Constraints on Horndeski Theory Using the Observations of Nordtvedt Effect, Shapiro Time Delay and Binary Pulsars

Constraints on Horndeski Theory Using the Observations of Nordtvedt Effect, Shapiro Time Delay and Binary Pulsars

Shaoqi Hou School of Physics, Huazhong University of Science and Technology, Wuhan, Hubei 430074, China    Yungui Gong School of Physics, Huazhong University of Science and Technology, Wuhan, Hubei 430074, China
July 10, 2019
Abstract

Alternative theories of gravity not only modify the polarization contents of the gravitational wave, but also affect the motions of the stars and the energy radiated away via the gravitational radiation. These aspects leave imprints in the observational data, which enables the test of General Relativity and its alternatives. In this work, the Nordtvedt effect and the Shapiro time delay are calculated in order to constrain Horndeski theory using the observations of lunar laser ranging experiments and Cassini time-delay data. The effective stress-energy tensor is also obtained using the method of Isaacson. Gravitational wave radiation of a binary system is calculated, and the change of the period of a binary system is deduced for the elliptical orbit. These results can be used to set constraints on Horndeski theory with the observations of binary systems, such as PSR J1738+0333. Constraints have been obtained for some subclasses of Horndeski theory, in particular, those satisfying the gravitational wave speed limits from GW170817 and GRB 170817A.

{CJK*}

GBgbsn

I Introduction

General Relativity (GR) is one of the cornerstones of modern physics. However, it faces several challenges. For example, GR cannot be quantized, and it cannot explain the present accelerating expansion of universe, i.e., the problem of dark energy. These challenges motivate the pursuit of the alternatives to GR, one of which is the scalar-tensor theory. The scalar-tensor theory contains a scalar field as well as a metric tensor to describe the gravity. It is the simplest alternative metric theory of gravity. It solves some of GR’s problems. For example, the extra degree of freedom of the scalar field might account for the dark energy and explain the accelerating expansion of the universe. Certain scalar-tensor theories can be viewed as the low energy limit of string theory, one of the candidates of quantum gravity Fujii and Maeda (2003).

The detection of gravitational waves by the Laser Interferometer Gravitational-Wave Observatory (LIGO) and Virgo confirms GR to an unprecedented precision Abbott et al. (2016a, b, 2017a, 2017b, 2017c, 2017d) and also provides the possibility to test GR in the dynamical, strong field limit. The recent GW170814 detected the polarizations for the first time, and the result showed that the pure tensor polarizations are favored against pure vector and pure scalar polarizations Abbott et al. (2017b). The newest GW170817 is the first neutron star-neutron star merger event, and the concomitant gamma-ray burst GRB 170817A was later observed by the Fermi Gamma-ray Burst Monitor and the Anti-Coincidence Shield for the Spectrometer for the International Gamma-Ray Astrophysics Laboratory, independently Abbott et al. (2017c); Goldstein et al. (2017); Savchenko et al. (2017). This opens the new era of multi-messenger astrophysics. It is thus interesting to study gravitational waves in alternative metric theories of gravity, especially the scalar-tensor theory.

In 1974, Horndeski Horndeski (1974) constructed the most general scalar-tensor theory whose action contains higher derivatives of and , but still yields at most the second order differential field equations, and thus has no Ostrogradsky instability Ostrogradsky (1850). Because of its generality, Horndeski theory includes several important specific theories, such as GR, Brans-Dicke theory Brans and Dicke (1961), and gravity Buchdahl (1970); O’Hanlon (1972); Teyssandier and Tourrenc (1983) etc..

In Refs. Liang et al. (2017); Hou et al. (2017); Gong and Hou (2018), we discussed the gravitational wave solutions in gravity and Horndeski theory, and their polarization contents. These works showed that in addition to the familiar + and polarizations in GR, there is a mixed state of the transverse breathing and longitudinal polarizations both excited by a massive scalar field, while a massless scalar field excites the transverse breathing polarization only. In this work, it will be shown that the presence of a dynamical scalar field also changes the amount of energy radiated away by the gravitational wave affecting, for example, the inspiral of binary systems. Gravitational radiation causes the damping of the energy of the binary system, leading to the change in the orbital period. In fact, the first indirect evidence for the existence of gravitational waves is the decay of the orbital period of the Hulse-Taylor pulsar (PSR 1913+16) Hulse and Taylor (1975).

