Tests of Gravitational Symmetries with Pulsar Binary J1713+0747

Symmetries play an important role in modern theories of gravity. The strong equivalence principle (SEP) constitutes a collection of gravitational symmetries which are all implemented by general relativity. Alternative theories, however, are generally expected to violate some aspects of SEP. We test three aspects of SEP using observed change rates in the orbital period and eccentricity of binary pulsar J1713+0747: 1. the gravitational constant's constancy as part of locational invariance of gravitation; 2. the post-Newtonian parameter $\hat{\alpha}_3$ in gravitational Lorentz invariance; 3. the universality of free fall (UFF) for strongly self-gravitating bodies. Based on the pulsar timing result of the combined dataset from the North American Nanohertz Gravitational Observatory (NANOGrav) and the European Pulsar Timing Array (EPTA), we find $\dot{G}/G = (-0.1 \pm 0.9) \times 10^{-12}\,{\rm yr}^{-1}$, which is weaker than Solar system limits, but applies for strongly self-gravitating objects. Furthermore, we obtain the constraints $|\Delta|<0.002$ for the UFF test and $-3\times10^{-20}<\hat{\alpha}_3<4\times10^{-20}$ at 95% confidence. These are the first direct UFF and $\hat{\alpha}_3$ tests based on pulsar binaries, and they overcome various limitations of previous tests.


INTRODUCTION
Einstein's equivalence principle (EEP) is one of the guiding ideas that aided Einstein to conceive the theory of general relativity (GR).EEP states that non-gravitational experiments in a local Lorentz frame should give the same result regardless of when and where they take place.This principle helped in establishing the idea that gravity is the manifestation of curved spacetime, which can be abstracted as a four-dimensional manifold endowed with a Lorentzian metric, where freely falling test bodies follow geodesics of that metric (universality of free fall), and the local non-gravitational laws of physics are those of special relativity.Gravity theories built upon this concept are called "metric theories of gravity", like GR and scalar-tensor theories.See Will (1993) for details.
The strong equivalence principle (SEP) extends the EEP by including local gravitational aspects of the test system (Will 2014).The universality of free fall is extended to self-gravitating bodies, which fall in an external gravitational field.Furthermore, local test experiments, including gravitational ones, should give the same results regardless of the location or velocity of the test system.It is conjectured that GR is the only gravity theory that fully embodies SEP. 1 Although in metric theories of gravity all matter fields couple only to one physical metric ("universal coupling"), alternatives to GR generally introduce auxiliary gravitational fields (e.g. one or more scalar fields) which ultimately lead to a violation of SEP at some point.For this reason, testing the symmetries related to SEP has strong potential to either exclude (or tightly constrain) alternative gravity theories or falsify GR.It is therefore a powerful tool in searching for new physics.To date, all experimental evidence supports SEP and, therefore, GR (Will 2014;Shao & Wex 2016).
The post-Newtonian parameterization (PPN) is a formalism introduced by Thorne & Will (1971) and Will & Nordtvedt (1972) to describe generically the potential deviation from GR in metric theories of gravitation at the post-Newtonian level.Through a set of simple assumptions, such as slow-motion, weak field, and no characteristic length scales in the gravitational interaction, the PPN formalism can encompass most metric theories using only ten param-⋆ E-mail:zhuww@mpifr-bonn.mpg.de 1 There is one known exception, which however is falsified by Solar system experiments (Deruelle 2011) eters.Most of these PPN parameters (or combinations of them) are directly related to a violation of specific aspects of SEP.For strongly self-gravitating bodies, like neutron stars, these PPN parameters become kind of body dependent quantities, which are functions of the compactness of the bodies of the system (see e.g.Damour 2009).Hence, one can have situations where a theory is in (nearly) perfect agreement with GR in the Solar system, but shows significant violations of SEP in the presence of strongly self-gravitating bodies.A particularly extreme example is spontaneous scalarization, which is a non-perturbative strong-gravity effect that is known for certain scalar-tensor theories (Damour & Esposito-Farese 1993).
Alternative theories of gravity, generally, also predict a temporal change in the locally measured Newtonian gravitational constant G, which is caused by the expansion of the Universe (Will 1993;Uzan 2011).Such a change in the local gravitational constant constitutes a violation of the local position invariance, which also refers to position in time.Hence, a varying G directly violates one of the three main pillars of SEP.One of the testable consequences of a change in G are changes in the orbital parameters of the Solar system and binary systems, in particular the size of an orbit and the orbital period.Again, the situation is more complicated in the presence of strongly self-gravitating bodies (Nordtvedt 1990;Wex 2014).
