Torsional oscillations in tensor-vector-scalar theory
aa r X i v : . [ a s t r o - ph . H E ] J un Torsional oscillations in tensor-vector-scalar theory
Hajime Sotani ∗ Division of Theoretical Astronomy, National AstronomicalObservatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan (Dated: November 18, 2018)With the Cowling approximation, the torsional oscillations on relativistic stars in tensor-vector-scalar (TeVeS) theory are examined. The spectrum features in TeVeS are very similar to those ingeneral relativity (GR), but the torsional frequencies in TeVeS become larger than those expectedin GR. We find that, compared with the fluid oscillations with polar parity, the torsional frequenciesdepend strongly on the gravitational theory. Since the dependences of fundamental frequencies onthe gravitational theory and on the equation of state are different from those of overtone, it couldbe possible to distinguish TeVeS from GR in the strong-field regime via observations of this type ofoscillations with the help of the observation of stellar mass.
PACS numbers: 04.40.Dg, 04.50.Kd, 04.80.Cc
I. INTRODUCTION
It is well-known that the theory of general relativity (GR) is valid in a weak gravitational field such as our SolarSystem, as has been shown via many experiments. Looking at the case in the strong gravitational field, the situationis quite different from the case in the weak-field regime, i.e., the tests of gravitational theory in the strong-fieldregime are still very poor. However, with the development of technology, it is becoming possible to observe compactobjects with high accuracy, and with these observations it will be possible to test the gravitational theory in a stronggravitational field [1]. In practice, one could observe compact objects not only via X-rays and γ -rays but also viagravitational waves emitted from the objects. So far, some possibilities for distinguishing the gravitational theoryin the strong-field regime have been suggested. For instance, it has been proposed that one could distinguish thescalar-tensor theory [2] from GR by using the surface atomic line redshifts [3] or gravitational waves radiated fromthe neutron stars [4]. The possibility of a definitive test for GR with the direct observation of gravitational waves hasalso been pointed out [5].As an alternative gravitational theory, tensor-vector-scalar (TeVeS) theory has attracted considerable attention inrecent years, as proposed by Bekenstein [6] to extend the modified Newtonian dynamics [7, 8] into a relativistic theory,i.e., a covariant theory. As well as the modified Newtonian dynamics, TeVeS can explain the galaxy rotational curveand Tully-Fisher law without the presence of dark matter [6]. This theory is also successful in explaining stronggravitational lensing [9] and galaxy distributions through the evolving Universe without cold dark matter [10]. Forthe strong gravitational region of TeVeS, Giannios found the Schwarzschild solution [11], Sagi and Bekenstein foundthe Reissner-Nordstr¨om solution [12], and Lasky et al. produced the static, spherically symmetric stellar models inTeVeS by deriving the Tolman-Oppenheimer-Volkoff (TOV) equations in TeVeS [13]. Additionally, Lasky and Donovaexamined the stability and quasinormal oscillations of the Schwarzshild solution in TeVeS [14].Furthermore, some have suggested how to distinguish TeVeS from GR observationally. For instance, one can revealthe gravitational theory in a strong gravitational field with the redshift of the atomic spectral line radiating fromthe neutron star surface [13], with Shapiro delays of gravitational waves and photons or neutrinos [15], with thespectrum of gravitational waves emitted from compact objects [16], and with the rotational effect of neutron stars[17]. In fact, via observations of gravitational waves due to stellar oscillations, one can obtain information about thestellar parameters, such as mass, radius, rotational rate, magnetic fields, and equation of state (e.g., [18–22]), whichis