Polarizations of gravitational waves in Horndeski theory
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Abstract
We analyze the polarization content of gravitational waves in Horndeski theory. Besides the familiar plus and cross polarizations in Einstein’s General Relativity, there is one more polarization state which is the mixture of the transverse breathing and longitudinal polarizations. The additional mode is excited by the massive scalar field. In the massless limit, the longitudinal polarization disappears, while the breathing one persists. The upper bound on the graviton mass severely constrains the amplitude of the longitudinal polarization, which makes its detection highly unlikely by the groundbased or spaceborne interferometers in the near future. However, pulsar timing arrays might be able to detect the polarization excited by the massive scalar field. Since additional polarization states appear in alternative theories of gravity, the measurement of the polarizations of gravitational waves can be used to probe the nature of gravity. In addition to the plus and cross states, the detection of the breathing polarization means that gravitation is mediated by massless spin 2 and spin 0 fields, and the detection of both the breathing and longitudinal states means that gravitation is propagated by the massless spin 2 and massive spin 0 fields.
1 Introduction
The detection of gravitational waves by the Laser Interferometer GravitationalWave Observatory (LIGO) Scientific Collaboration and VIRGO Collaboration further supports Einstein’s General Relativity (GR) and provides a new tool to study gravitational physics [1, 2, 3, 4, 5, 6]. In order to confirm gravitational waves predicted by GR, it is necessary to determine the polarizations of gravitational waves. This can be done by the network of groundbased Advanced LIGO (aLIGO) and VIRGO, the future spaceborne Laser Interferometer Space Antenna (LISA) [7] and TianQin [8], and pulsar timing arrays (e.g., the International Pulsar Timing Array and the European Pulsar Timing Array [9, 10]). In fact, in the recent GW170814 [4], the Advanced VIRGO detector joined the two aLIGO detectors, so they were able to test the polarization content of gravitational waves for the first time. The result showed that the pure tensor polarizations were favored against pure vector and pure scalar polarizations [4, 11]. Additionally, GW170817 is the first observation of a binary neutron star inspiral, and its electromagnetic counterpart, GRB 170817A, was later observed by the Fermi Gammaray Burst Monitor and the International GammaRay Astrophysics Laboratory [5, 12, 13]. The new era of multimessenger astrophysics comes.
In GR, the gravitational wave propagates at the speed of light and it has two polarization states, the plus and cross modes. In alternative metric theories of gravity, there may exist up to six polarizations, so the detection of the polarizations of gravitational waves can be used to distinguish different theories of gravity and probe the nature of gravity [14, 15]. For null plane gravitational waves, the six polarizations are classified by the little group E(2) of the Lorentz group with the help of the six independent NewmanPenrose (NP) variables \(\varPsi _2\), \(\varPsi _3\), \(\varPsi _4\) and \(\varPhi _{22}\) [16, 17, 18]. In particular, the complex variable \(\varPsi _4\) denotes the familiar plus and cross modes in GR, the variable \(\varPhi _{22}\) denotes the transverse breathing polarization, the complex variable \(\varPsi _3\) corresponds to the vectorx and vectory modes, and the variable \(\varPsi _2\) corresponds to the longitudinal mode. Under the E(2) transformation, all other modes can be generated from \(\varPsi _2\), so if \(\varPsi _2\ne 0\), then we may see all six modes in some coordinates. The E(2) classification tells us the general polarization states, but it fails to tell us the correspondence between the polarizations and gravitational theories. In BransDicke theory [19], in addition to the plus and cross modes \(\varPsi _4\) of the massless gravitons, there exists another breathing mode \(\varPhi _{22}\) due to the massless BransDicke scalar field [17].