Some alternative theories of gravity violate SEP by introducing a preferred frame of reference for the gravitational interaction.Generally, this preferred frame can be identified with the global mass distribution in the universe, which is the frame in which the cosmic microwave background is isotropic, i.e. has no dipole.In the PPN formalism, there are three parameters, α 1 , α 2 , and α 3 , which are related to such kinds of symmetry breaking.The parameter α 3 is linked to two effects, a preferred frame effect and a violation of conservation of total momentum (Will 1993).In this paper, we test the PPN parameter α 3 through the fact that it causes an anomalous self-acceleration of a spinning body, which is proportional to and perpendicular to the object's spin and motion with respect to the preferred frame.Such acceleration would lead to an observable effect in a binary system, such as an anomalous drift in the eccentricity of the binary.PSR J1713+0747, thanks to its high spin frequency and measurable proper motion, has the best figure of merit for testing α 3 in the present.More precisely, with binary pulsars one tests the quantity α3 , which is a generalization of α 3 to a situation with strongly self-gravitating objects.Therefore, α3 also contains preferred frame effects related to the binding energy of the neutron star (cf.discussion in Damour & Esposito-Farèse 1992b;Will 2018) Some pulsar binary systems are particularly useful for certain tests of SEP (see Shao & Wex 2016, for a recent review).In this paper, we measure the change rate in the orbital periodicity and eccentricity of the pulsar-white dwarf binary PSR J1713+0747 and use that to test the following three aspects of SEP: 1. the gravitational constant's constancy; 2. the PPN parameter α3 in the context of Lorentz invariance of gravitation and conservation of total momentum; 3. the universality of free fall (UFF) for strongly selfgravitating bodies.Section 2.1 describes the pulsar timing, Section 2.2, 2.3, 2.4 describe the test for the gravitational constant's constancy, the universality of free fall test, and the test for α3 .In Section 3 we give the conclusion and the summary.

METHOD AND RESULTS
PSR J1713+0747 is a millisecond pulsar orbiting a 0.29 M ⊙ white dwarf.The pulsar's short spin period (4.5 ms) and narrow pulse profile enable us to measure its pulse time of arrivals (TOAs) at submicrosecond precision.This pulsar is monitored by both the North American Nanohertz Gravitational Observatory (NANOGrav), the European Pulsar Timing Array (EPTA) and the Parkes Pulsar Timing Array (PPTA) for the purpose of detecting nHz gravitational waves.A major part of the data used in this work come from the NANOGrav2 and EPTA programs.The EPTA3 is a collaboration of European institutes to work towards the direct detection of lowfrequency gravitational waves and the running of the Large European Array for Pulsars (LEAP).These data were taken between 1993 and 2014 using observatories including William E. Gordon Telescope of Arecibo observatory, Robert C. Byrd Green Bank Telescope, Effelsberg Telescope, Lovell Telescope of Jodrell Bank observatory, Nançay Radio Telescope, and Westerbork Synthesis Radio Telescope.Splaver et al. (2005) published the early dataset for J1713+0747 from Arecibo and Green Bank Telescope (GBT).Subsequently, Zhu et al. (2015) published the combined J1713+0747 data from Splaver et al. (2005) and NANOGrav 9yr data release (Arzoumanian et al. 2015).Desvignes et al. (2016) published the data and timing analysis for the EPTA pulsars including J1713+0747.This work presents the first timing analysis of the combined J1713+0747 data published in Zhu et al. (2015) and Desvignes et al. (2016).We are able to model very accurately the binary system's orbit through various time-delaying effects like Römer delay, Shapiro delay, and annual-orbital parallax.From these we can measure the masses of the two stars, the sky orientation and inclination angle of the orbit, and the position and proper motion of the system.The modeling was performed using the pulsar timing software T 2 (Edwards et al. 2006) and the best-fit parameters are listed in Table 1.

Pulsar Binary Timing
For the timing of PSR J1713+0747, we employ an extended version of the ELL14 pulsar binary model (Lange et al. 2001), which is valid for e ≡ |e| ≪ 1, where e is the eccentricity vector of the orbit.ELL1 models a pulsar binary orbit with small eccentricity by decomposing e into two orthogonal vectors e x and e y , where e x ≡ e cos ω and e y ≡ e sin ω, and ω is the longitude of periastron, i.e. the angle between e and the ascending node.We use e x to represent the component of e pointing from the centre of the orbit to the ascending node and e y represents the part pointing from Earth to the pulsar.Lange et al. (2001) showed that the Römer delay of a small eccentricity orbit can be expressed simply as ∆ R = x[sin φ + (e y /2) sin 2φ − (e x /2) cos 2φ], omitting higher order terms proportional to O(xe 2 ).Here x is the projected semi-major axis of the pulsar orbit in units of light-seconds, and φ ≡ n b (T −T asc ), where n b = 2π/P b is the orbital frequency, T the time of the pulsar and T asc the so-called time of the ascending node (see Lange et al. (2001) for details).However, we find that the precision of this expression is insufficient for modeling PSR J1713+0747's Römer delay due to its timing precision.To increase the precision of our timing model, we extend the ELL1 model by including the second order terms: to model the effect of a changing eccentricity, where e x and e y represent the change rate of e in time.The higher order terms of eq. ( 1) could then straightforwardly be added to the existing implementation of the ELL1 model in TEMPO2 (Edwards et al. 2006).
