GW170817 constraints analyzed with Gogny forces and momentum-dependent interactions

A set of equations of state obtained from finite-range Gogny forces and momentum-dependent interactions is used to investigate the recent observation of gravitational waves from the binary neutron star merger GW170817 event. For this set of interactions, we have calculated the neutron star tidal deformabilities (related to the second Love number), the mass-radius diagram, and the moment of inertia~($I$). The $I$-Love relation has been verified. We also have found strong correlations among the tidal deformability of the canonical neutron star, its radius, and the derivatives of the nuclear symmetry energy at the saturation density. Most of the obtained results are located within the constraints of the tidal deformabilities extracted from the GW170817 detection.


Introduction
The theoretical analysis of astronomical observations of very dense matter in the Universe has disclosed that various properties of neutron stars (NSs), such as the mass-radius relation, the moment of inertia, and the tidal deformability, are very sensitive to the properties of nuclear matter at saturation and also at supra-nuclear densities. Hence, it is important to study theoretically the core of NSs, which in the center can attain densities of several times ρ 0 , where ρ 0 is the saturation density. Different nuclear models have since long been applied to analyze the properties of matter at these supra-nuclear densities through the equation of state (EoS), see e.g. [1][2][3][4][5] and references therein. The recent LIGO and Virgo observation of GW170817 [6][7][8], accounting for a merger of two NSs, has enhanced the present interest to examine the sensitivity of the EoS at large values of density and isospin asymmetry. Furthermore, the extracted NS tidal deformabilities of the binary system with 90% and 50% confidence limits, see Fig. 1 of Ref. [7], have provided a new constraint for the nuclear EoS. Many EoSs have been studied under β-equilibrium to formulate various connections between the tidal deformability (related to the second Love number) and other quantities [9][10][11][12][13][14] as, for instance, the moment of inertia (I). This specific correlation is named as I-Love relation, and it seems to be universal, i.e., EoS-model independent.
The total mass is one of the best established observables of NSs from many observational studies. Among them, there are the recent accurate observations of highly massive NSs, corresponding to (1.928±0.017)M ⊙ [15,16] and (2.01±0.04)M ⊙ [17] for the PSR J1614-2230 and PSR J0348+0432 pulsars, respectively, with M ⊙ being the solar mass. As a result, a great ef-fort has been addressed to derive nuclear models able to generate EoSs that predict such massive objects (see [18,19] and references therein). However, a precise mass measurement is not enough to completely constrain the underlying EoS. One would also need a precise measurement of the radius of the NS whose mass has been obtained. The uncertainties in the determination of the NS radius are still an open question for observational studies [20][21][22]. The Neutron Star Interior Composition Explorer (NICER) mission is already set up with the aim to provide a measurement of the radius with accuracy of order 5%. Moreover, the recent GW170817 observation of a NS merger [6][7][8] has dramatically changed the present status of the nuclear matter models by adding new constraints.
For several decades, a large amount of theoretical studies have been performed using different models to describe strongly interacting cold matter at high baryon densities in NSs [22][23][24][25]. Very recently, many hadronic matter models have been tested and tried to constrain their outcomes with the observed tidal boost in the gravitational wave (GW) detection from the binary NS merger GW170817 [6][7][8]. Among these studies, nonrelativistic models [9,[26][27][28][29], effective field theories [30,31], and the relativistic mean-field description of nucleons interacting via meson exchange [9,14,24], have significantly contributed to correlate nuclear observables at saturation density with the cold nuclear matter at very high densities. On the other hand, studies based on effective quark models are focused on the existence of a possible quark matter phase and on the role of color superconductivity at high densities [18,[32][33][34][35][36]. Other recent studies relating the tidal deformability to dense matter properties can also be found, for instance, in Refs. [37][38][39][40]. Our aim in this work is to try to understand the behavior of NS matter at high densities using finite-range interactions of Gogny type [41,42] and of momentum-dependent interaction (MDI) type [43,44], which are very successful in nuclear physics as we point out below. Our investigations will focus on the examination of the applicability of the presented interactions in describing the recent data on the GW170817 event [7], on the search for possible correlations between matter bulk parameters and the tidal deformability, and on the verification of the I-Love relation.
The paper is organized as follows. Sec. 2 gives a brief description for the Gogny and MDI interactions applied to stellar matter. The results of our calculations are presented in Sec. 3. Sec. 4 includes a short summary and the concluding remarks.

