Testing universal relations of neutron stars with a nonlinear matter-gravity coupling theory
Due to our ignorance of the equation of state (EOS) beyond nuclear density, there is still no unique theoretical model for neutron stars (NSs). It is therefore surprising that universal EOS-independent relations connecting different physical quantities of NSs can exist. Lau et al. found that the frequency of the -mode oscillation, the mass, and the moment of inertia are connected by universal relations. More recently, Yagi and Yunes discovered the I-Love-Q universal relations among the mass, the moment of inertia, the Love number, and the quadrupole moment. In this paper, we study these universal relations in the Eddington-inspired Born-Infeld (EiBI) gravity. This theory differs from general relativity (GR) significantly only at high densities due to the nonlinear coupling between matter and gravity. It thus provides us an ideal case to test how robust the universal relations of NSs are with respect to the change of the gravity theory. Thanks to the apparent EOS formulation of EiBI gravity developed recently by Delsate and Steinhoff, we are able to study the universal relations in EiBI gravity using the same techniques as those in GR. We find that the universal relations in EiBI gravity are essentially the same as those in GR. Our work shows that, within the currently viable coupling constant, there exists at least one modified gravity theory that is indistinguishable from GR in view of the unexpected universal relations.
Subject headings:dense matter - equation of state - stars: neutron
Neutron stars (NSs) are compact objects that have densities several times the normal nuclear density in their cores. Due to their internal high-density environment, NSs have long been regarded as the most natural cosmic laboratories for studying dense nuclear matter, which is not well understood and poorly constrained by experiments performed on Earth. It is hoped that inferring global properties of NSs from observations may allow one to select the correct equation of state (EOS) among many theoretical possibilities. However, achieving this goal in practice is non-trivial. The reason is that it is in general not possible to extract all global properties of a NS simultaneously from observations. It is thus of great interest to search for empirical relationships, which are in general EOS dependent (e.g., the mass-radius relation of NSs), connecting only a few physical quantities of a NS so that one might hope to put constraints on the EOS by measuring these quantities. On the other hand, from a fundamental physics point of view, it would even be more interesting if the relationships are insensitive to the NS EOS models, since one could then use the relations to test the underlying gravitational theory despite our ignorance of dense nuclear matter (Yagi & Yunes, 2013a, b).
In the past decade, several empirical relations connecting different physical parameters of NSs have been proposed. Bejger & Haensel (2002) and Lattimer & Schutz (2005) discovered relationships relating the scaled moment of inertia and compactness , where is the moment of inertia, is the mass, and is the radius. Making use of the discovered relation, Lattimer & Schutz (2005) suggested that the moment of inertia of star A in the double pulsar system J0737-3039 (Burgay et al., 2003; Lyne et al., 2004) could be determined to about 10% accuracy.
On the other hand, pulsating NSs are expected to be promising sources of gravitational waves. It is expected that studying the gravitational-wave signals emitted by oscillating NSs can yield useful information about the internal structure of the stars. Several universal behaviors of the quadrupolar -mode have been established (Andersson & Kokkotas, 1996, 1998; Benhar et al., 1999, 2004; Tsui & Leung, 2005; Lau et al., 2010). In particular, Lau et al. (2010) found a pair of nearly EOS-independent relations to connect the frequency and damping rate of the -mode to the global properties and of the stars (see Section 4). It has been furthermore shown (Lau et al., 2010) that the values of , , and of a NS can be inferred accurately from the -mode gravitational wave signals.
More recently, Urbanec et al. (2013) discovered a universal relation between and , where is the spin-induced quadrupole moment and is the angular momentum. Yagi & Yunes (2013a, b) discovered universal relations relating , , the tidal Love number , and the rotational Love number . These newly discovered I-Love-Q relations will be directly relevant to the understanding of the gravitational wave signals emitted during the late stages of NS-NS binary mergers (see Section 5). Finally, Bauböck et al. (2013) found universal relations among , , , , and the ellipticity of the stellar surface.
While Yagi & Yunes (2013a, b) have also studied the I-Love relation in an alternative theory of gravity (see Section 5), most of the universal relations of NSs discussed above are based on the assumption that gravity is described by the theory of general relativity (GR). But how well do we understand gravity? So far the most successful theory of gravity is GR, and it has been well tested in weak-field situations (for a review see Will, 2006). However, whether gravity behaves as GR predicts in strong-field situations, such as NS-NS binary mergers, is still an open question. It is hopeful that testing GR in the strong-field limit will soon become possible in the coming decade through gravitational wave observations by ground based detectors such as Advanced LIGO, Advanced VIRGO, and KAGRA (Will, 1993, 2006; Gair et al., 2013; Yagi, 2013; Yunes & Siemens, 2013; Mirshekari, 2013). On the other hand, we also know that GR is not complete because of its prediction of singularities in the Big Bang and those inside black holes. While it is generally believed that quantum gravity is needed to resolve these problems, it is still interesting to search for alternative theories of gravity that could avoid the singularity problems within the classical level.
