# Strong equivalence principle and gravitational wave polarizations in Horndeski theory

## Abstract

The relative acceleration between two nearby particles moving along accelerated trajectories is studied, which generalizes the geodesic deviation equation. The polarization content of the gravitational wave in Horndeski theory is investigated by examining the relative acceleration between two self-gravitating particles. It is found out that the apparent longitudinal polarization exists no matter whether the scalar field is massive or not. It would be still very difficult to detect the enhanced/apparent longitudinal polarization with the interferometer, as the violation of the strong equivalence principle of mirrors used by interferometers is extremely small. However, the pulsar timing array is promised relatively easily to detect the effect of the violation as neutron stars have large self-gravitating energies. The advantage of using this method to test the violation of the strong equivalence principle is that neutron stars are not required to be present in the binary systems.

## 1 Introduction

Soon after the birth of General Relativity (GR), several alternative theories of gravity were proposed. The discovery of the accelerated expansion of the Universe [1, 2] revives the pursuit of these alternatives because the extra fields might account for the dark energy. Since Sep 14th, 2015, LIGO/Virgo collaborations have detected ten gravitational wave (GW) events [3, 4, 5, 6, 7, 8, 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 there are extra GW polarizations in addition to the plus and cross ones in GR. So the detection of 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 the pure tensor polarizations were favored against pure vector and pure scalar polarizations [6]. Similar results were reached in the recent analysis on GW170817 [10]. More interferometers are needed to finally pin down the polarization content. Other detection methods might also determine the polarizations of GWs such as pulsar timing arrays (PTAs) [11, 12, 13, 14].

Alternative metric theories of gravity may not only introduce extra GW polarizations, but also violate the strong equivalence principle (SEP) [15].^{1} The violation of strong equivalence principle (vSEP) is due to the extra degrees of freedom, which indirectly interact with 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 no longer move along geodesics, even if there is only gravity acting on them, and the relative acceleration 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 along geodesics, so their relative acceleration is given by the geodesic deviation equation. Since the polarization content of GWs is determined by examining the relative acceleration, the departure from the geodesic motion might effectively result in different polarization contents, which can be detected by PTAs. Thus, the main topic of this work is to investigate the effects of vSEP on the polarization content of GWs and the observation of PTAs.

To be more specific, the focus is on the vSEP in the scalar-tensor theory, which is the simplest alternative metric theory of gravity. The scalar-tensor theory contains one scalar field \(\phi \) besides the metric tensor field \(g_{\mu \nu }\) to mediate the gravitational interaction. Because of the trivial transformation of the scalar field under the diffeomorphism, there are a plethora of scalar-tensor theories, such as Brans–Dicke theory [18], Einstein-dilaton-Gauss–Bonnet gravity (EdGB) [19] and *f*(*R*) gravity [20, 21, 22]. In 1974, Horndeski constructed the most general scalar-tensor theory [23]. Its action contains higher derivatives of \(\phi \) and \(g_{\mu \nu }\), but still gives rise to at most the second order differential field equations. So the Ostrogradsky instability is absent in this theory [24]. In fact, Horndeski theory includes previously mentioned theories as its subclasses. In this work, the vSEP in Horndeski theory will be studied.

*G*[34]. The dipole gravitational radiation for Horndeski theory has been studied in Ref. [35], and constraints on this theory were obtained. The pulsar timing observation of the binary system J1713+0747 has leads to \(\dot{G}/G=(-0.1\pm 0.9)\times 10^{-12}\text { yr}^{-1}\) and \(|\varDelta |<0.002\) [36].

As discussed above, none of the previous limits on vSEP was obtained directly using the GW. So probing vSEP by measuring the GW polarizations provides a novel way to test GR in the high speed and dynamical regime. It will become clear that although the vSEP will effectively enhance the longitudinal polarization, it is still very difficult for aLIGO to detect the effects of the longitudinal polarization, as the vSEP by the mirror is extremely weak. In contrast, neutron stars are compact objects with non-negligible self-gravitating energies. The vSEP by neutron stars is strong enough that the stochastic GW background will affect their motions, which is reflected in the cross-correlation function for PTAs [37, 38, 39, 40]. By measuring the cross-correlation function, it is probably easier to detect the presence of vSEP. For this purpose, one only has to observe the change in the arriving time of radial pulses from neutron stars without requiring the neutron stars be in binary systems.

