Testing velocitydependent violating gravitational forces with radio pulsars
Abstract
In the spirit of effective field theory, the StandardModel Extension (SME) provides a comprehensive framework to systematically probe the possibility of Lorentz/CPT violation. In the pure gravity sector, operators with mass dimension larger than 4, while in general being advantageous to shortrange experiments, are hard to investigate with systems of astronomical size. However, there is exception if the leadingorder effects are CPTviolating and velocitydependent. Here we study the lowestorder operators in the pure gravity sector that violate the CPT symmetry with carefully chosen relativistic binary pulsar systems. Applying the existing analytical results to the dynamics of a binary orbit, we put constraints on various coefficients for Lorentz/CPT violation with mass dimension 5. These constraints, being derived from the postNewtonian dynamics for the first time, are complementary to those obtained from the kinematics in the propagation of gravitational waves.
pacs:
04.80.Cc, 11.30.Cp, 11.30.Er, 95.30.Sf, 97.60.GbI Introduction
There is a great deal of theoretical interest to probe new physics beyond the Standard Model of particle physics, and the General Relativity (GR) theory of gravitation Goenner (2004); Liberati (2013); Tasson (2014); Will (2014a). Most of them stem from the need for a theory of quantum gravity, namely, to unify quantum field theories and GR, or in other words, to describe the four fundamental forces within a single mathematical setting Kostelecký and Samuel (1989a); Burgess (2004); Hossenfelder (2013). Up to now, although there are achievements at different levels, not one proposal has been singled out as the widely accepted final theory for quantum gravity. On the other hand, observational evidence that was accumulated during the past decades — with intriguing puzzles from dark matter, dark energy, and inflationary cosmology, just to name a few — points to the need going beyond the current paradigm of modern theoretical physics Clifton et al. (2012); Will (2014b); Tasson (2016).
Broadly speaking, there are two ways to investigate new physics beyond our current understanding: theory specific and theory agnostic. Effective field theory (EFT) is a natural candidate framework for the latter Weinberg (2009); Burgess (2004). In the spirit of EFT, Kostelecký and collaborators have developed a comprehensive framework, dubbed the StandardModel Extension (SME), to catalogue all possible operators that are gauge invariant, Lorentz covariant, and energymomentum conserving Kostelecký and Samuel (1989a); Colladay and Kostelecký (1997, 1998); Kostelecký (2004); Bailey and Kostelecký (2006); Kostelecký and Tasson (2011); Kostelecký and Mewes (2018). In general, a violation in CPT implies a violation in the Lorentz symmetry Greenberg (2002). In a practical way, we will collectively call the coefficients of new operators beyond the Standard Model and GR coefficients for Lorentz/CPT violation Kostelecký and Russell (2011). During the past decades, the SME has been successfully applied in various experiments, and many constraints were set on the coefficients for Lorentz/CPT violation Kostelecký and Russell (2011); Hees et al. (2016); Shao and Wex (2016). No statistically convincing violation has been found yet Kostelecký and Russell (2011).
We here focus on the pure gravity sector of SME Kostelecký (2004); Bailey and Kostelecký (2006); Kostelecký and Mewes (2017); Shao et al. (2016a); Bailey (2016); Bailey and Havert (2017); Kostelecký and Mewes (2018). The general framework for RiemannCartan spacetime was described in Ref. Kostelecký (2004). To be mathematically compatible with the RiemannCartan geometry, Lorentz/CPT breaking can be considered to be spontaneous, instead of explicit Bluhm (2015). Extra dynamical fields in the framework obtain their vacuum expectation values through symmetry breaking cosmologically, in analog with the Higgs mechanism in the Standard Model. However in SME these fields are not necessarily to be scalar fields, but can take on nontrivial spacetime indices and therefore have tensorial nature. Therefore, after symmetry breaking, the effective Lagrangian is observer Lorentz invariant, but particle Lorentz violating Kostelecký (2004); Bailey and Kostelecký (2006); Tasson (2016). To be fully compatible with geometrical requirements at desired orders, the underlying fluctuating NambuGoldstone modes that arise from the symmetry breaking need to be propoerly accounted for Kostelecký (2004); Bailey and Kostelecký (2006). In Ref. Bailey and Kostelecký (2006) the postNewtonian behaviours