Signature of Lorentz Violation in Continuous Gravitational-Wave Spectra of Ellipsoidal Neutron Stars

We study effects of Lorentz-invariance violation on the rotation of neutron stars (NSs) in the minimal gravitational Standard-Model Extension framework, and calculate the quadrupole radiation generated by them. Aiming at testing Lorentz invariance with observations of continuous gravitational waves (GWs) from rotating NSs in the future, we compare the GW spectra of a rotating ellipsoidal NS under Lorentz-violating gravity with those of a Lorentz-invariant one. The former are found to possess frequency components higher than the second harmonic, which does not happen for the latter, indicating those higher frequency components to be potential signatures of Lorentz violation in continuous GW spectra of rotating NSs.


I. INTRODUCTION
The observation of gravitational waves (GWs) from the compact binary system GW170817 initiates the era of multimessenger astronomy [1,2]. Gravitational theories, including the renowned general relativity (GR), are exposed to unprecedented tests utilizing GW signals [3]. Lorentz invariance, incorporated locally in GR as well as many other alternative gravitational theories, is certainly one of the fundamental principles subjected to these tests [4][5][6][7][8]. By employing the Standard-Model Extension (SME) framework [9][10][11][12][13], which is widely used to investigate consequences from possible violations of Lorentz invariance in terrestrial experiments and astrophysical observations [14], stringent bounds have been set for the coefficients for Lorentz violation in the gravitational sector of the SME framework after analyzing the observed GW data [2,[5][6][7].
Besides the coalescence of compact binary systems, another type of GW sources are deformed rotating neutron stars (NSs). Especially, when the angular velocity of a deformed NS is misaligned with its angular momentum, the star precesses about the direction of the angular momentum, radiating out GWs continuously [15,16]. The search for such continuous GW signals is ongoing [17][18][19][20][21][22]. Once detected, the continuous GW signals will tell us a substantial piece of information on NS structure and deformability. Furthermore, they will bring new tests for the laws of physics, among which lies Lorentz invariance as one of the fundamental principles (see e.g. Ref. [8]).
To test Lorentz invariance, an investigation of the scenario where it is violated is necessary. The effects of Lorentz violation on rotating spheroidal stars are studied in detail in Ref. [23] under the minimal gravitational SME framework. The modification to the free precession of a deformed star is depicted by the name twofold precession, as briefly speaking, Lorentz violation causes the angular momentum to precess about a fixed direction while at the same time the star still precesses about the instantaneous direction of the angular momentum. The correction in the quadrupole radiation due to the modification of the rotation of the star is calculated in Ref. [23], and it is found that the quadrupole radiation from a spheroidal star affected by Lorentz violation has frequency components higher than twice of the fundamental one.
In this work, we are going to extend the numerical results in Ref. [23] to ellipsoidal NSs. The characteristic higher harmonics due to Lorentz violation remain in the GW spectra as we expect. But more importantly, our numerical calculation for the quadrupole radiation from an ellipsoidal NS in the absence of Lorentz violation indicates that though the nonaxisymmetry of the star modulates the first and the second harmonics in the GW spectra as discussed in Refs. [16,24,25], it does not generate harmonics higher than the second for freely precessing NSs. Therefore, harmonics higher than the second are indeed possible signatures for Lorentz violation in the GW spectra of rotating solitary NSs.
We organize the paper as follows. In Sec. II, we present the analytical equations to construct the quadrupole radiation from a rotating ellipsoid under Lorentz-violating gravity. Then in Sec. III, numerical solutions to the rotation equations for ellipsoids with uniform density are obtained and used to construct examples of the quadrupole radiation.
FIG. 1. Euler angles transforming the X-Y -Z inertial frame to the x-y-z body frame. First, rotate the X-Y -Z frame about the Z axis with angle α so that the X-axis aligns with the intersection line M N . Then, rotate the just obtained X-Y -Z frame about the line M N with angle β so that the Z-axis aligns with the z-axis. Last, rotate the new X-Y -Z frame about the z-axis with angle γ so that it overlaps with the x-y-z frame.
Subsequently, Fourier transformations are performed to extract the frequency components of the quadrupole GWs, and we will see that while the GW from an ellipsoid under the twofold precession contains harmonics higher than the second, the GW from an ellipsoid under free precession only has frequencies around the first and the second harmonics. In the end, conclusions are summarized in Sec. IV. For simplicity in writing equations, we use the geometrized unit system where G = c = 1. However, standard units do appear when numerical estimations are desired for realistic NSs.