Previously, the effective stress energy tensor was obtained by Nutku Nutku (1969) using the method of Landau and Lifshitz Landau and Lifshitz (1975). The damping of a compact binary system due to gravitational radiation in Brans-Dicke theory was calculated in Refs. Wagoner (1970); Will and Zaglauer (1989); Damour and Esposito-Farèse (1998); Brunetti et al. (1999), then Alsing et al. Alsing et al. (2012) extended the analysis to the massive scalar-tensor theory. Refs. Stein and Yunes (2011); Saffer et al. (2018) surveyed the effective stress-energy tensor for a wide class of alternative theories of gravity using several methods. However, they did not consider Horndeski theory. Refs. Zhang et al. (2017); Liu et al. (2018) studied the gravitational radiation in screened modified gravity and gravity. Hohman Hohmann (2015) developed parameterized post-Newtonian (PPN) formalism for Horndeski theory. In this work, the method of Isaacson is used to obtain the effective stress-energy tensor for Horndeski theory. Then the effective stress-energy tensor is applied to calculate the rate of energy damping and the period change of a binary system, which can be compared with the observations on binary systems to constrain Horndeski theory. Nordtvedt effect and Shapiro time delay effect will also be considered to put further constraints. Ashtekar and Bonga pointed out in Refs. Ashtekar and Bonga (2017a, b) a subtle difference between the transverse-traceless part of defined by and the one defined by using the spatial transverse projector, but this difference does not affect the energy flux calculated in this work.

There were constraints on Horndeski theory and its subclasses in the past. The observations of GW170817 and GRB 170817A put severe constraints on the speed of gravitational waves Abbott et al. (2017e). Using this limit, Ref. Creminelli and Vernizzi (2017) required that and , while Ref. Ezquiaga and Zumalacárregui (2017) required and . Ref. Baker et al. (2017) obtained the similar results as Ref. Ezquiaga and Zumalacárregui (2017), and also pointed out that the self-accelerating theories should be shift symmetric. Arai and Nishizawa found that Horndeski theory with arbitrary functions and needs fine-tuning to account for the cosmic accelerating expansion Arai and Nishizawa (2017). For more constraints derived from the gravitational wave speed limit, please refer to Refs. Sakstein and Jain (2017); Gong et al. (2017); Crisostomi and Koyama (2018), and for more discussions on the constraints on the subclasses of Horndeski theory, please refer to Refs. Will (2014); Lambiase et al. (2015); Bhattacharya and Chakraborty (2017); Banerjee et al. (2017); Shao et al. (2017).

In this work, the calculation will be done in the Jordan frame, and the screening mechanisms, such as the chameleon Khoury and Weltman (2004a, b) and the symmetron Hinterbichler and Khoury (2010); Hinterbichler et al. (2011), are not considered. Vainshtein mechanism was first discovered to solve the vDVZ discontinuity problem for massive gravity Vainshtein (1972), and later found to also appear in theories containing the derivative self-couplings of the scalar field, such as some subclasses of Horndeski theory Deffayet et al. (2002); Babichev et al. (2009); Koyama et al. (2013); Babichev and Deffayet (2013); Winther and Ferreira (2015). When Vainshtein mechanism is in effect, the effect of nonlinearity cannot be ignored within the so-called Vainshtein radius from the center of the matter source. Well beyond , the linearization can be applied. The radius depends on the parameters defining Horndeski theory, and can be much smaller than the size of a celestial object. So in this work, we consider Horndeski theories which predict small , if it exists, compared to the sizes of the Sun and neutron stars. The linearization can thus be done even deep inside the stars. In this case, one can safely ignore Vainshtein mechanism.

