Post-Newtonian, Quasi-Circular Binary Inspirals in Quadratic Modified Gravity

TL;DR

This paper analytically investigates the effects of quadratic modified gravity theories on binary black hole inspirals, revealing scalar hair for black holes, no monopole charge for neutron stars, and potential observable deviations in gravitational wave signals.

Contribution

It derives and solves post-Newtonian field equations in quadratic gravity, models compact objects with effective sources, and predicts gravitational wave modifications detectable by observations.

Findings

Black holes can have scalar hair in these theories.

Neutron stars lack scalar monopole charge.

Scalar radiation dominates energy flux, affecting gravitational wave signals.

Abstract

We consider a general class of quantum gravity-inspired, modified gravity theories, where the Einstein-Hilbert action is extended through the addition of all terms quadratic in the curvature tensor coupled to scalar fields with standard kinetic energy. This class of theories includes Einstein-Dilaton-Gauss-Bonnet and Chern-Simons modified gravity as special cases. We analytically derive and solve the coupled field equations in the post-Newtonian approximation, assuming a comparable-mass, spinning black hole binary source in a quasi-circular, weak-field/slow-motion orbit. We find that a naive subtraction of divergent piece associated with the point-particle approximation is ill-suited to represent compact objects in these theories. Instead, we model them by appropriate effective sources built so that known strong-field solutions are reproduced in the far-field limit. In doing so, we prove that black holes in Einstein-Dilaton-Gauss-Bonnet and Chern-Simons theory can have hair, while neutron stars have no scalar monopole charge, in diametrical opposition to results in scalar-tensor theories. We then employ techniques similar to the direct integration of the relaxed Einstein equations to obtain analytic expressions for the scalar field, metric perturbation, and the associated gravitational wave luminosity measured at infinity. We find that scalar field emission mainly dominates the energy flux budget, sourcing electric-type (even-parity) dipole scalar radiation and magnetic-type (odd-parity) quadrupole scalar radiation, correcting the General Relativistic prediction at relative -1PN and 2PN orders. Such modifications lead to corrections in the emitted gravitational waves that can be mapped to the parameterized post-Einsteinian framework. Such modifications could be strongly constrained with gravitational wave observations.

Figures

Tables

Table 1. Table 1: Scalar field parameters, as defined in Eqs. ( 93 ) and ( 95 ). The quantities q 1 subscript 𝑞 1 q_{1} and μ 1 i superscript subscript 𝜇 1 𝑖 \mu_{1}^{i} are defined in Eqs. ( 65 ) and ( 68 ), while σ NZ p ​ q superscript subscript 𝜎 NZ 𝑝 𝑞 \sigma_{{\mbox{\tiny NZ}}}^{pq} is defined in Eq. ( 94 ). The quantities D i subscript 𝐷 𝑖 D_{i} and μ i subscript 𝜇 𝑖 \mu_{i} are defined in Eqs. ( 77 ) and ( 80 ), while σ FZ p ​ q superscript subscript 𝜎 FZ 𝑝 𝑞 \sigma_{{\mbox{\tiny FZ}}}^{pq} is given in Eq. ( 96 ).

		$A$	$B$	$C$	$a$	$b$	$c$	$d$
Even-P		$q_1$	$0$	${\dot{D}}_i{n}^i$	$1$	$0$	$-$	$-$
Odd-P, Spins		${\mu }_1^i{n}_1^i$	$0$	${\ddot{\mu }}_i{n}^{ij}$	$2$	$0$	$-$	$-$
Odd-P, No Spins		${\sigma }_{\text{NZ}}^{pq}{n}_{12}^p{n}_1^q$	$\frac{1}{2}{\sigma }_{\text{NZ}}^{pq}{n}_1^p{n}_2^q$	${\sigma }_{\text{FZ}}^{pq}{n}^{pq}$	$2$	$2$	$2$	$2$

Table 2. Table 2: Coefficients of the relative energy flux.

