Estimating the nuclear saturation parameter via low-mass neutron star asteroseismology

Neutron stars provided via supernova explosion, which is the last moment of massive star’s life, are a suitable environment for probing the physics in extreme conditions. The density inside the star easily exceeds the standard nuclear density, while the gravitational and magnetic fields become much stronger than that observed in our solar system. Through the observations of neutron star itself and/or the phenomena associated with the neutron stars, one would obtain the aspect under such extreme conditions. For example, the discovery of the $2M_{\odot}$ neutron stars D10 ; A13 ; C20 enables us to exclude some of soft equations of state (EOSs), with which the expected maximum mass does not approach the observed neutron star mass. This constraint consequently brings up an issue about the appearance of hyperon inside the neutron stars, the so-called hyperon puzzle. In addition, it is known that the light propagating in the strong gravitational field can bend due to the relativistic effects. By taking into account this effect, the light curve from a rotating neutron star with hot spot(s) mainly depends on the stellar compactness, which is the ratio of stellar mass to the radius (e.g., PFC83 ; LL95 ; PG03 ; PO14 ; SM18 ; Sotani20a ). In practice, via the X-ray observation by the Neutron star Interior Composition Explorer (NICER) mission, the properties of the millisecond pulsar PSR J0030+0451 has observationally been estimated Riley19 ; Miller19 . The further constraints on the neutron star mass and radius will enable us to constrain the EOS for neutron star matter.

Furthermore, the oscillation frequencies of the central objects are also important information. Since the frequency of the objects strongly depends on their interior properties, one would extract such an information via the observation of the corresponding frequency as an inverse problem. This technique is known as seismology on the Earth, helioseismology on the Sun, and asteroseisomogy on astronomical objects. In fact, by identifying the frequencies of the quasi-periodic oscillations observed in the giant flares with the neutron star crustal oscillations, the crust EOS (especially the nuclear saturation parameters) has been constrained (e.g., GNHL2011 ; SNIO2012 ; SIO2016 ). In a similar way, once the gravitational waves from compact objects would be observed in the future, one could extract the information about the mass ($M$), radius ($R$), and EOSs for the source objects (e.g., AK1996 ; AK1998 ; STM2001 ; SH2003 ; SYMT2011 ; PA2012 ; DGKK2013 ; Sotani20b ). Additionally, in order to identify the origin of gravitational wave signals found in the numerical simulations for core-collapse supernova, the gravitational wave asteroseismology on protoneutron stars newly born via supernova explosions recently attracts attention (e.g., FMP2003 ; FKAO2015 ; ST2016 ; SKTK2017 ; MRBV2018 ; TCPOF19 ; SS2019 ).

Among these attempts, the asteroseismology on cold neutron stars has extensively been studied up to now. For considering the asteroseismology, one first prepares the background models and then makes a linear analysis on the provided background models. Depending on the input physics, the corresponding oscillations would be excited as eigenmodes. That is, if one would identify a frequency from the neutron stars with a specific eigenmode, one could extract the physics behind the oscillation. In particular, the fundamental ($f$-) and the 1st pressure ($p_{1}$-) modes are observationally important gravitational wave signals from neutron stars composed of perfect fluid, because those frequencies are relatively low among the various eigenmodes KS1999 . Since the $p$-modes are overtone of the $f$-mode, these modes could be excited in the astrophysical scenarios involving dynamical processes such as neutron star mergers or core-collapse supernovae. A starquake associated with a pulsar glitch may also be considered as another excitation scenario. In addition, if a kind of mini-collapse due to the phase transition inside the neutron star would happen, the $f$- and $p$-modes may be excited. Anyway, since the amplitude of each mode excited in these scenarios is still unknown, which cannot be determined with the linear analysis, one has to carefully estimate it by numerical simulations to discuss the detectability for each gravitational wave mode.

Meanwhile, it has been also shown that the $f$-mode frequency is strongly associated with the neutron star average density, $M/R^{3}$. This is because that the $f$-mode is a kind of the acoustic oscillations, whose propagation speed, i.e., sound velocity, is related to the stellar average density. In fact, the $f$-mode frequency is expressed as a linear function of the square root of the stellar average density ($x$) almost independently of the EOS AK1996 ; AK1998 , even though there is still a little dependence on the EOS. Moreover, in our previous study for low-mass neutron stars, whose central density is less than two times the nuclear saturation density, we derived the empirical formula for the $f$-mode frequency as a function of $x$ and the parameter $\eta$, which is a combination of the nuclear saturation parameters, mainly adopting the phenomenological EOSs Sotani20b . That is, we could clearly separate the EOS dependence from the $x$ dependence in the $f$-mode frequency. Then, we proposed that the value of $\eta$ could be estimated via the observation of the $f$-mode gravitational wave from a low-mass neutron star, whose mass is known, by using the empirical formula for the $f$-mode frequency together with the mass formula derived in Ref. SIOO14 . This attempt partially works well, while we also found that it does not work for lower value of $\eta$.

