Strong Equivalence Principle and Gravitational Wave Polarizations in Horndeski Theory
SStrong Equivalence Principle and Gravitational WavePolarizations in Horndeski Theory
Shaoqi Hou ∗ and Yungui Gong † School of Physics, Huazhong University of Scienceand Technology, Wuhan, Hubei 430074, China (Dated: March 8, 2019)
Abstract
The relative acceleration between two nearby particles moving along accelerated trajectories isstudied, which generalizes the geodesic deviation equation. The polarization content of the grav-itational wave in Horndeski theory is investigated by examining the relative acceleration betweentwo self-gravitating particles. It is found out that the longitudinal polarization exists no matterwhether the scalar field is massive or not. It would be still very difficult to detect the enhancedlongitudinal polarization with the interferometer, as the violation of the strong equivalence prin-ciple of mirrors used by interferometers is extremely small. However, the pulsar timing array ispromised relatively easily to detect the effect of the violation as neutron stars have large self-energy.The advantage of using this method to test the violation of the strong equivalence principle is thatneutron stars are not required to be present in the binary systems. ∗ [email protected] † [email protected] a r X i v : . [ g r- q c ] M a r . INTRODUCTION Soon after the birth of General Relativity (GR), several alternative theories of gravitywere proposed. The discovery of the accelerated expansion of the Universe [1, 2] revives thepursuit of these alternatives because the extra fields might account for the dark energy. SinceSep. 14th, 2015, LIGO/Virgo collaborations have detected ten gravitational wave (GW)events [3–9]. This opens a new era of probing the nature of gravity in the highly dynamical,strong-field regime. Due to the extra fields, alternatives to GR generally predict that thereare extra GW polarizations in addition to the plus and cross ones in GR. So the detectionof the polarization content is very essential to test whether GR is the theory of gravity.In GW170814, the polarization content of GWs was measured for the first time, and thepure tensor polarizations were favored against pure vector and pure scalar polarizations [6].Similar results were reached in the recent analysis on GW170817 [10]. More interferometersare needed to finally pin down the polarization content. Other detection methods mightalso determine the polarizations of GWs such as pulsar timing arrays (PTAs) [11–14].Alternative metric theories of gravity may not only introduce extra GW polarizations,but also violate the strong equivalence principle (SEP) [15] [16]. The violation of strongequivalence principle (vSEP) is due to the extra degrees of freedom, which indirectly interactwith the matter fields via the metric tensor. This indirect interaction modifies the self-gravitating energy of the objects and leads to vSEP [17]. The self-gravitating objects nolonger move along geodesics, even if there is only gravity acting on them, and the relativeacceleration 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 theaLIGO, move along geodesics, so their relative acceleration is given by the geodesic deviationequation. Since the polarization content of GWs is determined by examining the relativeacceleration, the departure from the geodesic motion might effectively result in differentpolarization contents, which can be detected by PTAs. Thus, the main topic of this work isto investigate the effects of vSEP on the polarization content of GWs and the observationof PTAs.To be more specific, the focus is on the vSEP in the scalar-tensor theory, which is thesimplest alternative metric theory of gravity. The scalar-tensor theory contains one scalarfield φ besides the metric tensor field g µν to mediate the gravitational interaction. Because of2he trivial transformation of the scalar field under the diffeomorphism, there are a plethoraof scalar-tensor theories, such as Brans-Dicke theory [18], Einstein-dilaton-Gauss-Bonnetgravity (EdGB) [19] and f ( R ) gravity [20–22]. In 1974, Horndeski constructed the mostgeneral scalar-tensor theory [23]. Its action contains higher derivatives of φ and g µν , butstill gives rise to at most the second order differential field equations. So the Ostrograd-sky instability is absent in this theory [24]. In fact, Horndeski theory includes previouslymentioned theories as its subclasses. In this work, the vSEP in Horndeski theory will bestudied.Among the effects of vSEP, Nordtvedt effect is well-known for a long time [25, 26], andhappens in the near zone of the source of the gravitational field. It leads to observableeffects. For example, the Moon’s orbit around the Earth will be polarized when they aremoving in the