# Constraints on Horndeski theory using the observations of Nordtvedt effect, Shapiro time delay and binary pulsars

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## 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.

## 1 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 \(\phi \) as well as a metric tensor \(g_{\mu \nu }\) 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 [1].

The detection of gravitational waves by the Laser Interferometer Gravitational-Wave Observatory (LIGO) and Virgo confirms GR to an unprecedented precision [2, 3, 4, 5, 6, 7] 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 [5]. 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 [6, 8, 9]. 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 [10] constructed the most general scalar-tensor theory whose action contains higher derivatives of \(\phi \) and \(g_{\mu \nu }\), but still yields at most the second order differential field equations, and thus has no Ostrogradsky instability [11]. Because of its generality, Horndeski theory includes several important specific theories, such as GR, Brans–Dicke theory [12], and *f*(*R*) gravity [13, 14, 15] etc..

In Refs. [16, 17, 18], we discussed the gravitational wave solutions in *f*(*R*) gravity and Horndeski theory, and their polarization contents. These works showed that in addition to the familiar + and \(\times \) 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) [19].

Previously, the effective stress energy tensor was obtained by Nutku [20] using the method of Landau and Lifshitz [21]. The damping of a compact binary system due to gravitational radiation in Brans–Dicke theory was calculated in Refs. [22, 23, 24, 25], then Alsing et al. [26] extended the analysis to the massive scalar-tensor theory. Refs. [27, 28] 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. [29, 30] studied the gravitational radiation in screened modified gravity and *f*(*R*) gravity. Hohman [31] 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. [32, 33] a subtle difference between the transverse-traceless part of \(h_{\mu \nu }\) defined by \(\partial ^\nu h_{\mu \nu }=0,\,\eta ^{\mu \nu }h_{\mu \nu }=0\) 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 [34]. Using this limit, Ref. [35] required that \(\partial G_5/\partial X\) \(=0\) and \(2\partial G_4/\partial X+\partial G_5/\partial \phi =0\), while Ref. [36] required \(\partial G_4/\partial X\approx 0\) and \(G_5\approx \text {constant}\). Ref. [37] obtained the similar results as Ref. [36], and also pointed out that the self-accelerating theories should be shift symmetric. Arai and Nishizawa found that Horndeski theory with arbitrary functions \(G_4\) and \(G_5\) needs fine-tuning to account for the cosmic accelerating expansion [38]. For more constraints derived from the gravitational wave speed limit, please refer to Refs. [39, 40, 41], and for more discussions on the constraints on the subclasses of Horndeski theory, please refer to Refs. [42, 43, 44, 45, 46].

In this work, the calculation will be done in the Jordan frame, and the screening mechanisms, such as the chameleon [47, 48] and the symmetron [49, 50], are not considered. Vainshtein mechanism was first discovered to solve the vDVZ discontinuity problem for massive gravity [51], and later found to also appear in theories containing the derivative self-couplings of the scalar field, such as some subclasses of Horndeski theory [52, 53, 54, 55, 56]. When Vainshtein mechanism is in effect, the effect of nonlinearity cannot be ignored within the so-called Vainshtein radius \(r_\text {V}\) from the center of the matter source. Well beyond \(r_\text {V}\), the linearization can be applied. The radius \(r_\text {V}\) 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 \(r_\text {V}\), 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 Sect. 2, 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 3 derives the effective stress-energy tensor according to the procedure given by Isaacson. Section 4 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 Sect. 5, the metric and scalar perturbations are calculated in the far zone up to the quadratic order, and in Sect. 6, these solutions are applied to a compact binary system to calculate the energy emission rate and the period change. Section 7 discusses the constraints on Horndeski theory based on the observations. Finally, Sect. 8 summarizes the results. Throughout the paper, the speed of light in vacuum is taken to be \(c=1\).

## 2 Horndeski theory

*X*.