Apart from the changes in the binary model, the rest of the timing analysis follows those in Zhu et al. (2015) and Desvignes et al. (2016).For a high-timing precision pulsar such as PSR J1713+0747, it is necessary to employ a comprehensive noise model including dispersion measure (DM) variation, jitter noise, and red noise.Here we use two different approaches.The first based on the noise analysis technique described in van Haasteren & Levin (2013); Ellis (2013); Dolch et al. (2014); Zhu et al. (2015); Arzoumanian et al. (2015) and Dolch et al. (2016).In Dolch et al. (2016), the PSR J1713+0747 noise model enabled the strongest PTA-based νHz gravitational limit.Our analysis uses the DMX model that groups TOAs into epochs and fit for DM for each group respectively.We model jitter noise as a correlated noises between TOAs from the same observations and the red noise as a stationary Gaussian process with a power law spectrum.This analysis is conducted using the PAL2 software package5 (Ellis & van Haasteren 2017).The second approach is similar to the first one excepts that it models DM variation as a power-law Gaussian process (Lentati et al. 2013).This analysis is conducted using the T N6 software package (Lentati et al. 2014).We find consistent results in the best-fit timing and noise model from both approaches.In this paper, we are mostly interested in testing theories of gravitation.Therefore, we choose to base our GR tests on the result from the first approach (presented in Table 1) because it yields slightly more conservative uncertainty on e.
We use the solar system ephemeris DE421 (Folkner et al. 2009) in the timing analysis instead of the more recent DE436 (Folkner & Park 2016).Arzoumanian et al. (2018) showed that using DE436 leads to some marginally different timing results from using DE421.The discrepancies in different solar system ephemerides are in the masses and orbits of the outer solar system bodies, they cause extra timing residuals in time scales of the orbital periods of these bodies.PSR J1713+0747's orbital period is substantially smaller than those solar system bodies.Therefore, we argue that using different solar system ephemerides will have only marginal impact on our primary parameters of interest: P b and e.

Testing the Time Variation of G
Through pulsar timing, we measure the apparent change rate of the binary's orbital period ( P b = (0.34 ± 0.15) × 10 −12 s s −1 , Table 1).Despite being not statistically significant, the observed P b is consistent with what one expects from the apparent orbital period change caused by the binary's transverse motion (a.k.a.Shklovskii effect; Shklovskii 1970) and line-of-sight acceleration: Here, µ α and µ δ are the proper motion in right ascension and declination respectively, d is the distance from timing parallax, c is the speed of light, and A G is the system's line-of-sight acceleration computed using the Galactic potential in McMillan (2017) (see discussion in Appendix A).Subtracting these two (external) contributions from the observed P Obs b yields the residual change rate of the orbital period which is consistent with the much smaller and undetectable intrinsic change P GR b = −6 × 10 −18 s s −1 from quadrupolar gravitational radiation as predicted by GR.
This apparent consistency allows us to test the change rate of the (local) gravitational constant ( G) over the time span of the observation, since a G could lead to an observable change in P b (Damour et al. 1988), which has already been used to constrain G with binary pulsars (Damour et al. 1988;Kaspi et al. 1994;Nice et al. 2005;Verbiest et al. 2008;Deller et al. 2008;Lazaridis et al. 2009;Freire et al. 2012b;Zhu et al. 2015).Nordtvedt (1990) pointed out that a change of the gravitational constant also leads to changes in the neutron star's compactness and mass, and consequently leads to an additional contribution to P b which needs to be incorporated in our G tests with binary pulsars.If the companion is a weakly self-gravitating body, like in the case of PSR J1713+0747, then one finds to leading order (Nordtvedt 1990, cf. eq. ( 18) in) where M p , M c are the pulsar mass and the companion mass, respectively.The quantity s p denotes the "sensitivity" of the neutron star and the white dwarf and are given by (cf.Will 1993) respectively, where the number of baryons N is held fixed.The sensitivity s p of a neutron star depends on its mass, its equation of state (EoS), and the theory of gravity under consideration.As a reference, for Jordan-Fierz-Brans-Dicke (JFBD) gravity and the EoS AP4 in We will use this (simplified) relation in our generic calculations below but will keep in mind that depending on the EoS and the theory of gravity eq. ( 7) might only be a rough estimate.Furthermore, it is important to note, that the usage of the sensitivity (6) and eq.( 7) comes with certain assumptions about how gravity can deviate from GR in the strong field of neutron stars.It is evident that such a description cannot capture non-perturbative strong-field effects, like those discussed by Damour & Esposito-Farese (1993).For a weakly self-gravitating body A, one has is the gravitational binding energy of the body.Hence, one has s c ≃ 3 × 10 −5 for the white-dwarf companion to PSR J1713+0747 -negligible in eq. ( 5).
