Strong Equivalence Principle and Gravitational Wave Polarizations in Horndeski Theory

Eur. Phys. J. C (2019) 79:197 https://doi.org/10.1140/epjc/s10052-019-6684-9

Regular Article - Theoretical Physics

Strong equivalence principle and gravitational wave polarizations in Horndeski theory

Shaoqi Houa , Yungui Gongb School of Physics, Huazhong University of Science and Technology, Wuhan 430074, Hubei, China

Received: 4 October 2018 / Accepted: 14 February 2019 / Published online: 5 March 2019 © The Author(s) 2019

Abstract The relative acceleration between two nearby of GWs was measured for the first time, and the pure ten- particles moving along accelerated trajectories is studied, sor polarizations were favored against pure vector and pure which generalizes the geodesic deviation equation. The scalar polarizations [6]. Similar results were reached in the polarization content of the gravitational wave in Horndeski recent analysis on GW170817 [10]. More interferometers are theory is investigated by examining the relative accelera- needed to finally pin down the polarization content. Other tion between two self-gravitating particles. It is found out detection methods might also determine the polarizations of that the apparent longitudinal polarization exists no matter GWs such as pulsar timing arrays (PTAs) [11Ð14]. whether the scalar field is massive or not. It would be still Alternative metric theories of gravity may not only intro- very difficult to detect the enhanced/apparent longitudinal duce extra GW polarizations, but also violate the strong polarization with the interferometer, as the violation of the equivalence principle (SEP) [15].1 The violation of strong strong equivalence principle of mirrors used by interferom- equivalence principle (vSEP) is due to the extra degrees of eters is extremely small. However, the pulsar timing array freedom, which indirectly interact with the matter fields via is promised relatively easily to detect the effect of the vio- the metric tensor. This indirect interaction modifies the self- lation as neutron stars have large self-gravitating energies. gravitating energy of the objects and leads to vSEP [17]. The The advantage of using this method to test the violation of self-gravitating objects no longer move along geodesics, even the strong equivalence principle is that neutron stars are not if there is only gravity acting on them, and the relative acceler- required to be present in the binary systems. ation between the nearby objects does not follow the geodesic deviation equation. In the usual approach, one assumes that the test particles, such as the mirrors in the aLIGO, move 1 Introduction along geodesics, so their relative acceleration is given by the geodesic deviation equation. Since the polarization content Soon after the birth of General Relativity (GR), several alter- of GWs is determined by examining the relative accelera- native theories of gravity were proposed. The discovery tion, the departure from the geodesic motion might effec- of the accelerated expansion of the Universe [1,2]revives tively result in different polarization contents, which can be the pursuit of these alternatives because the extra fields detected by PTAs. Thus, the main topic of this work is to might account for the dark energy. Since Sep 14th, 2015, investigate the effects of vSEP on the polarization content of LIGO/Virgo collaborations have detected ten gravitational GWs and the observation of PTAs. wave (GW) events [3Ð9]. This opens a new era of probing To be more specific, the focus is on the vSEP in the scalar- the nature of gravity in the highly dynamical, strong-field tensor theory, which is the simplest alternative metric theory regime. Due to the extra fields, alternatives to GR gener- of gravity. The scalar-tensor theory contains one scalar field φ ally predict that there are extra GW polarizations in addition besides the metric tensor field gμν to mediate the gravitational to the plus and cross ones in GR. So the detection of the interaction. Because of the trivial transformation of the scalar polarization content is very essential to test whether GR is field under the diffeomorphism, there are a plethora of scalar- the theory of gravity. In GW170814, the polarization content tensor theories, such as BransÐDicke theory [18], Einstein-

1 Nordstrom’s scalar theory of gravity satisfies SEP, but is not a metric a e-mail: [email protected] theory, not to mention that it has been excluded by the observations b e-mail: [email protected] [16]. 123 197 Page 2 of 12 Eur. Phys. J. C (2019) 79 :197 dilaton-GaussÐBonnet gravity (EdGB) [19] and f (R) grav- only has to observe the change in the arriving time of radial ity [20Ð22]. In 1974, Horndeski constructed the most general pulses from neutron stars without requiring the neutron stars scalar-tensor theory [23]. Its action contains higher deriva- be in binary systems. tives of φ and gμν, but still gives rise to at most the second This work is organized as follows. Section 2 reviews the order differential field equations. So the Ostrogradsky insta- derivation of the geodesic deviation equation, and a general- bility is absent in this theory [24]. In fact, Horndeski theory ized deviation equation for accelerated particles is discussed includes previously mentioned theories as its subclasses. In in Sect. 3. Section 4 derives the motion of a self-gravitating this work, the vSEP in Horndeski theory will be studied. object in presence of GWs in Horndeski theory. The polar- Among the effects of vSEP, Nordtvedt effect is well- ization content of GWs in Horndeski theory is revisited by known for a long time [25,26], and happens in the near zone taking the vSEP into account in Sect. 5. The generalized devi- of the source of the gravitational field. It leads to observation equation is computed to reveal the polarization content able effects. For example, the Moon’s orbit around the Earth of GWs. Section 6 calculates the cross-correlation function will be polarized when they are moving in the gravitational for PTAs due to GWs. Finally, Sect. 7 briefly summarizes field generated by the Sun [27,28]. The polarization of the this work. Penrose’s abstract index notation is used [41]. The Moon’s orbit has been constrained by the lunar laser ranging units is chosen such that the speed of light c = 1 in vacuum. experiments [29], which gave the Nordtvedt parameter [30] − η = (0.6 ± 5.2) × 10 4, (1) N 2 Geodesic deviation equation which measures vSEP in the following way, This section serves to review the idea to derive the geodesic mg = 1 + η ε + O(ε2 ), (2) deviation equation following Ref. [42]. In the next sec- m N grav. grav. i tion, the derivation will be generalized to accelerated objects with mg and mi the gravitational and the inertial masses, straightforwardly. ε and grav. the ratio of the gravitational binding energy to the Let γs(t) represent a geodesic congruence, in which each inertial energy. A similar polarization of the orbit of the mil- geodesic is parameterized by t and labeled by s. Define the lisecond pulsar-white dwarf (MSP-WD) system also hap- following tangent vector fields, pens due to the gravitational field of the Milky Way [31,32]. ∂ a ∂ a In contrast with the Moon and the Earth, pulsars have large T a = , Sa = . ∂ ∂ (3) gravitational binding energies, so the observation of the orbit t s polarization of MSP-WD systems set constraints on vSEP in Sa is called the deviation vector. Their commutator vanishes, the strong field regime, which was discussed in Ref. [32]. The b a b a observation of a triple pulsar PSR J0337+1715 was used T ∇b S = S ∇bT . (4) to set Δ = (−1.09 ± 0.74) × 10−6 [33]. The vSEP also With a suitable parametrization, one requires that T b∇ T a = leads to the dipole gravitational radiation, and the variation b 0 so that t is an affine parameter. Note that it is not necessary of Newton’s constant G [34]. The dipole gravitational radia- to set T a T =−1 for the following discussion. Whenever tion for Horndeski theory has been studied in Ref. [35], and a desired, one can always reparameterize to normalize it. It is constraints on this theory were obtained. The pulsar timing now ready to derive the geodesic deviation equation, observation of the binary system J1713+0747 has leads to ˙ / = (− . ± . ) × −12 −1 |Δ| < . a = c∇ ( b∇ a) =− a c b d , G G 0 1 0 9 10 yr and 0 002 [36]. Arel T c T b S Rcbd T S T (5) As discussed above, none of the previous limits on vSEP was obtained directly using the GW. So probing vSEP by using Eq. (4). For details of derivation, please refer to measuring the GW polarizations provides a novel way to test Ref. [42]. a GR in the high speed and dynamical regime. It will become The deviation vector S is not unique. A new parametriza- clear that although the vSEP will effectively enhance the tion of the geodesics, longitudinal polarization, it is still very difficult for aLIGO t → t = α(s)t + β(s), (6) to detect the effects of the longitudinal polarization, as the vSEP by the mirror is extremely weak. In contrast, neutron results in the change in Sa by a multiple of T a, stars are compact objects with non-negligible self-gravitating a T d t − β(s ) energies. The vSEP by neutron stars is strong enough that the T a = , S a = Sa + T a. (7) α(s) ds α(s) stochastic GW background will affect their motions, which is reflected in the cross-correlation function for PTAs [37Ð40]. Therefore, there is a gauge freedom in choosing the deviation By measuring the cross-correlation function, it is probably vector field Sa. This gauge freedom will be used frequently easier to detect the presence of vSEP. For this purpose, one below to simplify the analysis. 123 Eur. Phys. J. C (2019) 79 :197 Page 3 of 12 197

