Lorentz-violating matter-gravity couplings in small-eccentricity binary pulsars
LLorentz-violating matter-gravity couplings in small-eccentricity binary pulsars ∗ Lijing Shao
1, 2, † Kavli Institute for Astronomy and Astrophysics, Peking University, Beijing 100871, China Max-Planck-Institut f¨ur Radioastronomie, Auf dem H¨ugel 69, D-53121 Bonn, Germany (Dated: August 28, 2019)Lorentz symmetry is an important concept in modern physics. Precision pulsar timing was usedto put tight constraints on the coefficients for Lorentz violation in the pure-gravity sector of theStandard-Model Extension (SME). We extend the analysis to Lorentz-violating matter-gravity cou-plings, utilizing three small-eccentricity relativistic neutron star (NS) – white dwarf (WD) binaries.We obtain compelling limits on various SME coefficients related to the neutron, the proton, and theelectron. These results are complementary to limits obtained from lunar laser ranging and clockexperiments.
I. INTRODUCTION
The theory of general relativity (GR) and the Stan-dard Model (SM) of particle physics represent our con-temporary condensed wisdom in the search of fundamen-tal laws in physics. Nevertheless, there exist variousmotivations to look for new physics. Among them, thepossibility of Lorentz violation is a well developed con-cept [1]. Lorentz violation could be resulted from a deepunderlying theory of quantum gravity [2]. At low energy,it is believed to be described by an effective field the-ory (EFT). An EFT framework, the so-called Standard-Model Extension (SME), systematically incorporates allLorentz-covariant, gauge-invariant, energy-momentum-conserving operators that are associated with GR andSM fields [3–5]. Field operators are sorted according totheir mass dimension, and for some certain species, op-erators of arbitrary mass dimensions are classified [6–9].The SME is supposed to be an effectively low-energytheory for the quantum gravity, thus the gravitationalaspect of the SME is of particular interests. Kosteleck´y[5] presented the general structure of the SME when thecurved spacetime is considered. Bailey and Kosteleck´y[10] worked out different kinds of observational phenom-ena associated with the minimal operators in the pure-gravity sector of the SME whose mass dimension d ≤ d ≤
4. Phenomeno-logical aspects and relevant experiments are identified.Moreover, the nonminimal SME with gravitational op-erators whose mass dimension d > et al. [19]have a comprehensive summary on this topic; see also the
Data Tables for Lorentz and CPT Violation , compiled by ∗ Invited article to special issue “Symmetry in Special and GeneralRelativity” in MDPI journal
Symmetry . † Corresponding author: [email protected]
Kosteleck´y and Russell [20]. In the pure-gravity sector,binary pulsars turn out to be among the best experimentsin constraining, (i) the d ≤ II. MATTER-GRAVITY COUPLINGS IN THESME
In order to incorporate fermion-gravity couplings, weuse the vierbein formalism [5]. In the SME, the actionfor a massive Dirac fermion ψ reads [11], S ψ = (cid:90) e (cid:18) ie µa ψ Γ a ←→ D µ ψ − ψM ψ (cid:19) d x , (1) a r X i v : . [ h e p - ph ] A ug where, for spin-independent cases,Γ a ≡ γ a − c µν e νa e µb γ b − e µ e µa , (2) M ≡ m + a µ e µa γ a . (3)Here e aµ is the vierbein with e its determinant; m is themass of the fermion; γ a is the Dirac matrix; a µ , c µν ,and e µ are species-dependent, spin-independent coeffi-cient fields for Lorentz violation [see Eq. (7) and Eq. (8)in Ref. [11] for spin-dependent terms].While being kept to the leading order, a field redefini-tion via a position-dependent component mixing in thespinor space can be used to show that, the CPT-odd coef-ficients a µ and e µ always appear in the combination [11],( a eff ) µ ≡ a µ − me µ . (4)Therefore, we shall consider only ( a eff ) µ and c µν in thefollowing.At leading order, the point-particle action is [11], S u = (cid:90) d λ (cid:20) − m (cid:113) − ( g µν + 2 c µν ) u µ u ν − ( a eff ) µ u µ (cid:21) , (5)where u µ ≡ d x µ / d λ . For a macroscopic composite ob-ject, the action (5) is still applicable with the replace-ments [11], m → (cid:88) w N w m w , (6) c µν → (cid:80) w N w m w ( c w ) µν (cid:80) w N w m w , (7)( a eff ) µ → (cid:88) w N w ( a w eff ) µ , (8)where w denotes the particle species, and N w is thenumber of particles of type w . We have neglected thecontribution from binding energies which could be atmost ∼