gravitational field generated by the Sun [27, 28]. The polarization of theMoon’s orbit has been constrained by the lunar laser ranging experiments [29], which gavethe Nordtvedt parameter [30] η N = (0 . ± . × − , (1)which measures vSEP in the following way, m g m i = 1 + η N ε grav. + O ( ε ) , (2)with m g and m i the gravitational and the inertial masses, and ε grav. the ratio of the grav-itational binding energy to the inertial energy. A similar polarization of the orbit of themillisecond pulsar-white dwarf (MSP-WD) system also happens due to the gravitationalfield of the Milky Way [31, 32]. In contrast with the Moon and the Earth, pulsars have largegravitational binding energies, so the observation of the orbit polarization of MSP-WD sys-tems set constraints on vSEP in the strong field regime, which was discussed in Ref. [32]. Theobservation of a triple pulsar PSR J0337+1715 was used to set ∆ = ( − . ± . × − [33]. The vSEP also leads to the dipole gravitational radiation, and the variation of Newton’sconstant G [34]. The dipole gravitational radiation for Horndeski theory has been studied inRef. [35], and constraints on this theory were obtained. The pulsar timing observation of thebinary system J1713+0747 has leads to ˙ G/G = ( − . ± . × − yr − and | ∆ | < . c = 1 in vacuum. II. GEODESIC DEVIATION EQUATION
This section serves to review the idea to derive the geodesic deviation equation followingRef. [42]. In the next section, the derivation will be generalized to accelerated objectsstraightforwardly.Let γ s ( t ) represent a geodesic congruence, in which each geodesic is parameterized by t and labeled by s . Define the following tangent vector fields, T a = (cid:18) ∂∂t (cid:19) a , S a = (cid:18) ∂∂s (cid:19) a . (3) S a is called the deviation vector. Their commutator vanishes, T b ∇ b S a = S b ∇ b T a . (4)With a suitable parametrization, one requires that T b ∇ b T a = 0 so that t is an affine param-eter. Note that it is not necessary to set T a T a = − A a rel = T c ∇ c ( T b ∇ b S a ) = − R cbda T c S b T d , (5)using Eq. (4). For details of derivation, please refer to Ref. [42].The deviation vector S a is not unique. A new parametrization of the geodesics, t → t (cid:48) = α ( s ) t + β ( s ) , (6)results in the change in S a by a multiple of T a , T (cid:48) a = T a α ( s ) , S (cid:48) a = S a + dd s (cid:48) (cid:18) t (cid:48) − β ( s (cid:48) ) α ( s (cid:48) ) (cid:19) T a . (7)Therefore, there is a gauge freedom in choosing the deviation vector field S a . This gaugefreedom will be used frequently below to simplify the analysis.Firstly, there is a parametrization such that T a T a is a constant along the coordinate linesof the constant t , i.e., the integral curves of S a . In fact, one knows that, S b ∇ b ( T a T a ) = 2 T a S b ∇ b T a = 2 T a T b ∇ b S a , (8)and under the reparameterization (6), one gets S (cid:48) b ∇ b ( T (cid:48) a T (cid:48) a ) = 2 α ( s ) T a T b ∇ b (cid:20) S a + ∂∂s (cid:18) t (cid:48) − β ( s ) α ( s ) (cid:19) T a (cid:21) , (9)so it is always possible to choose a parametrization to achieve that S (cid:48) b ∇ b ( T (cid:48) a T (cid:48) a ) = 0. Phys-ically, this means that all geodesics are parameterized by the “same” affine parameter t (cid:48) .Secondly, under the above parametrization, the inner product T a S a can be made constantalong the geodesics, T b ∇ b ( T a S a ) = T a T b ∇ b S a = T a S b ∇ b T a = 12 S b ∇ b ( T a T a ) = 0 . (10)An initial choice of T a S a = 0 will be preserved along the t coordinate line, so that S a is always a spatial vector field for an observer with 4-velocity u a = T a / √− T b T b along itstrajectory.From the derivation, one should be aware that the geodesic deviation equation (5) isindependent of the gauge choices made above, which only serves to make sure S a is alwaysa spatial vector relative to an observer with u a . In this way, there is no deviation in thetime coordinate, that is, no time dilatation. This is because one concerns the change in thespatial distance between two nearby particles measured by either one of them.5 II. NON-GEODESIC DEVIATION EQUATION