In this study, by adopting the unified realistic EOSs, first we will update the empirical formula for the $f$-mode frequency. With the updated formula, we can overcome the defect in the previous study for estimating lower value of $\eta$. In practice, one could estimate the value of $\eta$ within $\sim 20\%$ accuracy, once the $f$-mode frequency would be observed from the neutron star whose mass is known, if the mass is less than $M=1.174M_{\odot}$. In addition, we will newly derive the empirical formula for the $p_{1}$-mode frequency from low-mass cold neutron stars. Then, we will propose that the value of $\eta$ could be estimated via the simultaneous observation of the $f$- and $p_{1}$-mode gravitational waves by using our empirical formulae. Since the empirical formula for the $p_{1}$-mode frequencies is only applicable for low-mass neutron stars with the mass less than $\sim 0.9M_{\odot}$, which is significantly lower than the observed minimum mass of neutrons star, i.e., $M=1.174M_{\odot}$ for PSR J0453+1559 Martinez15 , our method is workable only if the neutron stars whose masses are less than $\sim 0.9M_{\odot}$ would exist. Furthermore, we find the strong correlation between the maximum frequency of the $f$-mode and the minimum radius of neutron star, which corresponds to the neutrons star model with the maximum mass.

This paper is organized as follows. In Sec. II, we mention the unified EOS considered in this study. In Sec. III, we calculate the eigenfrequencies from the neutron stars constructed with various EOSs. With the resultant eigenfrequencies, we update the empirical formula for the $f$-mode frequency derived in the previous study, and newly derive the empirical formula for the $p_{1}$-mode frequency. Then, we will show how well the empirical formulae work for estimating the value of $\eta$. Finally, we conclude this study in Sec. IV. The details for the update of the empirical formula for the $f$-mode is shown in Appendix A. Unless otherwise mentioned, we adopt geometric units in the following, $c=G=1$, where $c$ denotes the speed of light, and the metric signature is $(-,+,+,+)$.

In this study, we simply consider cold neutron stars without rotation as in Ref. Sotani20b . The neutron star models are constructed by integrating the Tolman-Oppenheimer-Volkoff equations together with appropriate EOSs. In particular, we adopt the unified realistic EOSs in this study. That is, we consider only the EOSs, which are consistently constructed with the same framework for both the core and crust region inside the neutron stars. Table 1 lists the EOSs considered in this study, where the EOS parameters, such as the incompressibility, $K_{0}$, the slope parameter for the nuclear symmetry energy, $L$, and an auxiliary parameter, $\eta$, defined as $\eta\equiv(K_{0}L^{2})^{1/3}$ SIOO14 , and the maximum mass of the neutron star constructed with each EOS are shown. Here, DD2 DD2 ; note_DD2 , Miyatsu Miyatsu , and Shen Shen are the EOSs based on the relativistic framework, FPS FPS , SKa SKa , SLy4 SLy4 , and SLy9 SLy9 are the EOSs based on the Skyrme-type effective interaction, and Togashi Togashi17 is the EOS derived with variational method. Owing to the nature of nuclear saturation properties, $K_{0}$ and $L$ have been more or less constrained via the terrestrial experiments, i.e., $K_{0}=230\pm 40$ MeV KM13 and $L=58.9\pm 16$ MeV Li19 , which leads to the fiducial value of $\eta$ as $70.5\,\,\raisebox{-3.44444pt}{$\stackrel{{\scriptstyle\textstyle<}}{{\sim}}$ }\eta\,\,\raisebox{-3.44444pt}{$\stackrel{{\scriptstyle\textstyle<}}{{\sim}}$ }114.8$ MeV.

In the left panel of Fig. 1, we show the relation between the mass ($M$) and radius ($R$) of the neutron stars constructed with the EOSs considered in this study, where the dotted, solid, and dashed lines correspond to the EOSs based on the relativistic framework, the Skyrme-type effective interaction, and derived with variational method, respectively. In the right pane of Fig. 1, the neutron star mass is shown as a function of the square root of the normalized stellar average density defined as

In the both panels, we plot the neutron star models up to that with the maximum mass. From this figure, it is clearly observed that the neutron star model with the maximum mass has the minimum radius and the maximum value of $x$ for each EOS.