called “gravitational wave asteroseismology.” In addition to this, the direct detection of gravitational waves couldpossibly be used to determine the radius of the accretion disk around a supermassive black hole [23] or to know themagnetic effect during a stellar collapse [24]. The test of gravitational theory in the strong-field regime is also one ofthe significant benefits of directly detecting gravitational waves.In this article, we focus on the torsional oscillations in neutron stars, which are incompressible oscillations. Usually,the frequency of this type of oscillations should degenerate to zero if the neutron stars consist of matter withoutelasticity. However, realistic stellar models have the solid crust region near the stellar surface, and in this region shear ∗ Electronic address: [email protected] torsional oscillations could exist. The boundary between the solid crust region and the fluid core is still uncertainbecause that depends strongly on the nuclear symmetric energy [25], but the density of this boundary is proposed tobe 2 . × g/cm [26] or 1 . × g/cm [27]. In this article, we adopt the density of 2 . × g/cm to see thedependence of torsional frequencies on the gravitational theory. From the observational point of view, the observedquasiperiodic oscillations in giant flares [28] are considered to come from this type of oscillations [29].In order to see the dependence of torsional frequencies on the gravitational theory, we adopt the relativistic Cowlingapproximation as a first step in this article, i.e., the perturbations except for the fluid one are neglected. It is knownthat this approximation is quite good at least in GR, because the torsional oscillations do not involve the densityvariation. A more detailed study including the perturbations of other fields will be done in the near future.This article is organized as follows. In the next section, we mention the stellar model in TeVeS, and in Sec. III wederive the perturbation equations for torsional oscillations with the Cowling approximation. In Sec. IV, the concretecalculations of the obtained perturbation equations will be performed to show the dependence on the gravitationaltheory. Finally, we conclude in Sec. V. In this article, we adopt the unit of c = G = 1, where c and G denote thespeed of light and the gravitational constant, respectively, and the metric signature is ( − , + , + , +). II. STELLAR MODELS IN TEVES
In this section, we only mention the fundamental parts of the theory. (See [6] for details of TeVeS.) TeVeS is basedon three dynamical gravitational fields: an Einstein metric g µν , a timelike four-vector field U µ , and a scalar field ϕ ,in addition to a nondynamical scalar field σ . The vector field fulfills the normalization condition with the Einsteinmetric as g µν U µ U ν = −
1, and the physical metric ˜ g µν is defined as˜ g µν = e − ϕ g µν − U µ U ν sinh(2 ϕ ) . (2.1)All quantities in the physical frame are denoted with a tilde, and any quantity without a tilde is in the Einsteinframe. The total action of TeVeS, S , contains contributions from the three dynamical fields mentioned above as wellas the matter contribution [6], which includes two positive dimensionless parameters, k and K , corresponding to thecoupling parameters for the scalar and vector fields. The field equations for the tensor, vector, and scalar fields canbe obtained by varying the total action with respect to g µν , U µ , and ϕ , respectively. (See [6] for the explicit forms.)Since a previous study for the neutron star structures in TeVeS has shown that the stellar properties are almostindependent from the scalar coupling k [13], in this article we focus only on the dependence of vector coupling K .The restriction on K has not been discussed in great detail in the literature, but the authors of[13] mentioned that K has to be less than 2 to construct the stellar models, and also that K should be less than 1 to produce a realisticstellar mass. According to this restriction on K , in this article we