BransDicke theory is a simple extension to GR. In BransDicke theory, gravitational interaction is mediated by both the metric tensor and the BransDicke scalar field, and the BransDicke field plays the role of Newton’s gravitational constant. In more general scalartensor theories of gravity, the scalar field \(\phi \) has selfinteraction and it is usually massive. In 1974, Horndeski found the most general scalartensor theory of gravity whose action has higher derivatives of \(g_{\mu \nu }\) and \(\phi \), but the equations of motion are at most the second order [20]. Even though there are higher derivative terms, there is no Ostrogradsky instability [21], so there are three physical degrees of freedom in Horndeski theory, and we expect that there is an extra polarization state in addition to the plus and cross modes. If the scalar field is massless, then the additional polarization state should be the breathing mode \(\varPhi _{22}\).
When the interaction between the quantized matter fields and the classical gravitational field is considered, the quadratic terms \(R_{\mu \nu \alpha \beta } R^{\mu \nu \alpha \beta }\) and \(R^2\) are needed as counterterms to remove the singularities in the energymomentum tensor [22]. Although the quadratic gravitational theory is renormalizable [23], the theory has ghost due to the presence of higher derivatives [23, 24]. However, the general nonlinear f(R) gravity [25] is free of ghost and it is equivalent to a scalartensor theory of gravity [26, 27]. The effective mass squared of the equivalent scalar field is \(f'(0)/3f''(0)\), and the massive scalar field excites both the longitudinal and transverse breathing modes [28, 29, 30]. The polarizations of gravitational waves in f(R) gravity were previously discussed in [31, 32, 33, 34, 35, 36, 37]. The authors in [33, 34] calculated the NP variables and found that \(\varPsi _2\), \(\varPsi _4\) and \(\varPhi _{22}\) are nonvanishing. They then claimed that there are at least four polarization states in f(R) gravity. Recently, it was pointed out that the direct application of the framework of Eardley et. al. (ELLWW framework) [17, 18] derived for the null plane gravitational waves to the massive field is not correct, and the polarization state of the equivalent massive field in f(R) gravity is the mixture of the longitudinal and the transverse breathing modes [37]. Furthermore, the longitudinal polarization is independent of the effective mass of the equivalent scalar field, so it cannot be used to understand how the polarization reduces to the transverse breathing mode in the massless limit.
Since the polarizations of gravitational waves in alternative theories of gravity are not known in general, so we need to study it [38, 39, 40]. In this paper, the focus is on the polarizations of gravitational waves in the most general scalartensor theory, Horndeski theory. It is assumed that matter minimally couples to the metric, so that test particles follow geodesics. The gravitational wave solutions are obtained from the linearized equations of motion around the flat spacetime background, and the geodesic deviation equations are used to reveal the polarizations of the massive scalar field. The analysis shows that in Horndeski theory, the massive scalar field excites both breathing and longitudinal polarizations. The effect of the longitudinal polarization on the geodesic deviation depends on the mass and it is much smaller than that of the transverse polarization in the aLIGO and LISA frequency bands. In the massless limit, the longitudinal mode disappears, while the breathing mode persists.
The paper is organized as follows. Section 2 briefly reviews ELLWW framework for classifying the polarizations of null gravitational waves. In Sect. 3, the linearized equations of motion and the plane gravitational wave solution are obtained. In Sect. 3.1, the polarization states of gravitational waves in Horndeski theory are discussed by examining the geodesic deviation equations, and Sect. 3.2 discusses the failure of ELLWW framework for the massive Horndeski theory. Section 4 discusses the possible experimental tests of the extra polarizations. In particular, Sect. 4.1 mainly calculates the interferometer response functions for aLIGO, and Section 4.2 determines the crosscorrelation functions for the longitudinal and transverse breathing polarizations. Finally, this work is briefly summarized in Sect. 5. In this work, we use the natural units and the speed of light in vacuum \(c=1\).
2 Review of ELLWW Framework
 Class II\(_6\)