Sector		$A$	$a$
Even-Parity		$\frac{5}{96}{\zeta }_3\frac{1}{{\eta }^4}\frac{\delta m^2}{m^2}$	$-2$
Odd-P, Spins ($\vartheta$),			$+4$
Odd-P, Spins ($h$),			$+4$
Odd-P, No Spin		$\frac{2}{3}{\zeta }_4\frac{\delta m^2}{m^2}$	$+14$

Equations

$$S$$

$$S_{\text{mbox}GR}\equiv$$

$$S_0\equiv$$

$$+{\alpha }_3^{(0)}{R}_{\mu \nu \delta \sigma }{R}^{\mu \nu \delta \sigma }\},$$

$$S_1\equiv$$

$$+{\alpha }_3^{(1)}\vartheta R_{\mu \nu \delta \sigma }{R}^{\mu \nu \delta \sigma }+{\alpha }_4^{(1)}\vartheta R_{\mu \nu \delta \sigma }{}^{\ast }{R}^{\mu \nu \delta \sigma }$$

$$-\frac{\beta }{2}\left[{\nabla }_{\mu }\vartheta {\nabla }^{\mu }\vartheta +2V(\vartheta )\right]\},$$

$$G_{\mu \nu }$$

$${\mathcal{H}}_{\mu \nu }^{(0)}\equiv$$

$${\mathcal{I}}_{\mu \nu }^{(0)}\equiv$$

$$+\frac{1}{2}{g}_{\mu \nu }\square R-{\nabla }_{\mu \nu }R,$$

$${\mathcal{J}}_{\mu \nu }^{(0)}\equiv$$

$$-2R{R}_{\mu \nu }+\frac{1}{2}{g}_{\mu \nu }{R}^2-2{\nabla }_{\mu \nu }R,$$

$${\mathcal{H}}_{\mu \nu }^{(1)}\equiv$$

$$+g_{\mu \nu }\left[2R\square \vartheta +4({\nabla }^{\delta }\vartheta ){\nabla }_{\delta }R\right],$$

$${\mathcal{I}}_{\mu \nu }^{(1)}\equiv$$

$$+2{\nabla }^{\delta }\vartheta {\nabla }_{\delta }{R}_{\mu \nu }+R_{\mu \nu }\square \vartheta -2R_{\delta (\mu }{\nabla }^{\delta }{\nabla }_{\nu )}\vartheta$$

$$+\text{mbox}{g}_{\mu \nu }\left({\nabla }^{\delta }\vartheta {\nabla }_{\delta }R+R^{\delta \sigma }{\nabla }_{\delta \sigma }\vartheta \right),$$

$${\mathcal{J}}_{\mu \nu }^{(1)}\equiv$$

$${\mathcal{K}}_{\mu \nu }^{(1)}\equiv$$

$$T_{\mu \nu }^{(\vartheta )}=\beta \left[({\nabla }_{\mu }\vartheta )({\nabla }_{\nu }\vartheta )-\frac{1}{2}{g}_{\mu \nu }\left({\nabla }_{\delta }\vartheta {\nabla }^{\delta }\vartheta -2V(\vartheta )\right)\right].$$

$$\beta \square \vartheta -\beta \frac{dV}{d\vartheta }=$$

$$-{\alpha }_3{R}_{\mu \nu \delta \sigma }{R}^{\mu \nu \delta \sigma }-{\alpha }_4{R}_{\mu \nu \delta \sigma }{}^{\ast }{R}^{\mu \nu \delta \sigma }.$$

$$g_{\mu \nu }=g_{\mu \nu }^{\text{mbox}GR}+\varsigma {\mathfrak{h}}_{\mu \nu }+O({\varsigma }^2),$$

$$\beta \square \vartheta =-{\alpha }_i\mathcal{S}(R_{\text{mbox}GR}^2)+\mathcal{O}(\varsigma ),$$

$$G_{\mu \nu }[{\mathfrak{h}}_{\mu \nu }]=-\frac{{\alpha }_i}{\kappa }{C}_{\mu \nu }[\vartheta ,g_{\mu \nu }^{\text{mbox}GR}]+\frac{1}{2\kappa }{T}_{\mu \nu }^{(\vartheta )}[\vartheta ],$$