A time-varying gravitational constant generally indicates a violation of SEP.On the other hand, most gravitational theories that violate SEP also predict the existence of dipolar gravitational radiation (DGR).Such waves are very efficient in draining orbital energy from an (asymmetric) binary.The gravitational-wave damping due to DGR enters the equations-of-motion of a binary already at the 1.5 post-Newtonian (v 3 /c 3 ) level (see e.g.Mirshekari & Will 2013), and to leading order adds the following change to the orbital period where n b ≡ 2π/P b (Will 1993).The quantity κ D is a bodyindependent constant, which depends on the fundamental parameters of the gravity theory under consideration.In JFBD gravity, for instance, one finds that where ω BD is the Brans-Dicke parameter (Will 1993). 7For completeness, we have kept s c in eq. ( 8) although, as mentioned above, it is negligible in our case.
As emphasized by Lazaridis et al. (2009), in a theory-agnostic approach the test of G and κ D requires at least two pulsar binary systems with different orbital periods to break the degeneracy between the two contributions eq. ( 5) and eq.( 8).This is because the extra variation in the orbital period due to dipolar gravitational radiation is stronger in binaries with shorter orbits (∝ P −1 b ) while the that caused by G increases with orbital period (∝ P b ).Therefore testing G/G using binaries of significantly different orbital periods breaks the degeneracy between the two effects (see Lazaridis et al. 2009, for further details).Here we adapt the method of Lazaridis et al. (2009) and incorporate results from three different pulsar-white dwarf systems, namely PSRs J0437−4715 (Reardon et al. 2016), and J1738+0333 (Freire et al. 2012b), in combination with the results for PSR J1713+0747 obtained in this work.Each pulsar provides a constraint on P D b + P G b , and hence via eqs.( 5) and ( 8) excludes certain regions in the G/G-κ D plane.As a result of the large difference in orbital period, these constraints are complementary and consequently lead to a small region of allowed values in the G/G-κ D plane (see Figure 1).The individual constraints on G and κ D are G/G = (−0.1 ± 0.9) × 10 −12 yr −1 , (10)  The shaded area and gray area mark the 95% confidence limit from LLR (Hofmann et al. 2010) and planetary ephemerides (Fienga et al. 2015), respectively.

Testing the Universality of Free Fall for Strongly Self-gravitating Bodies
One of the important pillars of SEP is the extension of the universality of free fall (UFF) to objects with significant gravitational binding energy E grav , i.e. the weak equivalence principle (WEP) is valid for test particles as well as for self-gravitating bodies.Every metric theory of gravity, by definition, fulfills WEP for test particles.
On the other hand, alternatives to GR are usually expected to violate WEP in the interaction of self-gravitating bodies (Will 1993(Will , 2014)).According to such theories, objects with different binding energy feel different accelerations in an external gravitational field g.More specifically, a binary system composed of two stars with different compactness would undergo a "gravitational Stark effect" that polarizes the binary orbit in a characteristic way.In the Earth-Moon system, this is called the Nordtvedt effect and has been tightly constrained by Lunar Laser Ranging (Nordtvedt 1968;Müller et al. 2012;Williams et al. 2012).Pulsar binary systems falling in the gravitational field of our Galaxy would (slowly) oscillate between a more and less eccentric configuration.Damour & Schäfer (1991) showed that the observed eccentricity is a combination of an intrinsic eccentricity and a forced eccentricity: Here F is a theory-dependent (and "sensitivity"-dependent) factor that accounts for potential deviations from GR in the rate of periastron advance ω.By definition, F = 1 in GR and indeed, it is constrained to be close to 1 by observations.For instance, from the Double Pulsar PSR J0737-3039A/B (Burgay et al. 2003;Lyne et al. 2004) one can quite generically infer that |F − 1| 10 −3 (Kramer & Wex 2009).Therefore, we can safely assume F = 1 in our analysis, in particular since the mass of PSR J1713+0747 is comparable to the masses in the Double Pulsar.G ≈ G is the effective gravitational constant in the interaction between the pulsar and the white dwarf.The vector g ⊥ is the projection of the Galactic acceleration g onto the orbital plane, and ∆ is the fractional difference in the accelerations between the pulsar and white dwarf, and therefore a dimensionless measure of the significance of the UFF violation.Damour & Schäfer (1991) have put forward a method to constrain ∆ from small-eccentricity binary pulsars with white dwarf companions, utilizing in probabilistic considerations the smallness of the observed eccentricities.This so-called "Damour-Schäfer" test has been extended to make use of an ensemble of suitable pulsarwhite dwarf systems (Wex 1997;Stairs et al. 2005) of which the precise orbital orientations and proper motions are unknown.The currently best limits from this method are |∆| < 5.6× 10 −3 (Stairs et al. 2005) and |∆| < 4.6 × 10 −3 (Gonzalez et al. 2011) 8 .The validity and effectiveness of the Damour-Schäfer test rely on some (strongfield and probabilistic) assumptions.It does not improve with timing precision and is not capable of actually detecting a violation of the UFF (see discussions in Damour 2009;Freire et al. 2012a).For this reason, it is desirable to have a direct test with a single binary pulsar.Freire et al. (2012a) already identified PSR J1713+0747 as a potential candidate for such a test.In Zhu et al. 2015, PSR J1713+0747 had been used in a (single system) Damour-Schäfer test.In this work, we utilize the exquisite timing precision of PSR J1713+0747 and the tight limit on e to directly test the violation of UFF in the strong-field regime.