\(\varPsi _2\ne 0\). All observers measure the same nonzero amplitude of the \(\varPsi _2\) mode, but the presence or absence of all other modes is observerdependent.
 Class III\(_5\)

\(\varPsi _2=0\), \(\varPsi _3\ne 0\). All observers measure the absence of the \(\varPsi _2\) mode and the presence of the \(\varPsi _3\) mode, but the presence or absence of \(\varPsi _4\) and \(\varPhi _{22}\) is observerdependent.
 Class N\(_3\)

\(\varPsi _2=\varPsi _3=0,\,\varPsi _4\ne 0\ne \varPhi _{22}\). The presence or absence of all modes is observerindependent.
 Class N\(_2\)

\(\varPsi _2=\varPsi _3=\varPhi _{22}=0,\,\varPsi _4\ne 0\). The presence or absence of all modes is observerindependent.
 Class O\(_1\)

\(\varPsi _2=\varPsi _3=\varPsi _4=0,\,\varPhi _{22}\ne 0\). The presence or absence of all modes is observerindependent.
 Class O\(_0\)

\(\varPsi _2=\varPsi _3=\varPsi _4=\varPhi _{22}=0\). No wave is observed.
3 Gravitational wave polarizations in Horndeski theory
3.1 Polarizations
In summary, the polarizations of gravitational waves in Horndeski theory include the plus and cross polarizations induced by the spin 2 field \(\tilde{h}_{\mu \nu }\). The scalar field excites both the transverse breathing and longitudinal polarizations if it is massive. However, if it is massless, the scalar field excites merely the transverse breathing polarization. In terms of the basis introduced for the massless fields in [17], there are three polarizations: the plus state \(\hat{P}_+\), the cross state \(\hat{P}_\times \) and the mix state of \(\hat{P}_b\) and \(\hat{P}_l\). In the massless limit, the mix state reduces to the pure state \(\hat{P}_b\). Note that the vector modes \(\hat{P}_{xz}\) and \(\hat{P}_{yz}\) are absent in Horndeski theory, this seems to be in conflict with the E(2) classification because the longitudinal mode \(\hat{P}_l\) is present, so we need to discuss the application of E(2) classification.
3.2 NewmanPenrose variables
These discussion tells us that the detection of polarizations probes the nature of gravity. If only the plus and cross modes are detected, then gravitation is mediated by massless spin 2 field and GR is confirmed. The detection of the breathing mode in addition to the plus and cross modes means that gravitation is mediated by massless spin 2 and spin 0 fields. If the breathing, plus, cross and longitudinal modes are detected, then gravitation is mediated by massless spin 2 and massive spin 0 fields. For the discussion on the detection of polarizations, please see Refs. [42, 43].
4 Experimental Tests
4.1 Interferometers
In the interferometer, photons emanate from the beam splitter, are bounced back by the mirror and received by the beam splitter again. The roundtrip propagation time when the gravitational wave is present is different from that when the gravitational wave is absent. To simplify the calculation of the response functions, the beam splitter is placed at the origin of the coordinate system. Then the change in the roundtrip propagation time comes from two effects: the change in the relative distance between the beam splitter and the mirror due to the geodesic deviation, and the distributed gravitational redshift suffered by the photon in the field of the gravitational wave [44].
The dependence of the magnitudes of \(\ddot{x}^j/x^j_0\) for the longitudinal and transverse modes on the frequencies of gravitational waves in units of Hz\(^2\sigma \upvarphi \) assuming the scalar mass is \(m_b\)
100 Hz  \(10^{3}\) Hz  \(10^{7}\) Hz  