In recent years, a new theory of gravity called Eddington-inspired Born-Infeld (EiBI) gravity proposed by Bañados & Ferreira (2010) has been gaining attention (see also Deser & Gibbons, 1998; Vollick, 2004). EiBI gravity is appealing because it reduces to GR in vacuum and can avoid the Big Bang singularity (Bañados & Ferreira, 2010). The deviation between EiBI gravity and GR becomes significant only at high densities. The implications of EiBI gravity in cosmological (Bañados & Ferreira, 2010; Scargill et al., 2012; Avelino & Ferreira, 2012; Escamilla-Rivera et al., 2012; Liu et al., 2012; Cho et al., 2012, 2013; Harko et al., 2013a; Bouhmadi-Lopez et al., 2013) and astrophysical (Pani et al., 2011, 2012; Pani & Sotiriou, 2012; Sham et al., 2012, 2013; Harko et al., 2013b) contexts have been widely investigated. Unlike GR, where gravity couples to matter linearly in the sense that the Einstein tensor is proportional to the stress-energy tensor in the Einstein field equations, EiBI gravity introduces nonlinear coupling between matter and gravity (see Section 2). It is in fact the nonlinear matter-gravity coupling in this theory that is responsible for avoiding some of the singularities that plague GR (Delsate & Steinhoff, 2012). However, it has also been shown recently that the same nonlinear coupling leads to some pathologies, such as surface singularities (Pani & Sotiriou, 2012) and anomalies associated with phase transitions (Sham et al., 2013), for compact stars in EiBI gravity.
While the works of Pani & Sotiriou (2012) and Sham et al. (2013) cast doubt on its viability, EiBI gravity certainly stands as an interesting example of a more general class of nonlinear matter-gravity coupling theories, due to its equivalence to GR in vacuum and its ability to avoid singularities. As discussed above, the theory differs from GR significantly only at high densities. Since NSs are the most dense stellar objects in the universe, EiBI gravity thus provides us with an ideal test case to investigate how the universal relations of NSs would change if the underlying gravitational theory is such that matter and gravity are nonlinearly coupled together. In this paper, we study the -mode universality relations and the I-Love-Q relations in EiBI gravity. Thanks to the apparent-EOS formulation of EiBI gravity developed by Delsate & Steinhoff (2012), we are able to study the universal relations in EiBI gravity using the same techniques as those in GR because in this formulation EiBI gravity coupled to a perfect fluid is equivalent to GR with the matter field described by an effective EOS (see Section 3). We find that these universal relations are indeed very robust and independent of whether gravity is described by GR or EiBI theory, as long as the coupling constant in EiBI gravity is within the range that is already constrained astrophysically. As we shall discuss, the results are not too surprising since the apparent EOS, within the range constrained astrophysically, does not differ significantly from nuclear EOS models.
The plan of the paper is as follows. We first briefly summarize EiBI gravity in Section 2. Section 3 discusses the apparent EOS formulation of EiBI gravity. In Sections 4 and 5, we review the universal behaviors of -mode oscillations and the I-Love-Q relations for NSs discovered in GR, respectively. We then present our numerical results to show that these relations also work well in EiBI gravity. Finally, our conclusions are summarized in Section 6. Unless otherwise noted, we use geometric units where .
2. Eddington-Inspired Born-Infeld Gravity
The EiBI theory is based on a Palatini formulation of the action (Bañados & Ferreira, 2010)
where is the symmetric part of the Ricci tensor constructed solely by the connection , is the matter action, and denotes the determinant of a tensor field . The parameters and are related to the cosmological constant by . In the limit , it can be shown that the action (Equation (2)) reduces to the Einstein-Hilbert action for GR. We shall set hereafter and consider as the only parameter of the theory. The current tightest constraint on (Avelino, 2012) is set by the existence of NSs and is given by , where .
Varying the action (Equation (2)) with respect to the metric and separately yields
where is an auxiliary metric compatible with the connection:
Since the matter action is assumed to depend only on the metric and the matter fields, but not on the connection , the stress-energy tensor still satisfies the same conservation equations as in GR
where the covariant derivative refers to the physical metric . It should be noted that is identical to the physical metric when . In fact, it can be shown that the action (Equation (2)) reduces to the Einstein-Hilbert action when the matter action vanishes (Bañados & Ferreira, 2010). Hence, EiBI gravity is identical to GR in vacuum.