20% for neutron stars (NSs), unless some un-known nonperturbative effects take place (see discussionsin Sec. V) [30]. In general the role of binding energy couldfurther aid the analysis of signals for Lorentz violation;see Sec. VI B in Ref. [11] for more details. Hereafter, forsimplicity we only consider three types of fermions, (i) theelectron w = e , (ii) the proton w = p , and (iii) the neu-tron w = n . In Table I, we list the estimated compositionof these three species for NSs and white dwarfs (WDs),and their corresponding composite coefficient fields forLorentz violation.In general, the coefficient fields, ( a eff ) µ and c µν , aredynamical fields. In the Riemann-Cartan spacetime, theLorentz violation often needs to be spontaneous [36], in-stead of explicit [5]. The coefficient fields obtain their vac-uum expectation values via the Higgs-like spontaneoussymmetry breaking mechanism. We denote the vacuumexpectation values of ( a eff ) µ and c µν , as ( a eff ) µ and c µν ,respectively. The barred quantities are also known as the â periastron ĉ s k y p l a n e b ⌃ Ω ω i FIG. 1. Pulsar orbit and the coordinate system (cid:16) ˆa , ˆb , ˆc (cid:17) [10,22, 23]. coefficients for Lorentz violation [20]. In asymptoticallyinertial Cartesian coordinates, they are assumed to besmall and satisfy [11], ∂ α ( a eff ) µ = 0 , (9) ∂ α c µν = 0 . (10)The coefficients for Lorentz violation, ( a eff ) µ and c µν [20],are the quantities that we want to investigate with pulsartiming experiments [37, 38] in this work. III. BINARY PULSARS WITHLORENTZ-VIOLATING MATTER-GRAVITYCOUPLINGS
Jennings et al. [31] worked out the osculating elementsfor a binary system, composed of masses M and M ,in the presence of the Lorentz-violating matter-gravitycouplings. We consistently use the subscript “1” to de-note the pulsar, and use the subscript “2” to denotethe companion which is a WD in our study. We define q ≡ M /M and M ≡ M + M . To simplify some ex-pressions, we also define X ≡ M /M = q/ (1 + q ); then M /M = 1 − X = 1 / (1 + q ).Neglecting the finite-size effects, the Newtonian rela-tive acceleration for a binary is a N = − GM M /r ˆ r ,where r is the relative separation and ˆ r ≡ r /r . In theNewtonian gravity, a two-body system with a negativetotal orbital energy forms an elliptical orbit. An ellipti-cal orbit in the celestial mechanics is usually describedby six orbital elements, (i) the semimajor axis a , (ii) theorbital eccentricity e , (iii) the epoch of periastron pas-sage T , (iv) the inclination of orbit i , (v) the longitudeof periastron ω , and (vi) the longitude of ascending nodeΩ. The last three angles are illustrated in Figure 1.When there is a perturbing acceleration to a N , say, δ a ,the orbit is changed perturbatively. In the osculating- TABLE I. Estimated composition for NSs and WDs. Composite coefficient fields for Lorentz violation are estimated accordingto Eqs. (6–8). In the table, M NS and M WD are the masses for NS and WD, respectively, and m n ( (cid:39) m p ) is the mass for aneutron (proton) particle. We define N NS ≡ M NS /m n and N WD ≡ M WD /m n . Neutron Stars White Dwarfs
