Inspiral gravitational waveforms from compact binary systems in Horndeski gravity

The discovery of gravitational waves (GWs) Abbott et al. (2016) opened up a new window for probing the physics in strong gravity regimes Berti et al. (2015); Barack et al. (2019); Berti et al. (2018). Up to now, there have been many observed events of the coalescence of two black holes (BHs) Abbott et al. (2019, 2021). The merger of two neutron stars (NSs) was first detected as the event GW170817 Abbott et al. (2017a), together with an electromagnetic counterpart Goldstein et al. (2017). This latter event constrained the speed of gravity to be very close to that of light Abbott et al. (2017b). There was also a possible NS-BH coalescence event GW190426-152155, albeit its highest false alarm rate Abbott et al. (2020); Broekgaarden et al. (2021); Li et al. (2020); Niu et al. (2021). The increasing sensitivity in future GW observations will allow one to detect more promising events of the NS-BH coalescence.

General Relativity (GR) is a fundamental theory of gravity consistent with submillimeter laboratory tests Hoyle et al. (2001); Adelberger et al. (2003), solar system constraints Will (2014), and binary pulsar measurements Taylor and Weisberg (1982); Stairs (2003); Antoniadis et al. (2013). However, there are several cosmological problems like the origins of inflation, dark energy, and dark matter, which are difficult to be resolved within the framework of GR and standard model of particle physics Lyth and Riotto (1999); Copeland et al. (2006); Bertone et al. (2005). To address these problems, one usually introduces additional degrees of freedom beyond those appearing in GR (two tensor polarizations). The simplest example is a scalar field $\phi$ minimally or nonminimally coupled to gravity. If there is a nonminimal coupling with the Ricci scalar $R$ of the form $F(\phi)R$, the gravitational interaction is generally modified from that in GR. Brans-Dicke (BD) theories Brans and Dicke (1961) with a scalar potential $V(\phi)$ belong to such a class, which accommodates $f(R)$ gravity Starobinsky (1980) as a special case. The applications of BD theories and $f(R)$ gravity to inflation and dark energy have been extensively performed in the literature (see the reviews Sotiriou and Faraoni (2010); De Felice and Tsujikawa (2010)).

In BD theories, the NS can have a scalar hair through the coupling between the scalar field and matter mediated by gravity. On the other hand, the no-hair property of BHs was proven in BD theories Hawking (1972); Bekenstein (1995); Sotiriou and Faraoni (2012). This means that the binary system containing at least one scalarized NS may leave some signatures for the deviation from GR in inspiral gravitational waveforms. In massless BD theories, Eardley Eardley (1975) estimated the change of an orbital period induced by dipole gravitational radiation in a compact binary system. In the same theories, Will Will (1994) computed gravitational waveforms radiated during an inspiral phase of the compact binary up to the Newtonian quadrupole order (see also Refs. Shibata et al. (1994); Harada et al. (1997); Brunetti et al. (1999); Berti et al. (2005); Scharre and Will (2002); Chatziioannou et al. (2012) for related works). Under a so-called post-Newtonian (PN) approximation Blanchet (2014) based on slow velocities of the binary system relative to the speed of light $c$, the gravitational radiation was also calculated up to 2PN Lang (2014); Mirshekari and Will (2013); Lang (2015); Sennett et al. (2016) and 2.5PN Bernard et al. (2022) orders, with the equations of motion up to 3PN order Bernard (2018).