When particles are accelerated, they are not moving on geodesics. This happens whenthere are forces acting on these particles. This also happens for self-gravitating particles inthe modified gravity theories, such as the scalar-tensor theory. Suppose a bunch of particlesare accelerated and therefore, their velocities satisfy the following relations, T b ∇ b T a = A a , (11)with A a the 4-acceleration and not proportional to T a . In the following, T a is assumedto be some arbitrary timelike vector field which is not necessarily the 4-velocity of someparticle. In this general discussion, the only assumption is that T a satisfies Eq. (11). Now,the non-geodesic deviation equation can be derived similarly, A a rel = − R cbda T c S b T d + S b ∇ b A a . (12)Again, the derivation of this result does not reply on the gauge fixing made similarly inthe previous section or the one to be discussed below. Compared with Eq. (5), there isone extra term, which is due to the fact that the trajectories are no longer geodesics. Thisequation and a more general one were derived in Ref. [43] using the definitions of curvatureand torsion. The authors did not discuss the suitable gauge for extracting physical resultswhich will be presented below.If T a A a (cid:54) = 0, one can reparameterize the integral curves of T a to make it vanish. Indeed,a reparameterization t → t (cid:48) = κ ( t ) leads to A (cid:48) a = T (cid:48) b ∇ b T (cid:48) a = A a ˙ κ − ¨ κ ˙ κ T a , (13)where dot denotes the derivative with respect to t . So one can always find a new parametriza-tion which annihilates T (cid:48) a A (cid:48) a , that is, κ ( t ) = α (cid:90) exp (cid:18) A a T a T b T b t (cid:19) d t + β, (14)with α, β integration constants. From now on, T a A a = 0 is assumed which implies that T b ∇ b ( T a T a ) = 0 . (15)So although t may not be the proper time τ , it is a linear function of τ . A further reparam-eterization t (cid:48) = α (cid:48) t + β (cid:48) does not change the above relation.6ow, pick a congruence of these trajectories σ s ( t ). So as in the previous section, σ s ( t )’salso lie on a 2-dimensional surface Σ parameterized by ( t, s ). There also exists the similargauge freedom to that discussed in Section II, except that A a depends on the gauge choice.For example, a reparametrization t → t (cid:48) = α ( s ) t + β ( s ) results in changes in S a (given byEq. (7)) and A a , i.e., A a → A a /α ( s ).With this gauge freedom, one also chooses a suitable gauge such that T a S a remainsconstant along each trajectory. In fact, it can be shown that T b ∇ b ( T a S a ) = S a A a + 12 S b ∇ b ( T a T a ) . (16)One requires that T a S a = 0 along the integral curves of T a , i.e., T b ∇ b ( T a S a ) = 0. Thisimplies that S b ∇ b ( T a T a ) = − S a A a . (17)This expression means that if the trajectory σ ( t ) is parameterized by the proper time t = τ ,a nearby trajectory σ s ( t ) with s (cid:54) = 0 will not be parameterized by its proper time, in general.It is necessary to choose this particular gauge as S a can be viewed as a spatial vector fieldrelative to T a as long as T a can be interpreted as the 4-velocity of an observer. A. Fermi normal coordinates
In this subsection, the relative acceleration will be expressed in the Fermi normal coor-dinate system of the observer σ ( τ ) with τ the proper time. Let the observer σ ( τ ) carry apseudo-orthonomal tetrad { ( e ˆ0 ) a = u a , ( e ˆ1 ) a , ( e ˆ2 ) a , ( e ˆ3 ) a } , which satisfies g ab ( e ˆ µ ) a ( e ˆ ν ) b = η ˆ µ ˆ ν and is Fermi-Walker transported along σ ( τ ). The observer σ ( τ ) will measure the deviationin its own proper reference frame, in which the metric takes the following form [44],d s = − (1 + 2 A ˆ j x ˆ j )d τ + δ ˆ j ˆ k d x ˆ j d x ˆ k + O ( | x ˆ j | ) , (18)where j, k = 1 , , σ ( τ ) has no time component ( A ˆ0 = − u a A a = 0).Similarly, S a = S ˆ j ( e ˆ j ) a , so the relative acceleration has the following spatial components A ˆ j rel = − R ˆ0ˆ k ˆ0ˆ j S ˆ k + S ˆ k ∇ ˆ k A ˆ j = − R ˆ0ˆ k ˆ0ˆ j S ˆ k + S ˆ k ∂ ˆ k A ˆ j , (19)since the only nonvanishing components of the Christoffel symbol areΓ ˆ0ˆ0ˆ j = Γ ˆ j ˆ0ˆ0 = A ˆ j . (20)7he relative acceleration can also be expanded as A ˆ j rel = u ˆ µ ∇ ˆ µ ( u ˆ ν ∇ ˆ ν S ˆ j ) = d S ˆ j d τ + A ˆ j A ˆ k S ˆ k . (21)Therefore, one gets d S ˆ j d τ = − R ˆ0ˆ k ˆ0ˆ j S ˆ k + S ˆ k ∂ ˆ k A ˆ j − A ˆ j A ˆ k S ˆ k . (22)Similar expression was also found in Ref. [45]. Due to the requirement T b ∇ b ( S a T a ) = 0,one knows that d S ˆ0 / d τ = T c ∇ c [ T b ∇ b ( S a T a )] = 0, so S ˆ0 = 0 is really preserved while S a is propagated along the integral curves of T a . Whenever the observer σ ( τ ) is moving ona geodesic, A a = 0, then Eq. (22) becomes the usual geodesic deviation equation used toanalyze the polarizations of GWs [44]. IV. THE TRAJECTORY OF A SELF-GRAVITATING OBJECT IN HORNDESKITHEORY
The most general scalar-tensor theory with second order equations of motion is the Horn-deski theory [23], whose action is given by [46], S = (cid:90) d x √− g ( L + L + L + L ) + S m [ ψ m , g µν ] , (23)where S m [ ψ m , g µν ] is the action for the matter field ψ m , and it is assumed that ψ m non-minimally couples with the metric only. The individual terms in the integrand are L = K ( φ, X ) , (24) L = − G ( φ, X ) (cid:50) φ, (25) L = G ( φ, X ) R + G X [( (cid:50) φ ) − ( φ ; µν ) ] , (26) L = G ( φ, X ) G µν φ ; µν − G X (cid:50) φ ) − (cid:50) φ )( φ ; µν ) + 2( φ ; µν ) ] . (27)In these expressions, X = − φ ; µ φ ; µ / φ ; µ = ∇ µ φ , φ ; µν = ∇ ν ∇ µ φ , (cid:50) φ = g µν φ ; µν ,( φ ; µν ) = φ ; µν φ ; µν and ( φ ; µν ) = φ ; µν φ ; µρ φ ; ν ; ρ for simplicity. K, G , G , G are arbitrary ana-lytic functions of φ and X , and G iX = ∂ X G i , i = 3 , ,