Electron number, N e ∼ N WD Proton number, N p ∼ N WD Neutron number, N n N NS 12 N WD Composite m M NS M WD Composite c µν c nµν (cid:0) c nµν + c pµν + 0 . c eµν (cid:1) Composite ( a eff ) µ N NS ( a n eff ) µ N WD (cid:104) ( a n eff ) µ + ( a p eff ) µ + ( a e eff ) µ (cid:105) element approach, one assumes that at any instant mo-ment, the orbit is still an ellipse, but the six orbital el-ements become functions of the time t [39]. The timederivatives of these six functions are derived from theextra acceleration δ a [39]. In the current case, after av-eraging over an orbital period P b , the secular changesread [31], (cid:28) d a d t (cid:29) = 0 , (11) (cid:28) d e d t (cid:29) = n b M γ (cid:18) e − εe A ˆ a ˆ b + n b aεe B ˆ a (cid:19) , (12) (cid:28) d i d t (cid:29) = n b M γ × (cid:18) εe A ˆ a ˆ c cos ω − e − εe A ˆ b ˆ c sin ω − n b εae B ˆ c sin ω (cid:19) , (13) (cid:28) d ω d t (cid:29) = − n b M γ tan i × (cid:18) εe A ˆ a ˆ c sin ω + e − εe A ˆ b ˆ c cos ω + n b εae B ˆ c cos ω (cid:19) + n b M (cid:20) e − ε e (cid:0) A ˆ b ˆ b − A ˆ a ˆ a (cid:1) + n b a (1 − γ ) e B ˆ b (cid:21) , (14)where we have defined γ ≡ √ − e , ε ≡ − γ =1 − √ − e , and n b ≡ π/P b . From Eq. (11), we cansee that the energy of the orbit is conserved at leadingorder, which is compatible with the action formulationof the system in the absence of gravitational waves. Theexpressions for (cid:104) dΩ / d t (cid:105) and (cid:104) d T / d t (cid:105) are not importantin the present context, thus not shown. The 3-vector B j and the 3 × A jl are defined as [31], A jl = (cid:88) w n w m w c w ( jl ) , (15) B j = − (cid:88) w (cid:104) n w ( a w eff ) j + ( n w − n w ) m w c w (0 j ) (cid:105) , (16)where n wi ( i = 1 , · · · ,
8) are defined in Eq. (9) of Ref. [31], and their approximated values for NS-NS and NS-WDbinaries are given in Table II for convenience.In the above two equations, only n wi with i = 2 , , , A jl M =(2 − X ) c n ( jl ) + X (cid:104) c p ( jl ) + 0 . c e ( jl ) (cid:105) , (17) B j M = 1 − Xm n (cid:104) ( a p eff ) j + ( a e eff ) j (cid:105) + 1 − Xm n ( a n eff ) j + (cid:0) X − X + 4 (cid:1) c n (0 j ) − X (1 + X ) (cid:104) c p (0 j ) + 0 . c e (0 j ) (cid:105) . (18)We can easily obtain the following conclusion from theabove two equations. (I) The sensitivity to c e ( jl ) and c e (0 j ) [compared with c p ( jl ) and c p (0 j ) respectively] is sup-pressed by the mass ratio of the electron to the proton( m e /m p (cid:39) . a e eff ) j [com-pared with ( a p eff ) j ] is not suppressed. (II) We have nosensitivity to ( a w eff ) nor c w ( w ∈ { n, p, e } ) from binarypulsars in this simplified situation. This is similar tothe case of s (the time-time component of the Lorentz-violating field s µν ) in the pure-gravity sector of the SMEwith dimension 4 operators [10, 21]; nevertheless, thesecoefficients can be probed with the help of the “boost ef-fect” introduced by the systematic velocity of the binary( v sys /c ∼ − ) with respect to the Solar system [22]. Wedefer the investigation along this line to future studies.In Eqs. (11–14), B j and A jl are projected to the co-ordinate system (cid:16) ˆa , ˆb , ˆc (cid:17) [10, 22, 23] where ˆa is the unitvector points from the center of binary towards the pe-riastron, ˆc is the unit vector points along the orbital an-gular momentum, and ˆb ≡ ˆc × ˆa (see Figure 1).We are interested in the small-eccentricity binaries. Inthe limiting case of small eccentricity e →
0, we have γ = 1 − e − e + O (cid:0) e (cid:1) , (19) ε = 12 e + 18 e + O (cid:0) e (cid:1) . (20) TABLE II. Expressions of n wi /N ( i = 1 , · · · , w ∈ { n, p, e } ) for NS-NS and NS-WD systems (see Eq. (9) in Ref. [31]), where N ≡ N + N (cid:39) M/m n . Results in Table I are adopted for the calculation here. Neutron Star – Neutron Star Neutron Star – White Dwarf n p e n p en w /N (1 + X ) (1 − X ) (1 − X ) n w /N X − (3 X − − (1 − X ) − (1 − X ) n w /N
32 12 12 n w /N −