a Firstly, there is a parametrization such that Ta T is a con- equation can be derived similarly, stant along the coordinate lines of the constant t, i.e., the a =− a c b d + b∇ a. integral curves of Sa. In fact, one knows that, Arel Rcbd T S T S b A (12) Again, the derivation of this result does not reply on the Sb∇ (T T a) = 2T Sb∇ T a = 2T T b∇ Sa, (8) b a a b a b gauge fixing made similarly in the previous section or the and under the reparameterization (6), one gets one to be discussed below. Compared with Eq. (5), there is one extra term, which is due to the fact that the trajectories are ∂ − β( ) b a 2 b a t s a no longer geodesics. This equation and a more general one S ∇b(T T )= Ta T ∇b S + T , a α(s)2 ∂s α(s) were derived in Ref. [43] using the definitions of curvature (9) and torsion. The authors did not discuss the suitable gauge for extracting physical results which will be presented below. a so it is always possible to choose a parametrization to achieve If T Aa = 0, one can reparameterize the integral curves b∇ ( a) = that S b Ta T 0. Physically, this means that all of T a to make it vanish. Indeed, a reparameterization t → geodesics are parameterized by the “same” affine parame- t = κ(t) leads to ter t. Secondly, under the above parametrization, the inner a a = b∇ a product T Sa can be made constant along the geodesics, A T bT Aa κ¨ (13) b∇ ( a ) = b∇ a = − T a, T b T Sa Ta T b S κ˙ 2 κ˙ 3 = b∇ a Ta S bT where dot denotes the derivative with respect to t. So one can (10) a 1 b a always find a new parametrization which annihilates T Aa, = S ∇b(T Ta) 2 that is, = 0. Aa T κ(t) = α exp a t dt + β, (14) a b An initial choice of T Sa = 0 will be preserved along the T Tb a t coordinate line, so that S is always a spatial vector field a with α, β integration constants. From now on, T Aa = 0is a = a/ − b for an observer with 4-velocity u T TbT along its assumed which implies that trajectory. b a From the derivation, one should be aware that the geodesic T ∇b(T Ta) = 0. (15) deviation equation (5) is independent of the gauge choices τ made above, which only serves to make sure Sa is always So although t may not be the proper time , it is a linear τ = α + β a spatial vector relative to an observer with ua. In this way, function of . A further reparameterization t t there is no deviation in the time coordinate, that is, no time does not change the above relation. σ ( ) dilatation. This is because one concerns the change in the Now, pick a congruence of these trajectories s t .Soas σ ( ) spatial distance between two nearby particles measured by in the previous section, s t ’s also lie on a 2-dimensional ( , ) either one of them. surface parameterized by t s . There also exists the similar gauge freedom to that discussed in Sect. 2, except that Aa depends on the gauge choice. For example, a reparametriza- tion t → t = α(s)t + β(s) results in changes in Sa (given 3 Non-geodesic deviation equation by Eq. (7)) and Aa, i.e., Aa → Aa/α2(s). With this gauge freedom, one also chooses a suitable When particles are accelerated, they are not moving on a gauge such that T Sa remains constant along each trajec- geodesics. This happens when there are forces acting on these tory. In fact, it can be shown that particles. This also happens for self-gravitating particles in the modified gravity theories, such as the scalarÐtensor the- b a a 1 b a T ∇b(T Sa) = Sa A + S ∇b(T Ta). (16) ory. Suppose a bunch of particles are accelerated and there- 2 a a fore, their velocities satisfy the following relations, One requires that T Sa = 0 along the integral curves of T , b∇ ( a ) = b a a i.e., T b T Sa 0. This implies that T ∇bT = A , (11) b a a S ∇b(T Ta) =−2Sa A . (17) with Aa the 4-acceleration and not proportional to T a.In a the following, T is assumed to be some arbitrary timelike This expression means that if the trajectory σ0(t) is param- vector field which is not necessarily the 4-velocity of some eterized by the proper time t = τ, a nearby trajectory σs(t) particle. In this general discussion, the only assumption is with s = 0 will not be parameterized by its proper time, in that T a satisfies Eq. (11). Now, the non-geodesic deviation general. It is necessary to choose this particular gauge as Sa 123 197 Page 4 of 12 Eur. Phys. J. C (2019) 79 :197 can be viewed as a spatial vector field relative to T a as long action is given by [46], a as T can be interpreted as the 4-velocity of an observer. √ 4 S = d x −g(L2 + L3 + L4 + L5) + Sm[ψm, gμν], 3.1 Fermi normal coordinates (23)