5. For any binary function f ( φ, X ),define the following symbol f ( m,n ) = ∂ m + n f ( φ, X ) ∂φ m ∂X n (cid:12)(cid:12)(cid:12) φ = φ ,X =0 , (28)8here φ is a constant value for the scalar field evaluated at infinity. Varying the action (23)with respect to g µν and φ gives rise to the equations of motion, which are too complicatedto write down. Please refer to Refs [46, 47].There have been experimental constraints on Horndeski theory. Ref. [35] discussed thebounds on it from some solar system tests and the observations on pulsars. GW170817 andits electromagnetic counterpart GRB 170817A together set a strong constraint on the speedof GWs [7, 48]. Based on this result, the Lagrangian takes a simpler form [49–56], L = K ( φ, X ) − G ( φ, X ) (cid:50) φ + G ( φ ) R. (29)Although Horndeski theory is highly constrained, we will still work with the original theoryin the following discussion.In this theory, WEP is respected due to the non-minimal coupling between ψ m and g µν .However, due to the indirect interaction between ψ m and φ mediated by g µν via the equationsof motion, SEP is violated. In fact, calculations have shown that the effective gravitational“constant” actually depends on φ [57]. Therefore, the gravitational binding energy of acompact object, viewed as a system of point particles, will also depend on the local value of φ . Because of the mass-energy equivalence E = m , the mass of the compact object, i.e., thetotal mass of the system of point particles, also depends on φ . This would affect the motionof the compact object. Following Eardley’s suggestion, the matter action can be describedby [58] S m = − (cid:90) m ( φ ( x ρ )) (cid:113) − g µν ( x ρ ) ˙ x µ ˙ x ν d λ, (30)with ˙ x µ = d x µ / d λ , when the compact object can be treated as a self-gravitating particle.In this action, φ and g µν also depend on the trajectory. In this treatment, the spin andthe multipole moment structure are ignored. To obtain the equation of motion, one appliesEuler-Lagrange equation and at the same time, assumes that the parameter λ parameterizesthe trajectory such that g µν ˙ x µ ˙ x ν is a constant along the trajectory. Usually, one parameter-izes particle trajectories with the proper time τ . This is not necessary, as one can alwaysreparameterize. A generic parametrization is convenient for the following discussion.The Euler-Lagrange equation reads, A a = u b ∇ b u a = − d ln m d ln φ ( − g ab u c u c + u a u b ) ∇ b ln φ, (31)9here u a = ( ∂/∂λ ) a . Therefore, the self-gravitating particle no longer moves on a geodesic.The failure of its trajectory being a geodesic is described by d ln m d ln φ , which is called the ”sen-sitivity”. One can check that u a u b ∇ b u a = 0, which is consistent with the parametrization.This means that the 4-acceleration of the particle is a spatial vector with respect to u a .If one chooses the proper time τ to parameterize the trajectory, the above expression getssimplified, A a = − d ln m d ln φ ( δ ab + u a u b ) ∇ b ln φ, (32)where δ ab + u a u b is actually the projection operator for u a . Therefore, a self-gravitating objectmoves along an accelerated trajectory when only gravity acts on it, and its acceleration isdue to the gradient in the scalar field φ .Now, consider two infinitesimally nearby self-gravitating particles, one of which travelsalong σ ( λ ). The deviation vector connecting σ ( λ ) to its nearby company is S a . It is usefulto parameterize σ ( λ ) by its proper time τ so that u a is a unit timelike vector associatedwith an observer. The relative acceleration is thus given by A a rel = − R cbda u c S b u d − S b ∇ b (cid:20) d ln m d ln φ ( − g ac u d u d + u a u c ) ∇ c ln φ (cid:21) . (33)Note that the right hand side is evaluated at σ ( τ ). The deviation vector S a should satisfy u b ∇ b S a = S b ∇ b u a , (34) S b ∇ b ( u a u a ) = 2 d ln m d ln φ S a ∇ a ln φ, (35)according to Eq. (17), which explains why u d u d inside of the brackets of Eq. (33) is not setto −