from the puregravity sector of SME for operators with mass dimension up to 4 were studied. The leadingorder postNewtonian effects are described by a tensor field, , where the “bar” indicates that it is the vacuum expectation value of the underlying dynamical field . Different experiments, including lunar laser ranging Battat et al. (2007); Bourgoin et al. (2017), atom interferometers Mueller et al. (2008); Chung et al. (2009); Flowers et al. (2017), cosmic rays Kostelecký and Tasson (2015), pulsar timing Shao and Wex (2012); Shao et al. (2013); Shao (2014a, b); Xie (2013); Shao (2016), planetary orbital dynamics Hees et al. (2015), and gravitational waves Abbott et al. (2017a, b) were used to constrain (see Hees et al. (2016) for a review).
Recently, higherdimensional operators with mass dimension larger than 4 in the gravity sector of SME were investigated, and shortrange gravity experiments in laboratory were identified to be the best to constrain these terms due to the extra powers in for the gravitational forces derived from these operators Bailey et al. (2015); Shao et al. (2016a, b); Kostelecký and Mewes (2017). However, there is an exception. Bailey and Havert (2017) found that the leadingorder CPTviolating operators with mass dimension 5 produce a gravitational force, between two objects and , proportional to . For shortrange gravity experiments, is very close to zero, thus these experiments are very hard, if ever possible, to probe these terms. Estimated sensitivities of different experiments to these new operators were tabulated (see Table III in Ref. Bailey and Havert (2017)), where binary pulsars turn out to be among the most sensitive probes. This motivates us to take a closer look at these new operators, and to collect the best binary pulsars in order to derive constraints on the coefficients for Lorentz/CPT violation.
The paper is organized as follows. In the next section, we review the structure of the gravity sector of SME at leading orders, and give the expressions for secular changes for elements of a binary orbit Bailey and Kostelecký (2006); Bailey and Havert (2017). Then in section III we carefully choose the binary pulsars that are suitable for the test, and discuss our approach to evade difficulties related to observationally unknown angles and the consistency in using timing parameters with a priori unknown component masses. Our direct constraints are summarised in Table 4, and they are properly converted to constraints on the coefficients in the Lagrangian in Tables 5 and 6. In the last section we point out the perturbative nature of SME and the postNewtonian approach, thus we should keep caveats in mind when dealing with strongly selfgravitating bodies like neutron stars (NSs) Damour and EspositoFarèse (1993); Shao et al. (2017). Throughout the paper, unless explicitly stated, we use units where .
Ii Theory
At present there are two approaches to the gravity sector of the SME. The first is a general coordinate invariant version Kostelecký (2004), while the second focuses on a spacetime that can be expanded around a Minkowski metric Kostelecký and Mewes (2018). These two approaches have distinct underlying methodology, but are interrelated. We use the latter in this work. We restrict ourselves to the discussion of the part of spacetime where, after fixing the gauge (say, the harmonic gauge), linearized gravity is a good approximation. The metric is decomposed into a flatspacetime metric, , and a perturbation, ,
(1) 
where . With this assumption, it is possible to write down the generic Lagrangian density for a spin2 massless particle, organized by the order of the mass dimension of the coupling coefficients Kostelecký (2004); Bailey and Kostelecký (2006); Bailey et al. (2015); Kostelecký and Mewes (2016); Bailey and Havert (2017),
(2) 
where the GR terms are,
(3) 
with the linearized Einstein tensor, and the matters’ energymomentum tensor.
The leadingorder corrections in Eq. (2) are Bailey and Havert (2017),
(4)  
(5) 
