Density Discontinuity of a Neutron Star and Gravitational Waves
We calculate quasi-normal f- and g-modes of a neutron star with density discontinuity, which may appear in a phase transition at extreme high density. We find that discontinuity will reflect largely on the f-mode, and that the g-mode could also be important for a less massive star.
pacs:PACS number(s): 04.25.Nx, 04.30.-w, 04.40.Dg
Gravitational wave interferometers such as LIGO, VIRGO, GEO600, and TAMA300 on earth, and LISA in space  are almost ready to observe directly the gravitational waves from the Universe. The significance of direct observation of gravitational waves is that we obtain new method to observe the Universe by gravitational waves (the so-called gravitational wave astronomy), and that we may be able to verify the theory of gravity, and that new physics at high density or high energy region can be found by observing a merger of a binary neutron star.
The one of the sources of the gravitational waves is supposed to be supernova explosion, which occurs at the last phase of a massive star. After explosion, a newly formed compact object may oscillate violently, and the emitted gravitational waves carry the information of the sources. The oscillation will eventually damp out because that the gravitational waves carry the energy. The merger of two coalescing black holes or neutron stars is another source. It is naturally expected that the final object will also oscillate. If a final compact object is a neutron star, its structure will reflect on the emitted gravitational waves. There should be some relation between a structure of a neutron star and the emitted gravitational waves. The structure of a neutron star not yet well-known because the equation of state is uncertain at the high density over the nuclear density. Therefore, although main composition of matter in a neutron star is neutron, it is possible that there appear new phase in extremely high density such as a pion condensation, or a kaon condensation. In such a situation, a phase transition may occur, and if it is strongly first order, one may expect that there appears the discontinuity of density at some density as the case of the kaon condensation . If we know in advance the relation between the structure of a neutron star and the emitted gravitational waves, we will get the information or the restriction on the equation of state over the nuclear density. In particular, if we find any specific feature in the emitted gravitational waves from a neutron star with discontinuity, we may be able to predict the existence of a phase transition in a neutron star. This is the main subject of the present paper. We will just focus into quasi-normal modes of a neutron star.
Although the gravitational waves emitted in a supernova explosion or from a merger of a binary may show very complicated wave form, only quasi-normal modes will eventually remain. We expect that these quasi normal modes contain a lot of information about a neutron star. Hence, it is very important for us to study those modes for a possible situation. The quasi normal modes are classified into various types. For the modes coupled to the fluid of a non-rotating neutron star, we have f-, p- and g-modes.
The damping rate of these modes is very small compared with the oscillation period of fluid because matter couples to gravity very weakly. There exists only one f-mode for each . The p-mode which is caused by the pressure of fluid. The g-mode is originated by density discontinuity or temperature of a star. Its damping rate is usually quite small, so few calculation has been done. There also exists another mode (the so-called w-mode) which is related to metric perturbations of spacetime rather than fluid oscillation. This mode has high damping rate in comparison with modes related to fluid oscillation.
For the case without density discontinuity, the relation between quasi normal modes of a star and equation of state has been studied by many authors, , . For g-mode, there are a few works. Newtonian case is studied in  and also in , which includes the effect of temperature of neutron stars by use of the Cowling approximation. For the relativistic case, Finn adopted an approximate method which is called the “slow motion formalism” and calculated the imaginary part by the energy loss for the case of density discontinuity , .
Although these works are very important, they have not discussed density discontinuity at high density, which may appear via a phase transition. In this paper, we shall study a neutron star with such a discontinuity, and investigate its quasi normal modes. Just for simplicity, we assume that equation of state is given by a polytrope (). We discuss only even parity modes, which are coupled to oscillation of matter fluid in a neutron star. We then assume that there exists density discontinuity due to a phase transition and density discontinuity.
First we present the basic equations to solve in §II, including the equation for a background neutron star with density discontinuity. In §III, we show an explicit structure of a neutron star with the density discontinuity and numerical results for quasi normal modes. We also compare our results with those obtained by the relativistic Cowling approximation. Conclusion is given in §IV. In Appendix A, we present the explicit boundary condition at the center of a star. In Appendix B, we briefly summarize the method of how to calculate the quasi normal modes.
In this paper we adopt the unit of , where and denote the speed of light and the gravitational constant, respectively, and the metric signature of .
Ii basic equation
ii.1 stellar model with discontinuity
We first construct a neutron star model with discontinuity of density to calculate its quasi-normal modes. A static and spherically symmetric spacetime is described by
where are metric functions with respect to . A mass function , which is defined as
The equilibrium condition of a stellar model is given by the Tolman-Oppenheimer-Volkoff equation,
where the function and are the energy density and the pressure of the fluid. The potential is given by
With an additional equation, i.e. the equation of state, we can solve these equations.
As for the boundary conditions, there are two boundaries; the center of a star and the surface. The stellar radius is determined by the condition that the pressure vanishes at the surface, i.e. . The total mass is given by , and the potential is smoothly connected at the surface, i.e.
At the center, a regularity condition should be imposed.