Based on the equation of motion, Damour & Schäfer (1991) derived that the eccentricity vector of the pulsar orbit would change according to (neglecting terms of order e 2 and smaller) for given violation parameter of UFF, ∆.Here the vector e points at periastron, k is a unit vector parallel to orbital angular momentum, is the relative orbital velocity between the two stars.The second term describes the post-Newtonian periastron advance rate: The extended ELL1 timing model allows us to measure the change rate of the eccentricity vector, and we do detect an apparent e y = (−1.7 ± 0.4) × 10 −17 s −1 along the line of sight, likely coming from the periastron advance of the orbit.After removing the contributions of periastron advance ( e PN x = −0.07× 10 −17 s −1 , e PN y = −1 × 10 −17 s −1 ) according to the measured system masses and orbital parameter (Table 1) and eq.( 14), the resulting excess eccentricity change rate [ e exc x = (0.4 ± 0.4) × 10 −17 s −1 , e exc y = (0.7 ± 0.4) × 10 −17 s −1 ] is consistent with zero.We perform a Monte-Carlo Markov Chain (MCMC) simulation that simultaneously fits both the timing model and the dispersion, jitter and red noises, using the PAL2 software.The MCMC allows us to obtain a large sample of possible timing parameter values along with their likelihood.We then use these MCMC results to calculate the ∆ needed to account for the residual e excess.Figure 2 shows the posterior distribution of ∆.From this result we derive that −0.0007 < ∆ < 0.0023 with 95% confidence level (C.L.).The -0.002 -0.001 0 0.001 0.002 0.003 0.004

∆ Normalized Likelihood
Figure 2. The normalized likelihood distribution of ∆ derived from the MCMC of PSR J1713+0747's timing and noise parameters.Solid line is the 95% confidence limit, dashed line marks the 99% confidence limit.
deviation from GR is insignificant.For the ease of comparison with previous results, we also derive from the above results: This constraint improves the previous best pulsar test of UFF in gravitation (Gonzalez et al. 2011) by more than a factor of two.More importantly, it is a direct test and therefore, as discussed above, more robust than previous Damour-Schäfer test based limits.We discuss the theoretical meaning of limit in eq. ( 15) in Section 3.

Testing the Lorentz Invariance and Conservation of Momentum in Gravitation
The PPN parameters α 1 , α 2 , and α 3 describe different symmetry breaking effects in gravitational theories that violate local Lorentz invariance in the gravitational sector.The parameters α 1 and α 2 have already been tightly constrained through binary and isolated pulsars (Shao & Wex 2012;Shao et al. 2013).In this paper, we focus on the test of α 3 , but we also account for those effects related to α 1 and α 2 which in principle could have an influence on our α 3 limits.The parameter α 3 describes a gravitational symmetry breaking that leads to both the existence of a preferred frame and a violation of conservation of momentum (Will 1993).A non-zero α 3 would give rise to an anomalous self-acceleration for a spinning self-gravitating body that moves in the preferred reference frame.One often assumes that the universal matter distribution selects the rest frame of the Cosmic Microwave Background (CMB) as the preferred frame.Following the same idea, we choose CMB frame as the preferred frame in our analysis.However, our constraint on α3 should be robust for preferred frames that moves at low velocity with respect to the pulsar, and it gets stronger when the selected frame is moving very fast with respect to the pulsar.A weakly self-gravitating body with mass M, gravitational binding energy E grav , rotational frequency ν, and velocity v CMB in the CMB frame, would undergo acceleration induced by α 3 effects: Here ns is a unit vector in the direction of the body's spin (Will 1993).For strongly self-gravitating bodies, following Bell & Damour (1996), we replace E grav /Mc 2 by the sensitivity of the pulsar s p . 9 Furthermore, we replace α 3 by α3 , where the hat symbol serves as a reminder that we are testing the (body-dependent) strong-field extension of α 3 , which may be different from the weakfield α 3 , but is expected to be of the same order (see Section 3 for a more detailed discussion).Similar to the violation of UFF, the acceleration caused by α3 would lead to a polarization of the orbit of a rapidly rotating pulsar (Bell & Damour 1996).For fully recycled millisecond pulsars like PSR J1713+0747, ns likely aligns with the orbital angular momentum.For this reason, we can set ns = k in our calculations.
Besides the influence of a non-vanishing α 3 , one also needs to account for contributions of α 1 and α 2 to the orbital dynamics, since in general one cannot assume α 1 and α 2 to be zero in a theory that breaks local Lorentz invariance in the gravitational sector and gives rise to a α 3 .In a near-circular binary, a non-vanishing α 2 would lead to a precession of the orbital angular momentum around the direction of w, which in turn leads to a temporal change in the projected semi-major axis x (Shao & Wex 2012).On the other hand |α 2 | is constrained by pulsar experiments to be less than about 10 −9 , which corresponds to a change in x that is about six orders of magnitude smaller than the contribution from proper motion, which is used to determine Ω. Therefore effects from α 2 can be safely neglected in our α 3 test.