12 12 12 n w /N X (1 − X ) 0 0 X (1 − X ) X (1 − X ) X (1 − X ) n w /N − X (1 − X ) X (1 − X ) X (1 − X ) n w /N − X X Xn w /N − X X − X + 1 − X − X Therefore, Eqs. (12–14) are simplified to, (cid:28) d e d t (cid:29) (cid:39) n b a M B ˆ a , (21) (cid:28) d i d t (cid:29) (cid:39) n b M (cid:0) A ˆ a ˆ c cos ω − A ˆ b ˆ c sin ω (cid:1) , (22) (cid:28) d ω d t (cid:29) (cid:39) n b a eM B ˆ b . (23)We can convert the derivatives of e , i , and ω , into deriva-tives of the projected semimajor axis of the pulsar orbit x p , and the Laplace-Lagrange parameters, η ≡ e sin ω and κ ≡ e cos ω , (cid:28) d x p d t (cid:29) = M cos i M ( GM n b ) / (cid:0) A ˆ a ˆ c cos ω − A ˆ b ˆ c sin ω (cid:1) , (24) (cid:28) d η d t (cid:29) = n b M ( GM n b ) / (cid:0) B ˆ a sin ω + B ˆ b cos ω (cid:1) , (25) (cid:28) d κ d t (cid:29) = n b M ( GM n b ) / (cid:0) B ˆ a cos ω − B ˆ b sin ω (cid:1) , (26)where we have used n b a = ( GM n b ) / . IV. BOUNDS ON THE SME COEFFICIENTS
We use the time derivatives of x p , η , and κ in Eqs. (24–26) to constrain the coefficients for Lorentz violation. Itis clear that the more relativistic the binary (namely,the larger n b ), the better the tests. Therefore, we usethree well-timed NS-WD binaries whose orbital periodsare shorter than half a day [32–34]. Relevant parame-ters of these binaries are collected in Table III. Due tothe binary interaction and matter exchange in the evolu-tionary history, these NS-WD binaries have small orbitaleccentricity e ≤ − , thus Eqs. (24–26) are sufficient toperform the tests. From Table III, we see that the time derivatives of η and κ are not reported in literature, as well as thetime derivative of x p for PSR J0348+0432. The reasonis usually the following. In fitting the times of arrival ofpulse signals, these quantities would be measured to beconsistent with zero if they were included in the timingformalism. To have a simpler timing model, these quan-tities are considered unnecessary for a good fit. Actually,the insignificance of these quantities is consistent withthe spirit of our tests to put upper limits on the Lorentzviolation. We estimate the upper limits for these quanti-ties using ˙ X ∼ √ σ X /T obs ( X ∈ { x p , η, κ } ) [21], where σ X is the measured uncertainty for the quantity X and T obs is the observational span of the data from wherethese quantities were derived. The factor “ √
12” approx-imately takes a linear-in-time evolution of the quantity X into account [21]. It is verified that this approximationworks reasonably well [21, 23]. For PSRs J0751+1807and J1738+0333, (cid:104) d x p / d t (cid:105) was measured to be nonzero.Because the proper motion of the binary in the skycould contribute to a nonzero (cid:104) d x p / d t (cid:105) for nearby pul-sars [37, 40], we use the measured value of (cid:104) d x p / d t (cid:105) asan upper limit for the effects from Lorentz violation. Fornearby pulsars, the contribution to (cid:104) d x p / d t (cid:105) from theproper motion depends sinusoidally on Ω [37, 40]; al-though Ω is not measured, we do not expect the Na-ture’s conspiracy in assigning certain values of Ω, caseby case to different binary pulsars, in order to hide theLorentz symmetry breaking. Therefore, we believe theabove treatments introduce uncertainties no larger thana multiplicative factor of a few.In order to use Eqs. (24–26), one also needs the abso-lute geometry of the orbit to properly project the vec-tor B j and the tensor A jl onto the coordinate system (cid:16) ˆa , ˆb , ˆc (cid:17) . In general, the longitude of the ascending nodeΩ is not an observable in pulsar timing [37]. Nevertheless,the procedure to randomize the value of Ω ∈ [0 , ◦ )and to systematically project vectors and tensors onto TABLE III. Relevant parameters for PSRs J0348+0432 [32], J0751+1807 [33], and J1738+0333 [34]. Parenthesized numbersrepresent the 1- σ uncertainty in the last digit(s) quoted. The parameter η is the intrinsic value, after subtraction of thecontribution from the Shapiro delay [35]. Masses are derived without using information related to (cid:104) d x p / d t (cid:105) , (cid:104) d η/ d t (cid:105) , nor (cid:104) d κ/ d t (cid:105) for consistency. For PSRs J0348+0432 and J1738+0333, masses were derived independently of gravity theories [32, 34], whilefor PSR J0751+1807 we have used observed quantities related to the Shapiro delay and orbital decay, assuming the validity ofGR [33]. Pulsar J0348+0432 J0751+1807 J1738+0333