As for the equation of state, just for simplicity, we adopt a polytropic equation of state except at a discontinuous surface of density, which is assumed to exist inside of the star. Then, the equation of state is given by two equations;
where is the radius of a discontinuous surface. The values of two adiabatic indecies ( and ), and those of two coefficients ( and ) may be different. However, since the pressure should be continuos at the discontinuous surface , we have one constraint on those parameters. For example, is fixed as
Now we have six parameters; the central density of the star , the inner density at the discontinuous surface , the ratio of the outer density to the inner density at the discontinuous surface , the inner polytropic index , the outer polytropic index and the outer polytropic coefficient . Some examples of this equation of state with discontinuous density are given in Figure 1.
ii.2 perturbation equations inside the star
Here we present the basic equations for perturbations against the background metric of a spherically symmetric star (1). The perturbation is given by
The perturbations against a spherical background can be decomposed by the spherical harmonics , and are classified into two by its parity, i.e. odd or even. Because we are interested in to discuss relation between a discontinuity in stellar density and the gravitational waves emitted from such a neutron star, in this paper, we only study even-parity perturbations, in which gravitational waves are coupled to fluid oscillation.
Following the method by Lindblom and Detweiler, is described as
where and are the perturbed metric function with respect to . The fluid perturbations are described by the Lagrangian displacement vectors as follows:
where and are functions with respect to . Furthermore we introduce the new variable instead of to make the boundary condition at the surface of a star simple (see below);
The perturbation equations of the Einstein equations are given by
where is the adiabatic index of the unperturbed stellar model defined by
ii.3 the perturbations outside the stars
In the region outside the star, the perturbations are described by the Zerilli equations. Setting , and replacing with new variables , defined by
we recover the Zerilli equations. Here is the tortoise coordinates and .
ii.4 boundary conditions
In order to solve the above perturbation equations, we have to impose the boundary conditions. At the center of the star, we have regularity condition for the perturbative variables. We expand all variables by Taylor power-series near in a similar way of Detweiler and Lindblom  (see Appendix A). Another boundary condition is that the pressure vanishes at the stellar surface. Then the perturbed pressure , is given by
also vanishes there. This boundary condition at the surface of star is equivalent to . The last condition is that there exist only outgoing gravitational waves at infinity. We also have a junction condition at the surface of discontinuity, where the variables of (and then ) must be continuous. Note that, and are determined by Eqs. (20) and (21). is continuos but is discontinuous.
Iii numerical results
iii.1 models of neutron star
As a concrete example of density discontinuity, we know kaon condensation , which may occur at g/cm with the discontinuity for a cold neutron star. Hence we consider density discontinuity around this value, i.e. we study the following three cases: , and g/cm. First we briefly discuss a models of a neutron star with density discontinuity. As mentioned before, we have six unknown parameters: and . We set and , which are usually adopted in a neutron star model. The rest four are and .
As for inner polytropic index, we set 1.0 (Fig. 2), 0.7 (Fig. 3), 1.5 (Fig. 4). For each polytropic index, we consider two models with different density at discontinuity: (a) and g/cm (b). We depict the - diagram for each value of = 1.0 (no discontinuity), 0.9, 0.7, 0.5, and 0.3. Fig. 2 for shows that the maximum masses for each value of do not depend on density discontinuity . As gets small, the maximum mass decreases and the stable region () becomes narrower and moves to high density region.
For , the maximum mass depends on . In fact, for (Fig. 3), the maximum mass decreases as increases, while for (Fig. 4), the maximum mass increases. As the equation of state of inner region becomes more stiff, i.e. as decreases, the maximum mass gets large. The other properties about maximum mass for and are same as those of . Note that for the case of , all stellar models are unstable if discontinuity gets large.
iii.2 Quasi-normal modes for a stellar model without discontinuity
In order to find a specific property in quasi-normal modes of a neutron star with discontinuity, first we calculate of in the case of a stellar model without discontinuity as reference. In this case, we have to set two parameters; the polytropic index and polytropic coefficient , if we fix the total mass . We adopt the following values; , , , for . We also the case of with to examine the mass dependence. The properties of the neutron star models are summarized in Table 4.
The quasi-normal modes of the gravitational wave emitted from such stars are plotted in Figure 5 and listed in Table 5. The plot point corresponds to the so-called f-mode of each stellar model. In this table we show the mode frequency by two normalizations: one is and the other is . Although the f-mode is determined by the radius as well as the mass, we may not know the radius from observation. Then we can plot the data by . In the figure, however, we only show . In a neutron star, we usually find many p-modes as well as f-mode, but we only show the lowest f-mode here because that we are interested in new mode which appears in low frequency for a star with discontinuity.