In massless BD theories given by the Lagrangian $L=(1/2)\phi R+\omega_{\rm BD}X/\phi$, where $X=-(1/2)\nabla^{\mu}\phi\nabla_{\mu}\phi$ is a field kinetic term with the covariant derivative operator $\nabla^{\mu}$, the solar-system tests of gravity put a tight bound on the BD parameter, $\omega_{\rm BD}>4.0\times 10^{4}$ Will (2014). With this constraint, the deviation from GR in strong gravity regimes is limited to be small. In other words, the GW measurements need to reach high sensitivities to distinguish between BD theories and GR from the observed gravitational waveform. If the scalar field has a potential $V(\phi)$ with a heavy mass inside a nonrelativistic star, while having a light mass outside the star, it is possible to suppress the propagation of fifth forces in the solar system through a so-called chameleon mechanism Khoury and Weltman (2004a, b). In such cases the constraint on $\omega_{\rm BD}$ is loosened Faulkner et al. (2007); Hu and Sawicki (2007); Capozziello and Tsujikawa (2008); Tsujikawa et al. (2008); Berti et al. (2012), so there is more freedom to probe signatures of the modification of gravity in strong gravity environments. The gravitational radiation and tensor waves emitted from a compact binary have been computed in massive BD theories Alsing et al. (2012); Berti et al. (2012); Sagunski et al. (2018); Liu et al. (2020) and screened modified gravity in the Einstein frame Zhang et al. (2017); Liu et al. (2018); Niu et al. (2019).

In the presence of a nonminimal coupling of the form $F(\phi)R$, there is yet the other scenario dubbed spontaneous scalarization of NSs Damour and Esposito-Farese (1993, 1996) in which the modification of gravity manifests itself only on the strong gravity background. Provided that $F(\phi)$ contains even power-law functions of $\phi$, the theory admits the existence of a nonvanishing field branch besides the GR branch ($\phi=0$). Since the Ricci scalar $R$ coupled to the scalar field is large inside the NS, the GR branch can be unstable to trigger tachyonic growth of $\phi$ toward the other nontrivial branch. For the nonminimal coupling $F(\phi)=e^{-\beta\phi^{2}/(2M_{\rm Pl}^{2})}$ chosen by Damour and Esposito-Farese Damour and Esposito-Farese (1993), spontaneous scalarization can occur for the coupling constant $\beta\leq-4.35$ Harada (1998); Novak (1998); Sotani and Kokkotas (2004); Silva et al. (2015); Barausse et al. (2013). On the other hand, the presence of a scalar charge for the scalarized solution leads to an energy loss through dipolar gravitational radiation. This results in time variation of the orbital period of binary systems. Indeed, binary-pulsar observations put the bound $\beta\geq-4.5$ Freire et al. (2012); Shao et al. (2017); Anderson et al. (2019), so the coupling constant $\beta$ is confined in a limited range. Since the gravitational waveform emitted from compact binaries containing a NS should give an independent constraint on the scalar charge, it remains to be seen how the future GW observations place the bound on $\beta$. In Ref. Niu et al. (2021), the authors started to derive constraints on $\beta$ by using the possible NS-BH coalescence event GW190426-152155.

Theory of spontaneous scalarization does not belong to a framework of BD theories, but it can be accommodated as a generalized class of BD theories by promoting the BD parameter $\omega_{\rm BD}$ to a scalar-field dependent function $\omega(\phi)$. Indeed, the gravitational radiation and waveforms in theories given by the Lagrangian $L=(1/2)\phi R+\omega(\phi)X/\phi$ have been investigated in Refs. Alsing et al. (2012); Yunes et al. (2012); Berti et al. (2018); Liu et al. (2020). Such theories belong to a scheme of Horndeski theories Horndeski (1974)– most general scalar-tensor theories with second-order Euler-Lagrange equations of motion Deffayet et al. (2011); Kobayashi et al. (2011); Charmousis et al. (2012). The subclass of Horndeski theories with the speed of gravity equivalent to that of light is given by the Lagrangian $L=G_{2}(\phi,X)-G_{3}(\phi,X)\square\phi+G_{4}(\phi)R$ Kobayashi et al. (2011); De Felice and Tsujikawa (2012); Kase and Tsujikawa (2019), where $G_{2}$ and $G_{3}$ are functions of $\phi$ and $X$, and $G_{4}$ is a function of $\phi$. This class of theories automatically evades the observational bound on the speed of gravity constrained by the GW170817 event Abbott et al. (2017b).