^{1}For notational simplicity and clarity, we define the following symbol for the function \(f(\phi ,X)\),

Suitable choices of \(G_i\) reproduce interesting subclasses of Horndeski theory. For instance, one obtains GR by choosing \(G_4=(16\pi G_\mathrm {N})^{-1}\) and the remaining \(G_i=0\), with \(G_\mathrm {N}\) Newton’s constant. Brans–Dicke theory is recovered with \(G_2=2\omega _\text {BD} X/\phi , G_4=\phi , G_3=G_5=0\), while the massive scalar-tensor theory with a potential \(U(\phi )\) [26] is obtained with \(G_2=2\omega _\text {BD} X/\phi -U(\phi ),\,G_4=\phi ,\,G_3=G_5=0\), where \(\omega _\text {BD}\) is a constant; or with \(G_2=X-U(\phi )\), \(G_4=g(\phi )\), \(G_3=G_5=0\). Finally, *f*(*R*) gravity is given by \(G_2=f(\phi )-\phi f'(\phi )\), \(G_4=f'(\phi )\), \(G_3=G_5=0\) with \(f'(\phi )=\mathrm {d} f(\phi )/\mathrm {d}\phi \).

### 2.1 Matter action

*E*is related to the inertial mass

*m*, then

*m*depends on \(\phi \), too. When their spins and multipole moments can be ignored, the gravitating objects can be described by point like particles, and the effect of \(\phi \) can be taken into account by the following matter action according to Eardley’s prescription [58],

*a*and \(u^\mu =\mathrm {d} x^\mu (\tau )/\mathrm {d}\tau \). 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).

*f*(

*R*) gravity. \(s_a\) and \(s'_a\) are the first and second sensitivities of the mass \(m_a\),

### 2.2 Linearized equations of motion

*xAct*package [59, 60, 61, 62, 63]. Because of their tremendous complexity, the full equations of motion will not be presented. Interested readers are referred to Refs. [57, 64]. It is checked,

*xAct*package gives the same equations of motion as Refs. [57, 64]. For the purpose of this work, the equations of motion are expanded up to the second order in perturbations defined as

^{2}with \(T^{(1)}=\eta ^{\mu \nu }T_{\mu \nu }^{(1)}\), and the mass of the scalar field is

From the equations of motion (17) and (18), one concludes that the scalar field is generally massive unless \(G_{2(2,0)}\) is zero, and the auxiliary field \(\tilde{h}_{\mu \nu }\) resembles the spin-2 graviton field \(\bar{h}_{\mu \nu }=h_{\mu \nu }-\eta _{\mu \nu }h/2\) in GR. \(\tilde{h}_{\mu \nu }\) is sourced by the matter stress-energy tensor, while the source of the scalar perturbation \(\varphi \) is a linear combination of the trace of the matter stress-energy tensor and the partial derivative of the trace with respect to \(\phi \). This is because of the indirect interaction between the scalar field and the matter field via the metric tensor.

## 3 Effective stress-energy tensor

*R*representing the typical value of the background Riemann tensor components. This approximation is trivially satisfied in our case, as the background is flat and \(R=0\). In averaging over several wavelengths, the following rules are utilized [67]:

- 1.
The average of a gradient is zero, e.g., \(\langle \partial _\mu (\tilde{h}_{\rho \sigma }\partial _\nu \tilde{h})\rangle =0\),

- 2.
One can integrate by parts, e.g., \(\langle \tilde{h}\partial _{\rho }\partial _{\sigma }\tilde{h}_{\mu \nu }\rangle =\) \(-\langle \partial _\rho \tilde{h}\) \(\partial _\sigma \tilde{h}_{\mu \nu }\rangle \),

*xAct*and given by,

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.

## 4 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 \(\mathbf {x}\) represent the field point whose length is denoted by \(r=|\mathbf {x}|\).