From Fig. 5, we find that the real part of eigenfrequency gets large as equation of state is softer. For example, the frequency in the case ’d4’ is by larger than that in the case ’d1’. However, this tendency becomes reverse if we use another normalization () because the radius of a neutron star constructed by soft matter is larger than one constructed by stiff matter (see Table 4). We also find that the imaginary part of eigenfrequency increases as the stellar mass increases, but the real part is insensitive to the mass.
iii.3 Quasi-normal modes for a stellar model with discontinuity
In the case with discontinuity, we have three unknown parameters: , and . First we show the eigenfrequency of quasi-normal modes of for the case of . We fix the mass of a neutron star as or . Now two free parameters are and .
For f-mode, we find that the mode frequency for less massive star (0.5 ) is more sensitive to discontinuity than that for massive star (1.2 ). For example, the difference of the model ’b5’ from the model without discontinuity is larger than 28% for . It is also found that the imaginary part of eigenfrequencies in the model with discontinuity is larger than one without discontinuity, and this difference becomes larger as increases.
The results of g-mode for and are given in Fig. 7 and in Table 8 and Table 9. For g-mode, we also find the similar properties to the f-mode. The mode frequency in a less massive star is more sensitive than that in a massive star, in particular the imaginary part changes by the factor for the same depending on the amount of discontinuity (see Fig.7). Furthermore, the existence of g-mode is due to discontinuity, and it turns out that its imaginary part of g-mode is not always so small if the discontinuity is very large. We conclude that the damping rate of g-mode may not be ignored in a particular situation.
Next we calculate the eigenfrequencies of f-mode and g-mode for . The results of f-modes and g-mode are shown in Fig. 8(a) with the case of the model without discontinuity, and in Fig. 8(b), respectively. The numerical data are given in Table 10 and Table 11, respectively. By comparing the upper panel in Fig. 6(a), we find that the effect of change of equation of state is quite small. On the other hand, the imaginary part of g-mode becomes by 10 times larger than the case of due to the existence of the inner stiff matter. Furthermore, in this case, even if the density is continuous (), one may expect the appearance of g-mode because of discontinuity of the equation of state ( and ). However, we could not find g-mode in our calculation. This may be because the g-mode does not exist or is too small to be found.
iii.4 Comparison with the Cowling approximation
Finally, by using relativistic Cowling approximation , we shall calculate the f- and g-modes, and compare them to our results. In the Cowling approximation, metric perturbation is ignored and only stellar oscillation is taken into account. Hence, the eigenfrequency has only the real part. The result is given in Table 12 for (g/cm). The error (the difference from our numerical value) estimated by
is also shown. Here is the eigenfrequency found by the Cowling approximation, and and denote those of g- and f-modes, respectively. From Table 12, we find that the Cowling approximation is very accurate for g-mode if is at least in this region, although it is not so good for the f-mode.
In this paper we calculate the eigenfrequencies of f-mode and g-mode to examine the dependence of the discontinuity of the density if there exists. We investigate with the various discontinuity, by changing and , for the case that the total mass is fixed for , and for . In consequence we find that the dependence of the eigenfrequencies of f-mode on the discontinuity of the density is more strongly for the case that the total mass is smaller. The effect that the inner matter is stiff, and , is quite small for f-mode. And it is found that the imaginary part of the eigenfrequencies of g-mode become a little large for the case that inner matter is stiff. In general imaginary part of g-mode is quite small, but we find that in particular situation such that an inner polytropic index is the same as an outer one and total mass is rather small, the imaginary part of g-mode may not be ignored.
For the considering the circumstances mentioned above, it might be possible to judge whether the discontinuity of the density exist or not by the direct observation of the gravitational waves, which might be practicable in future observation, for even f-mode in the particular situation which the total mass is small. Of course, if the gravitational waves of g-mode is directly observed for such stars as the small massive ones, it is found that the discontinuity of the density exist. So we can judge the existence of the phase transition of the density in the neutron stars by the direct observation of the gravitational waves, though we need to collect the detailed data of the eigenfrequencies for some characteristic mode.
Finally we find also that Cowling approximation is quite accurate for g-mode.
Acknowledgements.We would like to thank M. Saijo for useful discussion. HS acknowledges T. Harada for valuable comments. This work was supported partially by JSPS Grants-in-Aid (No. 1205705) and by the Waseda University Grant for Special Research Projects.
Appendix A the Taylor expansion at the center of a neutron star
is the adiabatic index at , which is defined by the equation of state. The perturbation variables , , and are expanded at follows,
where and are defined by
and are also expanded at as
where the coefficients are given by
Appendix B How to calculate the quasi normal modes
To calculate the quasi normal modes of a neutron star, we follow the method developed by Leins, Nollert and Soffel . First, we change the variables from Zerilli’s one () to the Regge-Wheeler’s one (), because the effective potential in the Regge-Wheeler equation is very simple. The relation between those two variables given by
To find the solution of the Regge-Wheeler with appropriate boundary condition at infinity, we expand the variable at a finite radius as
By substituting this into the Regge-Wheeler equation, we obtain a four-term recurrence relation for () ;
The coefficients , , and are given by the value of and at as
The convergence condition () is now . If , we set , while if , we integrate the Regge-Wheeler equation from to to find the values of and . Furthermore, defining new coefficients , , and as