$${\zeta }_i\equiv {\xi }_i/m^4=\mathcal{O}(\varsigma ),$$

$$g_{\mu \nu }^{\text{mbox}GR}={\eta }_{\mu \nu }+h_{\mu \nu }.$$

$${\bar{h}}^{\mu \nu }\equiv {\eta }^{\mu \nu }-\sqrt{-g_{\text{mbox}GR}}{g}_{\text{mbox}GR}^{\mu \nu }.$$

$${\bar{\mathfrak{h}}}^{\mu \nu }\equiv \left({\eta }^{\mu \nu }-\sqrt{-g}{g}^{\mu \nu }\right)-{\bar{h}}^{\mu \nu }.$$

$${\square }_{\eta }\vartheta$$

$$\frac{\kappa }{2}{\square }_{\eta }{\mathfrak{h}}_{\mu \nu }=$$

$$+{\alpha }_1{\tilde{\mathcal{H}}}_{\mu \nu }^{(1)}+{\alpha }_2{\tilde{\mathcal{I}}}_{\mu \nu }^{(1)}+{\alpha }_3{\tilde{\mathcal{J}}}_{\mu \nu }^{(1)}+{\alpha }_4{\tilde{\mathcal{K}}}_{\mu \nu }^{(1)}$$

$$-\frac{1}{2}\delta T_{\mu \nu }^{\text{mbox}mat}-\frac{1}{2}{T}_{\mu \nu }^{(\vartheta )},$$

$${\tilde{\mathcal{H}}}_{\mu \nu }^{(0)}$$

$${\tilde{\mathcal{I}}}_{\mu \nu }^{(0)}$$

$${\tilde{\mathcal{J}}}_{\mu \nu }^{(0)}$$

$${\tilde{\mathcal{H}}}_{\mu \nu }^{(1)}$$

$${\tilde{\mathcal{I}}}_{\mu \nu }^{(1)}$$

$${\tilde{\mathcal{J}}}_{\mu \nu }^{(1)}$$

$${\tilde{\mathcal{K}}}_{\mu \nu }^{(1)}$$

$$T_{\mu \nu }^{(\vartheta )}=\beta \left({\vartheta }_{,\mu }{\vartheta }_{,\nu }-\frac{1}{2}{\eta }_{\mu \nu }{\vartheta }_{,\delta }\vartheta {}^{,\delta }\right).$$

$${\mathbf{x}}_1\equiv x_1^i$$

Full text

I Introduction

The validity of Einstein’s theory in the strong-gravity regime will soon be put to the most stringent tests yet, through the observation of gravitational waves (GWs) from compact object binary inspirals Will (2006); Schutz et al. (2009); Sopuerta (2010). Such waves carry detailed information about their source and the underlying gravitational theory in play. This information is primarily encoded in the evolution of the GW frequency, which in turn depends directly on the rate of energy transport away from the binary Piotr Jaranowski (2005). In general relativity (GR), this transport is performed exclusively by GWs. In modified gravity theories, however, additional (scalar, vectorial or tensorial) degrees of freedom can also carry energy and angular momentum away as they propagate.

Calculating how gravitational waves are corrected in modified gravity theories can be a gargantuan task as the modification can increase the number of propagating degrees of freedom and the non-linearity of the equations that control their propagation. For example, the amount of energy-momentum transported away from a binary system must be computed both from the GWs excited by the corresponding sources, as well as any additional waves associated with extra degrees of freedom Misner et al. (1973). The sources that drive such waves can depend both on derivatives of the metric perturbation and the extra degrees of freedom, which, in turn, are specified by the solution to their own equations of motion. The situation worsens if these are non-linearly coupled, e.g. a scalar field equation of motion that depends on the metric tensor, whose evolution in turn depends on derivatives of the scalar field.

Such calculations, however, are feasible if one treats any GR deviations as small deformations Yunes and Pretorius (2009a), which can be formalized through the small-coupling approximation, a common technique in perturbation theory to isolate physically relevant solutions in higher-derivative theories Campanelli et al. (1994); Woodard (2007); Cooney et al. (2010). This is a reasonable approximation given that GR has passed a large number of tests, albeit in the weak-gravity regime. Even in the GW regime, signals will slowly transition from sampling weak fields to moderately strong fields during a full binary inspiral. The strongest GW events will not be able to sample anywhere close to the Plank regime, where one would expect completely new physics. The largest gravitational fields experienced by binaries occur when these merge, and even then, the metric curvature cannot exceed $m^{-2}$, where $m$ is the total mass of the binary. Earth-based detectors, such as LIGO lig , VIRGO vir and LCGT lcg , and future space-borne detectors, such as LISA lis , will only be able to sample gravitational fields up to this strength.