II. THEORETICAL BASICS
To proceed with the calculation, we neglect relativistic corrections to NS structure and motion, and solve its motion from the rotation equations for rigid bodies. 1 Assuming that in the body frame x-y-z, the surface of the star is described by with semi-axes a x , a y and a z , then the Lagrangian for the rotation of the star can be written as where I xx , I yy and I zz are the eigenvalues of the moment of inertia tensor along the principal axes, and Ω x , Ω y and Ω z are the components of the angular velocity of the star in the x-y-z frame. The orientation-dependent self-energy δU is calculated from the anisotropic correction δΦ to the Newtonian potential Φ in the minimal gravitational SME, namely [13] wheres ij , with i, j = x, y, z, are the coefficients for Lorentz violation in the body frame [12,13], and ρ is the density of the star.
In the SME framework, the coefficients for Lorentz violation are assumed to be constant in inertial frames. Therefore, as the star rotates, the coefficientss ij depend on the orientation of the star according tō where R iI represents the rotation matrix transforming an inertial frame X-Y -Z to the body frame x-y-z. The capital indices run over X, Y and Z, ands IJ are constant coefficients for Lorentz violation. The orientation dependence of s ij , originated from the rotation matrix, can be easily described by the Euler angles (α, β, γ) in Fig. 1, as the rotation matrix in terms of the Euler angles is Together with the relations between the velocity components and the Euler angles [26], where dots denote time derivatives, the Euler-Lagrange equations for the Euler angles can be obtained from the Lagrangian (2). Given the shape and density of the star, the moment of inertia tensor and the integrals in δU can be calculated, and then the Euler angles can be solved to describe the rotation of the star.
Once the rotation of the star is known, its gravitational quadrupole radiation can be calculated via the metric perturbation where r is the distance from the distant star to the observer, and the double dots denote the second time derivative. In Ref. [23], it is shown thatÏ IJ can be written as with the body-frame quantities A ij being for any rigid body subjected to arbitrary rotations. The quantities ∆ 1 , ∆ 2 and ∆ 3 are defined as and the components of the torque, Γ x , Γ y and Γ z , are calculated from the orientation-dependent self-energy δU via Γ x = − sin γ sin β ∂ α δU − cos γ ∂ β δU + cot β sin γ ∂ γ δU, Γ y = − cos γ sin β ∂ α δU + sin γ ∂ β δU + cot β cos γ ∂ γ δU, Finally, the two physical degrees of freedom in the GW can be extracted from h IJ by defining the plus and the cross modes for an observer whose colatitude and azimuth are θ o and φ o in the X-Y -Z frame [27], whereθ I o andφ I o are the XY Z-components of the transverse unit vectorŝ Note thatê X ,ê Y andê Z are the unit vectors of the X-Y -Z frame, whileê x ,ê y andê z will be used as the unit vectors of the x-y-z frame.