Gogny forces and MDI interactions for stellar matter
The Gogny forces were introduced by D. Gogny in order to describe the mean field and the pairing field within the same interaction. They consist of a density-dependent zero-range term and two finite-range Gaussian terms that generate the momentum dependence of the Gogny potential. Gogny forces are well adapted for describing the ground-state systematics as well as deformation and excitation properties of finite nuclei. An exhaustive compilation of nuclear properties computed with the D1S Gogny force [45] can be found in [46]. New Gogny forces such as D1N [47] and the highly accurate D1M [48] have been proposed. D1N and D1M take into account the microscopic neutron matter EoS of Friedman and Pandharipande [49] in the fit of their parameters and improve on the description of isovector properties. It has been found, however, that when one applies the usual parametrizations of the D1 family to studies of NSs, they can not reach masses of ∼ 2M ⊙ . To remedy this situation, a reparametrization of the D1M force called D1M * , which predicts maximum masses of 2M ⊙ when the associated EoS is used to solve the Tolman-Oppenheimer-Volkoff (TOV) equations, has been formulated very recently [42]. The Gogny parametrizations chosen to be analyzed here are those studied in Ref. [42], namely, the D1M, D1M* and D2 forces. D2 [50] is another recent Gogny interaction where the density-dependent contact term has been replaced by a finite-range term, and which also delivers NS masses of 2M ⊙ . Besides, we include in the study the D1M** force [51], which is fitted similarly to D1M* but it is required to predict a NS maximum mass of 1.91M ⊙ , corresponding to the lowest bound of the astronomical observations.
The MDI interactions [43] have been extensively used to study heavy-ion collisions [43,52]. Similar to the Gogny case, the MDI force may be expressed as a zero-range contribution plus a finite-range term, but with a Yukawa form factor instead of a Gaussian. A parameter x of the MDI interaction can be varied to modify the uncertain density dependence of the symmetry energy and of the neutron matter EoS, without changing the EoS of symmetric nuclear matter and the symmetry energy at the saturation density. The MDI model with the tuned isospin dependence has also been used to describe hot asymmetric matter [52] and the properties of NSs [44] and most recently it has been applied to study the tidal deformability of NSs [53]. In our study, we select MDI parametrizations with −1 x 0.2. This leads to a symmetry energy slope L 0 at the saturation density (namely, L 0 = 3ρ 0 ∂E sym (ρ)/∂ρ| ρ 0 ) in the range of 51 MeV L 0 106 MeV for the MDI forces, whereas the range predicted by the Gogny parametrizations used here is 25 MeV L 0 45 MeV (see Table 1 below). One can verify that these two boundaries for L 0 are compatible with those found in Ref. [54] (L 0 = 58.9±16.5 MeV) and used in Ref. [55], in Ref. [56] (40.5 MeV L 0 61.9 MeV), in Ref. [18] (L 0 = 58.7 ± 28.1 MeV), and in Ref. [57] (L 0 = 60 ± 15 MeV). We note, however, that some of the considered MDI forces have a slope parameter L 0 above the upper boundary suggested in these references.
In stellar matter under charge neutrality and β-equilibrium, the weak process and its inverse reaction, namely, n → p + e − + ν e and p + e − → n + ν e , take place simultaneously. At densities for which the electron chemical potential µ e exceeds the muon mass (m µ ), i.e., µ e > m µ , the appearance of muons in the system is energetically favorable. By considering only these two types of leptons in the NS matter, since we assume that neutrinos can escape due to their extremely small cross-sections, one can write the total energy density and pressure of the system as ǫ = ǫ had + l ǫ l and p = p had + l p l , respectively, where the indexes had and l stand for the hadrons and leptons. In this work the hadronic quantities are calculated using the aforementioned Gogny and MDI interactions, taking into account the following conditions: µ n −µ p = µ e = µ µ and ρ p −ρ e = ρ µ , where ρ p = y p ρ, and ρ l = [(µ 2 l − m 2 l ) 3/2 ]/(3π 2 ) for l = e, µ (we use the physical values for the electron mass m e and the muon mass m µ ). The chemical potentials and densities of protons, neutrons, electrons and muons are given, respectively, by µ p , µ n , µ e , µ µ , and ρ p , ρ n , ρ e , ρ µ , whereas y p denotes the proton fraction of the system. In this work we are dealing with NSs old enough to assume that neutrinos have left the star and therefore the β-equilibrium is established between the nucleons, electrons, and muons.