Longitudinal  2.81\(\times 10^{15}\)  2.81\(\times 10^{15}\)  2.81\(\times 10^{15}\) 
Transverse  \(1.97\times 10^5\)  1.97\(\times 10^{5}\)  1.97\(\times 10^{13}\) 
Table 1 shows that the effect of the longitudinal mode on the geodesic deviation is smaller than that of the transverse mode by 19 to 9 orders of magnitude at higher frequencies. But at the lower frequencies, i.e., at \(10^{7}\) Hz, the two modes have similar amplitudes. Therefore, aLIGO/VIRGO and LISA might find it difficult to detect the longitudinal mode, but pulsar time arrays should be able to detect the mix polarization state with both the longitudinal and transverse modes. The results are consistent with those for the massive graviton in a specific bimetric theory [46].
4.2 Pulsar timing arrays
A pulsar is a rotating neutron star or a white dwarf with a very strong magnetic field. It emits a beam of the electromagnetic radiation. When the beam points towards the Earth, the radiation can be observed, which explains the pulsed appearance of the radiation. Millisecond pulsars can be used as stable clocks [47]. When there is no gravitational wave, one can observe the pulses at a steady rate. The presence of the gravitational wave will alter this rate, because it will affect the propagation time of the radiation. This will lead to a change in the timeofarrival (TOA), called time residual R(t). Time residuals caused by the gravitational wave will be correlated between pulsars, and the crosscorrelation function is \(C(\theta )=\langle R_a(t)R_b(t)\rangle \), where \(\theta \) is the angular separation of pulsars a and b, and the brackets \(\langle \,\rangle \) imply the ensemble average over the stochastic background. This enables the detection of gravitational waves and the probe of the polarizations.
The effects of the gravitational wave in GR on the time residuals were first considered in Refs. [48, 49, 50]. Hellings and Downs [51] proposed a method to detect the effects by using the crosscorrelation of the time derivative of the time residuals between pulsars, while Jenet et. al. [52] directly worked with the time residuals instead of the time derivative. The later work was generalized to massless gravitational waves in alternative metric theories of gravity in Ref. [53], and further to massive gravitational waves in Refs. [54, 55]. More works have been done, for example, Refs. [56, 57, 58, 59] and references therein.
In Ref. [55], Lee also analyzed the time residual of TOA caused by massive gravitational waves and calculated the crosscorrelation functions. His results (the right two panels in his Fig. 1) differ from those on the right panel in Fig. 5, because in his treatment, the longitudinal and the transverse polarizations were assumed to be independent. In Horndeski theory, however, it is not allowed to calculate the crosscorrelation function separately for the longitudinal and the transverse polarizations, as they are both excited by the same field \(\upvarphi \) and the polarization state is a single mode.
5 Conclusion
This work analyzes the gravitational wave polarizations in the most general scalartensor theory of gravity, Horndeski theory. It reveals that there are three independent polarization modes: the mixture state of the transverse breathing \(\hat{P}_b=R_{txtx}+R_{tyty}\) and longitudinal \(\hat{P}_l=R_{tztz}\) polarizations for the massive scalar field, and the usual plus \(\hat{P}_+=R_{txtx}+R_{tyty}\) and cross \(\hat{P}_\times =R_{txty}\) polarizations for the massless gravitons. These results are consistent with the three propagating degrees of freedom in Horndeski theory. Since the propagation speed of the massive gravitational wave depends on the frequency and is smaller than the speed of light, the massive mode will arrive at the detector later than the massless gravitons. In addition to the difference of the propagation speed, the presence of both the longitudinal and breathing states without the vectorx and vectory states are also the distinct signature of massive scalar degree of freedom for graviton.
Using the NP variables, we find that \(\varPsi _2=0\). For null gravitational waves, this means that the longitudinal mode does not exist. However, our results show that the longitudinal mode exists in the massive case even though \(\varPsi _2=0\). We also find that the NP variable \(\varPhi _{00}\ne 0\), and \(\varPhi _{00}\), \(\varPhi _{11}\) and \(\varLambda \) are all proportional to \(\varPhi _{22}\). These results are in conflict with those for massless gravitational waves in [17], so the results further support the conclusion that the classification of the six polarizations for null gravitational waves derived from the little group E(2) of the Lorentz group is not applicable to the massive case. Although the longitudinal mode exists for the massive scalar field, it is difficult to be detected in the high frequency band because of the suppression by the extremely small graviton mass upper bound. Compared with aLIGO/VIRGO and LISA, pulsar timing arrays might be the primary tool to detect both the transverse breathing and longitudinal modes due to the massive scalar field. In the massless case, the longitudinal mode disappears and the mix state reduces to the pure transverse breathing mode. BransDicke theory and f(R) gravity are subclasses of Horndeski theory, so the general results obtained can be applied to those theories. Despite the fact that in f(R) gravity, the longitudinal mode does not depend on the mass of graviton, its magnitude is much smaller than the transverse one in the high frequency band, which makes its detection unlikely by the network of aLIGO/VIGO and LISA, too. It is interesting that the transverse mode becomes stronger for smaller graviton mass in the f(R) gravity, so the detection of the mixture state can place strong constraint on f(R) gravity.
For null gravitational waves, the presence of the longitudinal mode means that all six polarizations can be detected in some coordinate systems. For Horndeski theory, we find that the vector modes are absent even though the longitudinal mode is present. Since the massive scalar field excites both the breathing and longitudinal polarizations, while the massless scalar field excites the breathing mode only, the detection of polarizations can be used to understand the nature of gravity. If only the plus and cross modes are detected, then gravitation is mediated by massless spin 2 field and GR is confirmed. The detection of the breathing mode in addition to the plus and cross modes means that gravitation is mediated by massless spin 2 and spin 0 fields. If the breathing, plus, cross and longitudinal modes are detected, then gravitation is mediated by massless spin 2 and massive spin 0 fields.
Footnotes
Notes
Acknowledgements
We would like to thank Ke Jia Lee for useful discussions. This research was supported in part by the Major Program of the National Natural Science Foundation of China under Grant No. 11690021 and the National Natural Science Foundation of China under Grant No. 11475065.
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