The paper is organized as follows. In Section II, Horndeski theory is briefly introduced and the equations of motion are derived up to the second order in perturbations around the flat spacetime background. Section III derives the effective stress-energy tensor according to the procedure given by Isaacson. Section IV is devoted to the computation of the metric and scalar perturbations in the near zone up to Newtonian order and the discussion of the motion of self-gravitating objects that source gravitational waves. In particular, Nordtvedt effect and Shapiro time delay are discussed. In Section V, the metric and scalar perturbations are calculated in the far zone up to the quadratic order, and in Section VI, these solutions are applied to a compact binary system to calculate the energy emission rate and the period change. Section VII discusses the constraints on Horndeski theory based on the observations. Finally, Section VIII summarizes the results. Throughout the paper, the speed of light in vacuum is taken to be .

Ii Horndeski Theory

The action of Horndeski theory is given by Kobayashi et al. (2011),

 S=∫d4x√−g(L2+L3+L4+L5)+Sm[gμν,ψm], (1)

where represents matter fields, is the action for , and the terms in the integrand are

 L2=G2(ϕ,X),L3=−G3(ϕ,X)□ϕ, (2) L4=G4(ϕ,X)R+G4X[(□ϕ)2−(ϕ;μν)2], (3) L5=G5(ϕ,X)Gμνϕ;μν−G5X6[(□ϕ)3−3(□ϕ)(ϕ;μν)2 +2(ϕ;μν)3]. (4)

In these expressions, with , , , and for simplicity. are arbitrary functions of and 111 is usually called in literature.. For notational simplicity and clarity, we define the following symbol for the function ,

 (5)

so in particular, with the value of in the flat spacetime background.

Suitable choices of reproduce interesting subclasses of Horndeski theory. For instance, one obtains GR by choosing and the remaining , with Newton’s constant. Brans-Dicke theory is recovered with , while the massive scalar-tensor theory with a potential Alsing et al. (2012) is obtained with , where is a constant; or with , , . Finally, gravity is given by , , with .

ii.1 Matter action

Although there are no coupling terms between matter fields and , matter fields indirectly interact with via the metric tensor. For example, in Brans-Dicke theory, acts effectively like the gravitational constant, which influences the internal structure and motion of a gravitating object, so the binding energy of the object depends on . Since the total energy is related to the inertial mass , then depends on , too. When their spins and multipole moments can be ignored, the gravitating objects can be described by point like particles, and the effect of can be taken into account by the following matter action according to Eardley’s prescription Eardley (1975),

 Sm=−∑a∫ma(ϕ)dτa, (6)

whose stress-energy tensor is

 Tμν=1√−g∑ama(ϕ)uμuνu0δ(4)(xλ−xλa(τ)), (7)

where describes the worldline of particle and . Therefore, if there is no force other than gravity acting on a self-gravitating object, this object will not follow the geodesic. This causes the violation of the strong equivalence principle (SEP).

In this work, the gravitational wave is studied in the flat spacetime background with and , so we expand the masses around the value in the following way,

 ma(ϕ)=ma[1+φϕ0sa−12(φϕ0)2(s′a−s2a+sa)+O(φ3)]. (8)

Here, is the perturbation, and for simplicity. This expansion also requires that , so the present discussion does not apply to gravity. and are the first and second sensitivities of the mass ,

 sa=dlnma(ϕ)dlnϕ∣∣ϕ0,s′a=−d2lnma(ϕ)d(lnϕ)2∣∣ϕ0. (9)

The sensitivities measure the violation of SEP.

ii.2 Linearized equations of motion

The equations of motion can be obtained and simplified using xAct package Martín-García et al. (2007, 2008); Martín-García (2008); Brizuela et al. (2009); Martín-García (). Because of their tremendous complexity, the full equations of motion will not be presented. Interested readers are referred to Refs. Kobayashi et al. (2011); Gao (2011). As we checked, xAct package gives the same equations of motion as Refs. Kobayashi et al. (2011); Gao (2011). For the purpose of this work, the equations of motion are expanded up to the second order in perturbations defined as

 gμν=ημν+hμν,ϕ=ϕ0+φ. (10)

These equations are given in A.