This work is organized as follows. Section 2 reviews the derivation of the geodesic deviation equation, and a generalized deviation equation for accelerated particles is discussed in Sect. 3. Section 4 derives the motion of a self-gravitating object in presence of GWs in Horndeski theory. The polarization content of GWs in Horndeski theory is revisited by taking the vSEP into account in Sect. 5. The generalized deviation equation is computed to reveal the polarization content of GWs. Section 6 calculates the cross-correlation function for PTAs due to GWs. Finally, Sect. 7 briefly summarizes this work. Penrose’s abstract index notation is used [41]. The units is chosen such that the speed of light \(c=1\) in vacuum.

## 2 Geodesic deviation equation

This section serves to review the idea to derive the geodesic deviation equation following Ref. [42]. In the next section, the derivation will be generalized to accelerated objects straightforwardly.

*t*and labeled by

*s*. Define the following tangent vector fields,

*t*is an affine parameter. Note that it is not necessary to set \(T^aT_a=-1\) for the following discussion. Whenever desired, one can always reparameterize to normalize it. It is now ready to derive the geodesic deviation equation,

*t*, i.e., the integral curves of \(S^a\). In fact, one knows that,

*t*coordinate line, so that \(S^a\) is always a spatial vector field for an observer with 4-velocity \(u^a=T^a/\sqrt{-T_bT^b}\) along its trajectory.

From the derivation, one should be aware that the geodesic deviation equation (5) is independent of the gauge choices made above, which only serves to make sure \(S^a\) is always a spatial vector relative to an observer with \(u^a\). In this way, there is no deviation in the time coordinate, that is, no time dilatation. This is because one concerns the change in the spatial distance between two nearby particles measured by either one of them.

## 3 Non-geodesic deviation equation

*t*. So one can always find a new parametrization which annihilates \(T'^aA'_a\), that is,

*t*may not be the proper time \(\tau \), it is a linear function of \(\tau \). A further reparameterization \(t'=\alpha 't+\beta '\) does not change the above relation.

Now, pick a congruence of these trajectories \(\sigma _s(t)\). So as in the previous section, \(\sigma _s(t)\)’s also lie on a 2-dimensional surface \(\Sigma \) parameterized by (*t*, *s*). There also exists the similar gauge freedom to that discussed in Sect. 2, except that \(A^a \) depends on the gauge choice. For example, a reparametrization \(t\rightarrow t'=\alpha (s)t+\beta (s)\) results in changes in \(S^a\) (given by Eq. (7)) and \(A^a \), i.e., \(A^a \rightarrow A^a /\alpha ^2(s)\).

### 3.1 Fermi normal coordinates

## 4 The trajectory of a self-gravitating object in Horndeski theory

*X*, and \(G_{iX}=\partial _XG_{i}, i=3,4,5\). For any binary function \(f(\phi ,X)\), define the following symbol

## 5 The polarizations of gravitational waves in Horndeski gravity

In Ref. [59], the GW solutions for Horndeski theory [23] in the vacuum background have been obtained. The polarization content of the theory was also determined using the linearized 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 the scalar field on the trajectories of self-gravitating test particles. This will lead to a different polarization content of GWs in Horndeski theory.

### 5.1 The relative acceleration in the Fermi normal coordinates

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 \(s\sim 10^{-4}\), 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 sensitivity could be about 0.2 [29, 62]. They violate SEP relatively strongly, which might be detected by PTAs.

## 6 Pulsar timing arrays

In this section, the cross-correlation function will be calculated for PTAs. The possibility to detect the vSEP is thus inferred. A pulsar is a strongly magnetized, rotating neutron star or a white dwarf, which emits a beam of the radio wave along its magnetic pole. When the beam points towards the Earth, the radiation is observed, and this leads to the pulsed appearance of the radiation. The rotation of some “recycled” pulsars is stable enough so that they can be used as “cosmic 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 a steady rate. The presence of the GW will affect the propagation time of the radiation and thus alter this rate. This results in a change in the time-of-arrival (TOA), called timing residual *R*(*t*). Timing residuals caused by the stochastic GW background is correlated between pulsars, and the cross-correlation function is \(C(\theta )=\langle R_a(t)R_b(t)\rangle \) with \(\theta \) the angular separation of pulsars *a* and *b*, and the brackets \(\langle \,\rangle \) implying the ensemble average over the stochastic background. This makes it possible to detect GWs and probe the polarizations [37, 38, 39, 40, 66, 67, 68, 69, 70, 71, 72, 73]. The effect of vSEP can also be detected, as the longitudinal polarization of the scalar–tensor theory is enhanced due to vSEP.