Observational span, T obs (year) ∼ . ∼ . ∼ . P b (day) 0.102424062722(7) 0.263144270792(7) 0.3547907398724(13)Pulsar’s projected semimajor axis, x p (lt-s) 0.14097938(7) 0.3966158(3) 0.343429130(17) η ≡ e sin ω (10 − ) 19(10) 33(5) − . κ ≡ e cos ω (10 − ) 14(10) 3.8(50) 3.1(11)Time derivative of x p , ˙ x p – ( − . ± . × − (0 . ± . × − NS mass, m ( M (cid:12) ) 2.01(4) 1.64(15) 1 . +0 . − . WD mass, m ( M (cid:12) ) 0.172(3) 0.16(1) 0 . +0 . − . (cid:16) ˆa , ˆb , ˆc (cid:17) was worked out in Ref. [21]. It was success-fully applied to binary pulsars in previous studies [21–24]. Since here (i) we have already introduced an uncer-tainty with a factor of a few, and (ii) we are interestedin the “maximal-reach” limits in absence of the Lorentzviolation, we take a simplified approach and treat theseprojections as O (1) operators. The “maximal-reach” ap-proach [18] assumes that only one component of Lorentz-violating coefficients is nonzero in a test. We think ourapproach reasonable at the stage of setting upper lim-its to the coefficients for Lorentz violation. Neverthe-less, when people start to discover some evidence for theLorentz violation, it is absolutely needed to take into ac-count more sophisticated analysis, for example, to use the3-D orientation of the orbit (possibly in a probabilisticway with an unknown Ω) as was done in Refs. [21–24]. Inaddition, if one wants to explore the correlation betweendifferent coefficients for Lorentz violation, more sophisti-cated analysis is needed as well. These improvements laybeyond the scope of this work.In Table IV we list the “maximal-reach” [18] limitson the coefficients for Lorentz violation with matter-gravity couplings obtained from binary pulsars. As wecan see, the best limits on c wjk ( w ∈ { n, p, e } ) come fromPSR J1738+0333 due to its very good measurement onthe ˙ x p [34]. For c w k and ( a w eff ) k , the best limits comefrom PSR J0751+1807 due to its good measurement ofthe Lagrange-Laplace parameters [33]. V. DISCUSSIONS
Besides the streamlined theoretical analysis, themaximal-reach limits in Table IV are the main resultsof this paper. As far as we are aware, Altschul [41]was the first to put preliminary limits on the SMEneutron-sector coefficients with pulsar rotations. The pure-gravity sector of the SME at different mass dimen-sions was systematically tested with binary pulsars inRefs. [21–24]. Early limits on ( a w eff ) k were given withK/He magnetometer and torsion-strip balance [42, 43];but these limits, while constraining different linear com-binations of the Lorentz violating coefficients, are ratherweak. Later the maximal-reach limits on ( a w eff ) k wereobtained systematically with superconducting gravime-ters [44] and lunar laser ranging (LLR) experiments [45].The former got ( a w eff ) k ≤ O (cid:0) − GeV (cid:1) , while the lat-ter got ( a w eff ) k ≤ O (cid:0) − GeV (cid:1) . Our best limits fromPSR J0751+1807 for the proton and the electron areweaker than the LLR limits, while our limit for the neu-tron is slightly better. There is also a limits from the ob-servation of gravitational waves, but being weaker thanour limits by almost 30 orders of magnitude [46]. Thelimits on ( a w eff ) were cast by analyzing nuclear bindingenergy, Cs interferometer, torsion pendulum, and weakequivalence principle experiments [11, 47–49]. The analy-sis with binary pulsars in this work could not bound theseSME coefficients. The limits on ¯ c wµν from other experi-ments (e.g. clock experiments [50]) are much better thanthe limits from