3. Apparent EOS Formulation of EiBI gravity
Solving the field equations (Equations (4) and (5)) can be more difficult than in GR. The reason is that the auxiliary metric involves both and and thus the Ricci tensor contains second derivatives of the matter field444Indeed, the dependence of on derivatives of the matter field is the cause of the pathologies associated with compact stars in EiBI gravity found by Pani & Sotiriou (2012) and Sham et al. (2013). . However, Delsate & Steinhoff (2012) have recently shown that the field equations (Equations (4) and (5)) can be written in a form that resembles the Einstein equations in GR:
where and . The Einstein tensor is defined by the auxiliary metric and is defined as the apparent stress-energy tensor. The scalar is defined by and can be expressed as
Note that Equation (8) in general still depends on the physical metric through the standard stress-energy tensor . However, for the case of a perfect fluid
where is the energy density, is the pressure, and is the four velocity of the fluid, Equation (8) can be made to decouple from completely. In this special case, can be computed in a frame comoving with the fluid and is given by
The apparent stress-energy tensor can be written as
with the apparent energy density and pressure defined by
where . The apparent four velocity satisfies the conditions
Note that the indices of and are lowered by and , respectively.
The advantage of reformulating EiBI gravity in the form of Equation (8) has now become clear. For a given coupling parameter and a given physical EOS , one can solve a problem in EiBI gravity by solving the same Einstein equations (Equation 8)) as in GR, but with as the fundamental metric and an apparent EOS given by Equations (13) and (14). This implies that many theoretical ideas and numerical codes developed in GR can readily be transferred to EiBI gravity. For instance, instead of solving the field equations (Equations (4) and (5)) for the structure of compact stars, as was originally done (Pani et al., 2011, 2012; Sham et al., 2012, 2013; Harko et al., 2013b), one can simply solve the well-known Tolman-Oppenheimer-Volkov (TOV) equations in GR with an apparent EOS.
The advantage of this reformulation of EiBI gravity becomes even more transparent when one considers dynamical problems in EiBI gravity, where the algebraic complexity in the analysis grows quite rapidly with the problem size. For example, it is non-trivial to calculate non-radial oscillation modes of NSs even in GR. Formulating the same calculation in EiBI gravity starting with the field equations (Equations (4) and (5)) would be a more tedious task than in GR. However, thanks to the apparent EOS approach, we are able to make use of the numerical codes we previously developed in GR (Lau et al., 2010) to study the -mode of NSs in EiBI gravity. As a side remark, we have indeed verified that the frequencies of the radial oscillation modes of NSs in EiBI gravity we obtained previously (Sham et al., 2012), by solving Equations (4) and (5), agree with the results (within numerical accuracy) obtained by solving the corresponding eigenvalue equation in GR (see, e.g., Kokkotas & Ruoff, 2001) with apparent EOSs.
4. -mode universality
The oscillation modes of NSs are damped by the emission of gravitational waves. They are in general called quasi-normal modes and each mode has a complex eigenfrequency . The imaginary part corresponds to the damping rate of the oscillation mode. As discussed in Section 1, various attempts have been made to find relationships relating and of the -mode to the global parameters of the star. Most of these works (Andersson & Kokkotas, 1996, 1998; Benhar et al., 1999, 2004; Tsui & Leung, 2005) used and as the parameters in the analysis. However, Lau et al. (2010) have recently established two EOS-independent relations using the parameters and . The physical motivation for replacing by in the study of Lau et al. (2010) is that is sensitive to the low-density part of the EOS, while measures the mass distribution of the star globally. As the dynamics of -mode oscillations are signigicantly affected by the mass distribution, it is thus expected that should relate more directly to the -mode.
where the dimensionless factor . Equations (16) and (17) are much improved universal relations in the sense that these relations are less sensitive to the EOS, comparing with previous universal relations that use as a parameter. We refer the reader to Table 1 in Lau et al. (2010) for the accuracy of Equations (16) and (17).
Our aim in this section is to study whether the universal relations (Equations (16) and (17)) found for -mode oscillations in GR remain valid in EiBI gravity. To this end, we calculate the -mode frequency for NSs in EiBI gravity using the apparent-EOS approach as discussed in Section 3. The procedure is to (1) construct an apparent EOS for a given coupling parameter and physical EOS ; (2) construct an equilibrium background stellar model using the TOV equations in GR with the apparent EOS; and (3) calculate the -mode frequency for the perturbed background model using the numerical codes that we previously developed in GR (Lau et al., 2010), but with as the fundamental metric. The method for calculating oscillation modes of NSs in GR is well established and documented (for a review see Kokkotas & Schmidt, 1999). We refer the reader in particular to Lindblom & Detweiler (1983) and Detweiler & Lindblom (1985) for the formulation we used in the calculations.