examine K as varying in the range of 0 ≤ K ≤ K = 0 and k = 0 in TeVeS.As a background stellar model, we consider nonrotating relativistic stars, which have been investigated in [13]. Astatic, spherically symmetric stellar model can be expressed with the following metric: ds = − e ν dt + e ζ dr + r (cid:0) dθ + sin θdφ (cid:1) , (2.2)where ν and ζ are functions of the radial coordinate r . Although the vector field on a static, spherically symmetricspacetime can be generally described as U µ = ( U t , U r , , U r = 0, which is the same assumption adopted in [13, 16, 17]. Then, withthe normalization condition, the vector field can be fully determined as U µ = ( e − ν/ , , , d ˜ s = − e ν +2 ϕ dt + e ζ − ϕ dr + r e − ϕ (cid:0) dθ + sin θdφ (cid:1) , (2.3)and the fluid four-velocity is ˜ u µ = e ϕ U µ . Regarding the stellar matter, we assume a perfect fluid described by theenergy-momentum tensor ˜ T µν = (˜ p + ˜ ρ ) ˜ u µ ˜ u ν + ˜ p ˜ g µν , (2.4)where ˜ p and ˜ ρ are the pressure and energy density in the physical frame, respectively. Furthermore, in order toconstruct the stellar model, one needs to prepare the relation between ˜ p and ˜ ρ , i.e., the equation of state (EOS). Inthis article, we adopt the same EOS’s as in [4], which are polytropic ones derived by fitting functions to tabulateddata of realistic EOS’s known as EOS A and EOS II. With these EOS’s, the maximum masses of a neutron star inGR are M = 1 . M ⊙ for EOS A and M = 1 . M ⊙ for EOS II. That is, EOS A and EOS II are considered as soft andintermediate EOS’s, respectively. At last, the stellar modes in TeVeS can be constructed by using the recipe shownin [13]. III. TORSIONAL OSCILLATIONS
As mentioned above, in this article, we focus on the torsional oscillations with the relativistic Cowling approximation,where we consider only the fluid perturbation with axial parity and the other perturbations of vector and tensor fieldsare neglected. It should be noted that the perturbation of scalar field exists only for polar perturbation. Consideringthe torsional oscillations, the Lagrangian displacement vector for the fluid perturbation can be expressed as˜ ξ i = (cid:18) , , Z ( t, r ) 1sin θ ∂ θ P ℓ (cid:19) , (3.1)where ∂ θ denotes the partial derivative with respect to θ , while P ℓ = P ℓ (cos θ ) is the Legendre polynomial of order ℓ .Then, the nonzero component of the perturbed fluid four-velocity in the physical frame can be written as δ ˜ u φ = e − ϕ − ν/ ∂ t Z θ ∂ θ P ℓ , (3.2)where ∂ t denotes the partial derivative with respect to t . The perturbed energy-momentum tensor including thecontribution from shear is given by δ ˜ T µν = (˜ p + ˜ ρ ) ( δ ˜ u µ ˜ u ν + ˜ u µ δ ˜ u ν ) − µδ ˜ S µν , (3.3)where ˜ µ is the shear modulus and ˜ S µν is the shear tensor defined via ˜ σ µν = L ˜ u ˜ S µν [30, 31]. Here ˜ σ µν is the rate ofshear tensor, which is defined as ˜ σ µν = 12 (cid:16) ˜ P αν ˜ ∇ α ˜ u µ + ˜ P αµ ˜ ∇ α ˜ u ν (cid:17) −
13 ˜ P µν ˜ ∇ α ˜ u α , (3.4)where ˜ P µν is the projection tensor ˜ P µν = ˜ g µν + ˜ u µ ˜ u ν . (3.5)The speed of shear waves can be expressed as ˜ v s = ˜ µ/ (˜ p + ˜ ρ ) [31], where a typical value of ˜ v s is around 10 cm/s in thecrust of neutron stars. In this article, we adopt the shear modulus inside the crust determined by this simple relation.Finally, one can get the equations describing the fluid perturbations by taking a variation of the energy-momentumconservation law, i.e., δ ( ˜ ∇ β ˜ T αβ ) = 0, which reduces to ˜ ∇ β δ ˜ T αβ = 0 with the Cowling approximation. The explicitform with α = φ becomes(˜ p + ˜ ρ ) e − ν − ϕ ¨ Z − ˜ µe − ζ Z ′′ − (cid:20) ˜ µ ′ + (cid:18) r − ϕ ′ + ν ′ − ζ ′ (cid:19) ˜ µ (cid:21) e − ζ Z ′ + ( ℓ + 2)( ℓ − r ˜ µZ = 0 , (3.6)where ˙( ) and ( ′ ) denote the partial derivative with respect to t and r , respectively. Assuming that the perturbedvariable has a harmonic time dependence, such as Z ( t, r ) = Z ( r ) e iωt , Eq. (3.6) reduces to˜ µZ ′′ + (cid:20) ˜ µ ′ + (cid:18) r − ϕ ′ + ν ′ − ζ ′ (cid:19) ˜ µ (cid:21) Z ′ + (cid:20) (˜ p + ˜ ρ ) ω e − ν − ϕ − ( ℓ + 2)( ℓ − r ˜ µ (cid:21) e ζ Z = 0 . (3.7)Imposing appropriate boundary conditions on this equation, the problem to solve becomes the eigenvalue problem.In practice, the above equation will be integrated only in the crust region, i.e., the boundary conditions are imposedat the basis of crust ( r = R c ) and at the stellar surface ( r = R ), because ˜ µ = 0 in the fluid core as mentioned in theIntroduction. In this article, we impose a zero traction condition at r = R c and the zero-torque condition at r = R [31]. Both conditions correspond to Z ′ = 0. IV. OSCILLATION SPECTRA
In this section, we examine the torsional oscillations in the crust region of neutron stars both in GR and in TeVeS.With respect to the torsional modes in GR, it is known that one can see the dependence of ℓ in the fundamentalmodes, while the frequencies of overtone are almost independent of ℓ , as shown in the left panel of Fig. 1, wherethe specific frequencies in GR are plotted as a function of the stellar mass. On the other hand, we have done thenumerical calculations to determine the frequencies in TeVeS with different value of K . As an example, the result M ADM / M ! f ( H z ) (2,0)(3,0)(4,0)(5,0)1st GR M ADM / M ! f ( H z ) K = 0.5 FIG. 1: Frequencies of fundamental modes and first overtone are plotted as functions of the ADM mass with EOS A in GR(left panel) and in TeVeS with K = 0 . ℓ = 2, 3, 4, and 5, i.e., ( ℓ, n ) = (2 , n is the number of nodes in the eigenfunctions, whilethe broken lines correspond to the first overtone with ℓ = 2, 3, 4, and 5. with K = 0 . ℓ = 2 oscillation modes. Figure 2 shows the fundamental frequencies (left panel) and the frequencies of first overtone(right panel), in both GR (solid lines) and TeVeS (broken lines) for the stellar models with EOS A. Similar to Fig. 1,the frequencies are shown as functions of the stellar mass. From this figure, one can easily observe that the frequenciesexpected in TeVeS are quite different from those in GR. In practice, depending on the value of K , the frequencies inTeVeS become 34% larger for the fundamental oscillations and 150% larger for the first overtone than those in GR.In Fig. 3, we draw a figure similar to Fig. 2 but EOS II. One can observe that the relative augmentation of thefrequencies in TeVeS compared with those in GR becomes 28% for the fundamental oscillations and 116% for thefirst overtone. That is, the stellar models with softer EOS seem to be more sensitive about the gravitational theorythan those with stiffer EOS. Additionally, it should be emphasized that, compared with the fluid oscillations of polarparity accompanied by variation of density, the frequencies of torsional modes depend strongly on the gravitationaltheory, where the frequencies of polar parity in TeVeS become around 20% larger than those expected in GR [16].And, as mentioned in [16], since this deviation between the frequencies in GR and in TeVeS results from the existenceof a scalar field, observing the torsional oscillations as well as the fluid oscillations of polar parity could tell us theexistence of the scalar field. M ADM / M ! f ( H z ) GR K = 0.2 K = 0.5 K = 1.0 M ADM / M ! f ( H z ) GR0.20.51.0
FIG. 2: Frequencies of fundamental modes and first overtone with ℓ = 2 are plotted as functions of the ADM mass with EOSA. Left and right panels correspond to the fundamental and first overtone frequencies, respectively. In both figures, the solidand broken lines correspond to the frequencies in GR and TeVeS, respectively. Furthermore, in Fig. 4, we plot the fundamental frequencies (left panel) and the first overtone (right panel) oftorsional modes as functions of parameter K , where the ADM masses are fixed to be 1 . M ⊙ . Here, the frequencieswith ℓ = 2, 3, 4, and 5 are shown in the left panel, while those with only ℓ = 2 are shown in the right panel, becausethe frequencies of overtone are almost independent of ℓ , as we saw in Fig. 1. Additionally, in both panels, the resultsfor EOS A and EOS II are plotted using the solid lines with circle marks and the dotted lines with square marks,respectively, and the results in GR are shown at K = 0. From these figures, one can observe that the qualitative M ADM / M ! f ( H z ) GR K = 0.2 K = 0.5 K = 1.0 M ADM / M ! f ( H z ) GR0.20.51.0