binary pulsars [20]. However, our limitsare the best ones from gravitational systems. In a shortsummary, our limits are compelling, and being comple-mentary to limits obtained from other experiments.In using the SME, we have assumed the validity of theEFT and the smallness of the Lorentz violation. Thisis true for most ordinary objects. However, we shall beaware of a caveat for NSs, because of the possible non-perturbative behaviors which might be triggered by theirstrongly self-gravitating nature [38]. It was shown ex-plicitly that, in a class of scalar-tensor theories, highlynonlinear phenomena are possible within NSs and theymay result in large deviations from GR [51, 52]. Al-though the nonperturbative behaviors were constrainedtightly with binary pulsars and the binary neutron star TABLE IV. “Maximal-reach” limits from binary pulsars on the coefficients for Lorentz violation with matter-gravity couplingswhere, only one component is assumed to be nonzero at a time. The limits on c wjk ( w ∈ { n, p, e } ) come from (cid:104) d x p / d t (cid:105) , whilethe limits on c w k and ( a w eff ) k come from (cid:104) d η/ d t (cid:105) or (cid:104) d κ/ d t (cid:105) , and only the stronger one is listed in the table. For each row, thestrongest limit is shown in boldface. SME Coefficients PSR J0348+0432 PSR J0751+1807 PSR J1738+0333 c njk × − × − × − c pjk × − × − × − c ejk × − × − × − c n k × − × − × − c p k × − × − × − c e k × − × − × − ( a n eff ) k × − GeV × − GeV × − GeV( a p eff ) k × − GeV × − GeV × − GeV( a e eff ) k × − GeV × − GeV × − GeV inspiral GW170817 [34, 53, 54], the possibility is not com-pletely ruled out yet [55–57]. With this caveat in mind,conservatively speaking, the tests in this paper are ba-sically testing the strong-field counterparts of the weak-field SME coefficients. Usually, when the strong-field ef-fects are considered, the constraints become even tighter.Therefore, we treat the limits here conservative ones [30].The tests of Lorentz violation with binary pulsars im-prove with a longer baseline for data [21]. Specifically,even pessimistically assuming no advance in the qualityof binary-pulsar observation for the future, the tests inEqs. (24–26) improve as T − . where T obs is the totalobservational span. In reality, the quality of observationimproves rapidly, especially with the newly built and up-coming telescopes, like the Five-hundred-meter ApertureSpherical Telescope (FAST), the MeerKAT telescope,and the Square Kilometre Array (SKA) [58–61]. There-fore, we expect better tests than the T − . scaling intesting the Lorentz violation in the future. ACKNOWLEDGMENTS
We are grateful to Jay Tasson for the invitationand stimulating discussions. We thank Norbert Wexfor carefully reading the manuscript, Adrien Bourgoin,Zhi Xiao and Rui Xu for communication. This workwas supported by the Young Elite Scientists Sponsor-ship Program by the China Association for Science andTechnology (2018QNRC001), and partially supportedby the National Natural Science Foundation of China(11721303), the Strategic Priority Research Program ofthe Chinese Academy of Sciences through the GrantNo. XDB23010200, and the European Research Council(ERC) for the ERC Synergy Grant BlackHoleCam underContract No. 610058. [1] J. D. Tasson, Rept. Prog. Phys. , 062901 (2014),arXiv:1403.7785 [hep-ph].[2] V. A. Kosteleck´y and S. Samuel, Phys. Rev. D39 , 683(1989).[3] D. Colladay and V. A. Kosteleck´y, Phys. Rev.
D55 , 6760(1997), arXiv:hep-ph/9703464 [hep-ph].[4] D. Colladay and V. A. Kosteleck´y, Phys. Rev.
D58 ,116002 (1998), arXiv:hep-ph/9809521 [hep-ph].[5] V. A. Kosteleck´y, Phys. Rev.
D69 , 105009 (2004),arXiv:hep-th/0312310 [hep-th].[6] V. A. Kosteleck´y and M. Mewes, Phys. Rev.
D80 , 015020(2009), arXiv:0905.0031 [hep-ph].[7] V. A. Kosteleck´y and M. Mewes, Phys. Rev.
D85 , 096005(2012), arXiv:1112.6395 [hep-ph].[8] A. Kosteleck´y and M. Mewes, Phys. Rev.