In this subsection, the relative acceleration will be expressed where Sm[ψm, gμν] is the action for the matter field ψm, and σ (τ) in the Fermi normal coordinate system of the observer 0 it is assumed that ψm non-minimally couples with the metric with τ the proper time. Let the observer σ0(τ) carry a pseudo- only. The individual terms in the integrand are {( )a = a,( )a,( )a,( )a} orthonormal tetrad e0ˆ u e1ˆ e2ˆ e3ˆ , which a b L = K (φ, X), (24) satisfies gab(eμˆ ) (eνˆ ) = ημˆ νˆ and is Fermi-Walker trans- 2 ported along σ0(τ). The observer σ0(τ) will measure the L3 =−G3(φ, X)φ, (25) deviation in its own proper reference frame, in which the 2 2 L4 = G4(φ, X)R + G4X [(φ) − (φ;μν) ], (26) metric takes the following form [44], ;μν L5 = G5(φ, X)Gμνφ ˆ ˆ ˆ ˆ 2 =−( + j ) τ 2 + δ j k + (| j |2), G5X 3 2 3 ds 1 2A jˆx d jˆkˆdx dx O x (18) − [(φ) − 3(φ)(φ;μν) + 2(φ;μν) ]. (27) 6 , = , , σ (τ) where j k 1 2 3 and the acceleration of 0 has no ;μ ˆ In these expressions, X =−φ;μφ /2 with φ;μ =∇μφ, A0 =−u Aa = Sa = time component ( a 0). Similarly, φ =∇∇ φ φ = μνφ (φ )2 = φ φ;μν jˆ a ;μν ν μ , g ;μν, ;μν ;μν S (e ˆ) , so the relative acceleration has the following spatial ;μρ ;ν j and (φ;μν)3 = φ;μνφ φ for simplicity. K, G , G , G components ;ρ 3 4 5 are arbitrary analytic functions of φ and X, and GiX = ˆ ˆ ˆ ˆ ˆ j j k k j ∂X Gi , i = 3, 4, 5. For any binary function f (φ, X), define A =−Rˆ ˆ ˆ S + S ∇ˆ A rel 0k0 k (19) the following symbol jˆ kˆ kˆ jˆ =−Rˆ ˆ ˆ S + S ∂ˆ A , 0k0 k ∂m+n f (φ, X) ( , ) = , f m n m n (28) since the only nonvanishing components of the Christoffel ∂φ ∂ X φ=φ0,X=0 symbol are where φ0 is a constant value for the scalar field evaluated at ˆ ˆ Γ 0 = Γ j = . infinity. Varying the action (23) with respect to gμν and φ 0ˆ jˆ 0ˆ0ˆ A jˆ (20) gives rise to the equations of motion, which are too compli- The relative acceleration can also be expanded as cated to write down. Please refer to Refs. [46,47]. ˆ There have been experimental constraints on Horndeski j = μˆ ∇ ( νˆ ∇ jˆ) Arel u μˆ u νˆ S theory. Reference [35] discussed the bounds on it from 2 jˆ (21) some solar system tests and the observations on pul- d S jˆ kˆ = + A A ˆ S . sars. GW170817 and its electromagnetic counterpart GRB dτ 2 k 170817A together set a strong constraint on the speed of Therefore, one gets GWs [7,48]. Based on this result, the Lagrangian takes a 2 jˆ simpler form [49Ð53,53Ð56], d S jˆ kˆ kˆ jˆ jˆ kˆ =−R ˆ S + S ∂ˆ A − A A ˆ S . 2 0ˆk0ˆ k k (22) dτ L = K (φ, X) − G3(φ, X)φ + G4(φ)R. (29) Similar expression was also found in Ref. [45]. Due to the Although Horndeski theory is highly constrained, we will b∇ ( a ) = 2 0ˆ / τ 2 = requirement T b S Ta 0, one knows that d S d still work with the original theory in the following discussion. c b a 0ˆ T ∇c[T ∇b(S Ta)]=0, so S = 0 is really preserved while In this theory, WEP is respected due to the non-minimal a a S is propagated along the integral curves of T . Whenever coupling between ψm and gμν. However, due to the indi- σ (τ) a = the observer 0 is moving on a geodesic, A 0, then rect interaction between ψm and φ mediated by gμν via the Eq. (22) becomes the usual geodesic deviation equation used equations of motion, SEP is violated. In fact, calculations to analyze the polarizations of GWs [44]. have shown that the effective gravitational “constant” actually depends on φ [57]. Therefore, the gravitational binding energy of a compact object, viewed as a system of point par- 4 The trajectory of a self-gravitating object in ticles, will also depend on the local value of φ. Because of the Horndeski theory mass-energy equivalence E = m, the mass of the compact object, i.e., the total mass of the system of point particles, also The most general scalarÐtensor theory with second order depends on φ. This would affect the motion of the compact equations of motion is the Horndeski theory [23], whose object. Following Eardley’s suggestion, the matter action can 123 Eur. Phys. J. C (2019) 79 :197 Page 5 of 12 197

a be described by [58] can be expressed entirely in terms of u of the particle σ0(τ) by expanding the brackets and using Eq. (34) together with ρ ρ μ ν Sm =− m(φ(x )) −gμν(x )x˙ x˙ dλ, (30) Eq. (35), with x˙μ = dxμ/dλ, when the compact object can be treated Aa =−R auc Sbud as a self-gravitating particle. In this action, φ and gμν also rel cbd depend on the trajectory. In this treatment, the spin and the dlnm − (gac + uauc)Sb∇ ∇ φ b φ c ln multipole moment structure are ignored. To obtain the equa- dln tion of motion, one applies EulerÐLagrange equation and at dlnm c b a a b c (36) λ − (∇c ln φ) u u ∇b S + u u ∇b S the same time, assumes that the parameter parameterizes dlnφ μ ν the trajectory such that gμν x˙ x˙ is a constant along the tra- ac dlnm b jectory. Usually, one parameterizes particle trajectories with − 2g S ∇b ln φ . dlnφ the proper time τ. This is not necessary, as one can always reparameterize. A generic parametrization is convenient for Again, the right hand side is evaluated along σ0(τ). the following discussion. In the Fermi normal coordinates, the spatial components The EulerÐLagrange equation reads, of Aμ are given by a b a A = u ∇bu ˆ dlnm ˆ A j =− ∂ j ln φ, (37) dlnm (31) dlnφ =− (−gabu uc + uaub)∇ φ, φ c b ln dln according to Eq. (32). By Eq. (22), one obtains where ua = (∂/∂λ)a. Therefore, the self-gravitating particle 2 jˆ no longer moves on a geodesic. The failure of its trajectory d S jˆ kˆ kˆ dlnm jˆ =−Rˆ ˆ ˆ S − S ∂ˆ ∂ ln φ being a geodesic is described by dlnm , which is called the dτ 2 0k0 k dlnφ dlnφ (38) u ub∇ ua = 2 “sensitivity”. One can check that a b 0, which is dlnm jˆ kˆ + (∂ ln φ)S ∂ˆ ln φ. consistent with the parametrization. This means that the 4- dlnφ k acceleration of the particle is a spatial vector with respect φ = φ to ua. If one chooses the proper time τ to parameterize the When the scalar field is not excited,i.e., 0, a constant, trajectory, the above expression gets simplified, Eq. (38) reduces to the geodesic deviation equation, ˆ a dlnm a a b 2 j A =− (δ + u u )∇ ln φ, (32) d S jˆ kˆ φ b b =−Rˆ ˆ ˆ S . (39) dln dτ 2 0k0 δa + a a where b u ub is actually the projection operator for u . This is expected as vSEP is caused by a dynamical scalar Therefore, a self-gravitating object moves along an acceler- field. In the next section, Eq. (38) will be used to analyze the ated trajectory when only gravity acts on it, and its acceler- polarization content of GWs in Horndeski theory. ation is due to the gradient in the scalar field φ. Now, consider two infinitesimally nearby self-gravitating particles, one of which travels along σ0(λ). The deviation a 5 The polarizations of gravitational waves in Horndeski vector connecting σ0(λ) to its nearby company is S .Itis a gravity useful to parameterize σ0(λ) by its proper time τ so that u is a unit timelike vector associated with an observer. The In Ref. [59], the GW solutions for Horndeski theory [23]in relative acceleration is thus given by the vacuum background have been obtained. The polarization Aa =−R auc Sbud content of the theory was also determined using the linearized rel cbd geodesic deviation equation, as the vSEP was completely b dlnm ac d a c (33) − S ∇b (−g ud u + u u )∇c ln φ . dlnφ ignored. In this section, the GW solution will be substituted into Eq. (38) to take into account the effect of the scalar field Note that the right hand side is evaluated at σ0(τ). The devi- a on the trajectories of self-gravitating test particles. This will ation vector S should satisfy lead to a different polarization content of GWs in Horndeski b a b a u ∇b S = S ∇bu , (34) theory. Now, one expands the fields around the flat background b a dlnm a S ∇b(u ua) = 2 S ∇a ln φ, (35) = η + φ = φ + ϕ dlnφ such that gμν μν hμν and 0 . At the leading order, one obtains d according to Eq. (17), which explains why ud u inside of the brackets of Eq. (33)isnotsetto−1. The relative acceleration G2(0,0) = 0, G2(1,0) = 0. (40) 123 197 Page 6 of 12 Eur. Phys. J. C (2019) 79 :197