where is the linearized Riemann curvature tensor, and is its double dual; and are coefficients for Lorentz/CPT violation. Components of are dimensionless, while those of have the dimension of the length (or the inverse mass). In the operational counting in SME Kostelecký (2004), breaks the Lorentz symmetry, but preserves the CPT symmetry, while breaks both Lorentz and CPT symmetries Kostelecký (2004). is a symmetric, traceless tensor, thus it has 9 independent components. The first three indices of are completely antisymmetric, while the last four have the symmetry of the Riemann tensor. Thus, there are 60 independent coefficients in Kostelecký and Mewes (2016); Bailey and Havert (2017). Because has already been discussed in various literature Kostelecký (2004); Bailey and Kostelecký (2006); Kostelecký and Russell (2011), we will focus on in this paper. The contributions from are kept in some expressions in the text, only for interested readers for convenient comparisons; all numerical calculations in this paper have set . As mentioned by Bailey and Havert (2017), some specific models have direct or indirect mappings to the Lagrangian in Eqs. (4) and (5), like the vector field models with a potential term driving spontaneous Lorentz/diffeomorphism breaking Kostelecký and Samuel (1989b) and those with additional beyondMaxwell kinetic terms Jacoby et al. (2006), noncommutative geometry Carroll et al. (2001), quantum gravity Gambini and Pullin (1999), and so on.
Neglecting higherorder terms, the field equation derived from Eq. (2) reads Bailey and Havert (2017),
(6) 
where denotes the symmetrization of indices.
With postNewtonian techniques Will (1993), one can derive the leadingorder Lagrangian for two bodies and Bailey and Kostelecký (2006); Bailey and Havert (2017),
(7) 
where and are masses, and are velocities (a boldface indicates vectors), is the relative separation, and with , . As can be seen from the second line of the equation, while the terms depend on the “absolute” velocities of bodies, the terms (to be introduced below) only depend on the relative velocity of two bodies. When , the Lagrangian reduces to,
(8) 
In Eq. (7) we have defined,
(9) 
which is the linear combination of that enters the postNewtonian scheme at leading order Bailey and Havert (2017); “permutations” here mean all symmetric permutations in the last three indices . While the postNewtonian limit contains all 9 independent coefficients in , there are only 15 independent combinations of 30 irreducible pieces (out of 60) in appearing Bailey and Havert (2017). This is similar for the Lorentzviolating effects on the gravitationalwave propagation in SME, where a subset of 16 of these coefficients appear at leading order Kostelecký and Mewes (2016).
Using the EulerLagrange equation,
(10) 
we can obtain from Eq. (7) the acceleration of body Bailey and Havert (2017),
(11) 
where denotes the antisymmetrization of indices. The acceleration for body can be obtained by interchanging the body indices . Again, when , the equation reduces to,
(12) 
The second term of the above equation provides us with a nonstatic (namely velocitydependent) inverse cubic force between two masses. The behaviour of this term is vastly different from what occurs in GR and other Lorentzviolating terms that preserve the CPT symmetry Bailey et al. (2015); Kostelecký and Mewes (2017). There is no selfacceleration term in (12), which is consistent with the fact that SME is based on an action principle with energy and momentum conservation Shao and Wex (2016).
Now we discuss the secular changes for a bound orbit with the accleration (12). For an elliptical binary orbit, we use the notations in Damour and Taylor (1992). In particular, the coordinate systems and are defined in Figure 1. Notations are the same as that in Refs. Shao (2014b, a), but differ from Refs. Bailey and Kostelecký (2006); Bailey and Havert (2017) where was used. To connect the spatial frame with the cannonical Suncentered celestialequatorial frame, , one needs a spatial rotation, , to align the axes,^{1}^{1}1We neglect the boost between these two frames, which is small, with , where is the systematic velocity of the binary pulsar with respect to the Solar System Bailey and Kostelecký (2006); Shao (2014b).
(13) 
With the help of in Figure 1, one can decompose the full rotation into five simple parts, characterized by parameters in celestial mechanics Bailey and Kostelecký (2006); Shao (2014b, a),
(14) 
where
(15)  
(16)  
(17)  
(18)  
(19) 
In the rotation matrix, and are the right ascension and declination of the binary pulsar, is the orbital inclination, is the longitude of the periastron, and is the longitude of the ascending node (see Figure 1).
Using the techniques of osculating elements, Bailey and Havert (2017) obtained the secular changes of orbital elements after averaging over the orbitalperiod timescale,
(20)  
(21)  
(22)  
(23)  
(24) 
where is the semimajor axis, is the orbital eccentricity, and with the orbital period. In above equations, , , are defined by Bailey and Havert (2017),
(25)  
(26)  
(27) 
where the indices on the right hand sides denote the projection of in Eq. (9) onto the directions. More details can be found in Ref. Bailey and Havert (2017).
PSR B1913+16  PSR B1534+12  PSR B2127+11C  PSR J07373039A  