III. NUMERICAL EXAMPLES
Now we can use the above equations to numerically calculate the GW spectra of a rotating ellipsoidal NS affected by Lorentz violation. To simplify the calculation of δU , we assume the density of the star to be constant. Extension to realistic nonuniform NSs is straightforward. Then the angular parts of the integrals in δU can be carried out analytically. Specifically speaking, define then they are related to the Newtonian potential via The Newtonian potential of a uniform ellipsoid is known to be [27,28] where with i = x, y, z. Consequently, the nonvanishing U ij are found to be For NSs, the density varies from the center to the surface. For our purpose, we will take a uniform density of 10 15 g/cm 3 in numerical calculations. As for the semi-axes, because NSs are compact objects having tiny deformations if any, we can only say that they are all about 10 km, roughly the radius of a spherical NS predicted by GR. The often used parameters to characterize NS deformation are the oblateness and the nonaxisymmetry δ. They are defined as with an assumption that I zz is the largest eigenvalue of the moment of inertia tensor. NS models have suggested that is less than 10 −7 [29], while the magnitude of δ is hardly known. For demonstration, we take 0.1 for both and δ in the following numerical examples. In addition, we use 10 km for a z , and then the values of a x and a y are determined by noticing 15 ρa x a y a z (a 2 y + a 2 z ), I yy = 4π 15 ρa x a y a z (a 2 x + a 2 z ), I zz = 4π 15 ρa x a y a z (a 2 x + a 2 y ), for uniform ellipsoids.
Then to compute δU as a function of the Euler angles, we take numerical valuess XX = 0.02,s Y Y = 0.01,s ZZ = −0.04 ands XY =s XZ =s Y Z = 0 for the coefficients for Lorentz violation in the inertial frame. This means that the axes of the inertial frame are the principal axes of thes ij tensor. Note that this is a theoretical inertial frame fixed by the coefficients for Lorentz violation. It generally does not coincide with the widely used experimental inertial frame, namely the Sun-centered celestial-equatorial frame defined in Ref. [11].
All the parameters in the Lagrangian (2) have been set now. Numerical solutions for the Euler angles can be obtained once initial values are given. For numerical calculations, a dimensionless parametrization for the angular velocities is helpful. This can be achieved by employing a time unit. To be consistent with the choice in Ref. [23], it is taken to be For a uniform ellipsoid, keeping only the leading contribution from and δ, it is where the magnitude estimation is made for ρ = 10 15 g/cm 3 and = δ = 0.1. Therefore, a dimensionless angular velocity at order unity in our numerical results corresponds to about 1000 rad/s. Figure 2 shows the trajectories of the tail of the unit vectorê z in the inertial frame to intuitively illustrate the rotations of the star for a certain set of initial values. Our examples consist of two solutions: the plot on the left shows a twofold precession withs IJ taking the above said values, and the plot on the right shows a free precession without Lorentz violation for comparison. The distinction is also reflected by the trajectories of the tail of the angular momentum unit vector: in the left plot, there is a nontrivial trajectory for the angular momentum unit vector, while in the right plot, the angular momentum unit vector does not change with time.
With the two solutions, we calculate the GWs according to Eq. (12) for an observer at θ o = 0.8 rad and φ o = 0. The results are presented in Fig. 3. Their Fourier transformations are shown in Fig. 4; only the plus mode is shown as the cross mode has very much the same spectra. The spectra of the free precession shows a fundamental angular frequency at about 1.4/t c , and peaks around the second harmonic at about 2.9/t c . We know that if the star is axisymmetric, free precessions generate GWs having exactly two frequencies, with one being twice of the other. The nonaxisymmetry   Fig. 3. The two noticeable peaks at about 1.4 and 2.9 in the left plot are the first and the second harmonics for both twofold precession and free precession. The modulation due to nonaxisymmetry is clearly represented by the adjacent peak at about 2.7 close to the second harmonic for both kinds of motion. However, the barely visible tiny peaks, reflecting modulations due to Lorentz violation, only exist for twofold precession. The right plot, which zooms in on the tiny peak between 4 and 5, demonstrates the point. Note that in the plots the geometrized unit of Fourier amplitude is tc, while the geometrized unit of angular frequency is 1/tc.
here modulates both the fundamental frequency and the second harmonic. This has been discussed in Refs. [16,24,25]. What we are showing in the left plot of Fig. 4 tells us that the twofold precession, namely the rotation of an otherwise freely precessing NS under Lorentz-violating gravity, generates similar GW frequency components. However, more interestingly, in the enlarged plot on the right, we clearly see the distinction that while the twofold precession generates frequency components around the third harmonic, the free precession has no component of the third harmonic at all. Higher frequency components exist in the spectra of the Lorentz-violating twofold precession, but they can easily be missed as they are too small.

IV. CONCLUSION
We have presented the analytical formulae to calculate the rotation of NSs under Lorentz-violating gravity in the minimal gravitational SME framework, and to construct the quadrupole GWs emitted from these NSs. Numerical examples are plotted to demonstrate our conclusion that while freely precessing NSs in the Lorentz-invariant gravity do not emit quadrupole GWs at frequencies higher than the second harmonic, NSs undergoing the twofold precession due to Lorentz violation do. Therefore, harmonics higher than the second in the spectra of continuous GWs are appealing signatures of Lorentz violation. Once continuous GWs from rotating NSs are detected, a potential test of Lorentz invariance can be performed by examining harmonics higher than the second in the spectra. However, we do notice a possible difficulty in this test: there might be conventional torques, like the electromagnetic spin-down torque [30][31][32][33], acting on the NS to cause similar twofold precession motions and to generate higher harmonics in the GW spectra. Although the questions whether the twofold precession caused by Lorentz violation can be distinguished from rotations of NSs under the electromagnetic spin-down torque and whether the GW spectra of the latter have frequency components higher than the second harmonic lie beyond the scope of this work, they are certainly worth to be investigated further. Furthermore, a statistical study of continuous GWs from an ensemble of NSs might have the potential to distinguish between the two scenarios, as the Lorentz violation is universal for all NSs while the astrophysical torques are different for different systems.