Of the plethora of modified gravity theories, we choose to focus on a general class that is characterized by the addition of quadratic curvature invariants to the action, coupled to scalar fields with standard kinetic terms (see e.g. Eq. (1). Such theories are motivated from loop quantum gravity Ashtekar and Lewandowski (2004); Rovelli (2004) and heterotic string theory Polchinski (1998), arising generically upon four-dimensional compactification in the low-energy limit. Disjoint sub-classes of quadratic theories reduce to Einstein-Dilaton-Gauss-Bonnet (EDGB) theory Moura and Schiappa (2007); Pani and Cardoso (2009) and Dynamical Chern-Simons (CS) modified gravity Jackiw and Pi (2003); Alexander and Yunes (2008).

From a phenomenological standpoint, such quadratic gravity theories are also interesting as straw-men to study small deviations from GR. This is because the new quadratic terms are always small relative to the Einstein-Hilbert term when considering merging binaries. In such systems, the minimum radius of curvature is always larger than the new scale introduced by the scalar fields. If this were not the case, astrophysical observations would already have constrained quadratic gravity deviations.

Quadratic gravity introduces an equation of motion for the scalar field and modifies the metric field equations. The former is a driven wave equation, whose sources are quadratic curvature invariants. The latter contains new terms that depend on the product of the scalar field and its derivatives with the Riemann tensor, Ricci tensor, Ricci scalar and their derivatives. As such, one might worry that higher derivative terms in the field equations could render the system unstable. One must remember, however, that the action is a truncation (at quadratic order in the present case) of an effective theory derived by integrating out heavy degrees of freedom contained in a more complete theory. Since we truncate the effective action, its validity is limited only to leading-order in the coupling parameters. Accounting for higher-order terms in the coupling would require the inclusion of higher-order terms (cubic, quartic, etc.) in the action Woodard (2007). Therefore, the modified field equations should not be considered as an exact system, but rather as an effective one.

Given the above and using the small-coupling approximation, the field equations become driven differential equations for the metric deformation and the scalar field. The source of the latter depends only on derivatives of the GR metric perturbation, while the source of the former depends both on the GR metric perturbation and the scalar field. We solve these equations in the post-Newtonian (PN) limit, where in particular we consider comparable-mass, spinning black hole (BH) binaries (electromagnetically uncharged), spiraling in a quasi-circular orbit. This forces the driven differential equations into driven wave equations, which can be studied with PN techniques Damour (1983); Tagoshi and Sasaki (1994); Tanaka et al. (1996); Blanchet and Damour (1992); Blanchet (1995); Blanchet et al. (1995); Blanchet (2006) and then solved via retarded Green function methods.

A complication arises when attempting to solve these equations, as one must choose a prescription to describe BHs and neutron stars (NSs). In standard PN theory and up to a certain high PN order, one can choose a point-particle prescription, essentially because the exterior gravitational field of a compact object is the same as that induced by a point-particle. In modified quadratic gravity, however, both non-spinning Yunes and Stein (2011) and spinning Yunes and Pretorius (2009a), strong-field BH solutions differ from that generated by simple point particles with a mass-monopole and a current-dipole moment; BHs in these theories have additional scalar multipole moments. (See Refs. Ali-Haimoud (2011); Ali-Haimoud and Chen (2011) for similar discussions on NSs in CS gravity.) One can take these effects into account by constructing an effective point-particle source that reproduces known, strong-field solutions to leading order in the weak-field region, sufficiently far away from the compact objects. With this effective point-particle prescription, we can then evaluate the source of the driven wave equations and analytically solve them to find the radiative part of the scalar field and metric perturbation.