1. The relative acceleration can be expressed entirely in terms of u a of the particle σ ( τ ) by expanding the brackets and using Eq. (34) together with Eq. (35), A a rel = − R cbda u c S b u d − ( g ac + u a u c ) S b ∇ b (cid:18) d ln m d ln φ ∇ c ln φ (cid:19) − d ln m d ln φ ( ∇ c ln φ ) (cid:20) u c u b ∇ b S a + u a u b ∇ b S c − g ac d ln m d ln φ S b ∇ b ln φ (cid:21) . (36)Again, the right hand side is evaluated along σ ( τ ).In the Fermi normal coordinates, the spatial components of A µ are given by A ˆ j = − d ln m d ln φ ∂ ˆ j ln φ, (37)10ccording to Eq. (32). By Eq. (22), one obtainsd S ˆ j d τ = − R ˆ0ˆ k ˆ0ˆ j S ˆ k − S ˆ k ∂ ˆ k (cid:18) d ln m d ln φ ∂ ˆ j ln φ (cid:19) + (cid:18) d ln m d ln φ (cid:19) ( ∂ ˆ j ln φ ) S ˆ k ∂ ˆ k ln φ. (38)When the scalar field is not excited,i.e., φ = φ , a constant, Eq. (38) reduces to the geodesicdeviation equation, d S ˆ j d τ = − R ˆ0ˆ k ˆ0ˆ j S ˆ k . (39)This is expected as vSEP is caused by a dynamical scalar field. In the next section, Eq. (38)will be used to analyze the polarization content of GWs in Horndeski theory. V. THE POLARIZATIONS OF GRAVITATIONAL WAVES IN HORNDESKIGRAVITY
In Ref. [59], the GW solutions for Horndeski theory [23] in the vacuum backgroundhave been obtained. The polarization content of the theory was also determined using thelinearized geodesic deviation equation, as the vSEP was completely ignored. In this section,the GW solution will be substituted into Eq. (38) to take into account the effect of thescalar field on the trajectories of self-gravitating test particles. This will lead to a differentpolarization content of GWs in Horndeski theory.Now, one expands the fields around the flat background such that g µν = η µν + h µν and φ = φ + ϕ . At the leading order, one obtains G , = 0 , G , = 0 . (40)At the first order, the linearized equations of motion can be written in the following form,( (cid:50) − m s ) ϕ = 0 , (41) (cid:50) ˜ h µν = 0 , (42)where the scalar field ϕ is generally massive with the squared mass given by m s = − K (2 , K (0 , − G , + 3 G , /G , , (43)and ˜ h µν is an auxiliary field defined as˜ h µν = h µν − η µν η αβ h αβ − χη µν ϕ, (44)11ith χ = G , G , . Note that the transverse-traceless (TT) gauge ∂ µ ˜ h µν = 0, η µν ˜ h µν = 0 hasbeen made. A GW propagating in the + z direction is given below˜ h µν = e µν cos Ω( t − z ) , (45) ϕ = ϕ cos( ωt − kz ) , (46)where ω − k = m s and the only nonvanishing components of tensor wave amplitude e µν are e = − e and e . The coordinate system in which the TT gauge is chosen is calledthe TT coordinate system.One is interested in studying the relative acceleration of two nearby particles which wereat rest before the arrival of the GW. Because of the presence of the GW induced by thescalar field, one expects σ ( τ ) to deviate from a straight line in the TT coordinates, so oneassumes its 3-velocity is (cid:126)v and u µ = u (1 , (cid:126)v ). The normalization of u a implies that u = 1 + 12 h + O ( v ) . (47)The acceleration of σ ( τ ) can be approximated as A µ ≈ − sφ ( η µν + u µ u ν ) ∇ ν ϕ, (48)with s = (d ln m/ d ln φ ) | φ called the sensitivity and u µ = (1 , (cid:126)
0) the background value.Written in component form, the acceleration is given by A = 0 , (49) A j = − δ j ks ϕ φ sin( ωt − kz ) . (50)On the other hand, the left hand side of Eq. (48) is, in coordinate basis, A µ = d x µ d τ + Γ µρν d x ρ d τ d x ν d τ ≈ ( u ) (cid:18) d x µ d t + Γ µ (cid:19) + u d u d t d x µ d t . (51)Consider a trivial motion, i.e., x = y = 0. Then one obtains v ≈ − k ω (cid:18) χ − sφ (cid:19) ϕ cos ωt, (52) z ≈ − k ω (cid:18) χ − sφ (cid:19) ϕ sin ωt. (53)12ere, the initial position of σ ( τ ) is chosen to be x = y = z = 0. In addition, u = d t d τ ≈ χϕ cos ωt, (54)according to Eq. (47), which implies that τ ≈ t − χϕ ω sin ωt. (55)From this, one clearly sees that the TT coordinate system is not the proper reference framefor the observer σ ( τ ).Therefore, the trajectory of σ ( τ ) in the TT coordinate system is described by τ = t − χϕ ω sin ωt, (56) x = y = 0 , (57) z = − k ω (cid:18) χ − sφ (cid:19) ϕ sin ωt, (58)up to the linear order. Because of the scalar field, the observer oscillates with the samefrequency of the GW in the TT coordinate system according to Eq. (58). The time dilatationalso oscillates by Eq. (56).In the limit of GR ( χ = s = 0), the trajectory of σ ( τ ) is thus t = τ, x j = 0 up to thelinear order, i.e., a geodesic of the background metric. If vSEP is weak, i.e. s ≈
0, thetrajectory is τ = t − χϕ ω sin ωt, (59) x = y = 0 , (60) z = − kχϕ ω sin ωt. (61)This agrees with Ref. [59]. Although the particle σ ( τ ) does not follow a geodesic of thebackground metric, it still travels along a geodesic of the full metric. A. The relative acceleration in the Fermi normal coordinates