Observational span, (year)  
Right ascension, (J2000)  
Declination, (J2000)  
Orbital period, (day)  
Eccentricity,  
Pulsar’s projected semimajor axis, (lts)  
Longitude of periastron, (deg)  
Epoch of periastron, (MJD)  
Advance of periastron, (deg yr)  
Time derivative of ,  
Parameters used to derive masses  &  &  &  & 
Pulsar mass, ()  
Companion mass, ()  
Excess of , (deg yr) 
PSR J0348+0432  PSR J1738+0333  PSR J1012+5307  

Observational span, (year)  
Right ascension, (J2000)  
Declination, (J2000)  
Orbital period, (day)  
Pulsar’s projected semimajor axis, (lts)  
Time derivative of ,  
Pulsar mass, ()  
Companion mass, () 
PSR J0751+1807  PSR J18022124  PSR J19093744  PSR J2043+1711  

Observational span, (year)  
Right ascension, (J2000)  
Declination, (J2000)  
Orbital period, (day)  
Pulsar’s projected semimajor axis, (lts)  
Time derivative of ,  
Parameters used to derive masses  &  &  &  & 
Pulsar mass, ()  
Companion mass, () 
Iii Binary pulsars
Our starting point to put constraints on the SME coefficients with binary pulsars will be using the secular changes in orbital elements. In general, pulsar timing is insensitive to the longitude of the ascending node , unless the binary is very nearby Kopeikin (1996); Lorimer and Kramer (2005). Thus, the secular changes in the orbital inclination and the longitude of the periastron are the most relevant to our tests. A nonzero will be reflected in the accurately measured, projected semimajor axis of the pulsar orbit, , where is the semimajor axis of the pulsar orbit.^{2}^{2}2We hereafter use and to denote the masses of the pulsar and its companion, respectively. From Eq. (23), one has,
(28) 
In the following, we will make use of Eqs. (22) and (28), naming them as the test and the test respectively, to put bounds on the coefficients for Lorentz/CPT violation. It is apparent from Eqs. (22) and (28) that binary pulsars with small orbits will provide tight constraints. Besides the smallness of the orbit, there are other criteria to meet for binary pulsars, that will become clear later. According to the needs for the test and/or the test, we carefully pick 11 welltimed binary pulsars with relativistic orbits. We categorize them into three groups:

Group I: relativistic double NS binaries with orbital period smaller than 1 day. We pick 4 binary pulsars: PSRs B1913+16 Weisberg and Huang (2016), B1534+12 Fonseca et al. (2014), B2127+11C Jacoby et al. (2006), and J07373039A Kramer et al. (2006). Relevant timing parameters for our tests are listed in Table 1.

Group II: relativistic neutronstar–whitedwarf (NSWD) binaries with orbital period smaller than 1 day, and whose WD companions were well studied with optical observations. We pick 3 binary pulsars: PSRs J0348+0432 Antoniadis et al. (2013), J1738+0333 Freire et al. (2012), and J1012+5307 Lazaridis et al. (2009). Relevant timing parameters for our tests are listed in Table 2.

Group III: relativistic NSWD binaries with orbital period smaller than 2 days, and whose Shapiro delays were also identified in the timing observations. We pick 4 binary pulsars: PSRs J07511807 Desvignes et al. (2016), J18022124 Ferdman et al. (2010), J19093744 Desvignes et al. (2016), and J2043+1711 Arzoumanian et al. (2018). Relevant timing parameters for our tests are listed in Table 3.
Pulsar  Test  1 constraint 

PSR J0348+0432  
PSR J07373039A  
PSR J0751+1807  
PSR J1012+5307  
PSR B1534+12  
PSR J1738+0333  
PSR J18022124  
PSR J19093744  
PSR B1913+16  
PSR J2043+1711  
PSR B2127+11C  
Coefficient  1 limit [m]  Coefficient  1 limit [m]  Coefficient  1 limit [m] 

22  11  12  
10  5.7  9.7  
8.0  8.3  6.2  
8.3  3.7  5.3  
24  10  11  
6.2  4.8  18  
27  11  6.5  
8.8  29  14  
—  13  —  
14  13  29 
These 11 binary pulsars all have been monitored for years, most of which were regularly observed within the pulsartimingarray projects, including the Parks Pulsar Timing Array (PPTA) Hobbs (2013), the European Pulsar Timing Array (EPTA) Kramer and Champion (2013), and the North American Nanohertz Observatory for Gravitational Waves (NANOGrav) McLaughlin (2013). To successfully achieve the proposed test and/or test, we address the following concerns:

Because was not always fitted for in deriving the timing solution of binary pulsars, wherever it is inaccessible, we conservatively estimate a 1 upper limit from the uncertainty of , as Shao (2014a), where is the time span used in deriving the timing solution. The prefactor “” was inspired by a linearintime evolution. Actually as was already noticed for PSR B1534+12, this is a quite good estimation Shao (2014a). In addition, PSR B1913+16 was estimated by Shao (2014a) to have using the results of Weisberg et al. (2010) where was not reported. Recently, Weisberg and Huang (2016) fitted for , and obtained in excellent agreement with the estimation. This further gives us confidence in using the estimation formula. Estimated ’s are decorated with “” in Tables 1, 2, and 3.

Sometimes for nearby binary pulsars, there is a contribution to from the proper motion of the binary Kopeikin (1996),
(29) where and are proper motions in and directions respectively Lorimer and Kramer (2005). It could produce a nonzero , as was measured for several binary pulsars. Assuming GR as the theory of gravity, this piece of information can be used to constrain . Here we do not assume GR and stay agnostic about the longitude of ascending node. We randomly distribute it uniformly in the range ; thus the net effect from Eq. (29) after averaging over vanishes. For these pulsars with reported ’s, we take the uncertainty of the observed as an estimate for its upper limit.

Usually, for double NS binaries in Group I, the total mass of the binary is calculated from the very well measured Lorimer and Kramer (2005). For consistency, the test is invalid if masses were derived from the observed by assuming GR. Therefore, we need to recalculate masses without using the measured . We performed such calculations for PSRs B1913+16, B1534+12, B2127+11C, and J07373039A. Results are listed in Table 1. By using these independent masses, we recalculate the periastron advance rate with GR, and obtain the excess of by substracting it from the observed value. By doing so, we obtain a “clean” test. The uncertainties in the excess of are dominated by the uncertainties of the masses, and as a cost the clean test usually gives much worse limits than those from (see Table 4). This will be the bottleneck for our global analysis (see below).

One caution in directly using the secular change of in Lorentzviolating theories was pointed out by Wex and Kramer (2007), that a large can render the secular changes nonconstant. These effects cannot be too large based on the fact that all binaries were well fitted with simple timing models. In our samples, the biggest change in is for PSR B1913+16 Weisberg and Huang (2016). Therefore, we consider it safe to use timeaveraged values for related quantities as a rough approximation at current stage.^{3}^{3}3This will not be valid for the (unpublished) new timing solution of the double pulsar Kramer et al. (2006); Kramer (2016) where, assuming GR, up to now a change in is already. For example, in Eqs. (22) and (28), we use the value in the middle of the observational span. In principle, a timing model with nonlinearintime evolution of would be perfect in addressing this issue Wex and Kramer (2007), which is rather complicated and it is beyond the scope of this work (see Ref. Wex and Kramer (2007) for a simplified version when assuming an edgeon orbit, approximating the double pulsar).

As was pointed out several times, is in general not determined in pulsar timing. We will treat it a random variable uniformly distributed in . This choice makes our tests “probabilistic tests”.

To perform the test and the test, component masses of the binary are needed sometimes. We have discussed the situation for double NS binaries in Group I. For NSWD binaries in Group II, we use the masses derived from the optical observation of the WD. These masses are independent of the gravity theories Wex (2014); Shao and Wex (2016) (see Table 2). For NSWD binaries in Group III, we derive masses from the measurement of the Shapiro delay for PSRs J18022124, J19093744, and J2043+1711, while for PSR J0751+1807, we also used the orbital decay measurement for assistance (see Table 3). These calculation assumes that the deviations from GR are small, in consistent with the observational results, as well as the effectivefieldtheory framework. Nevertheless, we might overlook strongfield effects that arise in some specific theories Damour and EspositoFarèse (1993); Sennett et al. (2017); Shao et al. (2017) (see section IV).
Symbol  Definition  1 limit [ m] 

6.6  
3.1  
7.1  
2.7  
8.1  
20  
3.1  
6.6  
9.3  