Executive Summary of Results

Given the length of this paper, let us summarize the main results. We have devised a framework in the small-coupling approximation to solve for compact binary inspirals in modified quadratic gravity theories. One of the key ingredients in this framework is the calculation of effective source terms that allow us to use the point-particle approximation even for theories where such approximation is not valid. We applied this to modified quadratic gravity to find that both NSs and BHs have scalar hair, which leads to dipolar emission. EDGB and CS gravity are exceptions, where although BHs retain scalar monopole and dipole charge, respectively, NSs shed the scalar monopole charge. Therefore, BHs in EDGB generically contains dipolar GW emission, while CS gravity leads to modified quadrupolar emission.

The presence of scalar monopole and dipole hair, and in particular the flux of energy-momentum carried by this hair, leads to a modification in the rate of change of the binary’s binding energy. The even-parity sector of the theory leads to scalar hair, which modifies the energy flux at $-1$PN order relative to the GR quadrupole flux. Of course, such a modification is proportional to the coupling parameter of the theory, which is assumed small. The odd-parity sector leads to dipole hair for spinning BH binaries, which modifies the energy flux at $2$PN relative order. If the BH binary components are non-spinning, they have no dipole hair but the binary orbital interaction generates a modification in the energy flux that enters at relative $7$PN order. Figure 1 shows the energy flux carried by the even-parity scalar field (long dashed line), odd-parity scalar field (dot-dashed for spinning binaries and short dashed line for non-spinning binaries), and the GR quadrupole flux (solid line) as a function of orbital velocity. Observe that when one assumes that BHs are non-spinning, the scalar emission is greatly suppressed.

These energy flux corrections translate into changes to the waveform observables. We explicitly calculate these and map them to the parametrized post-Einsteinian (ppE) framework Yunes and Pretorius (2009b); Cornish et al. (2011). Using the results of Cornish et al. Cornish et al. (2011) we estimate that GW observations could constrain the new length scale introduced in quadratic gravity (related to the coupling constants of the theory) to roughly the BH horizon scale. With a typical Ad. LIGO stellar-mass BH inspiral observation, one should be able to constrain the even-parity sector to roughly $\mathcal{O}(10)$ km. With a typical LISA extreme-mass ratio inspiral (EMRI) observation, one should be able to constrain the odd-parity sector to roughly $\mathcal{O}(100)$ km. Such projected constraints are much stronger than current Solar System bounds Alexander and Yunes (2007); Smith et al. (2008); Amendola et al. (2007); Ali-Haimoud and Chen (2011).

This paper is organized as follows: Section II describes the action that will be considered in this paper and reviews the associated modified field equations and the scalar field equation of motion. Section III expands the field equations in the small-deformation approximation. Sections IV and V study the scalar field and metric deformation evolution, analytically solving the modified field equations. Section VI computes the energy flux carried by the scalar field and the metric deformation. Section VII considers the impact that such fluxes would have on gravitational waveform phase. Section VIII concludes and points to future research.

We have deferred many details of the computational techniques to the appendices. Appendix A shows the NSs in EDGB theory have no scalar monopole charge. Appendix B discusses specific integration techniques. Appendix C estimates the order of the metric correction from the regularized contribution for non-spinning BHs in the odd-parity sector of the modified theory. Appendix D discusses particular integrals that appear when solving the field equations.

Henceforth, we follow mostly the conventions of Misner, Thorne and Wheeler Misner et al. (1973): Greek letters stand for spacetime indices; Latin letters in the middle of the alphabet $i,j,\ldots$, stand for spatial indices only. Parenthesis, square brackets and angled brackets in index lists denote symmetrization, antisymmetrization and the symmetric and trace free (STF) operator, respectively. Capital Latin letters usually refer to a multi-index, such as $x^{Q}=x^{ijk\ldots}$, where $x^{ijk\ldots}=x^{i}x^{j}x^{k}\ldots$. Partial derivatives are denoted with $\partial_{i}A=A_{,i}=\partial A/\partial x^{i}$, while covariant derivatives are denoted with the nabla $\nabla_{i}A$, for any quantity $A$. Deformations are labeled with the order-counting parameter $\varsigma$. Finally, we use geometric units, where $G=c=1$, except when denoting the order of certain terms in the PN approximation. Throughout, we performed analytic calculations with the xTensor package for Mathematica Martín-García (2008); Brizuela et al. (2009).