The advantage for considering the unified EOSs is that the properties of low-mass neutron stars can be characterized well as a function of $\eta$ SIOO14 . In fact, the mass and gravitational redshift for low-mass neutron stars are written as a function of $\eta$ and the central density normalized by the saturation density, $\rho_{0}$. In addition, it has been shown that the properties for slowly rotating low-mass neutron stars are associated with $\eta$ SSB16 and that the possible maximum mass of neutron star is also discussed with $\eta$ Sotani17 ; SK17 . Furthermore, we have shown that $x$ for low-mass neutron stars is expressed as a function of $\eta$ Sotani20b , which is updated in Appendix A by using the EOSs considered in this study, i.e., Eqs. (16) - (19).

On the neutron star models constructed with the various EOSs, the gravitational wave frequencies are determined via linear analysis. In this study we simply adopt the relativistic Cowling approximation, where the metric perturbations are neglected during the fluid oscillations. The perturbation equations are derived by linearizing the energy-momentum conservation law. Then, imposing the appropriate boundary conditions at the stellar center and surface, the problem to solve becomes an eigenvalue problem, where the eigenvalue, $\omega$, is associated with the gravitational wave frequency, $f$, via $\omega=2\pi f$. The concrete perturbation equations and boundary conditions are the same as in Ref. SYMT2011 . In this study, we focus on only the $\ell=2$ modes, because the $\ell=2$ modes are considered to be more energetic in gravitational wave radiation. We remark that one can qualitatively discuss the behavior of the eigenfrequencies even with the relativistic Cowling approximation, while the $f$- and $p$-modes calculated with the approximation deviate from the frequencies calculated with metric perturbations (without relativistic Cowling approximation) within $\sim 20\%$ and $\sim 10\%$, respectively YK1997 .

Since we consider the gravitational waves from cold neutron stars composed of the perfect fluid in this study, only the $f$- and $p_{i}$-modes can be excited. In Fig. 2, as an example we show the $f$-, $p_{1}$-, and $p_{2}$-modes as a function of the central density $\rho_{c}$ normalized by the nuclear saturation density $\rho_{0}$ for the neutron star models constructed with Togashi EOS (solid lines) and Shen EOS (dashed lines). From this figure, one can observe that the central density when the avoided crossing happens strongly depends on the EOS. Even so, we have shown in the previous study Sotani20b that the $f$-mode frequency, $f_{f,{\rm AC}}$, and the square root of the normalized stellar average density, $x_{\rm AC}$, for the neutron star model at the avoided crossing between the $f$- and the $p_{1}$-modes are expressed well as a function of $\eta$, which are updated as Eqs. (9) and (10) by using the EOSs considered in this study in Appendix A. That is, $x_{\rm AC}$ depends on the EOS. The values of $x_{\rm AC}$ for various EOSs are shown in Table 1, which are also shown in Fig. 3 with marks. Additionally, using the dependence of $f_{f,{\rm AC}}$ and $x_{\rm AC}$ on $\eta$, we have also derived the empirical formula for the $f$-mode frequencies for low-mass neutrons stars as a function of $x$ and $\eta$. This empirical formula is also updated with the EOSs considered in this study (see in Appendix A for the details), such as

where $f_{0}(\eta)$ is given by Eq. (15).

In the previous study Sotani20b , we have proposed that the value of $\eta$ could be estimated by the observation of the $f$-mode frequency from the neutron star whose mass is known, using the empirical formula for the $f$-mode frequency together with the mass formula derived in Ref. SIOO14 , where the estimation of lower value of $\eta$ does not work well with the original empirical formula for the $f$-mode. Owing to the update in this study, we find that this defect can be removed (see Fig. 14 and in Appendix A for the details). In addition, we considered only the low-mass neutron stars whose central density is less than $2\rho_{0}$ in the previous study, but we find that the empirical formula given by Eq. (2) is applicable even for canonical neutron stars. To check the accuracy of this formula, we calculate the relative deviation of the $f$-mode frequency estimated with Eq. (2) from the frequency calculated for each stellar model, $f_{f}$, via

and the result is shown in Fig. 4. From this figure, we find that the $f$-mode frequency even for the neutron stars with maximum mass can be estimated with Eq. (2) less than $20\%$ accuracy.