In this subsection, one obtains the relative acceleration in the Fermi normal coordinatesusing Eq. (38). This discussion will also reveal the polarization content of GWs. The4-velocity of the observer is u a = ( e ˆ0 ) a = (1 + h / , , , v ) , (62)13o the following triad can be chosen,( e ˆ1 ) a = (0 , − h / , − h / , , (63)( e ˆ2 ) a = (0 , − h / , − h / , , (64)( e ˆ3 ) a = ( v , , , h / . (65)These basic vectors are Fermi-Walker transported and evaluated along σ ( τ ). The dual basisis denoted as { ( e ˆ µ ) a } and ( e ˆ µ ) ν ≈ δ ˆ µν is sufficient.Up to the linear order in perturbations, Eq. (22) is given byd S ˆ j d t = − R ˆ0ˆ k ˆ0ˆ j S ˆ k + S ˆ k ∂ ˆ k A ˆ j , (66)since the acceleration A ˆ j is of the linear order, and the last term in Eq. (22) should bedropped. Normally, one has to find the Fermi normal coordinates explicitly [60, 61]. How-ever, the Fermi normal coordinates differ from the TT coordinates by quantities of orderone, and the Riemann tensor and the 4-acceleration of the test particle are both of linear or-der, so any changes in their components caused by the coordinate transformation are of thesecond order in perturbations. Therefore, one only has to calculate the components of theRiemann tensor and the 4-acceleration in the TT coordinates, and then simply substitutesthem in Eq. (66).More explicitly, the driving force matrix is given by S ˆ k ˆ j = R ˆ0ˆ k ˆ0ˆ j − ∂ ˆ k A ˆ j ≈ R k j − ∂ k A j ≈ − ω χϕ + Ω ˜ h
11 Ω ˜ h Ω ˜ h − ω χϕ − Ω ˜ h
00 0 − m s χϕ − k sφ ϕ , (67)where ˜ h µν and ϕ are evaluated at ( t, (cid:126)x = 0). Comparing this matrix with the one (Eq. (29))in Ref. [59], one finds out that vSEP introduces an order one correction − k sϕ/φ to thelongitudinal polarization. This means that the longitudinal polarization gets enhanced.Even if the scalar field is massless, the longitudinal polarization persists because the testparticles are accelerated.However, the enhancement is very extremely small for objects such as the mirrors used indetectors such as LIGO. According to Refs. [29, 62], white dwarfs have typical sensitivities14 ∼ − , so a test particle, like the mirror used by LIGO, would have an even smallersensitivity. So it would be still very difficult to use interferometers to detect the enhancedlongitudinal polarization as in the previous case [59]. In contrast, neutron stars are compactobjects. Their sensitivity could be about 0.2 [29, 62]. They violate SEP relatively strongly,which might be detected by PTAs. VI. PULSAR TIMING ARRAYS
In this section, the cross-correlation function will be calculated for PTAs. The possibilityto detect the vSEP is thus inferred. A pulsar is a strongly magnetized, rotating neutronstar or a white dwarf, which emits a beam of the radio wave along its magnetic pole. Whenthe beam points towards the Earth, the radiation is observed, and this leads to the pulsedappearance of the radiation. The rotation of some “recycled” pulsars is stable enough sothat they can be used as “cosmic light-house” [63]. Among them, millisecond pulsars arefound to be more stable [64] and used as stable clocks [65]. When there is no GW, theradio pulses arrive at the Earth at a steady rate. The presence of the GW will affectthe propagation time of the radiation and thus alter this rate. This results in a changein the time-of-arrival (TOA), called timing residual R ( t ). Timing residuals caused by thestochastic GW background is correlated between pulsars, and the cross-correlation functionis C ( θ ) = (cid:104) R a ( t ) R b ( t ) (cid:105) with θ the angular separation of pulsars a and b , and the brackets (cid:104) (cid:105) implying the ensemble average over the stochastic background. This makes it possibleto detect GWs and probe the polarizations [37–40, 66–73]. The effect of vSEP can alsobe detected, as the longitudinal polarization of the scalar-tensor theory is enhanced due tovSEP.One sets up a coordinate system shown in Fig. 1 to calculate the timing residual R ( t )caused by the GW solution (45) and (46). Before the GW comes, the Earth is at the origin,and the distant pulsar is at rest at (cid:126)x p = ( L cos β, , L sin β ) in this coordinate system. TheGW is propagating in the direction of a unit vector ˆ k , and ˆ n is the unit vector pointingto the pulsar from the Earth. ˆ l = ˆ k ∧ (ˆ n ∧ ˆ k ) / cos β = [ˆ n − ˆ k (ˆ n · ˆ k )] / cos β is actually theunit vector parallel to the y axis. Where there is no GW, the photon is assumed to havea 4-velocity given by u µ = γ (1 , − cos β, , − sin β ) with γ = d t/ d λ a constant and λ anarbitrary affine parameter. Let the perturbed photon 4-velocity be u µ = u µ + v µ . The15 z l k Earth Pulsar n β FIG. 1. The GW is propagating in the direction of ˆ k , and the photon is traveling in − ˆ n directionat the leading order. ˆ l is perpendicular to ˆ k and in the same plane determined by ˆ k and ˆ n . Theangle between ˆ n and ˆ l is β . condition g µν u µ u ν = 0 together with the photon geodesic equation lead to v = γ (cid:110) χϕ cos[( ω + k sin β ) t − k ( L + t e ) sin β ] − e − sin β ) cos[Ω(1 + sin β ) t − Ω( L + t e ) sin β ] (cid:111) , (68) v = γ {− χϕ cos β cos[( ω + k sin β ) t − k ( L + t e ) sin β ]+ e cos β cos[Ω(1 + sin β ) t − Ω( L + t e ) sin β ] } , (69) v = γ e cos β cos Ω[(1 + sin β ) t − ( L + t e ) sin β ] , (70) v = γ (cid:110) − χϕ sin β cos[( ω + k sin β ) t − k ( L + t e ) sin β ] − e − sin β ) cos[Ω(1 + sin β ) t − Ω( L + t e ) sin β ] (cid:111) , (71)where t e is the time when the photon is emitted from the pulsar.The 4-velocity of an observer on the Earth has been obtained in Section V, which reads T µe = (cid:18) χϕ cos ωt, , , − k ω (cid:18) χ − s r φ (cid:19) ϕ cos ωt (cid:19) , (72)where s r is the sensitivity of the Earth. The 4-velocity of another observer comoving withthe pulsar can be derived in a similar way. In fact, the translational symmetry in thebackground spacetime (i.e., Minkowskian spacetime) gives T µp = (cid:18) χϕ cos( ωt − kL sin β ) , , , − k ω (cid:18) χ − s e φ (cid:19) ϕ cos( ωt − kL sin β ) (cid:19) , (73)16hich agrees with the result from the direct calculation. Here, s e is the sensitivity of thepulsar.So the measured frequency by the observer on the Earth is f r = − u µ T µe = γ (cid:34) (cid:18) ω − k sin β ω χ + s r kφ ω sin β (cid:19) ϕ cos ω ( t e + L ) − e − sin β ) cos Ω( t e + L ) (cid:35) , (74)and the one by the observer comoving with the pulsar is f e = − u µ T µp = γ (cid:34) (cid:18) ω − k sin β ω χ + s e kφ ω sin β (cid:19) ϕ cos( ωt e − kL sin β ) − e − sin β ) cos Ω( t e − L sin β ) (cid:35) . (75)Therefore, the frequency shift is given by f e − f r f r = ω − k ˆ k · ˆ n ω χ [ ϕ ( t − L, L ˆ n ) − ϕ ( t, − e jk ˆ n j ˆ n k k · ˆ n ) (cid:104) ˜ h jk ( t − L, L ˆ n ) − ˜ h jk ( t, (cid:105) + kωφ ˆ k · ˆ n [ s e ϕ ( t − L, L ˆ n ) − s r ϕ ( t, , (76)where t = t e + L is the time when the photon arrives at the Earth at the leading order. Thisequation has been expressed in a coordinate independent way, so it can be straightforwardlyused in any coordinate system with arbitrary orientation and at rest relative to the originalone. Note that the first two lines reproduce the result in Ref. [59], and the third line comesfrom the effect of vSEP. This effect is completely determined by the scalar perturbation ϕ ,as expected.Therefore, the focus will be on the cross-correlation function for the scalar GW in thefollowing discussion. Eq. (76) is the frequency shift due to a monochromatic wave. Now,consider the contribution of a stochastic GW background which consists of monochromaticGWs, ϕ ( t, (cid:126)x ) = (cid:90) ∞−∞ d ω π (cid:90) d ˆ k (cid:110) ϕ ( ω, ˆ k ) exp[ i ( ωt − k ˆ k · (cid:126)x )] (cid:111) , (77)17here ϕ ( ω, ˆ k ) is the amplitude for the scalar GW propagating in the direction ˆ k at theangular frequency ω . Usually, one assumes that the GW background is isotropic, stationaryand independently polarized, then one can define the characteristic strains ϕ c given by, (cid:104) ϕ ∗ ( ω, ˆ k ) ϕ ( ω (cid:48) , ˆ k (cid:48) ) (cid:105) = δ ( ω − ω ) δ (ˆ k − ˆ k (cid:48) ) | ϕ c ( ω ) | ω , (78)where the star ∗ implies the complex conjugation.The total timing residual in TOA due to the stochastic GW background is R ( T ) = (cid:90) ∞−∞ d ω π (cid:90) d ˆ k (cid:90) T d t f e − f r f r , (79)where the argument T is the total observation time. Insert Eq. (76) in, neglecting the secondline, to obtain R ( T ) = (cid:90) ∞−∞ d ω π (cid:90) d ˆ kϕ ( ω, ˆ k )( e iωT − (cid:40) ω − k ˆ k · ˆ ni ω χ × [ e − i ( ω + k ˆ k · n ) L −
1] + k ˆ k · ˆ niω φ [ s e e − i ( ω + k ˆ k · n ) L − s r ] (cid:41) . (80)With this result, consider the correlation between two pulsars a and b located at (cid:126)x a = L ˆ n and (cid:126)x b = L ˆ n , respectively. The angular separation is θ = arccos(ˆ n · ˆ n ). The cross-correlation function is thus given by C ( θ ) = (cid:104) R a ( T ) R b ( T ) (cid:105) = (cid:90) ∞ m s d ω (cid:90) d ˆ k | ϕ c ( ω ) | πω (cid:34) k ˆ k · ˆ n ˆ k · ˆ n φ P + k ˆ k · ˆ n ( ω − k ˆ k · ˆ n )2 φ χ P + k ˆ k · ˆ n ( ω − k ˆ k · ˆ n )2 φ χ P + ( ω − k ˆ k · ˆ n )( ω − k ˆ k · ˆ n )4 χ P (cid:35) , (81)where P , P , P and P are defined to be P = 1 − cos ∆ − cos ∆ + cos(∆ − ∆ ) , (82) P = s r − s r cos ∆ − s e cos ∆ + s e cos(∆ − ∆ ) , (83) P = s r − s r cos ∆ − s e cos ∆ + s e cos(∆ − ∆ ) , (84) P = s r − s r s e cos ∆ − s r s e cos ∆ + s e cos(∆ − ∆ ) , (85)with ∆ j = ( ω + k ˆ k · ˆ n j ) L j for j = 1 ,
2. To obtain this result, Eq. (78) is used, and the realpart is taken. In addition, T drops out, as the ensemble average also implies the averagingover the time [38]. 18ecause of the isotropy of the GW background, one setsˆ n = (0 , , , (86)ˆ n = (sin θ, , cos θ ) . (87)Also, let ˆ k = (sin θ g cos φ g , sin θ g sin φ g , cos θ g ), so∆ = ( ω + k cos θ g ) L , (88)∆ = [ ω + k (sin θ g cos φ g sin θ + cos θ g cos θ )] L , (89)Working in the limit that ωL j (cid:29)