The gravitational wave solutions are investigated in the flat spacetime background, which requires that

 G2(0,0)=0,G2(1,0)=0. (11)

This can be easily checked by a quick inspection of Eqs. (94) and (94). Then dropping higher order terms in Eqs. (94) and (94), the linearized equations of motion are thus given by

 (G2(0,1)−2G3(1,0))□φ+G2(2,0)φ+G4(1,0)R(1)=−(∂T∂ϕ)(1), (12) G4(0,0)G(1)μν−G4(1,0)(∂μ∂νφ−ημν□φ)=12T(1)μν, (13)

where is the trace, from now on, and the superscript implies the leading order part of the quantity.

The equations of motion can be decoupled by introducing an auxiliary field defined as following,

 ~hμν=hμν−12ημνh−G4(1,0)G4(0,0)ημνφ, (14)

where is the trace, and the original metric tensor perturbation is,

 hμν=~hμν−12ημν~h−G4(1,0)G4(0,0)ημνφ, (15)

with . The equations of motion are gauge invariant under the the following infinitesimal coordinate transformation,

 φ′=φ,~h′μν=~hμν−∂μξν−∂νξμ+ημν∂ρξρ, (16)

with . Therefore, one can choose the transverse gauge , and after some algebraic manipulations, the equations of motion become

 (□−m2s)φ=T(1)∗2G4(0,0)ζ, (17) □~hμν=−T(1)μνG4(0,0), (18)

where 222The way defining is different from the one defining in Ref. Alsing et al. (2012) in that the coefficient of is not 1. with , and the mass of the scalar field is

 m2s=−G2(2,0)/ζ, ζ=G2(0,1)−2G3(1,0)+3G24(1,0)/G4(0,0). (19)

Of course, , otherwise is non-dynamical.

From the equations of motion (17) and (18)), one concludes that the scalar field is generally massive unless is zero, and the auxiliary field resembles the spin-2 graviton field in GR. is sourced by the matter stress-energy tensor, while the source of the scalar perturbation is a linear combination of the trace of the matter stress-energy tensor and the partial derivative of the trace with respect to . This is because of the indirect interaction between the scalar field and the matter field via the metric tensor.

Iii Effective Stress-Energy Tensor

The method of Isaacson Isaacson (1967, 1968) will be used to obtain the effective stress-energy tensor for gravitational waves in Horndeski theory in the short-wavelength approximation, i.e., the wavelength with representing the typical value of the background Riemann tensor components. This approximation is trivially satisfied in our case, as the background is flat and . In averaging over several wavelengths, the following rules are utilized Misner et al. (1973):

1. The average of a gradient is zero, e.g., ,

2. One can integrate by parts, e.g., ,

where implies averaging. These rules apply to not only terms involving but also those involving . In the case of a curved background, these rules are supplemented by the one that covariant derivatives commute, which always holds in the flat background case.

With this method, the effective stress-energy tensor in an arbitrary gauge can be calculated straightforwardly using xAct and given by,

 TGWμν=⟨12G4(0,0)(∂μ~hρσ∂ν~hρσ−12∂μ~h∂ν~h−∂μ~hνρ∂σ~hσρ−∂ν~hμρ∂σ~hσρ)+ζ∂μφ∂νφ+G4(1,0)(m2sφ~hμν+∂μφ∂ρ~hρν+∂νφ∂ρ~hρμ−ημν∂σφ∂ρ~hρσ)⟩. (20)

It can be checked that this expression is gauge invariant under Eq. (16). In fact, the terms in the first around brackets take exactly the same forms as in GR excerpt for a different factor. The fourth line remains invariant, as in the gauge transformation. To show that the remaining lines are also gauge invariant, making the replacement gives

 Remaining lines=⟨G4(1,0)(m2sφ~hμν+∂μφ∂ρ~hρν+∂νφ∂ρ~hρμ−ημν∂σφ∂ρ~hρσ)⟩+⟨m2sG4(1,0)φ(−∂μξν−∂νξμ+ημν∂ρξρ)+G4(1,0)(−∂μφ∂ρ∂ρξν−∂νφ∂ρ∂ρξμ+ημν∂σφ∂ρ∂ρξσ)⟩. (21)

Far away from the matter, according to Eq. (17). Substituting this into the fourth line of Eq. (21), one immediately finds total derivatives of the forms and . So the first averaging rule implies that the last three lines of Eq. (21) vanish. Therefore, the effective stress-energy tensor (20) is indeed gauge invariant.

