normal modes have to be determined by a two stage process, one inside the star and the other outside. Inside the star we have to obtain the coefficients of the differential equations, which are defined at each point. Those coefficients depend directly on the mass, metric, pressure, energy density, etc., and these quantities can be obtained from the stellar structure equations. Then we continue the integration outside the star by the use of the Zerilli equations. All the procedure has to respect boundary conditions at the centre, surface of the star and at infinity. At the centre regularity conditions are required, at the surface of the star the Lagrangian perturbation on the pressure has to be zero and at infinity we have the outgoing gravitational radiation condition. All the equations are numerically integrated for quadrupole oscillations (l = 2). More details about the method can be find in Ref. [64]. This procedure allows us to obtain ω for each value of the central density of the star, or equivalently for each value of the stellar mass. The real part of ω is the pulsation frequency (f = Re(ω)/2π) and the imaginary part is the inverse of the damping time of the mode due to gravitational-wave emission (τ = 1/Im(ω)).
A family of modes exists and it offers a great opportunity to test the gravitational wave asteroseismology approach in neutron stars. If we consider detectability, the most promising modes are the crustal modes, g-modes, f -modes and r-modes. In this work we focus only on the f -mode because it is easily excited in astrophysical events while it is expected to be soon detected by third generation detectors. In the next lines we present and discuss our main results.
In Fig. 4 (a)-(b) we show the plots for the frequency of the quadrupole fundamental fluid mode as a function of the stellar mass and redshift, from where we observe that only the magnetic field of 3×10 18 G produces a small effect on the frequency and this result can be noted for both types of stars, with and without hyperons. This effect is related to the small increase in mass obtained with highly magnetized matter and this increase is larger for hyperonic than for nucleonic matter. The same qualitative behaviour is observed for the frequency as a function of the redshift. On the other hand, it is clear that the constitution of the star plays a very important role and the frequency generated by massive hyperonic stars (larger than 1.8 M ⊙ ) is greatly increased as compared with their nucleonic counterparts. It is also observed that the gravitational wave frequency of the fundamental mode for our models fall in the range of 1.4 -2 kHz for stars with masses between 1.4 -2.4 M ⊙ , this values corresponds with previous results in the literature [64,65] obtained with less realistic EOS. We can see that in general a high magnetic field produce little effect in this frequency window.
We also present, in Fig. 4(c)-(d) the plots of the damping time as a function of the mass and gravitational redshift. The damping time for a typical 1.4 M ⊙ neutron star is near 350 ms, as previously obtained in [64] for strange stars and a high magnetic field does not affect this value. Once again, we observe that only the strongest magnetic field produces non-negligible effects, better noticed in hyperonic stars. As can be seen in Fig. 4(c), the presence of hyperons produces a decrease in the damping time for stars with masses higher than the maximum mass, but those stars are not expected to be stable.

V. SUMMARY AND FINAL REMARKS
In the present work we have analysed the influence of strong magnetic fields on equations of state that describe both nucleonic and hyperonic matter and the resulting effects on the Love number, tidal polarizabilities, stellar radii and the fundamental quasi-normal oscillation mode. To compute the EOS, the chaotic field approximation has been used and an energy density dependent magnetic field prescription utilized so that the magnetic field of the crust never exceeds the observed 10 15 G.
We have seen that the constitution of the stars (nucleonic or hyperonic) plays a much more important role in the computation of the Love number and the tidal polarizability than the strength of the magnetic field. For very large magnetic field (3×10 18 G) a small effect is perceived in the Love number, but this effect becomes negligible in the dimensionless tidal polarizability Λ. As already discussed in [66], this behavior can be understood due to the fact that a cancellation between the Love number k 2 and the compactness of the star C takes place and the Λ behavior is mainly dictated by the EOS of the stellar core, which, as shown in Fig.1 depends very little on the intensity of the magnetic field.
Future third generation detectors, like the Einstein Telescope, will have enough sensitivity to observe the quasi-normal modes of neutron stars. Therefore, it is of paramount importance to fully investigate the gravitational wave frequencies of the most important modes of neutron stars. Amongst all the family of modes, we have studied the fundamental mode, which has been the focus of attention for many years, because it has a frequency of nearly 2 kHz and could be detected with an amplitude of ≃ 10 −23 at 10 Kpc. Also this mode can be easily excited in supernovae events, mergers of binary systems, pulsar glitches and magnetar flares. For this objective we have also calculated the effect of the magnetic field on the fundamental mode. We have observed that the frequencies practically coincide in all cases for stars with masses below 1.8M ⊙ . However, if more massive stars are considered, the ones constituted by nucleons only present frequencies lower than the ones with hyperonic cores and this feature might me a way of pointing out the real constituents of neutron stars.
In all cases, only the strongest magnetic field, i.e., 3×10 18 G alters the frequency. The same behavior is found if we consider the frequency as a function of the redshift. The damping time is typically above 250 ms for masses lower than 1.6 M ⊙ and for very massive star the damping time is between 100 -200 ms. We conclude that by the use of the fundamental mode, high magnetized stars could be discriminated only in the limiting case of magnetic field intensities of the order of 3×10 18 G, but we reinforce the statement that the different constitutions of the liquid core can be easily tracked. Acknowledgments