At the first order, the linearized equations of motion can be Consider a trivial motion, i.e., x = y = 0. Then one obtains written in the following form, k 2s v ≈− χ − ϕ cos ωt, (52) ( − 2)ϕ = , 3 2ω φ 0 ms 0 (41) 0 ˜ = , k 2s hμν 0 (42) z ≈− χ − ϕ ωt. 2 0 sin (53) 2ω φ0 where the scalar field ϕ is generally massive with the squared Here, the initial position of σ (τ) is chosen to be x = y = mass given by 0 0 0 z0 = 0. In addition, K(2,0) m2 =− , (43) 0 dt 1 s − + 2 / u = ≈ 1 + χϕ0 cos ωt, (54) K(0,1) 2G3(1,0) 3G4(1,0) G4(0,0) dτ 2 ˜ according to Eq. (47), which implies that and hμν is an auxiliary field defined as χϕ τ ≈ t − 0 sin ωt. (55) ˜ 1 αβ ω hμν = hμν − ημνη hαβ − χημνϕ, (44) 2 2 From this, one clearly sees that the TT coordinate system is σ (τ) with χ = G4(1,0) . Note that the transverse-traceless (TT) not the proper reference frame for the observer 0 . G4(0,0) Therefore, the trajectory of σ (τ) in the TT coordinate ∂ ˜μν = ημν ˜ = 0 gauge μh 0, hμν 0 has been made. A GW system is described by + propagating in the z direction is given below χϕ τ = − 0 ω , ˜ t sin t (56) hμν = eμν cos Ω(t − z), (45) 2ω x = y = 0, (57) ϕ = ϕ0 cos(ωt − kz), (46) k 2s z =− χ − ϕ ωt, ω2 − 2 = 2 2 0 sin (58) where k ms and the only nonvanishing components 2ω φ0 of tensor wave amplitude eμν are e =−e and e .The 11 22 12 up to the linear order. Because of the scalar field, the observer coordinate system in which the TT gauge is chosen is called oscillates with the same frequency of the GW in the TT coor- the TT coordinate system. dinate system according to Eq. (58). The time dilatation also One is interested in studying the relative acceleration of oscillates by Eq. (56). two nearby particles which were at rest before the arrival of In the limit of GR (χ = s = 0), the trajectory of σ (τ) the GW. Because of the presence of the GW induced by the 0 is thus t = τ,x j = 0 up to the linear order, i.e., a geodesic scalar field, one expects σ (τ) to deviate from a straight line 0 of the background metric. If vSEP is weak, i.e. s ≈ 0, the in the TT coordinates, so one assumes its 3-velocity is v and trajectory is uμ = u0(1, v). The normalization of ua implies that χϕ τ = t − 0 ωt, 1 ω sin (59) u0 = 1 + h + O(v2). (47) 2 2 00 x = y = 0, (60) σ (τ) kχϕ0 The acceleration of 0 can be approximated as z =− sin ωt. (61) 2ω2 μ s μν μ ν A ≈− (η + u u )∇νϕ, σ (τ) φ (48) This agrees with Ref. [59]. Although the particle 0 does 0 not follow a geodesic of the background metric, it still travels = ( / φ)| μ = with s dlnm dln φ0 called the sensitivity and u along a geodesic of the full metric. (1, 0) the background value. Written in component form, the acceleration is given by 5.1 The relative acceleration in the Fermi normal coordinates A0 = 0, (49) ϕ A j =−δ j ks 0 (ωt − kz). 3 φ sin (50) In this subsection, one obtains the relative acceleration in the 0 Fermi normal coordinates using Eq. (38). This discussion will On the other hand, the left hand side of Eq. (48) is, in coor- also reveal the polarization content of GWs. The 4-velocity dinate basis, of the observer is

2 μ ρ ν a a μ d x μ dx dx u = (eˆ ) = (1 + h00/2, 0, 0,v3), (62) A = + Γ ρν 0 τ 2 τ τ d d d 2 μ 0 μ (51) so the following triad can be chosen, 0 2 d x μ 0 du dx ≈ (u ) + Γ + u . a 2 00 ( ) = ( , − / , − / , ), dt dt dt e1ˆ 0 1 h11 2 h12 2 0 (63) 123 Eur. Phys. J. C (2019) 79 :197 Page 7 of 12 197

( )a = ( , − / , − / , ), e2ˆ 0 h12 2 1 h22 2 0 (64) 6 Pulsar timing arrays ( )a = (v , , , + / ). e3ˆ 3 0 0 1 h00 2 (65) In this section, the cross-correlation function will be calcu- lated for PTAs. The possibility to detect the vSEP is thus These basic vectors are FermiÐWalker transported and eval- inferred. A pulsar is a strongly magnetized, rotating neutron σ (τ) {( μˆ ) } uated along 0 . The dual basis is denoted as e a and star or a white dwarf, which emits a beam of the radio wave ( μˆ ) ≈ δμˆ e ν ν is sufﬁcient. Up to the linear order in perturba- along its magnetic pole. When the beam points towards the tions, Eq. (22) is given by Earth, the radiation is observed, and this leads to the pulsed appearance of the radiation. The rotation of some “recycled” 2 jˆ pulsars is stable enough so that they can be used as “cos- d S jˆ kˆ kˆ jˆ =−Rˆ ˆ ˆ S + S ∂ˆ A , (66) dt2 0k0 k mic light-house” [63]. Among them, millisecond pulsars are found to be more stable [64] and used as stable clocks [65]. When there is no GW, the radio pulses arrive at the Earth at jˆ since the acceleration A is of the linear order, and the last a steady rate. The presence of the GW will affect the prop- term in Eq. (22) should be dropped. Normally, one has to ﬁnd agation time of the radiation and thus alter this rate. This the Fermi normal coordinates explicitly [60,61]. However, results in a change in the time-of-arrival (TOA), called tim- the Fermi normal coordinates differ from the TT coordinates ing residual R(t). Timing residuals caused by the stochas- by quantities of order one, and the Riemann tensor and the tic GW background is correlated between pulsars, and the 4-acceleration of the test particle are both of linear order, cross-correlation function is C(θ) = Ra(t)Rb(t) with θ so any changes in their components caused by the coordi- the angular separation of pulsars a and b, and the brackets nate transformation are of the second order in perturbations. implying the ensemble average over the stochastic back- Therefore, one only has to calculate the components of the ground. This makes it possible to detect GWs and probe the Riemann tensor and the 4-acceleration in the TT coordinates, polarizations [37Ð40,66Ð73]. The effect of vSEP can also be and then simply substitutes them in Eq. (66). detected, as the longitudinal polarization of the scalarÐtensor More explicitly, the driving force matrix is given by theory is enhanced due to vSEP. One sets up a coordinate system shown in Fig. 1 to calculate the timing residual R(t) caused by the GW solution (45) jˆ = jˆ − ∂ jˆ Skˆ R0ˆkˆ0ˆ kˆ A and (46). Before the GW comes, the Earth is at the origin, and j j the distant pulsar is at rest at x = (L cos β,0, L sin β)in this ≈ R0k0 − ∂k A p ⎛ ⎞ coordinate system. The GW is propagating in the direction of − ω2 χϕ + Ω2 ˜ Ω2 ˜ 2 2 h11 2 h12 0 ˆ ˆ ⎜ 2 2 2 ⎟ a unit vector k, and n is the unit vector pointing to the pulsar ≈ ⎝ Ω ˜ − ω χϕ − Ω ˜ ⎠ , ˆ ˆ ˆ ˆ ˆ 2 h12 2 2 h11 0 from the Earth. l = k ∧(nˆ ∧k)/ cos β =[ˆn −k(nˆ ·k)]/ cos β 2 2 00− ms χϕ − k s ϕ is actually the unit vector parallel to the y axis. 2 φ0 (67)