In Lau et al. (2010), nine ordinary nuclear-matter and two quark-matter EOS models were considered in establishing the universal relations (Equations (16) and (17)). In this work, we consider four nuclear-matter EOS models: model APR (Akmal et al., 1998), model FPS (Lorenz et al., 1993), model SLy4 (Douchin & Haensel, 2000), and model WS (Lorenz et al., 1993; Wiringa et al., 1988). Among these four models, the model APR was also used in Lau et al. (2010). For the coupling parameter in EiBI gravity, we consider three different values defined by , where . These values are consistent with the constraint set by the existence of NSs (Avelino, 2012). In particular, corresponds to the GR limit. In order to show that the range of we consider is already large enough that the resulting NSs in EiBI gravity differ significantly from those in GR, we plot the gravitational mass as a function of the central density for the APR EOS in Figure 1. It can be seen from the figure that can change by as much as 30% when increases from to 0.1.
Figure 2 plots the real part of the scaled -mode frequency against for our chosen EOS models with different values of . It can be seen clearly that the data display universal relations that are essentially independent of the EOS models and the value of , as long as is within the range constrained by the existence of NSs as discussed above. We see that the data can be fit well by Equation (16). The relative fractional difference between the numerical results and Equation (16) is shown in the lower panel of the figure. Similarly, we plot against in Figure 3 and see that the data can be modeled well by Equation (17). We refer the reader to Lau et al. (2010) for the motivation of this way of plotting . In summary, we find that the universal relations (Equations (16) and (17)) for the -mode of NSs in GR still hold for EiBI gravity as long as the coupling parameter is in the range .
5. I-Love-Q relations
After discussing the universality of -mode oscillations, we now turn to the universal I-Love-Q relations discovered more recently by Yagi & Yunes (2013a, b). The moment of inertia of a star is defined by , where and are the angular momentum and angular velocity of the star, respectively. Physically, determines how fast a star can spin for a given angular momentum. It thus seems quite natural that should somehow relate to the spin-induced quadrupole moment of the star, since characterizes the deformation of the star due to self rotation. However, it is surprising that the relation between and found by Yagi & Yunes (2013a, b) is EOS independent. On the other hand, the tidal Love number measures the deformation of a NS due to the presence of a companion and is defined by , where is the traceless quadrupole moment tensor of the star and is the tidal tensor that induces the deformation (see, e.g., Flanagan & Hinderer, 2008). In general, there is no reason why there should exist EOS-independent universal relations relating the three quantities , , and . More specifically, the universal relations concern the dimensionless quantities , (with being the dimensionless spin parameter), and .
The relevance of the I-Love-Q relations to astrophysics, gravitational-wave, and fundamental physics has been discussed (Yagi & Yunes, 2013a, b). For instance, it was proposed that, for a detected gravitational wave signal emitted by an inspiralling NS binary, the relations could break the degeneracy between the NS quadrupole moment and the NS’s individual spins. It has also been shown by Maselli et al. (2013) that while the I-Love relation connecting and depends on the inspiral frequency during the inspiral, it nevertheless remains EOS independent. More recently, Haskell et al. (2013) showed that the I-Q universal relation fails for magnetized NSs with long spin periods ( s) and strong magnetic fields ( G) 666After we submitted the paper, there were three preprints discussing the I-Q universal relation in rapidly rotating stars (Doneva et al., 2013; Pappas & Apostolatos, 2013; Chakrabarti et al., 2013). .
As discussed in Section 1, our aim is to study whether the universal I-Love-Q relations remain valid in EiBI gravity. The methodologies to calculate , , and in GR are well established (Hartle, 1967; Hartle & Thorne, 1968; Flanagan & Hinderer, 2008; Hinderer, 2008; Damour & Nagar, 2009; Binnington & Poisson, 2009; Yagi & Yunes, 2013a; Urbanec et al., 2013). Here we determine these quantities in EiBI gravity by solving the relevant equations in GR with apparent EOSs.