D88 , 096006(2013), arXiv:1308.4973 [hep-ph]. [9] V. A. Kosteleck´y and Z. Li, Phys. Rev.
D99 , 056016(2019), arXiv:1812.11672 [hep-ph].[10] Q. G. Bailey and V. A. Kosteleck´y, Phys. Rev.
D74 ,045001 (2006), arXiv:gr-qc/0603030 [gr-qc].[11] V. A. Kosteleck´y and J. D. Tasson, Phys. Rev.
D83 ,016013 (2011), arXiv:1006.4106 [gr-qc].[12] Q. G. Bailey, V. A. Kosteleck´y, and R. Xu, Phys. Rev.
D91 , 022006 (2015), arXiv:1410.6162 [gr-qc].[13] C.-G. Shao et al. , Phys. Rev. Lett. , 071102 (2016),arXiv:1607.06095 [gr-qc].[14] V. A. Kosteleck´y and M. Mewes, Phys. Lett.
B779 , 136(2018), arXiv:1712.10268 [gr-qc].[15] J. D. Tasson, in
Proceedings, 7th Meeting on CPTand Lorentz Symmetry (CPT 16): Bloomington, In-diana, USA, June 20-24, 2016 (2017) pp. 13–16,arXiv:1612.09264 [hep-ph]. [16] Q. G. Bailey, in (2019) arXiv:1906.08657 [gr-qc].[17] L. Shao, in (2019) arXiv:1905.08405 [gr-qc].[18] J. D. Tasson, in (2019) arXiv:1907.08106 [hep-ph].[19] A. Hees, Q. G. Bailey, A. Bourgoin, H. P.-L. Bars,C. Guerlin, and C. Le Poncin-Lafitte, Universe , 30(2016), arXiv:1610.04682 [gr-qc].[20] V. A. Kosteleck´y and N. Russell, Rev. Mod. Phys. ,11 (2011), arXiv:0801.0287 [hep-ph].[21] L. Shao, Phys. Rev. Lett. , 111103 (2014),arXiv:1402.6452 [gr-qc].[22] L. Shao, Phys. Rev. D90 , 122009 (2014),arXiv:1412.2320 [gr-qc].[23] L. Shao and Q. G. Bailey, Phys. Rev.
D98 , 084049(2018), arXiv:1810.06332 [gr-qc].[24] L. Shao and Q. G. Bailey, Phys. Rev.
D99 , 084017(2019), arXiv:1903.11760 [gr-qc].[25] C. M. Will, Living Rev. Rel. , 4 (2014),arXiv:1403.7377 [gr-qc].[26] C. M. Will, Theory and Experiment in GravitationalPhysics (Cambridge University Press, 2018).[27] L. Shao and N. Wex, Class. Quant. Grav. , 215018(2012), arXiv:1209.4503 [gr-qc].[28] L. Shao, R. N. Caballero, M. Kramer, N. Wex, D. J.Champion, and A. Jessner, Class. Quant. Grav. ,165019 (2013), arXiv:1307.2552 [gr-qc].[29] L. Shao and N. Wex, Class. Quant. Grav. , 165020(2013), arXiv:1307.2637 [gr-qc].[30] L. Shao and N. Wex, Sci. China Phys. Mech. Astron. ,699501 (2016), arXiv:1604.03662 [gr-qc].[31] R. J. Jennings, J. D. Tasson, and S. Yang, Phys. Rev. D92 , 125028 (2015), arXiv:1510.03798 [gr-qc].[32] J. Antoniadis et al. , Science , 448 (2013),arXiv:1304.6875 [astro-ph.HE].[33] G. Desvignes et al. , Mon. Not. Roy. Astron. Soc. ,3341 (2016), arXiv:1602.08511 [astro-ph.HE].[34] P. C. C. Freire, N. Wex, G. Esposito-Far`ese, J. P. W.Verbiest, M. Bailes, B. A. Jacoby, M. Kramer, I. H.Stairs, J. Antoniadis, and G. H. Janssen, Mon. Not. Roy.Astron. Soc. , 3328 (2012), arXiv:1205.1450 [astro-ph.GA].[35] C. Lange, F. Camilo, N. Wex, M. Kramer, D. C. Backer,A. G. Lyne, and O. Doroshenko, Mon. Not. Roy. Astron.Soc. , 274 (2001), arXiv:astro-ph/0102309 [astro-ph].[36] R. Bluhm, H. Bossi, and Y. Wen, (2019),arXiv:1907.13209 [gr-qc].[37] D. R. Lorimer and M. Kramer,
Handbook of Pulsar As-tronomy (Cambridge University Press, Cambridge, Eng-land, 2005).[38] N. Wex, in
Frontiers in Relativistic Celestial Mechanics:Applications and Experiments , Vol. 2, edited by S. M.Kopeikin (Walter de Gruyter GmbH, Berlin/Boston, 2014) p. 39, arXiv:1402.5594 [gr-qc].[39] E. Poisson and C. M. Will,
Gravity: Newtonian, Post-Newtonian, Relativistic (Cambridge University Press,Cambridge, England, 2014).[40] S. M. Kopeikin, Astrophys. J. Lett. , L93 (1996).[41] B. Altschul, Phys. Rev.