On the other hand, in the left panel of Fig. 5 we show the $p_{1}$-mode frequency as a function of $x$. One can observe three avoided crossing in the $p_{1}$-mode, i.e., the first with the $p_{2}$-mode, the second with the $f$-mode, and the third with the $p_{2}$-mode again (see also Fig. 2), as the central density increases. Since the $p_{1}$-mode frequency and the value of $x$ for the neutron star model at the second avoided crossing (with the $f$-mode) are strongly associated with $f_{f,{\rm AC}}$ and $x_{\rm AC}$, in the middle panel of Fig. 5 we show the $p_{1}$-mode frequency shifted by $f_{f,{\rm AC}}$ as a function of $x-x_{\rm AC}$. From this figure, $f_{p_{1}}-f_{f,\rm AC}$ in the region between the second and third avoided crossing is written as a function of $x-x_{\rm AC}$ almost independently of the EOS, where the EOS dependence in this region is confined to $f_{f,{\rm AC}}$ and $x_{\rm AC}$. In practice, we derive the fitting line in this region as

which is also shown with the thick-solid line in the middle panel of Fig. 5. Since $f_{f,{\rm AC}}$ and $x_{\rm AC}$ in this equation are expressed as a function of $\eta$, i.e., Eqs. (9) and (10), we eventually obtain $f_{p_{1}}$ as a function of $x$ and $\eta$ (although the explicit function form is not shown here), such as

even though this fitting formula can be valid only for rather light neutron stars whose masses are less than $\sim 0.9M_{\odot}$, as seen the right panel of Fig. 5.

In order to check the accuracy of the obtained empirical formula for the $p_{1}$-mode frequency, in a similar way to the case of the $f$-mode, we calculate the relative deviation via

where $f_{p_{1}}$ denotes the $p_{1}$-mode frequency calculated for each stellar model, while $f_{p_{1}}(x,\eta)$ denotes the frequency estimated with the empirical formula given by Eq. (5) by using the value of $x$ and $\eta$ for each stellar model. The resultant value of $\Delta f_{p_{1}}$ is shown in Fig. 6 as a function of $x-x_{\rm AC}$. Since the $p_{1}$-mode frequencies for more massive neutron stars than that at the third avoided crossing with the $p_{2}$-mode significantly deviate from the empirical formula given by Eq. (5) as shown in the right panel of Fig. 5, the accuracy becomes worse for the massive neutron star models. Meanwhile, we find that the $p_{1}$-mode frequencies for the neutron star models between the second and third avoided crossing are well estimated with our empirical formula.

Now, we have derived two empirical formulae for the $f$- and $p_{1}$-mode frequencies as a function of $x$ and $\eta$, i.e., Eqs. (2) and (5). With using the empirical formula for the $f$-mode frequency, the $p_{1}$-mode can be estimated as a function of $f_{f}$ and $\eta$, i.e., $f_{p_{1}}=f_{p_{1}}(f_{f},\eta)$, with which one can in principle rewrite as $\eta=\eta(f_{f},f_{p_{1}})$. That is, one could constrain the value of $\eta$ via the simultaneous observation of the $f$- and $p_{1}$-mode gravitational waves from neutron stars. In fact, for the neutron star models between the second and third avoided crossing in the $p_{1}$-mode, as shown in Fig. 7 the ratio of the $p_{1}$- to the $f$-modes are almost linear function of the $f$-mode frequency and that relation seems to be ordered by the value of $\eta$, i.e., $\eta$ increases from right to left. In order to see how accurate the value of $\eta$ would be estimated by using the empirical formulae via the simultaneous observation of the $f$- and $p_{1}$-modes, we show the value of $\eta(f_{f},f_{p_{1}})$ estimated with the empirical formula as a function of the $f$-mode frequency for the case with $f_{p_{1}}/f_{f}=1.5$ (solid line), 2.0 (dashed line), and 2.5 (dotted line) in the left panel of Fig. 8, while the $f$-mode frequencies calculated for each stellar model (having own value of $\eta$) are also plotted with marks. We find that the value of $\eta$ is well estimated with the empirical formulae for the $f$- and $p_{1}$-modes except for the case with Shen EOS. Nevertheless, by considering the fact that the nuclear saturation parameters for Shen EOS have been excluded by the terrestrial experiments and that the neutron star models constructed with Shen EOS are also excluded by the gravitational wave observation in the GW170817 event Annala18 , maybe one should take care of the EOSs except for Shen EOS. So, with respect to the results shown in the left panel of Fig. 8 except for Shen EOS, the relative deviation of the value of $\eta$ estimated with the empirical formulae from the value of $\eta$ for each neutron star model is evaluated via