A non-vanishing α 1 adds to the polarization of the orbit in the same way as a non-vanishing α 3 , and analogous to eq. ( 13) one finds for the change of the orbital eccentricity vector where (Damour & Esposito-Farèse 1992b) and (Bell & Damour 1996).The velocity w ⊥ is the projection of the systemic CMB frame velocity into the orbital plane of the binary pulsar.To independently constrain α3 from PSR J1713+0747 timing, we will include the α1 limits obtained from PSR J1738+0333 by Shao & Wex (2012) in our analysis.Conversely, owning to its small orbital period, PSR J1738+0333's possible α3 effects would be much smaller than its α1 effects and would have insignificant impact on the α1 limits derived from this system.There is one assumption, however, that we have to make here.Since PSR J1713+0747 and PSR J1738+0333 have different masses, and therefore different sensitivities s p , we cannot assume that they would lead to identical α1 .For this reason, our analysis only applies to deviations from GR which exhibit only a moderate mass dependence of the strong-field parameter α1 , at least for neutron stars in the range of 1.3 to 1.5 M ⊙ .All parameters involved in the evaluation of e are measurable through the timing of PSR J1713+0747 (Table 1), except for the radial velocity of the pulsar binary with respect to the Solar system v r .To calculate e from eq. ( 17) one needs the system's threedimensional (3D) velocity v CMB in the CMB rest frame.v CMB can in principle be computed from the binary's 3D velocity v about our Solar System by adding the Solar System speed in the CMB rest frame, which is well known from the measurement of the CMB 9 Bell & Damour (1996) introduced the compactness c p = 2s p .dipole (see Aghanim et al. 2014, for the latest measurement).We measure the pulsar's proper motion and distance (Table 1), which allows us to determine the transverse component of v.The white dwarf companion of PSR J1713+0747 is relatively faint (Lundgren et al. 1996), and a measurement of the radial velocity v r through optical spectroscopy is currently not available.Therefore, we treat v r as a free parameter and calculate the limits on α3 as a function of v r .A limit for v r comes from the plausible assumption that the PSR J1713+0747 system is bound to the Galaxy and therefore must be slower than the Galactic escape velocity.Taking the Galactic potential of McMillan (2017), we find v r to be within the range of about −680 to +460 km/s.10As shown in Figure 3, the constraint on α3 tightens as |v r | gets larger because large v r lead to large v CMB , which enhance the polarization effects.We find that in the most conservative scenario by taking the minimum α3 value from the left-side of the 95% C.L. contour as the lower bound and the maximum value from the right-side of the contour as the upper bound (despite the fact that the two values correspond to different values of v r ): This result is better than the previous best constraint on α3 , | α3 | < 5.5 × 10 −20 from (Gonzalez et al. 2011), which is based on a Damour-Schäfer type of test using an ensemble of pulsars.

DISCUSSION
In this paper, we present new tests of the constancy of G, and the violations of UFF, Lorentz invariance, and conservation of momentum in gravitation.These violations of gravitational symmetries lead to changes in the orbital period and eccentricity particularly in binaries such as pulsar-white dwarf systems like J1713+0747.We conduct these tests through the measurement of an excess change in the orbital period and eccentricity from the timing analysis of PSR J1713+0747.
We repeat the combined G/G, κ D test presented in Zhu et al. (2015) by incorporating more PSR J1713+0747 timing data from EPTA and find the result improves [ G/G = (−0.1±0.9)×10−12 yr −1 and κ D = (−0.7 ± 2.2) × 10 −4 ].The improvement on G/G could be attributed to the inclusion of EPTA data which increased the number of TOAs used in the experiment, whereas the changes in κ D limits are mostly due to the change from using a fiducial sensitivity function that scales linearly with neutron star mass to using a more realistic non-linear neutron star sensitivity function.Our κ D test is a generic test of dipolar gravitational radiation, included for the purpose of generalizing the G/G test for general SEP-violating theories.More stringent tests of the dipolar gravitational radiation effects could be done with pulsar timing if one takes into account the nature of the SEP-violating theories (Freire et al. 2012a).In some theories, O(s 3 p ) terms cannot be neglected (cf.eq. ( 8)) even in a first order estimation.In some theory-specific tests, a more stringent limit on κ D could come from non-radiative tests, for instance from the Solar system measurements based on the Cassini experiment, like for JFBD gravity where Cassini implies |κ D | 9 × 10 −5 with 95% confidence (cf.eq. ( 9)).