D75 , 023001 (2007), arXiv:hep-ph/0608094 [hep-ph].[42] J. D. Tasson, Phys. Rev.
D86 , 124021 (2012),arXiv:1211.4850 [hep-ph].[43] H. Panjwani, L. Carbone, and C. C. Speake, in
Proceed-ings, 5th Meeting on CPT and Lorentz Symmetry (CPT10): Bloomington, Indiana, June 28-July 2, 2010 (2010)pp. 194–198.[44] N. A. Flowers, C. Goodge, and J. D. Tasson, Phys. Rev.Lett. , 201101 (2017), arXiv:1612.08495 [gr-qc].[45] A. Bourgoin, C. Le Poncin-Lafitte, A. Hees, S. Bouquil-lon, G. Francou, and M.-C. Angonin, Phys. Rev. Lett. , 201102 (2017), arXiv:1706.06294 [gr-qc].[46] M. Schreck, Class. Quant. Grav. , 135009 (2017),arXiv:1603.07452 [gr-qc].[47] V. A. Kosteleck´y and J. Tasson, Phys. Rev. Lett. ,010402 (2009), arXiv:0810.1459 [gr-qc].[48] M. A. Hohensee, S. Chu, A. Peters, and H. Mueller,Phys. Rev. Lett. , 151102 (2011), arXiv:1102.4362[gr-qc].[49] M. A. Hohensee, H. Mueller, and R. B. Wiringa, Phys.Rev. Lett. , 151102 (2013), arXiv:1308.2936 [gr-qc].[50] H. Pihan-Le Bars, C. Guerlin, R. D. Lasseri, J. P. Ebran,Q. G. Bailey, S. Bize, E. Khan, and P. Wolf, Phys. Rev. D95 , 075026 (2017), arXiv:1612.07390 [gr-qc].[51] T. Damour and G. Esposito-Far`ese, Phys. Rev. Lett. ,2220 (1993).[52] T. Damour and G. Esposito-Far`ese, Phys. Rev. D54 ,1474 (1996), arXiv:gr-qc/9602056 [gr-qc].[53] L. Shao, N. Sennett, A. Buonanno, M. Kramer, andN. Wex, Phys. Rev. X7 , 041025 (2017), arXiv:1704.07561[gr-qc].[54] J. Zhao, L. Shao, Z. Cao, and B.-Q. Ma, (2019),arXiv:1907.00780 [gr-qc].[55] N. Yunes, K. Yagi, and F. Pretorius, Phys. Rev. D94 ,084002 (2016), arXiv:1603.08955 [gr-qc].[56] B. S. Sathyaprakash et al. , (2019), arXiv:1903.09221[astro-ph.HE].[57] Z. Carson, B. C. Seymour, and K. Yagi, (2019),arXiv:1907.03897 [gr-qc].[58] M. Kramer, D. C. Backer, J. M. Cordes, T. J. W. Lazio,B. W. Stappers, and S. Johnston, New Astron. Rev. ,993 (2004), arXiv:astro-ph/0409379 [astro-ph].[59] L. Shao et al. , in Advancing Astrophysics with the SquareKilometre Array , Vol. AASKA14 (Proceedings of Science,2015) p. 042, arXiv:1501.00058 [astro-ph.HE].[60] P. Bull et al. , (2018), arXiv:1810.02680 [astro-ph.CO].[61] M. Bailes et al. , in