In vacuum, the transverse-traceless (TT) gauge ( and ) can be taken, and the effective stress-energy simplifies,

 TGWμν=⟨12G4(0,0)∂μ~hTTρσ∂ν~hρσTT+ζ∂μφ∂νφ+m2sG4(1,0)φ~hTTμν⟩, (22)

where denotes the transverse-traceless part. In the limit that and the remaining arbitrary functions vanish, Eq. (20) recovers the effective stress-energy tensor of GR Misner et al. (1973). One can also check that Eq. (20) reduces to the one given in Ref. Brunetti et al. (1999) for Brans-Dicke theory in the gauge of and .

In order to calculate the energy carried away by gravitational waves, one has to first study the motion of the source. This is the topic of the next section.

Iv The Motion of Gravitating Objects in the Newtonian Limit

The motion of the source will be calculated in the Newtonian limit. The source is modeled as a collection of gravitating objects with the action given by Eq. (6). In the slow motion, weak field limit, there exists a nearly global inertial reference frame. In this frame, a Cartesian coordinate system is established whose origin is chosen to be the center of mass of the matter source. Let represent the field point whose length is denoted by .

In the near zone Poisson and Will (2014), the metric and the scalar perturbations will be calculated at the Newtonian order. The stress-energy tensor of the matter source is given by 333The matter stress-energy tensor and the derivative of its trace with respect to , , are both expanded beyond the leading order, because the higher order contributions are need to calculate the scalar perturbations in Section V.,

 Tμν=∑amauμuν(1−12v2a−12hjj+saφϕ0+O(v4))δ(4)(xλ−xλa(τ)), (23)

and one obtains,

 ∂T∂ϕ=−∑amaϕ0[sa(1−12hjj−v2a2)−(s′a−s2a+sa)φϕ0+O(v4)]δ(4)(xλ−xλa(τ)). (24)

In these expressions, the 4-velocity of particle is and . With these results, the leading order of the source for the scalar field is

 T(1)∗=−∑amaSaδ(4)(xλ−xλa(τ)), (25)

with .

Now, the linearized equations (17, 18) take the following forms

 (□−m2s)φ=−12G4(0,0)ζ∑amaSaδ(4)(xλ−xλa(τ)), (26) □~hμν=−1G4(0,0)∑amauμuνδ(4)(xλ−xλa(τ)), (27)

and the leading order contributions to the perturbations are easily obtained,

 φ(t,→x)=18πG4(0,0)ζ∑amaSarae−msra, (28) ~h00(t,→x)=14πG4(0,0)∑amara, (29)

and at this order, where and the scalar field is given by a sum of Yukawa potentials. The leading order metric perturbation can be determined by Eq. (15),

 (30) hjk=δjk8πG4(0,0)∑amara(1−G4(1,0)G4(0,0)ζSae−msra), (31)

with .

iv.1 Static, spherically symmetric solutions

For the static, spherically symmetric solution with a single point mass at rest at the origin as the source, the time-time component of the metric tensor is

 g00=−1+18πG4(0,0)Mr(1+G4(1,0)G4(0,0)ζSMe−msr)+⋯, (32)

where and is the sensitivity of the point mass . From this, the “Newton’s constant” can be read off

 GN(r)=116πG4(0,0)(1+G4(1,0)G4(0,0)ζSMe−msr), (33)

which actually depends on the distance because the scalar field is massive. The measured Newtonian constant at the earth is with the radius of the Earth. The “post-Newtonian parameter” can also be read off by examining , which is

 gjk=δjk[1+18πG4(0,0)Mr(1−G4(1,0)G4(0,0)ζSMe−msr)]+⋯=δjk(1+2G4(0,0)ζ−G4(1,0)SMe−msrG4(0,0)ζ+G4(1,0)SMe−msrGN(r)Mr)+⋯. (34)