When compared directly to the Solar System tests (LLR (Hofmann et al. 2010) and planetary ephemerides (Fienga et al. 2015); Figure 1), our G/G constraint is not as tight.But the pulsar binary tests involve objects of much stronger self-gravitation than objects in the Solar System.The pulsar timing limits of G/G and κ D are testing SEP-violating effects beyond linear extrapolations from the weak-field limit.Wex (2014) demonstrated that in certain theories of gravitation, G/G effects could be greatly enhanced by a strongly self-gravitating body while remaining insignificant in the Solar System.
Previous pulsar UFF tests, such as Wex (1997); Stairs et al. (2005) and Gonzalez et al. (2011) employed the idea of Damour & Schäfer (1991), which uses an ensemble of wide-orbit small-eccentricity pulsar-white dwarf binaries.The effectiveness of that approach relies on the smallness of e and the statistical argument that the unknown orientations of the orbits from a collection of pulsars are randomly and uniformly distributed.Hence the Damour & Schäfer (1991) tests cannot directly detect SEP violation and may only improve when a new pulsar binary with a better figure of merit is found.There comes a further caveat with previous tests, as these tests are based on an ensemble of systems with different neutron-star masses.Hence, for this mix of neutron-star masses a priori assumptions about the strong-field behavior of gravity had to be made (Damour 2009).For our tests this is not the case, since these tests are based on a single neutron-star with well determined mass.Damour & Schäfer (1991) also pointed out the possibility of directly testing UFF violations by constraining temporal changes in the orbital eccentricity.Such a test has the advantage of not depending on a group of pulsar binaries, the smallness of their e and assumptions on their orbital orientations.The effectiveness of this test improves with timing precision.Freire et al. (2012a) identified PSR J1713+0747 as one of the best candidates for such a direct test.But at that time e was not directly modeled in the timing of that pulsar, and Freire et al. (2012a) used an estimate of the upper limit of e based on the uncertainties of the measured e.As a result, Freire et al. (2012a) could put some preliminary limits on UFF violations.In this paper, we conduct the first direct UFF test with a measured e, which in principle could detect a violation of UFF, should the effect be strong enough.
Using an extended version of the ELL1 timing model of Lange et al. (2001) in our analysis, we find e ≃ (0.4 ± 0.4, −1.7 ± 0.4)×10 −17 s −1 (see Table 1), which is consistent with being caused by post-Newtonian periastron advance predicted by GR.We find no evidence for a violation of UFF with |∆| < 0.002, a result that improves by more than a factor of 2 from the previous best constraint (Stairs et al. 2005;Gonzalez et al. 2011).Similarly, we find −3 × 10 −20 < α3 < 4 × 10 −20 , which is also better than the previous best result.These limits go beyond the PPN framework since they also capture strong field deviations.To illustrate this, for example for α 3 , we expand α3 with respect to the sensitivity then the limit in eq. ( 20) not only constrains the weak-field counterpart, α 3 , at the level of O(10 −20 ), but also poses strong constraints on higher-order terms, α 3 , α 3 , and so on.The same applies for the strong-field generalization of the Nordtvedt parameter, that is given by ∆ ≡ ηN s p .Detailed accurate mapping of strong-field generalization and weak-field counterpart needs explicit calculations in specified gravity theories.Note, equations like (21) ultimately fail to capture non-perturbative strong field deviations, like spontaneous scalarization (Damour & Esposito-Farese 1993).
Our UFF test using PSR J1713+0747 is expected to be surpassed by tests using the recently discovered pulsar in a stellar triple system (Ransom et al. 2014), as that system has a much stronger external gravitational field provided by the third star.Simulations suggest an improvement by at least three orders of magnitude (Berti et al. 2015;Shao 2016;Kramer 2016).However, PSR J1713+0747 will remain to be one of the best systems for testing G with pulsars, due to its wide orbit and high timing precision, and for testing α3 because of the fast rotation of this pulsar and its well-constrained orbit.

APPENDIX A: GALACTIC CORRECTIONS
In Section 2.2, we discuss how the changing Doppler effect related to the systemic motion of a binary pulsar causes apparent variations in the orbital parameters.An important part of the changing Doppler effect comes from the Galactic contribution to the line-ofsight acceleration between the pulsar binary and the Solar System.There are two orthogonal components in this Galactic contribution: the difference in the centrifugal acceleration in our circular motion around the Galactic center, and the difference in the vertical acceleration caused mainly by the Galactic disk.We can derive the expected P Gal b /P b based on our knowledge of the Galactic rotation curve and the local surface density of the Galactic disk.
Based on Damour & Taylor (1991), Nice & Taylor (1995) derived an analytical formula for calculating the apparent orbital variation using a flat rotation curve and the vertical acceleration model for the Solar vicinity.The subsequent pulsar timing works (Deller et al. 2008;Verbiest et al. 2008;Lazaridis et al. 2009;Freire et al. 2012b;Zhu et al. 2015) used the same formula with the most recent Galactic disk model coming from Holmberg &Flynn 2004 andReid et al. 2014.However, the analytical approach put forward by Damour & Taylor (1991); Nice & Taylor (1995) is a good approximation only when the pulsar systems are close to the Solar System or have a comparable distance to the Galactic center.As shown in Figure A1, J1713+0747 is about 1 kpc closer to the Galactic center than the Sun.Therefore, it experiences a slightly higher vertical gravity than modeled previously in Zhu et al. (2015).