II Modified gravity theories

In this Section, we introduce the class of modified gravity theories that we study, by writing down its action and equations of motion. We then proceed to define the small deformation approximation more precisely.

II.1 ABC of quadratic gravity

Consider the following $4$-dimensional effective action:

[TABLE]

Here, $g$ stands for the determinant of the metric $g_{\mu\nu}$. $R$, $R_{\mu\nu}$, $R_{\mu\nu\delta\sigma}$ and ${}^{*}R_{\mu\nu\delta\sigma}$ are the Ricci scalar and tensor, the Riemann tensor and its dual Alexander and Yunes (2009), respectively, with the latter defined as111This definition is correct, in agreement with Alexander and Yunes (2009), and fixing an inconsequential typo in Sopuerta and Yunes (2009). ${}^{*}R^{\mu}{}_{\nu\delta\sigma}=(1/2)\varepsilon_{\delta\sigma}{}^{\alpha\beta}R^{\mu}{}_{\nu\alpha\beta}$ and with $\varepsilon^{\mu\nu\delta\sigma}$ the Levi-Civita tensor. The quantity $\mathcal{L}_{\rm mat}$ is the external matter Lagrangian, $\vartheta$ is a field, $(\alpha_{i},\beta)$ are coupling constants and $\kappa=(16\pi)^{-1}$. This action contains all possible quadratic, algebraic curvature scalars with running (i.e. non-constant) couplings, where we assumed that all quadratic terms are coupled to the same field. All other quadratic curvature terms are linearly dependent, such as the Weyl tensor squared.

The theory defined by the action above is different from $f(R)$ theories on several counts. First, $f(R)$ theories depend only on the Ricci scalar, while the action above depends on the Ricci tensor, the Riemann tensor and a dynamical field $\vartheta$. Second, $f(R)$ theories are usually treated as exact, while the action presented above is an effective theory, truncated to quadratic order in the Riemann tensor. The consequence of this is insisting on the use of order-reduction in the field equations, where we treat all quantities that depend on $\alpha_{i}$ perturbatively. Such order reduction then leads to the absence of additional polarization modes Sopuerta and Yunes (2009); Stein and Yunes (2011), such as the longitudinal scalar mode that arises in $f(R)$ theories.

The field equations of dynamical quadratic gravity can be obtained by varying the action with respect to all fields. For simplicity, we restrict attention to coupling functions $f_{i}(\vartheta)$ that admit the Taylor expansion $f_{i}(\vartheta)=f_{i}(0)+f_{i}^{\prime}(0)\vartheta+\mathcal{O}(\vartheta^{2})$ about small $\vartheta$, where $f_{i}(0)$ and $f_{i}^{\prime}(0)$ are constants, and we assume that the asymptotic value of $\theta$ at spatial infinity vanishes. Let us further reabsorb $f_{i}(0)$ into the coupling constants $\alpha_{i}^{(0)}\equiv\alpha_{i}f_{i}(0)$ and $f_{i}^{\prime}(0)$ into the constants $\alpha_{i}^{(1)}\equiv\alpha_{i}f_{i}^{\prime}(0)$. Equation (1) then becomes $S=S_{{\mbox{\tiny GR}}}+S_{0}+S_{1}$:

[TABLE]

where clearly $S_{{\mbox{\tiny GR}}}$ is the Einstein-Hilbert plus matter action. Notice that $S_{0}$ defines a GR correction that is decoupled from $\theta$. The term proportional to $\alpha_{4}^{(0)}$ can not affect the classical field equations since it is topological, i.e. the second Chern form, so we have omitted it. Similarly, if $\alpha_{i}^{(0)}$ are chosen to reconstruct the Gauss-Bonnet invariant, $(\alpha_{1}^{(0)},\alpha_{2}^{(0)},\alpha_{3}^{(0)})=(1,-4,1)\alpha_{\mathrm{GB}}$, then these will not modify the field equations. On the other hand, $S_{1}$ defines a modification to GR with a direct (non-minimal) scalar field coupling, such that as the field goes to zero, the modified theory reduces to GR. We here restrict attention to the case $\alpha_{i}^{(0)}=0$. From this point forward, we will drop the superscript from $\alpha_{i}^{(1)}$.