In order to describe a spherically symmetric NS of mass M, we solve the TOV equations [58,59]

given (in units of
and dm(r)/dr = 4πr 2 ǫ(r), where f (r) = 1 − 2m(r)/r. The solution is constrained to p(0) = p c (central pressure) and m(0) = 0. At the star surface, one has p(R) = 0 and m(R) ≡ M, with R defining the NS radius. To describe the EoS of the matter in the NS core we use the Gogny and MDI interactions. For the NS crust we consider two regions, the outer and the inner crust. For the former, we use the EoS proposed by Baym, Pethick and Sutherland (BPS) [60] in a density region of 6.3 × 10 −12 fm −3 ρ outer 2.5 × 10 −4 fm −3 . Currently, microscopic calculations of the EOS of the inner crust for the different Gogny and MDI interactions are not available. Here, following previous literature [41,42,44,[61][62][63], we use for the inner crust a polytropic EoS of the form p(ǫ) = A + Bǫ 4/3 . The index 4/3 assumes that the pressure at these densities is dominated by the relativistic degenerate electrons. For each interaction, we match this polytropic formula continuously to the BPS EoS at the interface between the outer and the inner crust and to the EoS of the homogeneous core at the core-crust transition pressure and energy density computed with the thermodynam-ical method [41,44,64,65]. Thus, the core-crust matching occurs at the transition point that is predicted by each nuclear model. A similar procedure of description of the stellar matter EoS was performed in Ref. [39]. In this work, the authors analyzed more than 200 Skyrme parametrizations in the light of GW170817, also applying a relativistic Fermi gas EoS for the inner crust as in our case. Another approach used to describe stellar matter is the unified treatment such as the one developed in Ref. [66] in which possible thermodynamical inconsistencies are avoided. In our calculations, with the present models and with some models where the unified EoS is available, we verified by using different crust prescriptions that the crustal EoS has no significant influence on the results for the tidal deformability Λ. The same conclusion has been brought into notice recently [63].

Application to the GW170817 constraints
In a binary NS system, tidal forces originated from the gravitational field induce tidal deformabilities in each companion star analogously to the tides generated on Earth due to the Moon. Deformations in the stars related to the quadrupole moment generate GW in which the phase evolution depends on the tidal deformability [67][68][69]. In a recent work [6], the LIGO/Virgo Collaboration (LVC) published an analysis on the first detection of GW coming from a binary NS inspiral (the energy flux out of the binary due to the GW emission causes the inspiral motion). The measured data allowed the LVC to determine constraints on the dimensionless tidal deformabilities Λ 1 and Λ 2 for each NS in the binary system, as well as on that one related to a canonical star of 1.4M ⊙ (Λ 1.4 ). Later on, in Refs. [7,8], the constraints on the Λ 1 ×Λ 2 region and in the Λ 1.4 value were updated. Here we test the Gogny and MDI interactions against the LVC constraints. In order to do that, we calculate the dimensionless tidal deformability as Λ = 2k 2 /(3C 5 ), with the NS compactness given by C = M/R and the second Love number written as with y R ≡ y(R), where y(r) is obtained through the solution of r(dy/dr) + y 2 + yF(r) + r 2 Q(r) = 0, that has to be solved as part of a coupled system containing the TOV equations. The quantities F(r) and Q(r) are defined as  Figure 1: Neutron star mass-radius diagrams for Gogny and MDI parametrizations. Horizontal bands indicate the masses of PSR J1614-2230 [15] and PSR J0348+0432 [17] pulsars. Outer orange and inner grey bands: constraints extracted from Ref. [21]. Outer white and inner black bands: constraints extracted from Ref. [74].