^{3}

*a*is \(u_a^\mu =u_a^0(1,\mathbf {v}_a)\) and \(v_a^2=\mathbf {v}_a^2\). With these results, the leading order of the source for the scalar field is

### 4.1 Static, spherically symmetric solutions

*M*at rest at the origin as the source, the time-time component of the metric tensor is

*M*. From this, the “Newton’s constant” can be read off

*r*because the scalar field is massive. The measured Newtonian constant at the earth is \(G_\mathrm {N}(r_\otimes )\) with \(r_\otimes \) the radius of the Earth. The “post-Newtonian parameter” \(\gamma (r)\) can also be read off by examining \(g_{jk}\), which is

*f*(

*R*) gravity and general scalar-tensor theory [31, 69, 70, 71] if we keep the equivalence principle. In the massless case (\(G_{2(2,0)}=0\)), we get

### 4.2 Equations of motion of the matter

*a*and

*b*. The equation of motion for the mass \(m_a\) can thus be obtained using the Euler–Lagrange equation, yielding its acceleration,

*j*-th component of the canonical momentum of particle

*a*, and the total rest mass has been dropped. In particular, the Hamiltonian of a binary system is given by

### 4.3 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 [72, 73]. It results in measurable effects in the solar system, one of which is the polarization of the Moon’s orbit around the Earth [74, 75].

*c*and studies the relative acceleration of

*a*and

*b*in the field of

*c*. With Eq. (40) and assuming \(r_{ab}\ll r_{ac}\approx r_{bc}\), the relative acceleration is

*c*, and the last one describes the Nordtvedt effect. The effective Nordtvedt parameter is

### 4.4 Shapiro time delay effect

*M*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 \(u^\mu =u^0(1,\mathbf {v})\), then \(u^\mu u_\mu =0\) gives

*M*case. In the flat spacetime, the trajectory for a photon emitted from position \(\mathbf {x}_e\) at time \(t_e\) is a straight line \(\mathbf {x}(t)=\mathbf {x}_e+\hat{N}(t-t_e)\), where \(\hat{N}\) is the direction of the photon. The presence of the gravitational potential introduces a small perturbation \(\delta \mathbf {x}(t)\) so that \(\mathbf {x}(t)=\mathbf {x}_e+\hat{N}(t-t_e)+\delta \mathbf {x}(t)\). Substituting Eqs. (30) and (31) into Eq. (46), one obtains

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

*r*in the above equation should be 1 AU, as this is approximately the distance where the Keplerian mass \(M_\text {K}\) of the Sun is measured.

## 5 Gravitational wave solutions

*massless*for simplicity. The details to obtain the following results can be found in Appendix B. The leading order contribution to \(\varphi \) comes from the first term on the right hand side of Eq. (54), which is the mass monopole moment,

*n*] indicates the order of a quantity in terms of the speed

*v*, i.e., \(\varphi ^{[n]}\) is of the order \(O(v^{2n})\). \(\varphi ^{[1]}\) 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,

*a*and

*b*with \(a\ne b\), and

## 6 Gravitational radiation for a compact binary system

*massless*scalar field . According to Eq. (22), the energy carried away by the gravitational wave is at a rate of

*xOy*plane. In the polar coordinate system \((r,\theta , z)\), the relative distance is thus given by

*l*the angular momentum per unit mass and

*e*the eccentricity. The orbital period is

*X*only, the stellar sensitivity \(s_a\) vanishes [77], and in Brans–Dicke theory, the sensitivity of a black hole is 1/2 [26, 78, 79]. So if the binary system consists of, e.g., two neutron stars in SSHT or if the two stars are black holes in Brans–Dicke theory, the dipolar radiation vanishes.

Given the sensitivities (\(s_a, s'_a\)) of all kinds of celestial objects, Eq. (77) can be compared with the observed period change to set bounds on some of parameters characterizing a particular scalar-tensor theory (e.g., \(\phi _0,\,G_{4(0,0)},\,G_{4(1,0)},\,\zeta \) etc.) as done in Ref. [26].

## 7 Observational constraints

In this section, constraints on Horndeski theory are obtained using observations from lunar laser ranging experiments, Cassini time-delay measurement and binary pulsars. Since Horndeski theory contains many parameters, the following discussions start with generic constraints on the full Horndeski theory, and then specify to some concrete subclasses of Horndeski theory.