FIG. 3: Similar to Fig. 2, but with EOS II. dependencies of frequency on the value of K are independent of the adopted EOS and the value of ℓ . That is, thefrequencies of torsional modes are increasing as the value of K becomes large. From the left panel of Fig. 4, sincethe dependence of K on the fundamental frequencies is almost comparable to that of EOS, it might be difficult todistinguish the gravitational theory by using only the observation of fundamental modes. On the other hand, theright panel of Fig. 4 shows that the frequencies of overtone depend strongly on EOS rather than on the value of K .So, with observation of the frequencies of overtone, it could be possible to make a constraint in EOS independent ofthe gravitational theory. Then, after making a constraint in EOS with the help of observation of the stellar mass, itmight be possible to restrict on the value of K using the observations of fundamental oscillations, i.e., it might bepossible to probe the gravitational theory in the strong-field regime observationally. K f ( H z ) l = 2 l = 3 l = 4 l = 5 K f ( H z ) EOS IIEOS A
FIG. 4: For stellar models with M ADM = 1 . M ⊙ , the frequencies of fundamental and first overtone torsional modes are shownas functions of parameter K with EOS A (solid lines with circles) and EOS II (dotted lines with squares). The left panelcorresponds to the fundamental modes with ℓ = 2, 3, 4, and 5, while the right panel corresponds to the first overtone with ℓ = 2. V. CONCLUSION
In this article, focusing on the torsional oscillations inside the crust region, we have derived the perturbationequation of neutron stars in TeVeS and calculated their eigenfrequencies. We find that the frequencies of torsionalmodes in TeVeS become larger than those expected in GR, whose dependence on the gravitational theory is strongerthan that of fluid oscillations involving the variation of density. It is also found that the stellar models with softerEOS are more sensitive about the gravitational theory than those with stiffer EOS. Since the frequencies of overtonedepend strongly on the adopted EOS rather than on the gravitational theory, it is possible, via the observation of thetorsional frequencies of overtone, to restrict on the EOS independent of the gravitational theory. Then, with the helpof the observation of stellar mass, it might also be possible to make a constraint in the gravitational theory via theobservation of fundamental torsional modes. It should be noticed that, since these imprints of TeVeS come from thepresence of a scalar field, it could be also possible, via observations of such oscillations of neutron star, to probe theexistence of a scalar field.As a first step, we assume the Cowling approximation in this article, i.e., our examinations are restricted to onlyfluid oscillations, so we should make a more detailed study including the metric and vector field perturbations. Viathese oscillations, one could obtain additional information, and combining that with the results reported in this articlewould provide more accurate constraints on the gravitational theory in the strong-field regime. Additionally, since itis suggested that the nuclear structure in the bottom of a crust could be nonuniform, the so-called “pasta structure,”we might consider the effect of the pasta phase on the torsional oscillations [32]. Furthermore, we should take intoaccount the magnetic effects on the frequencies, because the torsional oscillation could be detected in the magnetarsas the quasiperiodic oscillations during the giant flares [33]. Considering these additional effects, one must obtainmore accurate constraints in a more realistic situation.
Acknowledgments
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