˜ where hμν and ϕ are evaluated at (t, x = 0). Comparing this matrix with the one (Eq. (29)) in Ref. [59], one finds out that 2 vSEP introduces an order one correction −k sϕ/φ0 to the longitudinal polarization. This means that the longitudinal polarization gets enhanced. Even if the scalar field is massless, the longitudinal polarization persists because the test particles are accelerated. However, the enhancement is very extremely small for objects such as the mirrors used in detectors such as LIGO. According to Refs. [29,62], white dwarfs have typical sensitivities s ∼ 10−4, so a test particle, like the mirror used by LIGO, would have an even smaller sensitivity. So it would be still very difficult to use interferometers to detect the enhanced longitudinal polarization as in the previous case [59]. In contrast, neutron stars are compact objects. Their Fig. 1 The GW is propagating in the direction of kˆ, and the photon is sensitivity could be about 0.2 [29,62]. They violate SEP rel- traveling in −ˆn direction at the leading order. lîs perpendicular to kˆ and ˆ ˆ atively strongly, which might be detected by PTAs. in the same plane determined by k and nˆ. The angle between nˆ and l is β 123 197 Page 8 of 12 Eur. Phys. J. C (2019) 79 :197

Where there is no GW, the photon is assumed to have and the one by the observer comoving with the pulsar is μ a 4-velocity given by u = γ0(1, − cos β,0, − sin β) with γ = / λ λ =− μ 0 dt d a constant and an arbitrary affine parameter. fe uμTp μ = μ + vμ Let the perturbed photon 4-velocity be u u .The ω − β μ ν k sin sek condition gμνu u = 0 together with the photon geodesic = γ 1 + χ + sin β ϕ cos(ωte 0 2ω φ ω 0 equation lead to 0 e11 − kLsin β) − (1 − sin β)cos Ω(te − L sin β) . 2 0 e11 v = γ {χϕ cos[(ω + k sin β)t − k(L + te) sin β]− 0 0 2 (75) ×(1 − sin β)cos[Ω(1 + sin β)t − Ω(L + te) sin β]}, (68) Therefore, the frequency shift is given by v = γ {−χϕ cos β cos[(ω + k sin β)t − k(L + t ) sin β] 1 0 0 e ˆ fe − fr ω − kk ·ˆn + e11 cos β cos[Ω(1 + sin β)t − Ω(L + te) sin β]}, = χ ϕ(t − L, Lnˆ) − ϕ(t, 0) fr 2ω (69) e nˆ j nˆk − jk ˜ ( − , ˆ) − ˜ ( , ) (76) v2 = γ0e12 cos β cos Ω[(1 + sin β)t − (L + te) sin β], h jk t L Ln h jk t 0 2(1 + kˆ ·ˆn) (70) + k ˆ ·ˆ[ ϕ( − , ˆ) − ϕ( , )], v = γ {−χϕ sin β cos[(ω + k sin β)t k n se t L Ln sr t 0 3 0 0 ωφ0 e11 −k(L + te) sin β]− 2 where t = t + L is the time when the photon arrives at the ( − β) [Ω( + β) − Ω( + ) β]}, e 1 sin cos 1 sin t L te sin Earth at the leading order. This equation has been expressed (71) in a coordinate independent way, so it can be straightforwardly used in any coordinate system with arbitrary orien- where te is the time when the photon is emitted from the tation and at rest relative to the original one. Note that the pulsar. first two lines reproduce the result in Ref. [59], and the third The 4-velocity of an observer on the Earth has been line comes from the effect of vSEP. This effect is completely obtained in Sect. 5, which reads determined by the scalar perturbation ϕ, as expected. Therefore, the focus will be on the cross-correlation func- μ 1 T = 1 + χϕ0 cos ωt, 0, tion for the scalar GW in the following discussion. Eq. (76)is e 2 (72) the frequency shift due to a monochromatic wave. Now, con- k 2s , − χ − r ϕ ωt , sider the contribution of a stochastic GW background which 0 ω φ 0 cos 2 0 consists of monochromatic GWs, where sr is the sensitivity of the Earth. The 4-velocity of ∞ dω another observer comoving with the pulsar can be derived ϕ(t, x) = 2kˆ{ϕ (ω, kˆ) [i(ωt − kkˆ · x)]}, π d 0 exp in a similar way. In fact, the translational symmetry in the −∞ 2 background spacetime (i.e., Minkowskian spacetime) gives (77) 1 ˆ μ = + χϕ (ω − β), , where ϕ0(ω, k) is the amplitude for the scalar GW propagat- Tp 1 0 cos t kLsin 0 2 ing in the direction kˆ at the angular frequency ω. Usually, (73) k 2se one assumes that the GW background is isotropic, stationary 0, − χ − ϕ0 cos(ωt − kLsin β) , 2ω φ0 and independently polarized, then one can define the charac- teristic strains ϕc given by, which agrees with the result from the direct calculation. Here, se is the sensitivity of the pulsar. |ϕ (ω)|2 ϕ∗(ω, kˆ)ϕ (ω, kˆ) =δ(ω − ω)δ(kˆ − kˆ) c , So the measured frequency by the observer on the Earth 0 0 ω (78) is =− μ where the star ∗ implies the complex conjugation. fr uμTe The total timing residual in TOA due to the stochastic GW ω − k sin β sr k = γ0 1 + χ + sin β ϕ0× background is 2ω φ0ω (74) ∞ ω T − e11 d 2 ˆ fe fr cos ω(te + L) − (1 − sin β)cos Ω(te + L) , R(T ) = d k dt , (79) 2 −∞ 2π 0 fr 123 Eur. Phys. J. C (2019) 79 :197 Page 9 of 12 197 where the argument T is the total observation time. Insert But for θ = 0, one considers the auto-correlation function, Eq. (76) in, neglecting the second line, to obtain so set nˆ1 =ˆn2 = (0, 0, 1) and L1 = L2 = L. The auto- correlation function is thus given by ∞ ω ω − ˆ ·ˆ d 2 ˆ ˆ iωT kk n R(T ) = d kϕ0(ω, k)(e − 1) χ ∞ 2 2 2 −∞ 2π i2ω2 |ϕc(ω)| k 2sr C(0) = dω χ 2 2 + 1 − ω3 ω2 φ χ ˆ ms 3 0 − (ω+ ˆ· ) kk ·ˆn − (ω+ ˆ· ) (91) ×[e i kk n L − ]+ [s e i kk n L −s ] . 2 2 1 2 e r k 2se iω φ0 + 1 − , 3ω2 φ χ (80) 0 With this result, consider the correlation between two pulsars a and b located at xa = L1nˆ1 and xb = L2nˆ2, respectively. The angular separation is θ = arccos(nˆ1 ·ˆn2). The cross- correlation function is thus given by