which is shown as a function of the $f$-mode frequency in the right panel of Fig. 8, where the solid, dashed, and dotted lines correspond to the results for $f_{p_{1}}/f_{f}=1.5$, 2.0, and 2.5, respectively. As a result, we find that via the simultaneous observation of the $f$- and $p_{1}$-mode gravitational waves for $1.5\,\,\raisebox{-3.44444pt}{$\stackrel{{\scriptstyle\textstyle<}}{{\sim}}$ }f_{p_{1}}/f_{f}\,\,\raisebox{-3.44444pt}{$\stackrel{{\scriptstyle\textstyle<}}{{\sim}}$ }2.5$, one could extract the value of $\eta$ less than $\sim 10\%$ accuracy. We remark that, since in Fig. 7 the minimum and maximum of $f_{p_{1}}/f_{f}$ correspond to the neutron star models at the second and third avoided crossing in the $p_{1}$-mode, maybe one can estimate the value of $\eta$ even for $1.1\,\,\raisebox{-3.44444pt}{$\stackrel{{\scriptstyle\textstyle<}}{{\sim}}$ }f_{p_{1}}/f_{f}\,\,\raisebox{-3.44444pt}{$\stackrel{{\scriptstyle\textstyle<}}{{\sim}}$ }1.5$ and $2.5\,\,\raisebox{-3.44444pt}{$\stackrel{{\scriptstyle\textstyle<}}{{\sim}}$ }f_{p_{1}}/f_{f}\,\,\raisebox{-3.44444pt}{$\stackrel{{\scriptstyle\textstyle<}}{{\sim}}$ }2.7$ with using our empirical formulae. But, in such a case since the $f$-mode frequency has two solutions with the fixed value of $f_{p_{1}}/f_{f}$, we only focus on the case with $1.5\,\,\raisebox{-3.44444pt}{$\stackrel{{\scriptstyle\textstyle<}}{{\sim}}$ }f_{p_{1}}/f_{f}\,\,\raisebox{-3.44444pt}{$\stackrel{{\scriptstyle\textstyle<}}{{\sim}}$ }2.5$ in this study.

Furthermore, as shown in Fig. 9 we also find that the maximum $f$-mode frequency, which comes from the neutron star model with the maximum mass, is strongly correlated to the radius of the neutron star model with the maximum mass, which corresponds to the minimum radius of neutron star, such as

That is, if one would observe a large frequency of the $f$-mode gravitational wave from a cold neutron star, one can constrain the upper limit of the minimum neutron star radius, which enables us to exclude some of stiff EOSs.

We examined the $f$- and $p_{1}$-mode frequencies from cold neutron stars constructed various unified EOSs. First, by using the realistic EOSs considered in this study, we updated the empirical formula for the $f$-mode frequency as a function of the square root of the normalized stellar average density, $x$, and a parameter, $\eta$, which is a combination of the nuclear saturation parameters. The similar empirical formula has already been derived in the previous study for low-mass neutron stars by mainly using the phenomenological EOS, where we proposed that the value of $\eta$ would observationally be estimated with the empirical formula for the $f$-mode together with the mass formula, but it does not work well in the estimation of lower value of $\eta$. Owing to the update in this study, not only this defect can be removed, but also we confirmed that the updated empirical formula is applicable within $\sim 20\%$ accuracy even for the neutron star model with the maximum mass. In addition, we newly derived the empirical formula for the $p_{1}$-mode frequency as a function of $x$ and $\eta$. As a result, we proposed that the value of $\eta$ could be estimated with the simultaneous observation of the $f$- and $p_{1}$-modes by using our empirical formulae. In fact, according to our proposal, we showed that $\eta$ could be estimated within $\sim 10\%$ accuracy, when the ratio of the $p_{1}$-mode to the $f$-mode is in the region between 1.5 and 2.5. Finally, we also found that the maximum $f$-mode frequency is strongly associated with the minimum radius of the neutron star (which corresponds to the stellar model with the maximum mass), and consequently derived the fitting formula for the maximum $f$-mode frequency as a function of the minimum stellar radius. That is, if one would observe a large frequency of the $f$-mode, one might constrain the upper limit of the minimum stellar radius, which helps us to understand the EOS for high density region. In this study we only consider the non-rotating neutron star models. That is, we do not know whether or not our approach can be applied even for rotating neutron stars. This is a very interesting topics and we will study somewhere in the future.