*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 \(\mathbf {x}_p=(L\cos \beta ,0,L\sin \beta )\) in this coordinate system. The GW is propagating in the direction of a unit vector \({\hat{k}}\), and \({\hat{n}}\) is the unit vector pointing to the pulsar from the Earth. \({\hat{l}}={\hat{k}}\wedge ({\hat{n}}\wedge {\hat{k}})/\cos \beta =[{\hat{n}}-{\hat{k}}(\hat{n}\cdot {\hat{k}})]/\cos \beta \) is actually the unit vector parallel to the

*y*axis.

*T*is the total observation time. Insert Eq. (76) in, neglecting the second line, to obtain

*a*and

*b*located at \(\mathbf {x}_a=L_1{\hat{n}}_1\) and \(\mathbf {x}_b=L_2{\hat{n}}_2\), respectively. The angular separation is \(\theta =\arccos ({\hat{n}}_1\cdot {\hat{n}}_2)\). The cross-correlation function is thus given by

*T*drops out, as the ensemble average also implies the averaging over the time [38].

*L*are dropped as they barely contribute according to the experience in Ref. [59]. Finally, the observation time

*T*sets a natural cutoff for the angular frequency, i.e., \(\omega \ge 2\pi /T\), so the lower integration limits in Eqs. (90) and (91) should be replaced by \(\text {Max}\{m_s,2\pi /T\}\).

As usual, assume \(\varphi _c(\omega )\propto (\omega /\omega _c)^\alpha \) with \(\omega _c\) the characteristic angular frequency. Here, \(\alpha \) is called the power-law index, and usually, \(\alpha =0,\,-2/3\) or \(-1\) [38, 74]. Numerically integrating Eqs. (90) and (91) gives the so-called normalized correlation function \(\zeta (\theta )=C(\theta )/C(0)\). In the integration, set the observation time \(T=5\) years. The sensitivities of the Earth and the pulsar are taken to be \(s_r=0\) and \(s_e=0.2\), respectively. This leads to Fig. 2, where the power-law index \(\alpha \) takes different values.

If the scalar field is massless, the results are shown in the upper panel which displays the normalized correlation functions for the plus and cross polarizations – Hellings–Downs curve (labeled by “GR”) [69]. The remaining two curves are for the breathing polarization: the dashed one is for the case where SEP is respected, while the dotted one is for the case where SEP is violated. They are independent of the power-law index \(\alpha \). As one can see that vSEP makes \(\zeta (\theta )\) bigger by about 5%. If the scalar field has a mass \(m_s=7.7\times 10^{-23}\,\mathrm {eV}/c^2\), the results are shown in the lower panel. In this panel, the cross-correlation functions for the scalar polarization are drawn for different values of \(\alpha \). The solid curves correspond to the case 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 not plot them again in the lower panel. In the massive case, vSEP also increases \(\zeta (\theta )\) by about 2–3%.

Reference [75] published the constraint on the stochastic GW background based on the recently released 11-year dataset from the North American Nanohertz Observatory for Gravitational Waves (NANOGrav). Assuming the background is isotropic and \(\alpha =-2/3\), the strain amplitude of the GW is less than \(1.45\times 10^{-15}\) at \(f=1\text { yr}^{-1}\). In addition, the top panel in Figure 6 shows the observed cross correlation. As one can clearly see, the error bars are very large.^{2} More observations are needed to improve the statistics.

## 7 Conclusion

This work discusses the effects of the vSEP on the polarization content of GWs in Horndeski 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 enhancement of the longitudinal polarization in Horndeski theory, so even if the scalar field is massless, the longitudinal polarization still exists. This is in contrast with the previous results [59, 76, 77, 78] that the massive scalar field excites the longitudinal polarization, while the massless scalar field does not. The enhanced longitudinal polarization is nevertheless difficult for aLIGO to detect, as the mirrors does not violate SEP enough. However, pulsars are highly compact objects with sufficient self-gravitating energy such that their trajectories deviate from geodesics enough. Using PTAs, one can measure the change in TOAs of electromagnetic radiation from pulsars and obtain the cross-correlation function to tell whether vSEP effect exits. The results show that the vSEP leads to large changes in the behaviors of the cross-correlation functions. In principle, PTAs are capable of detecting the vSEP if it exists.

## Footnotes

## Notes

### Acknowledgements

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. This was also a project funded by China Postdoctoral Science Foundation (no. 2018M632822).

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