### 7.1 Constraints from lunar laser ranging experiments

### 7.2 Constraints from Cassini time-delay data

### 7.3 Constraints from period change for circular motion

Orbital parameters of the binary system PSR J1738+0333 [85]

Eccentricity | \((3.4\pm 1.1)\times 10^{-7}\) |

Orbital period | 0.354 790 739 8724(13) |

Period change \(\dot{T}_\text {obs}\) | \((-25.9\pm 3.2)\times 10^{-15}\) |

Pulsar mass \(m_1(M_\odot )\) | \(1.46_{-0.05}^{+0.06}\) |

Companion mass \(m_2(M_\odot )\) | \(0.181_{-0.007}^{+0.008}\) |

The eccentricity of PSR J1738+0333 is \((3.4\pm 1.1)\times 10^{-7}\), so the orbit is nearly a circle, and one can use Eq. (86) to obtain the bounds on Horndeski theory. At 95% confidential level, one requires that \(|\dot{T}_\text {pred.}-\dot{T}_\text {obs.}|<2\sigma \) where \(\dot{T}_\text {pred.}\) is determined by Eq. (86) with Eq. (82) substituted in, \(\dot{T}_\text {obs.}\) is the observed period change and \(\sigma \) is the uncertainty for \(\dot{T}_{\text {obs.}}\). The expression for \(\dot{T}_\text {pred.}-\dot{T}_\text {obs.}\) is too complicated and will not be presented here.

### 7.4 Constraints on special examples

## Example 1

## Example 2

## Example 3

One may also consider the constraints set on a massive Horndeski theory. In this case, one can only use the constraints from the Nordtvedt effect and the Shapiro time delay. The mass \(m_s\) of the scalar field is expected to be very small. As suggested in Ref. [26], if \(10^{-21}\text { eV}<m_s<10^{-15}\) eV, the constraints can also be set on \(G_{4(0,0)}\) and \(\chi \), provided that they are independent of each other. The allowed parameter space \((G_{4(0,0)},\chi )\) is approximately given by the area enclosed by the two vertical dashed curves, and the dot dashed one in Fig. 1. The constraint on the combination \(G_{2(0,1)}-2G_{3(1,0)}\) is also approximately given by Eq. (90). If \(G_4\propto \phi \), the constraints are approximately given by Eqs. (91) and (92).

## 8 Conclusion

In this work, the observational constraints on Horndeski theory are obtained based on the observations from the Nordtvedt effect, Shapiro time delay and binary pulsars. For this purpose, the near zone metric and scalar perturbations are first calculated in order to obtain the equations of motion for the stars. These solutions are thus used to study the Nordtvedt effect and the Shapiro time delay. Then, the effective stress-energy tensor of Horndeski theory is derived using the method of Isaacson. It is then used to calculate the rate of energy radiated away by the gravitational wave and the period change of a binary system. For this end, in the far zone, the auxiliary metric perturbation is calculated using the familiar quadratic formula, and the scalar field is calculated with the monopole moment contribution dominating, although it does not contribute to the effective stress-energy tensor. The leading contribution of the scalar field to the energy damping is the dipolar radiation, which is related to the difference in the sensitivities of the stars in the binary system, so the dipolar radiation vanishes if the two stars have the same sensitivity. The energy damping is finally calculated with the far zone field perturbations, and the period change is derived. Finally, the observational constraints are discussed based on the data from lunar laser ranging experiments, the observations made by the Cassini spacecraft, and the observation on the PSR J1738+0333. Explicit constraints have been obtained for both the massless and massive Horndeski theory, and in particular, for the one satisfying the recent gravitational wave speed limits [6].

## Footnotes

- 1.
\(G_2\) is usually called \(K_2\) in literature.

- 2.
The way defining \(T_*^{(1)}\) is different from the one defining \(T^*\) in Ref. [26] in that the coefficient of \(T^{(1)}_*\) is not 1.

- 3.
The matter stress-energy tensor \(T_{\mu \nu }\) and the derivative of its trace

*T*with respect to \(\phi \), \(\partial T/\partial \phi \), are both expanded beyond the leading order, because the higher order contributions are need to calculate the scalar perturbations in Sect. 5.

## Notes

### Acknowledgements

We would like to thank Zhoujian Cao for helpful discussions. This research was supported in part by the Major Program of the National Natural Science Foundation of China under Grant No. 11475065 and the National Natural Science Foundation of China under Grant No. 11690021.

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