Theories with the Lagrangian $L=G_{2}(\phi,X)+G_{4}(\phi)R$ accommodate not only the generalized massive BD theories with $L=(1/2)\phi R+\omega(\phi)X/\phi-V(\phi)$ but also nonminimally coupled k-essence Armendariz-Picon et al. (1999); Chiba et al. (2000); Armendariz-Picon et al. (2000); Arkani-Hamed et al. (2004); Piazza and Tsujikawa (2004); ter Haar et al. (2021); Bezares et al. (2022) containing higher-order kinetic terms like $\mu_{2}X^{2}$ in $G_{2}$. When spontaneous scalarization occurs inside the NS, higher-order derivatives can be as large as the linear kinetic term around the surface of star. Indeed, we will propose a new scenario of spontaneous scalarization where the scalar charge is reduced by an additional term $\mu_{2}X^{2}$. This allows a possibility for alleviating the tension of the coupling constant $\beta$ mentioned above. The modified scalar-field solution also affects the gravitational waveform radiated from the binary system containing a NS. Thus, it is convenient to provide a general scheme for confronting such theories with future GW observations of the NS-BH or NS-NS coalescence.

In this paper, we compute the gravitational waveform emitted during the inspiral phase of compact binary systems in a subclass of Horndeski theories given by the Lagrangian $L=G_{2}(\phi,X)-G_{3}(\phi,X)\square\phi+G_{4}(\phi)R$. For this purpose, we perform the PN expansion of a source energy-momentum tensor and neglect nonlinear derivative terms arising from the cubic coupling $G_{3}(X)\square\phi$ outside the source. This amounts to neglecting nonlinear Galileon-type self-interactions Nicolis et al. (2009); Deffayet et al. (2009) relative to the linear kinetic term. Hence it is difficult to accommodate the case in which the field derivative is suppressed in the exterior region of NSs by the Vainshtein mechanism Vainshtein (1972); Burrage and Seery (2010); De Felice et al. (2012); Kimura et al. (2012); Koyama et al. (2013), unless some specific scaling methods McManus et al. (2016); Renevey et al. (2020, 2022) are employed. However, if the Vainshtein radius $r_{V}$ is of the same order as the NS radius $r_{s}$ ($\sim 10$ km), the PN analysis used for the derivation of solutions to scalar perturbations from $r=r_{s}$ to an observer does not lose its validity. In such a case, the field derivative and scalar charge can be suppressed by the Vainshtein screening inside the NS, analogous to the findings in Ref. Chagoya et al. (2014); Ogawa et al. (2020). Thus, the gravitational waveform derived in this paper can be applied to the case $r_{V}\lesssim r_{s}$.

This paper is organized as follows. In Sec. II, we review the field equations of motion and the matter action of two point-like sources in the subclass of Horndeski theories. In Sec. III, we perform the weak-field expansions of metric and scalar field to study the propagation of GWs from the binary system of a quasicircular orbit to the observer and derive solutions to tensor GWs. In Sec. IV, we obtain the time-domain gravitational waveforms corresponding to two transverse and longitudinal tensor polarizations as well as breathing and longitudinal modes. In Sec. V, we study the energy loss induced by the GW emission and derive the Fourier-transformed gravitational waveforms by using a stationary phase approximation. We show that the resulting waveform reduces to the one in a parameterized post-Einsteinian (ppE) framework Yunes and Pretorius (2009); Cornish et al. (2011); Chatziioannou et al. (2012); Tahura and Yagi (2018) and derive the ppE parameters in our theory. In Sec. VI, we apply our general results to several concrete theories and clarify the relations between ppE parameters and the scalar charge in the Einstein frame. In particular, we show that a new theory of spontaneous scalarization with the higher-order derivative term $\mu_{2}X^{2}$ in $G_{2}$ allows an interesting possibility for reducing the scalar charge in comparison to the case $\mu_{2}=0$, whose property can be probed in future GW observations. Sec. VII is devoted conclusions.

Throughout the paper, we use the metric signature $(-,+,+,+)$ and natural units $c=\hbar=1$, where $\hbar$ is the reduced Planck constant.