In this work, we employ a new Galactic model by McMillan (2017), which fits the rotation curve and the stellar dynamic data with an axisymmetric Galactic potential and provides a code for computing the Galactic gravitational acceleration.It is worth noting that the key input data for constraining their model regarding vertical forces were from Kuijken & Gilmore 1991 -an earlier study than Holmberg & Flynn 2004.But as shown in Figure A1, the later Holmberg & Flynn 2004result fits McMillan (2017) model better and the changes were relatively small.Table A1 shows J1713+0747's P Gal b computed using various Galactic potential models and the observed excess after removing Shklovskii and GR effects.One can see that most realistic models (Dehnen & Binney 1998;Binney & Tremaine 2008;Piffl et al. 2014;McMillan 2017) fit the observed excess better than the analytical approximation.
Figure A2 shows the ∆ P Gal b /P b from using the McMillan 2017 model instead of Nice & Taylor 1995 for a putative pulsar binary at 1 kpc distance as a function of Galactic longitude and latitude.The correction becomes of the same order of magnitudes as the observed effect | P b /P b | (∼ 10 −12 yr −1 ) for J1713+0747.PSR J0437-4715 is another pulsar binary system that is sensitive to this effect (Verbiest et al. 2008;Deller et al. 2008;Reardon et al. 2016).However, at a distance of only 0.16 kpc, this pulsar's Galactic acceleration could be computed fairly accurately by Nice & Taylor 1995 model, with a |∆ P Gal b /P b | < 0.2 × 10 −12 yr −1 .For the other pulsar binary used in our G/G analysis -PSR J1738+0333, the correction is similar to that of J1713+0747, but still relatively insignificant compared to the observational uncertainty δ P b /P b of ∼ 4 × 10 −12 (Freire et al. 2012b).
Since Nice & Taylor 1995 used the local disk surface density and assumed a flat rotation curve, the real Galactic potential deviates more from it at greater distances from the Sun.In Figure A3, we show that at a distance of 3 kpc the extra Galactic correction |∆ P b /P b | (or |∆ P/P| for the pulsar period) comes close to 10 −11 yr −1 for some directions, therefore, a more realistic model must be used for pulsars in this situation.
Presently, there are only a handful of pulsars that are sensitive to the Galactic acceleration.In the future, new telescopes such as the Five hundred meter Aperture Spherical Telescope (FAST;Nan et al. 2011;Li & Pan 2016) and the Square Kilometer Array (SKA; Kramer & Stappers 2015) could improve both the timing and distance measurement for many more pulsars such that this correction becomes significant for them.It might be possible to start using some pulsars as accelerometers for probing the Galactic gravity field and improving our knowledge of the Galactic potentials, independent to the improvements expected from GAIA 11 (Prusti et al. 2016) .

Figure 3 .
Figure 3.The likelihood distribution of α3 derived from the MCMC of PSR J1713+0747's timing and noise parameters as a function of the assumed line-of-sight velocity v r .The black solid curves corresponds to the 95% C.L. and the black dashed line to the 99% C.L. The red curves shows the same limits with α1 taken into account (see Section 2.4 for details).The two horizontal lines labels the escape velocity of the pulsar binary.

Figure A1 .
Figure A1.The solid curves show the models of Galactic disk surface density as a function of Galactic radius from McMillan (2017).The error bars indicate disk surface densities from studies of the dynamics of stars in the Solar vicinity by Kuijken & Gilmore (1991) (red diamond) and Holmberg & Flynn (2004) (blue circle).The vertical lines mark the positions of the pulsar and the Sun.

Figure A2 .Figure A3 .
Figure A2.The difference between the Galactic corrections ( P Gal /P and P Gal b /P b ) derived from McMillan (2017) and the Nice & Taylor (1995) approximation as a function of Galactic longitude (l) and latitude (b) for a pulsar binary at 1 kpc distance from us in units of 10 −12 yr −1 .

Table 1 .
Arzoumanian et al. (2015) from T 2. T2 binary model to include higher order corrections from the ELL1 model.Numbers in parentheses indicate the uncertainties on the last digit(s).Uncertainties on parameters are estimated from the result of MCMC process in which the timing and noise model were evaluated.bTheaveragedDMvaluebasedontheDMX model.cMostpulsartimingmodelparameterspresented in this paper are consistent with those reported inZhu et al. (2015)except for the orbital period P b .This is because P b is defined differently in ELL1 model used here from the DD model used inZhu et al. (2015).In DD model, P b is defined as the time between two periastron passings, while in ELL1 model, P b is the time between two ascending node passings.dSeeZhuetal.(2015)andArzoumanian et al. (2015)for the description and discussion of the FD model.
Damour & Esposito-Farèse (1992a)ito-Farèse (1992a), we further assume, as a first order approximation, that s p is proportional to the mass