where v 2 s (r) = ∂p(r)/∂ε(r) is the squared sound velocity. See for instance Refs. [67,[70][71][72][73] for the derivations. Table 1: Stellar matter properties obtained from the Gogny and MDI models used here: maximum neutron star mass (M max ) with its corresponding radius (R max ) and central energy density (ǫ c ), along with the radius (R 1.4 ) of a canonical neutron star of mass of M = 1.4M ⊙ , and its dimensionless tidal deformability (Λ 1.4 ). The value of the slope parameter L 0 of the symmetry energy for the considered models is shown in the last column. We show in Fig. 1 the NS mass-radius diagrams for the Gogny and MDI parametrizations used in this work. Some quantities related to the maximum NS mass and a canonical NS are shown in Table 1. Concerning the Gogny model, the parametrization D1M* [42] is compatible with the bands coming from constraints derived from data obtained, with some underlying assumptions for the used EoS, by Steiner [74] and Nättilä [21]. It also predicts NSs with M ∼ 2M ⊙ , in agreement with the well-known data from the PSR J1614-2230 [15] and PSR J0348+0432 [17] pulsars. These features for D1M * also hold for the D2 force [50]. Concerning the MDI models, we see that the model with x = 0.2 does not reach 1.9M ⊙ , whereas M max of the other models is above 1.9M ⊙ . A very recent value for the observed largest NS mass of 2.14 +0.20 −0.18 M ⊙ at 95.4% credible level and of 2.14 +0. 10 −0.09 M ⊙ at 68.3% credible level is proposed in Ref. [75] from the MSP J0740+6620 pulsar. D1M*, D2 and the MDI models with x ≤ −0.2 would be compatible with the lower limit of this mass at the 95.4% level and only D2 in the case of the 68.3% level. Table 1, we remark that it seems there is at least a trend of a certain "threshold value" of L 0 ∼ 40 MeV, the lower limit of the range proposed in Ref. [56], below which there are only configurations predicting M max < 2M ⊙ for the maximum NS mass. For instance, this is true for the D1M and D1M** parametrizations and for the 42 parametrizations of Skyrme and relativistic models analyzed in Ref. [76]. A more systematic study in that direction is needed in order to confirm this result for a larger set of models. For now on, we restrict our study only to parametrizations presenting M max ∼ 2M ⊙ , i.e., we exclude the EoSs of D1M and MDI(x = 0.2), which predict maximum masses clearly below the observations of Refs. [15][16][17].

Still, regarding the values shown in
In Fig. 2a we plot the dimensionless tidal deformability Λ as a function of the NS mass. We see that all parametrizations, except the MDI ones with x = −0.8 and x = −1, reproduce the data from LIGO/Virgo [7] of Λ 1.4 = 190 +390 −120 for the canonical star. Furthermore, by using the results presented in Fig. 2a and in Table 1, it is possible to extract the following fitting expression for Λ 1.4 as a function of the canonical star radius, Λ 1.4 = aR b 1.4 , with a = 6.97 × 10 −5 and b = 6.19. Although Λ is defined as proportional to k 2 R 5 , k 2 is obtained through the solution of the differential equation for y as a function of R, see Eq. (1). This is the origin of the power not equal to 5 in the Λ 1.4 × R 1.4 relationship. Different a and b values are found for different relativistic and nonrelativistic models, as one can verify for instance in Refs. [9,23,24]. Such a relation has also been derived in Ref. In Fig. 2b we show the tidal deformabilities Λ 1 and Λ 2 of a binary NS system calculated with the Gogny and MDI parametrizations along with the contour line of 90% credible level (dashed blue curve) related to the GW170817 NS merger event [7]. To generate the curves, we vary the mass of one of the stars in the binary system, m 1 , in the range 1.36 ≤ m 1 /M ⊙ ≤ 1.60 [7,8]. The mass of the companion star, m 2 , is related to m 1 through the so-called chirp mass M, which is given by M = (m 1 m 2 ) 3/5 /(m 1 + m 2 ) 1/5 . The GW170817 detection provides a chirp mass of M = 1.188 +0.004 −0.002 M ⊙ [6]. Hence, m 2 varies in the ranges 1.17 ≤ m 2 /M ⊙ ≤ 1.36 [6,7]. From Fig. 2b we find a good agreement between the results obtained with the Gogny and MDI parametrizations used in this work and the LIGO/Virgo data. We also compare the present results with the values provided by the widely-known Skyrme model [29]. In that work, the authors calculated the tidal deformabilities for a set of Skyrme forces that are consistent with the constraints in symmetric and asymmetric nuclear matter discussed The blue solid circle with error bars (panel a) is related to the constraint extracted from the GW170817 event [7]. Grey band: predictions from Skyrme parametrizations consistent with constraints from symmetric and asymmetric nuclear matter [29,78]. in Ref. [78]. We see that all the Gogny forces analyzed here lie inside the region defined by the Skyrme models, which is represented by the grey band in Fig. 2b. A similar kind of agreement is found for MDI with x = 0 and x = 0.2.