We consider a subclass of Horndeski theories Horndeski (1974) given by the action

where $g$ is the determinant of metric tensor $g_{\mu\nu}$, $X=-(1/2)\nabla^{\mu}\phi\nabla_{\mu}\phi$ is the kinetic term of a scalar field $\phi$, and $\square\equiv g^{\mu\nu}\nabla_{\mu}\nabla_{\nu}$ is the d’Alembertian. The action ${\cal S}_{m}$ contains matter fields $\Psi_{m}$ minimally coupled to gravity. In theories (1), there are no asymptotically flat spherically symmetric and static BH solutions with a scalar hair Hawking (1972); Bekenstein (1995); Sotiriou and Faraoni (2012); Graham and Jha (2014); Faraoni (2017); Minamitsuji et al. (2022). On the other hand, NSs can have scalar charges in the presence of nonminimal couplings $G_{4}(\phi)R$ Eardley (1975).

Our aim is to compute a gravitational waveform emitted from inspiraling compact binaries containing at least one NS. If the NS has a scalar hair, the waveform is subject to modifications in comparison to GR. This allows us to probe signatures for the possible existence of a scalar field nonminimally coupled to gravity. In theories given by the action (1), the speed of gravity on the cosmological background is identical to that of light Kobayashi et al. (2011); De Felice and Tsujikawa (2012); Kase and Tsujikawa (2019). We note that the equivalence principle can be generally violated in scalar-tensor theories including the action (1). However, in concrete models discussed in Sec. VI, we are interested in the case where the fifth force induced by scalar-gravitational couplings is suppressed on weak gravity backgrounds for the consistency with solar-system constraints. For the derivation of gravitational waveforms, we do not restrict the analysis to some specific models by the end of Sec. V.

We deal with the binary system of a quasicircular orbit as a collection of two point-like particles with masses $m_{I}(\phi)$, where $I=A,B$ for each particle. Since the existence of $\phi$ affects matter through gravitational field equations of motion, there is the $\phi$-dependence in $m_{I}$. The matter sector is expressed by the action Eardley (1975)

where $\tau_{I}$ is the proper time along the world line $x_{I}^{\mu}$ of particle $I$. The infinitesimal line element is given by

Then, the matter action (2) can be written as

where $\delta^{(4)}(x^{\mu}-x_{I}^{\mu})$ is the four dimensional delta function. Varying (4) with respect to $g_{\mu\nu}$, it follows that

where $u_{I}^{\mu}={\rm d}x_{I}^{\mu}/{\rm d}\tau_{I}$ is the four velocity of particle $I$. The matter energy-momentum tensor $T^{\mu\nu}$ is defined by

Comparing Eq. (6) with Eq. (5), we obtain

In the second line, we used ${\rm d}\tau_{I}={\rm d}x_{I}^{0}/u_{I}^{0}$ and integrated Eq. (7) with respect to $x_{I}^{0}$. Note that $\delta^{(3)}({\bm{x}}-{\bm{x}}_{I}(t))$ is a three dimensional delta function and that the time $t$ is determined by $t=x_{I}^{0}$.

On using the property $g_{\mu\nu}u_{I}^{\mu}u_{I}^{\nu}=-1$, the trace of Eq. (8) yields

On the other hand, the action (2) can be written in the form

Comparing Eq. (9) with Eq. (10), it follows that

whose form is used for the variation of ${\cal S}_{m}$ with respect to $\phi$.

Varying the action (1) with respect to $g^{\mu\nu}$, we obtain the covariant gravitational field equations of motion Kobayashi et al. (2011)

where we used the notations like $G_{2,X}=\partial G_{2}/\partial X$, $G_{4,\phi\phi}=\partial^{2}G_{4}/\partial\phi^{2}$, etc. The variation of (1) with respect to $\phi$ leads to the scalar-field equation of motion

where

The $\phi$ dependence in $T(\phi)$ influences the scalar-field equation through the last term in Eq. (15). The Ricci scalar $R$ in ${\cal P}_{\phi}$ is affected by the presence of matter through the gravitational Eq. (12). Taking the trace of Eq. (12), we obtain

Then, we can express ${\cal P}_{\phi}$ in the following form

The last two contributions to Eq. (17) work as matter source terms for the scalar-field equation.