C(θ) = Ra(T )Rb(T ) ∞ |ϕ (ω)|2 k2kˆ ·ˆn kˆ ·ˆn = dω d2kˆ c 1 2 P πω5 φ2 4 ms 0 ˆ ˆ kk ·ˆn1(ω − kk ·ˆn2) + χP2 2φ0 (81) ˆ ·ˆ (ω − ˆ ·ˆ ) + kk n2 kk n1 χP 2φ 3 0 (ω − ˆ ·ˆ )(ω − ˆ ·ˆ ) + kk n1 kk n2 χ 2P , 4 1 where P1, P2, P3 and P4 are deﬁned to be

P1 = 1 − cos Δ1 − cos Δ2 + cos(Δ1 − Δ2), (82)

P2 = sr − sr cos Δ2 − se cos Δ1 + se cos(Δ1 − Δ2), (83)

P3 = sr − sr cos Δ1 − se cos Δ2 + se cos(Δ1 − Δ2), (84) P = 2 − Δ − Δ + 2 (Δ − Δ ), 4 sr sr se cos 1 sr se cos 2 se cos 1 2 (85) ˆ with Δ j = (ω+kk ·ˆn j )L j for j = 1, 2. To obtain this result, Eq. (78) is used, and the real part is taken. In addition, T drops out, as the ensemble average also implies the averaging over the time [38]. Because of the isotropy of the GW background, one sets nˆ1 = (0, 0, 1), (86) nˆ2 = (sin θ,0, cos θ). (87) ˆ Also, let k = (sin θg cos φg, sin θg sin φg, cos θg),so Fig. 2 The normalized cross-correlation functions ζ(θ) = (θ)/ ( ) Δ1 = (ω + k cos θg)L1, (88) C C 0 . The upper panel shows the cross-correlations when Δ =[ω + ( θ φ θ + θ θ)] , the scalar field is massless, i.e., when there is no longitudinal polariza- 2 k sin g cos g sin cos g cos L2 (89) tion. The solid curve is for familiar GR polarizations (i.e., the plus or ω cross ones), the dashed red curve for the breathing polarization with Working in the limit that L j 1, one can drop the cosines SEP and the dotted purple curve for the breathing polarization with in the definitions (82)Ð(85)ofP j ( j = 1, 2, 3, 4), when vSEP.The lower panel shows the normalized cross-correlations induced θ = 0. The integration can be partially done, resulting in together by the transverse breathing and longitudinal polarizations = . × −23 / 2 when the mass of the scalar field is taken to be ms 7 7 10 eV c . ∞ |ϕ (ω)|2 k2 2s 2 The solid curves are for the cases where SEP is satisfied, while the C(θ)= dω c χ 2 1 + 1 − r cos θ . ω3 ω2 φ χ dashed curves are for those where SEP is violated. The power-law ms 3 0 index α = 0, −2/3, −1. The calculation was done assuming T = 5 (90) years 123 197 Page 10 of 12 Eur. Phys. J. C (2019) 79 :197 where the terms containing L are dropped as they barely less, the longitudinal polarization still exists. This is in con- contribute according to the experience in Ref. [59]. Finally, trast with the previous results [59,76Ð78] that the massive the observation time T sets a natural cutoff for the angular scalar field excites the longitudinal polarization, while the frequency, i.e., ω ≥ 2π/T , so the lower integration limits in massless scalar field does not. The enhanced longitudinal Eqs. (90) and (91) should be replaced by Max{ms, 2π/T }. polarization is nevertheless difficult for aLIGO to detect, as α As usual, assume ϕc(ω) ∝ (ω/ωc) with ωc the char- the mirrors does not violate SEP enough. However, pulsars acteristic angular frequency. Here, α is called the power-law are highly compact objects with sufficient self-gravitating index, and usually, α = 0, −2/3or−1[38,74]. Numerically energy such that their trajectories deviate from geodesics integrating Eqs. (90) and (91) gives the so-called normalized enough. Using PTAs, one can measure the change in TOAs of correlation function ζ(θ) = C(θ)/C(0). In the integration, electromagnetic radiation from pulsars and obtain the cross- set the observation time T = 5 years. The sensitivities of the correlation function to tell whether vSEP effect exits. The Earth and the pulsar are taken to be sr = 0 and se = 0.2, results show that the vSEP leads to large changes in the respectively. This leads to Fig. 2, where the power-law index behaviors of the cross-correlation functions. In principle, α takes different values. PTAs are capable of detecting the vSEP if it exists. If the scalar field is massless, the results are shown in the upper panel which displays the normalized correlation Acknowledgements This research was supported in part by the Major functions for the plus and cross polarizations Ð HellingsÐ Program of the National Natural Science Foundation of China under Grant no. 11475065 and the National Natural Science Foundation of Downs curve (labeled by “GR”) [69]. The remaining two China under Grant no. 11690021. This was also a project funded by curves are for the breathing polarization: the dashed one is China Postdoctoral Science Foundation (no. 2018M632822). for the case where SEP is respected, while the dotted one Data Availability Statement This manuscript has no associated data or is for the case where SEP is violated. They are independent the data will not be deposited. [Authors’ comment: This work discusses of the power-law index α. As one can see that vSEP makes the theoretical possibility to use PTAs to determine whether vSEP exists. ζ(θ)bigger by about 5%. If the scalar field has a mass ms = There is no data to be displayed.] 7.7 × 10−23 eV/c2, the results are shown in the lower panel. Open Access This article is distributed under the terms of the Creative In this panel, the cross-correlation functions for the scalar Commons Attribution 4.0 International License (http://creativecomm polarization are drawn for different values of α. The solid ons.org/licenses/by/4.0/), which permits unrestricted use, distribution, curves correspond to the case where SEP is satisfied, and the and reproduction in any medium, provided you give appropriate credit dashed curves are for the case where SEP is violated. Since to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. the cross correlation for the plus and cross polarizations does Funded by SCOAP3. not change, we do not plot them again in the lower panel. In the massive case, vSEP also increases ζ(θ) by about 2Ð3%. Reference [75] published the constraint on the stochastic GW background based on the recently released 11-year References dataset from the North American Nanohertz Observatory for Gravitational Waves (NANOGrav). Assuming the back- 1. S. Perlmutter et al., Measurements of Omega and Lambda from 42 α =− / high redshift supernovae. Astrophys. J. 517, 565Ð586 (1999) ground is isotropic and 2 3, the strain amplitude of the 2. Adam G. Riess et al., Observational evidence from supernovae for − − GW is less than 1.45 × 10 15 at f = 1yr 1. In addition, the an accelerating universe and a cosmological constant. Astron. J. top panel in Figure 6 shows the observed cross correlation. 116, 1009Ð1038 (1998) As one can clearly see, the error bars are very large.2 More 3. B.P.Abbott, Observation of gravitational waves from a binary black hole merger. Phys. Rev. Lett. 116(6), 061102 (2016) observations are needed to improve the statistics. 4. B.P. Abbott, GW151226: observation of gravitational waves from a 22-solar-mass binary black hole coalescence. Phys. Rev. Lett. 116(24), 241103 (2016) 5. B.P. Abbott, GW170104: observation of a 50-solar-mass binary 7 Conclusion black hole coalescence at redshift 0.2. Phys. Rev. Lett. 118(22), 221101 (2017) This work discusses the effects of the vSEP on the polariza- 6. B.P. Abbott, GW170814: a three-detector observation of gravitation content of GWs in Horndeski theory and calculates the tional waves from a binary black hole coalescence. Phys. Rev. Lett. 119, 141101 (2017) cross-correlation functions for PTAs. Because of the vSEP, 7. B.P. Abbott, GW170817: observation of gravitational waves from self-gravitating particles no longer travel along geodesics, a binary neutron star inspiral. Phys. Rev. Lett. 119(16), 161101 and this leads to the enhancement of the longitudinal polar- (2017) ization in Horndeski theory, so even if the scalar field is mass- 8. B.P. Abbott, GW170608: observation of a 19-solar-mass binary black hole coalescence. Astrophys. J. 851(2), L35 (2017) 9. B.P. Abbott et al., GWTC-1: a gravitational-wave transient catalog 2 Precisely due to the large errors, we do not plot the observed cross of compact binary mergers observed by LIGO and Virgo during correlation in our Fig. 2. the first and second observing runs (2018). arXiv:1811.12907 123 Eur. Phys. J. C (2019) 79 :197 Page 11 of 12 197