1, one can drop the cosines in the definitions (82)-(85) of P j ( j = 1 , , , θ (cid:54) = 0. The integration can be partially done, resulting in C ( θ ) = (cid:90) ∞ m s d ω | ϕ c ( ω ) | ω χ (cid:34) k ω (cid:18) − s r φ χ (cid:19) cos θ (cid:35) . (90)But for θ = 0, one considers the auto-correlation function, so set ˆ n = ˆ n = (0 , ,
1) and L = L = L . The auto-correlation function is thus given by C (0) = (cid:90) ∞ m s d ω | ϕ c ( ω ) | ω χ (cid:34) k ω (cid:18) − s r φ χ (cid:19) + k ω (cid:18) − s e φ χ (cid:19) (cid:35) , (91)where the terms containing L are dropped as they barely contribute according to the ex-perience in Ref. [59]. Finally, the observation time T sets a natural cutoff for the angularfrequency, i.e., ω ≥ π/T , so the lower integration limits in Eqs. (90) and (91) should bereplaced by Max { m s , π/T } .As usual, assume ϕ c ( ω ) ∝ ( ω/ω c ) α with ω c the characteristic angular frequency. Here, α iscalled the power-law index, and usually, α = 0 , − / − ζ ( θ ) = C ( θ ) /C (0). Inthe integration, set the observation time T = 5 years. The sensitivities of the Earth andthe pulsar are taken to be s r = 0 and s e = 0 .
2, respectively. This leads to Fig. 2, where thepower-law index α takes different values.If the scalar field is massless, the results are shown in the left panel which displays thenormalized correlation functions for the plus and cross polarizations – Hellings-Downs curve(labeled by “GR”) [69]. The remaining two curves are for the breathing polarization: thedashed one is for the case where SEP is respected, while the dotted one is for the case whereSEP is violated. They are independent of the power-law index α . As one can see that vSEP19 RSEPvSEP - θ ζ ( θ ) α = α = α =- / α =- / α =- α =- θ ζ ( θ ) FIG. 2. The normalized cross-correlation functions ζ ( θ ) = C ( θ ) /C (0). The left panel shows thecross-correlations when the scalar field is massless, i.e., when there is no longitudinal polarization.The solid curve is for familiar GR polarizations (i.e., the plus or cross ones), the dashed redcurve for the breathing polarization with SEP and the dotted purple curve for the breathingpolarization with vSEP. The right panel shows the normalized cross-correlations induced togetherby the transverse breathing and longitudinal polarizations when the mass of the scalar field is takento be m s = 7 . × − eV /c . The solid curves are for the cases where SEP is satisfied, while thedashed curves are for those where SEP is violated. The power-law index α = 0 , − / , −
1. Thecalculation was done assuming T = 5 yrs. makes ζ ( θ ) bigger by about 5%. If the scalar field has a mass m s = 7 . × − eV /c , theresults are shown in the right panel. In this panel, the cross-correlation functions for thescalar polarization are drawn for different values of α . The solid curves correspond to thecase where SEP is satisfied, and the dashed curves are for the case where SEP is violated.Since the cross correlation for the plus and cross polarizations does not change, we do notplot them again in the right panel. In the massive case, vSEP also increases ζ ( θ ) by about2% to 3%.Ref. [75] published the constraint on the stochastic GW background based on the recentlyreleased 11-year dataset from the North American Nanohertz Observatory for GravitationalWaves (NANOGrav). Assuming the background is isotropic and α = − /
3, the strain20mplitude of the GW is less than 1 . × − at f = 1 yr − . In addition, the top panelin Figure 6 shows the observed cross correlation. As one can clearly see, the error bars arevery large[76]. More observations are needed to improve the statistics. VII. CONCLUSION
This work discusses the effects of the vSEP on the polarization content of GWs in Horn-deski theory and calculates the cross-correlation functions for PTAs. Because of the vSEP,self-gravitating particles no longer travel along geodesics, and this leads to the enhancementof the longitudinal polarization in Horndeski theory, so even if the scalar field is massless, thelongitudinal polarization still exists. This is in contrast with the previous results [59, 77–79]that the massive scalar field excites the longitudinal polarization, while the massless scalarfield does not. The enhanced longitudinal polarization is nevertheless difficult for aLIGOto detect, as the mirrors does not violate SEP enough. However, pulsars are highly com-pact objects with sufficient self-gravitating energy such that their trajectories deviate fromgeodesics enough. Using PTAs, one can measure the change in TOAs of electromagneticradiation from pulsars and obtain the cross-correlation function to tell whether vSEP ef-fect exits. The results show that the vSEP leads to large changes in the behaviors of thecross-correlation functions. In principle, PTAs are capable of detecting the vSEP if it exists.
ACKNOWLEDGMENTS
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