The matter action (2) can be also expressed in the form

Varying this action with respect to $x_{I}^{\lambda}$ and integrating it by parts, we obtain

Then, the equation of motion for the $I$-th particle is given by

Multiplying Eq. (20) by $g^{\alpha\lambda}$ and using ${\rm d}m_{I}/{\rm d}\tau=m_{I,\phi}u^{\beta}\nabla_{\beta}\phi$ and ${\rm d}g_{\mu\lambda}/{\rm d}\tau=(\partial g_{\mu\lambda}/\partial x_{I}^{\nu})u_{I}^{\nu}/2+(\partial g_{\lambda\mu}/\partial x_{I}^{\nu})u_{I}^{\nu}/2$, it follows that

where $\Gamma^{\alpha}_{\mu\nu}$ is the Christoffel symbol. The $\phi$ dependence in $m_{I}$ modifies the standard geodesic equation. One can express Eq. (21) in a simple form Eardley (1975)

In terms of the matter energy-momentum tensor given by Eq. (7), Eq. (22) is equivalent to the continuity equation $\nabla_{\beta}T^{\alpha\beta}=T_{,\phi}\nabla^{\alpha}\phi$. This latter equation coincides with the one derived in Ref. Will and Zaglauer (1989) in BD theories.

Equations (12), (13), and (21) are the master equations used to describe the dynamics of gravity, scalar-field, and point-like particles, respectively.

To compute the gravitational waveform emitted from the inspiraling compact binary, we need to study the propagation of GWs from the binary to an observer. For this purpose, we expand the metric $g_{\mu\nu}$ about a Minkowski background and the scalar field $\phi$ around a constant asymptotic cosmological value $\phi_{0}$, as Will (1994)

where $\eta_{\mu\nu}={\rm diag}(-1,1,1,1)$, and $h_{\mu\nu}$ and $\varphi$ are the perturbed quantities. We would like to calculate the gravitational waveform associated with $h_{\mu\nu}$ and $\varphi$ up to quadrupole order. We perform the expansions of Eqs. (12) and (13) with respect to the perturbations $h_{\mu\nu}$ and $\varphi$ and derive a quadrupole formlula for tensor GWs in this section.

The scalar-field equation (17) contains the matter source term $T_{,\phi}-G_{4,\phi}T/(2G_{4})$. The trace $T$ acquires the $\phi$ dependence through the mass term $m_{I}(\phi)$ in Eq. (9). We expand $m_{I}(\phi)$ and $G_{4}(\phi)$ around the background field value $\phi=\phi_{0}$, respectively, as

where $M_{\rm Pl}=2.4354\times 10^{18}$ GeV is the reduced Planck mass, and

On using Eq. (9), the matter source terms in Eq. (17) evaluated on the Minkowski background are expressed as

where

As we will show in Sec. VI, the quantity $\hat{\alpha}_{I}$ is directly related to a scalar charge. In this sense, $\hat{\alpha}_{I}$ is a more fundamental physical quantity than $\alpha_{I}$. It is known that the theory (1) does not have hairy BH solutions, in which case $\hat{\alpha}_{I}=0$. On the other hand, the NS can have scalar hairs, in which case $\hat{\alpha}_{I}\neq 0$. As we will see in Sec. V, the scalar charge $\hat{\alpha}_{I}$ appears in the gravitational waveform as a quantity charactering the deviation from GR. One can also consider the following sensitivity parameter Eardley (1975)

In BD theories given by $G_{4}=\phi/(16\pi)$, we have $g_{4}=M_{\rm Pl}/\phi_{0}$ and hence $\hat{\alpha}_{I}=(M_{\rm Pl}/\phi_{0})(s_{I}-1/2)$. Then, the no-hair BH in BD theories corresponds to the sensitivity parameter $s_{I}=1/2$. Depending on the theories under consideration, however, $s_{I}$ defined in Eq. (30) can be affected by an ambiguity of the asymptotic value of $\phi_{0}$. In the following we will use $\hat{\alpha}_{I}$ instead of $s_{I}$, as the former has a direct relation with the scalar charge.