10. B.P. Abbott et al., Tests of general relativity with GW170817 35. Shaoqi Hou, Yungui Gong, Constraints on Horndeski theory using (2018). arXiv:1811.00364 the observations of Nordtvedt effect, Shapiro time delay and binary 11. M. Kramer, D.J. Champion, The European pulsar timing array pulsars. Eur. Phys. J. C 78, 247 (2018) and the large European array for pulsars. Class. Quantum Grav- 36. W.W. Zhu, Tests of gravitational symmetries with pulsar binary ity 30(22), 224009 (2013) J1713+0747. Mon. Not. R. Astron. Soc. 482, 3249 (2018) 12. G. Hobbs, The international pulsar timing array project: using pul- 37. Fredrick A Jenet, George B Hobbs, K .J. Lee, Richard N Manch- sars as a gravitational wave detector. Class. Quantum Gravity 27, ester, Detecting the stochastic gravitational wave background using 084013 (2010) pulsar timing. Astrophys. J. 625, L123ÐL126 (2005) 13. Maura A. McLaughlin, The North American Nanohertz observa- 38. K.J. Lee, F.A. Jenet, R.H. Price, Pulsar timing as a probe of non- tory for gravitational waves. Class. Quantum Gravity 30, 224008 Einsteinian polarizations of gravitational waves. Astrophys. J. 685, (2013) 1304Ð1319 (2008) 14. G. Hobbs, The Parkes pulsar timing array. Class. Quantum Gravity 39. Kejia Lee, Fredrick A. Jenet, Richard H. Price, Norbert Wex, 30, 224007 (2013) Michael Kramer, Detecting massive gravitons using pulsar timing 15. C.M. Will, Theory and experiment in gravitational physics (Uni- arrays. Astrophys. J. 722, 1589Ð1597 (2010) versity Press (1993), Cambridge, 1993), p. 380 40. K.J. Lee, Pulsar timing arrays and gravity tests in the radiative 16. Nathalie Deruelle, Nordstrom’s scalar theory of gravity and the regime. Class. Quantum Gravity 30(22), 224016 (2013) equivalence principle. Gen. Relativ. Grav. 43, 3337Ð3354 (2011) 41. R. Penrose, W. Rindler, Spinors and space-time. Cambridge Mono- 17. E. Barausse, K. Yagi, Gravitation-wave emission in shift- graphs on Mathematical Physics, vol. 1, 1st edn. Cambridge Uni- symmetric Horndeski theories. Phys. Rev. Lett. 115(21), 211105 versity Press, Cambridge (1984) (2015) 42. Robert M. Wald, General Relativity (University of Chicago Press, 18. C. Brans, R.H. Dicke, Mach’s principle and a relativistic theory of Chicago, 1984) gravitation. Phys. Rev. 124(3), 925Ð935 (1961) 43. N.S. Swaminarayan, J.L. Safko, A coordinate-free derivation of a 19. P. Kanti, N.E. Mavromatos, J. Rizos, K. Tamvakis, E. Winstanley, generalized geodesic deviation equation. J. Math. Phys. 24, 883Ð Dilatonic black holes in higher curvature string gravity. Phys. Rev. 885 (1983) D 54, 5049Ð5058 (1996) 44. Charles W. Misner, Kip S. Thorne, John Archibald Wheeler, Grav- 20. Hans A. Buchdahl, Non-linear Lagrangians and cosmological the- itation (W.H. Freeman, San Francisco, 1973) ory. Mon. Not. R. Astron. Soc. 150, 1 (1970) 45. S.W. Hawking, G.F.R. Ellis, Cambridge Monographs on Math- 21. John O’Hanlon, Intermediate-range gravity: a generally covariant ematical Physics, The large scale structure of space-time (Cam- model. Phys. Rev. Lett. 29(2), 137Ð138 (1972) bridge University Press, Cambridge, 2011) 22. P. Teyssandier, Ph Tourrenc, The Cauchy problem for the r + r 2 46. Tsutomu Kobayashi, Masahide Yamaguchi, Ju’ichi Yokoyama, theories of gravity without torsion. J. Math. Phys. 24(12), 2793Ð Generalized G-inflation: inflation with the most general second- 2799 (1983) order field equations. Prog. Theor. Phys. 126, 511Ð529 (2011) 23. Gregory Walter Horndeski, Second-order scalar-tensor field equa- 47. Xian Gao, Conserved cosmological perturbation in Galileon mod- tions in a four-dimensional space. Int. J. Theor. Phys. 10(6), 363Ð els. JCAP 1110, 021 (2011) 384 (1974) 48. B.P. Abbott, Gravitational waves and gamma-rays from a binary 24. M. Ostrogradsky, Mémoires sur les équations différentielles, rela- neutron star merger: GW170817 and GRB 170817A. Astrophys. tives au problème des isopérimètres. Mem. Acad. St. Petersb. 6(4), J. 848(2), L13 (2017) 385Ð517 (1850) 49. Lucas Lombriser, Andy Taylor, Breaking a dark degeneracy with 25. Kenneth Nordtvedt, Equivalence principle for massive bodies. 1. gravitational waves. JCAP 1603(03), 031 (2016) Phenomenol. Phys. Rev. 169, 1014Ð1016 (1968) 50. Lucas Lombriser, Nelson A. Lima, Challenges to self-acceleration 26. Kenneth Nordtvedt, Equivalence principle for massive bodies. 2. in modified gravity from gravitational waves and large-scale struc- Theory. Phys. Rev. 169, 1017Ð1025 (1968) ture. Phys. Lett. B 765, 382Ð385 (2017) 27. K. Nordtvedt, The fourth test of general relativity. Rep. Prog. Phys. 51. T. Baker, E. Bellini, P.G. Ferreira, M. Lagos, J. Noller, I. Sawicki, 45(6), 631 (1982) Strong constraints on cosmological gravity from GW170817 and 28. Clifford M. Will, The confrontation between general relativity and GRB 170817A. Phys. Rev. Lett. 119(25), 251301 (2017) experiment. Living Rev. Relativ. 17, 4 (2014) 52. Paolo Creminelli, Filippo Vernizzi, Dark energy after GW170817 29. Justin Alsing, Emanuele Berti, Clifford M. Will, Helmut Zaglauer, and GRB170817A. Phys. Rev. Lett. 119(25), 251302 (2017) Gravitational radiation from compact binary systems in the massive 53. Jeremy Sakstein, Bhuvnesh Jain, Implications of the neutron star BransÐDicke theory of gravity. Phys. Rev. D 85, 064041 (2012) merger GW170817 for cosmological scalarÐtensor theories. Phys. 30. F. Hofmann, J. Müller, L. Biskupek, Lunar laser ranging test of the Rev. Lett. 119(25), 251303 (2017) Nordtvedt parameter and a possible variation in the gravitational 54. Jose María Ezquiaga, Miguel Zumalacárregui, Dark energy after constant. Astron. Astrophys. 522, L5 (2010) GW170817: dead ends and the road ahead. Phys. Rev. Lett. 31. Thibault Damour, Gerhard Schäefer, New tests of the strong equiv- 119(25), 251304 (2017) alence principle using binary pulsar data. Phys. Rev. Lett. 66, 2549Ð 55. David Langlois, Ryo Saito, Daisuke Yamauchi, Karim Noui, 2552 (1991) ScalarÐtensor theories and modified gravity in the wake of 32. Paulo C .C. Freire, Michael Kramer, Norbert Wex, Tests of the GW170817. Phys. Rev. D 97(6), 061501 (2018) universality of free fall for strongly self-gravitating bodies with 56. YunguiGong, Eleftherios Papantonopoulos, Zhu Yi, Constraints on radio pulsars. Class. Quantum Gravity 29, 184007 (2012) scalarÐtensor theory of gravity by the recent observational results 33. Anne M Archibald, Nina V Gusinskaia, Jason W .T. Hessels, on gravitational waves. Eur. Phys. J. C 78(9), 738 (2018) Adam T Deller, David L Kaplan, Duncan R Lorimer, Ryan S Lynch, 57. Manuel Hohmann, Parametrized post-Newtonian limit of Horn- Scott M Ransom, Ingrid H Stairs, Universality of free fall from deski’s gravity theory. Phys. Rev. D 92(6), 064019 (2015) the orbital motion of a pulsar in a stellar triple system. Nature 58. D.M. Eardley, Observable effects of a scalar gravitational field in 559(7712), 73Ð76 (2018) a binary pulsar. Astrophys. J. 196, L59ÐL62 (1975) 34. Ingrid H. Stairs, Testing general relativity with pulsar timing. Liv- 59. Shaoqi Hou, Yungui Gong, Yunqi Liu, Polarizations of gravita- ing Rev. Relativ. 6, 5 (2003) tional waves in Horndeski theory. Eur. Phys. J. C 78, 378 (2018)