The action above defines a class of modified gravity theories that contains well-known GR extensions. For example, when $\alpha_{4}=-\frac{1}{4}\alpha_{{\mbox{\tiny CS}}}$ and all other $\alpha_{i}=0$, quadratic gravity reduces to dynamical CS gravity, where $\alpha_{{\mbox{\tiny CS}}}$ is the CS coupling parameter (see e.g. Alexander and Yunes (2009)). Alternatively, when $\alpha_{4}=0$, while $(\alpha_{1},\alpha_{2},\alpha_{3})=(1,-4,1)\alpha_{\mathrm{EDGB}}$, quadratic gravity reduces to Einstein-Dilaton-Gauss-Bonnet theory (see e.g. Pani and Cardoso (2009)). Both of these theories are motivated from fundamental physics; they unavoidably arise as low-energy expansions of heterotic string theory Green et al. (1987a, b); Alexander and Gates (2006); Burgess (2004). Dynamical CS gravity also arises in loop quantum gravity when the Barbero-Immirzi parameter is promoted to a field in the presence of fermions Taveras and Yunes (2008); Mercuri and Taveras (2009); Gates et al. (2009).

Variation of the action with respect to the metric yields the modified field equations:

[TABLE]

where we have defined the short-hands222This corrects an error in Eq. (5b) of Yunes and Stein (2011).

[TABLE]

where $\nabla_{\mu}$ is the covariant derivative, $\nabla_{\mu\nu}\equiv\nabla_{\mu}\nabla_{\nu}$, and $\square=\nabla_{\mu}\nabla^{\mu}$ is the d’Alembertian operator. The $\vartheta$ field’s stress-energy tensor is

[TABLE]

Variation of the action with respect to $\vartheta$ yields the $\vartheta$ equation of motion:

[TABLE]

Notice that when the spacetime is curved by some mass distribution, the right-hand side will be proportional to density squared.

The parity of the field $\vartheta$ can be inferred from its equation of motion. Since terms of the form $R^{2}$ are even-parity, while terms of the form $R_{\mu\nu\delta\sigma}\!{}^{*}R^{\mu\nu\delta\sigma}$ are odd-parity, the field $\vartheta$ is of mixed parity. Note however that the even and odd-parity couplings tend to have different origins from an underlying theory. In this paper we will consider the even and odd-parity cases separately.

The inclusion of dynamics for the $\vartheta$ field in the action guarantees that the field equations are covariantly conserved without having to include any additional constraints, i.e. the covariant divergence of Eq. (5) identically vanishes, upon imposition of Eq. (8). This is a consequence of the action being diffeomorphism invariant. Such invariance is in contrast to the preferred-frame effects present in a non-dynamical theory Jackiw and Pi (2003), i.e. in the theory defined by the action in Eq. (4) but with $\beta=0$. In the latter, the field $\vartheta$ must be prescribed a priori. Moreover, the theory requires the existence of an additional constraint (the right-hand side of (8) to vanish), which is an unphysical consequence of treating $\vartheta$ as prior structure Yunes and Sopuerta (2008); Grumiller and Yunes (2008).

Before proceeding, let us further discuss the scalar field potential $V(\vartheta)$. This potential allows us to introduce additional couplings, such as a mass term, to drive the evolution in Eq. (8). However, there are reasons one might restrict such a potential. If the mass is much larger than the inverse length scale of the system that we concern, the effect of such a field on the dynamics of binaries is strongly suppressed. To the contrary, if the mass is much smaller, the presence of mass does not give any significant effects. Therefore we cannot expect to observe the effects of a finite mass without fine tuning. No mass term may appear in a theory with a shift symmetry, which is invariance under $\vartheta\to\vartheta+{\rm{const}}$. Such theories are common in 4D, low-energy, effective string theories Boulware and Deser (1985); Green et al. (1987b, a); Campbell et al. (1992); Burgess (2004), such as dynamical CS and EDGB. For these reasons, and because the assumption makes the resulting equations analytically tractable, we will henceforth assume $V(\vartheta)=0$.