Another investigation performed in this work is the search for possible correlations between the tidal deformability Λ 1.4 and the bulk parameters of nuclear matter. Since Λ 1.4 is constrained by the LIGO/Virgo analysis of the GW170817 NS merger event, correlations between this quantity and a generic nuclear quantity A may help to establish boundaries on A, and, consequently, to better constrain the microphysics related to the EoS of the hadronic model. This is the case for the nuclear matter bulk parameters [79][80][81]. In Figs. 3a and 3b, we display Λ 1.4 as a function of L 0 and K 0 sym , namely, the symmetry energy slope and energy curvature (both at the saturation density) with the respective fitting curves applied to the considered Gogny and MDI parametrizations. In Fig. 3c we show how Λ 1.4 depends on the symmetry energy skewness at the saturation density, Q 0 sym . Here, the fitting curve is obtained only for the MDI parametrizations, since we have not enough points re- lated to the Gogny model. From the figure, we verify a strong correlation of Λ 1.4 with L 0 and K 0 sym , as the fitted curves show a high correlation coefficient. The increasing of Λ 1.4 with L 0 was also verified in Ref. [38], in which the authors used a set of unified equations of state based on the Skyrme model. We find in Fig. 3c a high correlation of Λ 1.4 with Q 0 sym within the MDI model. However, although there are only few Gogny points, the results suggest that the trend for Λ 1.4 with Q 0 sym is different for Gogny and MDI models. Analyzing in addition the predictions of several Skyrme parametrizations (not shown) we have also observed that the correlation of Λ 1.4 with Q 0 sym is weak among different models. One can assign this larger deviation to the higher-order derivative of the symmetry energy, i.e., small variations on the symmetry energy are enough to cause larger variations on Q 0 sym (third derivative of the symmetry energy). One can use the given correlations and the observational range for Λ 1.4 predicted in Ref. [7] to determine compatible ranges for L 0 and K 0 sym . By applying this procedure we find that the dimensionless tidal deformability of the canonical neutron star obtained by the LIGO/Virgo collaboration, namely, −120 , can be satisfactorily described by parametrizations presenting 15 MeV ≤ L 0 ≤ 88 MeV and −165 MeV ≤ K 0 sym ≤ 33 MeV. As we have discussed before, interactions with slope parameter L 0 40 MeV seem to be unable to provide NSs of M max of 2M ⊙ . This constraint, applied to Fig. 3a, points to a lower bound of Λ 1.4 244, which in turn implies a lower bound of −98 MeV for K 0 sym . We also remark that such constraints on the nuclear matter bulk parameters obtained from their relation with Λ 1.4 are an attempt to find a general trend for the values of these quantities. In that sense, they are model dependent and a more systematic study including other kind of hadronic models is needed in order to confirm or improve these numbers. Concerning the range for K 0 sym , for instance, it is worth to mention that it is similar to the one suggested in Ref. [57], in which the authors proposed a metamodeling approach to treat the EoS for dense nuclear mat-  [14]. Blue circle with error bars: predictions from Ref. [14] forĪ ⋆ , namely, the dimensionless moment of inertia of the PSR J0737-3039 primary component pulsar.
ter, namely, K 0 sym = −100 ± 100 MeV. Regarding the range of L 0 , on the other hand, there are more stringent bounds found from combined analyses of astrophysical and nuclear data, see Ref. [56] for instance.
All of the Gogny and MDI parametrizations analyzed in this work show similar bulk properties in symmetric nuclear matter, with saturation density ρ 0 ∼ 0.16 fm −3 , energy per particle E 0 ∼ −16 MeV, effective mass m * 0 /m ∼ 0.7, incompressibility K 0 between 209 and 225 MeV, and skewness Q 0 between −460 and −427 MeV. In the isovector sector, we find a similar situation for the symmetry energy E 0 sym at saturation, which shows a limited variation (E 0 sym = 28.5 − 31 MeV) in the considered parametrizations. Therefore, this fact can be the origin of the very small interference of these bulk parameters in possible correlations arising from Λ 1.4 as a function of L 0 , K 0 sym and Q 0 sym . Finally, we show in Fig. 4 the dimensionless moment of inertia,Ī ≡ I/M 3 , calculated from the Gogny and MDI models. We obtain this quantity by solving the Hartle's slow rotation equation given in Refs. [11,14,82], namely, a differential equation for one of the metric decomposition functions [11], ω(r), coupled to the TOV equations. The moment of inertia is given in terms of ω R ≡ ω(R) as I = R 3 (1−ω R )/2, where ω R is the framedragging function ω(r) [14] evaluated at the surface of the star in which r = R. In Fig. 4a, one can verify that all used Gogny and MDI interactions are indistinguishable in the plot ofĪ as a function of Λ. Such a feature is claimed to be universal and is known as the I-Love relation. In Refs. [10,11], using a set of 10 different EoSs, the authors showed that such a relation is independent of the neutron star (or quark star) structure for slowlyrotating stars. In Ref. [14], the same result was obtained for a set of 53 Skyrme and relativistic mean-field parametrizations. Here, we show that the same feature remains for the Gogny and MDI finite-range nuclear interactions, thereby confirming the universality of the I-Love relation also with this class of models. In Fig. 4a we display by a dashed line the fitting curve de-termined in Ref. [14]. We see that the Gogny and MDI models are compatible with this fitting.