123 197 Page 12 of 12 Eur. Phys. J. C (2019) 79 :197

60. P.L. Fortini, C. Gualdi, Fermi normal co-ordinate system and elec- 71. Nicolás Yunes,Xavier Siemens, Gravitational-wave tests of general tromagnetic detectors of gravitational waves. I- Calculation of the relativity with ground-based detectors and pulsar timing-arrays. metric. Nuovo Cim. B 71, 37Ð54 (1982) Living Rev. Relativ. 16, 9 (2013) 61. P.Fortini, A. Ortolan, Light phase shift in the ﬁeld of a gravitational 72. Jonathan Gair, Joseph D Romano, Stephen Taylor, Chiara M .F. wave. Nuovo Cim. B 106, 101 (1991) Mingarelli, Mapping gravitational-wave backgrounds using meth- 62. H.W. Zaglauer, Neutron stars and gravitational scalars. Astrophys. ods from CMB analysis: application to pulsar timing arrays. Phys. J. 393, 685Ð696 (1992) Rev. D 90(8), 082001 (2014) 63. Emanuele Berti, Testing general relativity with present and future 73. Jonathan R. Gair, Joseph D. Romano, Stephen R. Taylor, Mapping astrophysical observations. Class. Quantum Gravity 32, 243001 gravitational-wave backgrounds of arbitrary polarisation using pul- (2015) sar timing arrays. Phys. Rev. D 92(10), 102003 (2015) 64. D.R. Lorimer, M. Kramer, Handbook of Pulsar Astronomy (2004) 74. Joseph D. Romano, Neil J. Cornish, Detection methods for stochas- 65. J.P.W. Verbiest, Timing stability of millisecond pulsars and tic gravitational-wave backgrounds: a uniﬁed treatment. Living prospects for gravitational-wave detection. Mon. Not. R. Astron. Rev. Relativ. 20, 2 (2017) Soc. 400, 951Ð968 (2009) 75. Z. Arzoumanian, The NANOGrav 11-year data set: pulsar- 66. F.B. Estabrook, H.D. Wahlquist, Response of Doppler spacecraft timing constraints on the stochastic gravitational-wave back- tracking to gravitational radiation. Gen. Relativ. Grav. 6, 439Ð447 ground. Astrophys. J. 859(1), 47 (2018) (1975) 76. Dicong Liang, Yungui Gong, Shaoqi Hou, Yunqi Liu, Polarizations 67. M.V. Sazhin, Opportunities for detecting ultralong gravitational of gravitational waves in f (R) gravity. Phys. Rev. D 95, 104034 waves. Sov. Astron. 22, 36 (1978) (2017) 68. Steven L. Detweiler, Pulsar timing measurements and the search 77. Yungui Gong, Shaoqi Hou, Gravitational wave polarizations in for gravitational waves. Astrophys. J. 234, 1100Ð1104 (1979) f (R) gravity and scalarÐtensor theory. EPJ Web Conf. 168, 01003 69. R.W. Hellings, G.S. Downs, Upper limits on the isotropic gravi- (2018) tational radiation background from pulsar timing analysis. Astro- 78. Y.Gong, S. Hou, The polarizations of gravitational waves. Universe phys. J. 265, L39ÐL42 (1983) 4(8), 85 (2018) 70. Sydney J. Chamberlin, Xavier Siemens, Stochastic backgrounds in alternative theories of gravity: overlap reduction functions for pulsar timing arrays. Phys. Rev. D 85, 082001 (2012)

123