In the PPN formalism, the space-space components of the metric take the following form,

 gPPNjk=δjk(1+2γGNMr)+⋯, (35)

where the parameter is a constant. So

 γ(r)=G4(0,0)ζ−G4(1,0)SMe−msrG4(0,0)ζ+G4(1,0)SMe−msr. (36)

The above result can recover the results for gravity and general scalar-tensor theory Capone and Ruggiero (2010); Perivolaropoulos (2010); Hohmann et al. (2013); Hohmann (2015) if we keep the equivalence principle. In the massless case (), we get

 GN=116πG4(0,0)[1+G4(1,0)G4(0,0)ζSM], (37) γ=G4(0,0)ζ−G4(1,0)SMG4(0,0)ζ+G4(1,0)SM. (38)

Note that and both depend on which reflects the internal structure and motion of the gravitating object in question. Even if the scalar field is massless, this dependence still persists. Therefore, neither of them is universal due to the violation of SEP caused by the scalar field. It is obvious that should take the same value as .

iv.2 Equations of motion of the matter

With the near zone solutions (28), (30) and (31) one obtains the total matter Lagrangian up to the linear order,

 (39)

where is the distance between the particles and . The equation of motion for the mass can thus be obtained using the Euler-Lagrange equation, yielding its acceleration,

 aja=−116πG4(0,0)∑b≠ambr2ab^rjab×[1+SaSbG4(0,0)ζ(1+msrab)e−msrab], (40)

with . In particular, for a binary system, the relative acceleration is

 aj=−m^rj1216πG4(0,0)r212[1+SaSbG4(0,0)ζ(1+msr12)e−msr12], (41)

where is the total mass. The first term in the square brackets gives the result that resembles the familiar Newtonian gravitational acceleration, while the second one reflects the effect of the scalar field. In the massless case, the second term no longer depends on and can be absorbed into the first one, so the binary system moves in a similar way as in Newtonian gravity with a modified Newton’s constant.

The Hamiltonian of the matter is

 Hm=∑a→pa⋅→xa−Lm=∑ama[12v2a−132πG4(0,0)×∑b≠ambrab(1+SaSbG4(0,0)ζe−msrab)], (42)

where is the -th component of the canonical momentum of particle , and the total rest mass has been dropped. In particular, the Hamiltonian of a binary system is given by

 Hm=μv22−μm16πG4(0,0)r12×[1+S1S2G4(0,0)ζ(1+msr12)e−msr12], (43)

where , and is the reduced mass. This will be useful for calculating the total mechanical energy of a binary system and the ratio of energy loss due to the gravitational radiation.

iv.3 Nordtvedt effect

The presence of the scalar field modifies the trajectories of self-gravitating bodies. They will no longer follow geodesics. Therefore, SEP is violated in Horndeski theory. This effect is called the Nordtvedt effect Nordtvedt (1968a, b). It results in measurable effects in the solar system, one of which is the polarization of the Moon’s orbit around the Earth Nordtvedt (1982); Will (1993).

To study the Nordtvedt effect, one considers a system of three self-gravitating objects and and studies the relative acceleration of and in the field of . With Eq. (40) and assuming , the relative acceleration is

 ajab≈−116πG4(0,0)ma+mbr2ab^rjab×[1+SaSbG4(0,0)ζ(1+msrab)e−msrab]−mc16πG4(0,0)(^rjacr2ac−^rjbcr2bc)+Sc(sa−sb)8πG4(0,0)ϕ0ζmc^rjacr2ac(1+msrac)e−msrac, (44)

where the first term presents the Newtonian acceleration modified by the presence of the scalar field, the second is the tidal force caused by the gravitational gradient due to the object , and the last one describes the Nordtvedt effect. The effective Nordtvedt parameter is

 ηN=Sc8πGNG4(0,0)ϕ0ζ(1+msrac)e−msrac. (45)

This parameter depends on , so this effect is indeed caused by the violation of SEP.