In Figures 4 and 5, we plot and against , respectively, for our chosen EOS models and values of . As in Section 4, we consider three different values . Similarly, we plot against in Figure 6. It is seen clearly from the figures that the I-Love-Q relations are essentially independent of the chosen EOS models as discovered by Yagi & Yunes (2013a, b). Furthermore, as in the case of the -mode universal relations, we find that the I-Love-Q relations are insensitive to the coupling parameter as long as it is in the range . We also note that our numerical results can be fit well by the following relation (the solid line in each figure) as suggested by Yagi & Yunes (2013a, b):
where , , , , and are some fitting coefficients. The relative fractional difference between our numerical results and Equation (18) is shown in the lower panel of each figure.
Yagi & Yunes (2013a, b) also studied the universal relations in a modified theory called dynamical Chern-Simons (CS) gravity. They found that there also exists a universal I-Love relation connecting and , although the relation is different from the corresponding I-Love relation in GR. They suggested that if one can measure of the double-binary pulsar J0737-3039 to 10% accuracy and to 60% accuracy with future gravitational wave observations, the CS theory can then be constrained much better than current tests by six orders of magnitude. For comparison, the universal I-Love-Q relations in EiBI theory are the same as the GR ones for the range of the coupling parameter , which has already been constrained astrophysically. Hence, contrary to CS theory, EiBI gravity is an example of a modified theory where the I-Love-Q relations are degenerate with the corresponding GR relations and cannot be used to put a stronger constraint on the theory than that obtained by the current astrophysical one.
In this paper, we have studied the EOS-independent universal -mode (Lau et al., 2010) and I-Love-Q (Yagi & Yunes, 2013a, b) relations for NSs in EiBI gravity. With the coupling parameter of the theory in the range , which is constrained by the existence of NSs (Avelino, 2012), we find that the universal relations discovered in GR remain valid in EiBI gravity.
Naively, since EiBI gravity reduces to GR when the coupling constant vanishes, one might worry that the agreement between the universal relations in EiBI gravity and those in GR is simply because the values of we consider are so small that the effect of nonlinear matter-gravity coupling in EiBI gravity is not apparent. But this is not the case as we have seen in Figure 1. The effect of nonlinear matter-gravity coupling is not vanishingly small in our study. On the other hand, since EiBI gravity can be recast to GR with an apparent EOS, the nonlinear matter-gravity coupling in EiBI gravity is thus mimicked by the stiffness of the apparent EOS in GR. As long as the stiffness of the resulting apparent EOS is not significantly different from realistic nuclear EOS models, which is the case in our study, it is then not surprising that the universal relations in the two theories are identical. For comparison, there also exists a nearly EOS-independent I-Love relation in dynamical CS gravity (Yagi & Yunes, 2013a, b), although the relation is different from the GR one.
While we have only focused on the range for the coupling parameter in this paper, we have in fact also considered a much larger range of and found that the universal relations are still valid. For instance, for , the relative fractional differences between the numerical results and the fitted universal curves still remain about 0.5%. Nevertheless, it should be pointed out that such a large value of would lead to stellar configurations that are very different from typical NSs observed in the universe (e.g., their masses can be much larger than ). On the other hand, for a negative value of , we found that stellar configurations cannot be constructed if is too negative since cannot decrease continuously to zero as one integrates the stellar equations and hence the boundary conditions at the stellar surface cannot be fulfilled (see also Sham et al., 2013). The exact value of at which this would happen depends on the EOS, but it is typically for our chosen EOS models.
Finally, let us conclude with two remarks: (1) as discussed in Section 1, EiBI gravity is appealing because it reduces to GR in vacuum and can avoid some of the singularities that plague GR. If it was not due to the fact that EiBI gravity suffers from some pathologies associated with compact stars (Pani & Sotiriou, 2012; Sham et al., 2013), the theory would have one additional merit toward being a serious contender to GR because it has essentially the same non-trivial universal -mode and I-Love-Q relations in the high-density regime as those in GR. It leaves one to wonder whether there exists any variation of EiBI gravity or more generically a version of nonlinear matter-gravity coupling theory that could capture all the nice features of EiBI gravity without suffering from any pathologies. (2) It should be noted that the reason why there exist universal -mode and I-Love-Q relations is not yet understood even in GR. Yagi & Yunes (2013a, b) suggested two possible reasons for the I-Love-Q relations: (1) the physical quantities considered depend most strongly on the NS outer layer, where the ignorance of the EOS is smaller than that in the NS core and (2) the relations approach the black hole limit as the NS compactness increases, and hence the relations do not depend on the internal stellar structure. It is worth investigating whether these two suggestions are indeed the correct explanation for the I-Love-Q relations. It will also be interesting to study whether the two apparently different sets of -mode and I-Love-Q universal relations have a common origin or not. We hope to return to these issues in the future.
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