In Ref. [14], the authors also determined a range forĪ related to the PSR J0737-3039 primary component pulsar (a pulsar of mass M ⋆ = 1.338M ⊙ ), namely,Ī ⋆ ≡Ī(M ⋆ ) = 11.10 +3. 68 −2.28 . Measurements regarding this slowly rotating pulsar are expected to determine its moment of inertia in the next few years with hitherto unachieved accuracy (∼10%). Such a range of Ref. [14] was calculated, using the candidate Skyrme and relativistic mean-field EoSs, in two steps. First, the authors of [14] verified a relation between Λ ⋆ and Λ 1.4 , called as the binary-Love relation. Here, Λ ⋆ is the dimensionless tidal deformability related to the star of mass M ⋆ . Then, they used a suitable fitting for Λ ⋆ as a function of Λ 1.4 along with the I-Love relation fitting curve presented in Fig. 4a to establishĪ ⋆ as a function of Λ ⋆ . Finally, the observational range Λ 1.4 = 190 +390 −120 obtained by the LIGO/Virgo collaboration was used to determine a range for Λ ⋆ , and consequently, the limits ofĪ ⋆ = 11.10 +3.68 −2.28 . As one can see in Fig. 4b, the Gogny parametrizations used presentĪ ⋆ inside the predicted range. For the MDI interactions, those in which x > −0.6 are also in agreement with the limits. The fact that the Gogny and MDI forces that fulfill the LVC observational bounds Λ 1.4 = 190 +390 −120 also agree with the range of Ref. [14] forĪ, corroborates with another type of hadronic models, the limits for the moment of inertia of pulsar A of PSR J0737-3039 established by Landry and Kumar [14].

Summary and concluding remarks
We have studied the predictions of the EoS of stellar matter from finite-range effective interactions of the Gogny and MDI types for describing NS properties related with the information extracted from the GW170817 detection of gravitational waves by the LIGO/Virgo collaboration. The tidal deformability Λ 1.4 predicted by the Gogny and MDI interactions for a 1.4M ⊙ NS agrees with the corresponding data supplied by LIGO/Virgo, except for the MDI interactions with x = −0.8 and x = −1, which have the stiffest symmetry energies of the considered models. From the dependence of Λ 1.4 with the NS radius R 1.4 obtained with the Gogny and MDI interactions, the LVC observational value Λ 1.4 = 190 +390 −120 allows one to constrain the radius of a canonical NS in the range of 9.3 km ≤ R 1.4 ≤ 13.1 km. We have also calculated the tidal deformabilities of the two NSs in the binary system assuming the chirp mass that was measured in the GW170817 event. The results lie within the 90% confidence region extracted from the LIGO/Virgo data for most of the analyzed parametrizations.
We have seen that the isoscalar properties and the value of the symmetry energy at saturation of the underlying Gogny and MDI interactions used to build the stellar EoS have little impact on the tidal deformability Λ 1.4 computed with these models, while Λ 1.4 shows a clear dependence, roughly linear, with the isovector properties of the interaction, such as the slope (L 0 ), curvature (K 0 sym ) and skewness (Q 0 sym ) of the symmetry energy. We find that the LIGO/Virgo determination of Λ 1.4 can be reproduced by the effective interactions characterized by 15 MeV ≤ L 0 ≤ 88 MeV and −165 MeV ≤ K 0 sym ≤ 33 MeV.
In particular, the L 0 range is in harmony with other estimates of the slope of the symmetry energy derived from nuclear observables such as the neutron skin thickness, giant resonances, heavy-ion reactions, or the systematics of nuclear masses (see, for instance, [83] and references therein). Finally, we have analyzed the moment of inertia of a NS as a function of the tidal deformability. We have shown that the so-called I-Love relation [10,11], which is expected to be universal, is, indeed, also satisfied by the Gogny and MDI interactions. In addition, we obtain that the prediction of Landry and Kumar [14] of a dimensionless moment of inertiaĪ ⋆ = 11.10 +3.68 −2.28 for PSR J0737-3039 is supported by the Gogny and MDI models that, at the same time, are consistent with the LIGO/Virgo determination of Λ 1.4 .