iv.4 Shapiro time delay effect

Another effect useful for constraining Horndeski theory is the Shapiro time delay Shapiro (1964). In order to calculate this effect, one considers the photon propagation time in a static (or nearly static) gravitational field produced by a single mass at the origin. Due to the presence of gravitational potential, the 3-velocity of the photon in the nearly inertial coordinate system is no longer 1 and varies. The propagation time is thus different from that when the spacetime is flat. Let the 4 velocity of the photon be , then gives

 −1+h00+(δjk+hjk)vjvk=0, (46)

where and are given by Eqs. (30) and (31) specialized to a single mass case. In the flat spacetime, the trajectory for a photon emitted from position at time is a straight line , where is the direction of the photon. The presence of the gravitational potential introduces a small perturbation so that . Substituting Eqs. (30) and (31) into Eq. (46), one obtains

 ^N⋅dδ→xdt=−M8πG4(0,0)r(t), (47)

where . Suppose the photon emitted from position is bounced back at position and finally returns to . The total propagation time is

 Δt=2|→xp−→xe|+δt, (48)

where is caused by the Shapiro time delay effect,

 δt=2∫tpte^N⋅dδ→xdtdt=M4πG4(0,0)ln(re+^N⋅→xe)(rp−^N⋅→xp)r2b, (49)

where , and is the impact parameter of the photon relative to the source.

Since in Eq. (49) is not measurable, one replaces it with the Keplerian mass

 MK=M16πG4(0,0)GN(1+G4(1,0)G4(0,0)ζSMe−msr), (50)

with and the sensitivity of the source. In terms of , the Shapiro time delay is

 δt=2GNMK(1+γ(r))ln(re+^N⋅→xe)(rp−^N⋅→xp)r2b. (51)

For the Shapiro time delay occurring near the Sun, in the above equation should be 1 AU, as this is approximately the distance where the Keplerian mass of the Sun is measured.

V Gravitational Wave Solutions

In the far zone, only the space-space components of the metric perturbation are needed to calculate the effective stress-energy tensor. Since the equation of motion (18) for takes the similar form as in GR, the leading order contribution to is given by,

 ~hjk(t,→x)=18πG4(0,0)rd2Ijkdt2, (52)

where is the mass quadrupole moment. As in GR, the TT part of is also related to the reduced quadrupole moment ,

 ~hTTjk=18πG4(0,0)rd2JTTjkdt2. (53)

The leading order term for the scalar field is the mass monopole which does not contribute to the effective stress-energy tensor, so it is necessary to take higher order terms into account. To do so, the scalar equation (94) is rewritten with the linearized equations substituted in, which is given by

 (54)

In the following discussion, it is assumed that the scalar field is massless for simplicity. The details to obtain the following results can be found in B. The leading order contribution to comes from the first term on the right hand side of Eq. (54), which is the mass monopole moment,

 φ[1]=18πG4(0,0)ζr∑amaSa. (55)

From now on, the superscript indicates the order of a quantity in terms of the speed , i.e., is of the order . is independent of time, so it does not contribute to the effective stress-energy tensor. The next leading order term is the mass dipole moment,

 φ[1.5]=18πG4(0,0)ζr∑amaSa(^n⋅→va), (56)

in which . This gives the leading contribution to the effective stress-energy tensor. At the next next leading order, there are more contributions from the remaining terms on the right hand side of Eq. (54). First, there is the mass quadruple moment contribution,

 φ[2]1=18πG4(0,0)ζr∑amaSa[(^n⋅→aa)(^n⋅→xa)+(^n⋅→va)2]. (57)

And the remaining contribution to the scalar wave is

 φ[2]2=−116πG4(0,0)ζr∑amaSav2a+164π2G24(0,0)ζr′∑a,bmambrab(−Sa2+3G4(1,0)2G4(0,0)ζSaSb+S′aSbϕ0ζ)+164π2G24(0,0)ζ2r⎛⎝G4(2,0)−G24(1,0)G4(0,0)⎞⎠∑∑′a,bmambSbrab,−G2(3,0)256π2