II.2 Small deformations

The “unreasonable” accuracy of GR to explain all experimental data to date suggests that it is an excellent approximation to nature in situations where the gravitational field is very weak and velocities are very small relative to the speed of light. GW detectors will be sensitive to events in situations where the field is stronger than ever previously sampled. This, however, does not imply that GWs will ever sample the Planck/string regime, where one could expect large deviations from GR.

We will here be interested in binary compact object coalescences up until the binary reaches the innermost stable circular orbit (ISCO). Even during merger, the largest curvature that GWs will sample will be limited to the scale determined by the horizon sizes, proportional to $m^{-2}$. Such scales are far removed from high-energy ones, like the electroweak one, as GW detectors will not be sensitive to mergers of compact objects with masses below a solar mass. Even then, however, GWs can and will probe the strong field, which has not been tested before. One is then justified in modeling GWs that may contain deviations from GR as small deformations.

The small deformation scheme is also appealing for theoretical reasons. As mentioned earlier, the theories we consider are effective, valid only up to the truncation order. There are higher-order terms that we have here neglected in the action, such as cubic and quartic curvature combinations. Thus, one should not treat these theories as exact nor insist on solving the equations of motion to higher orders in $\alpha_{i}$. If this is desired, then higher-order curvature terms should also be included in the action.

One might be worried that such effective theories are unstable, since they lead to field equations with derivatives higher than second order. Such derivatives could lead to instabilities or ghost modes if the Hamiltonian is not bounded from below. Linearization in the coupling parameter, however, has the effect of recasting the field equations in Einstein form with an effective stress-energy tensor that depends on the GR solution, thus stabilizing the differential equations Campanelli et al. (1994). Linearization removes modes besides the two that arise in GR Sopuerta and Yunes (2009); Stein and Yunes (2011).

Small deformations can be treated similarly to how one models BH perturbations. That is, we expand the metric as

[TABLE]

where the GR superscript is to remind us that this quantity is a GR solution, while $\mathfrak{h}_{\mu\nu}$ is a metric deformation away from GR. The order-counting parameter $\varsigma$ is kept around only for book-keeping purposes and is to be set to unity in the end.

Applying such an expansion to Eq. (8), one finds

[TABLE]

where $\mathcal{S}(R^{2}_{{\mbox{\tiny GR}}})$ stands for all source terms evaluated on the GR background $g_{\mu\nu}^{{\mbox{\tiny GR}}}$. The solution to this equation will obviously scale as $\vartheta\propto\alpha_{i}/\beta$. Applying the decomposition and expansion of Eq. (9) to Eq. (5) in vacuum, one finds

[TABLE]

where the $\mathcal{O}(\varsigma^{0})$ terms automatically vanish, as $g_{\mu\nu}^{{\mbox{\tiny GR}}}$ satisfies the Einstein equations, and we have grouped modifications into the tensor $C_{\mu\nu}$. This tensor and $T_{\mu\nu}^{(\vartheta)}$ are to be evaluated on the GR metric and act as sources for the metric deformation. Notice that, as a differential operator acting on $\mathfrak{h}_{\mu\nu}$, the principal part of these differential equations continues to be strongly hyperbolic, as it is still given by the $G_{\mu\nu}$ differential operator, with the higher derivatives in $C_{\mu\nu}$ and the $T_{\mu\nu}^{(\vartheta)}$ acting as sources. Given this, the metric deformation is proportional to $\xi_{i}\equiv\alpha_{i}^{2}/(\beta\kappa)$, which is our actual perturbation parameter.

Proper perturbation or deformation parameters should be dimensionless, but the $\xi_{i}$ are dimensional. The dimensions of $\alpha$ and $\beta$, of course, depend on the choice of dimensions for the scalar field. We here take the viewpoint that $\vartheta$ is dimensionless, which then forces $\beta$ to be dimensionless as well as $\kappa$, and $\alpha$ to have dimensions of length squared. Then, the deformation parameter $\xi$ has units of length to the fourth power, which is why we define the dimensionless

[TABLE]

as our proper deformation parameter. One could choose different units for the scalar field, but in all cases one arrives at the conclusion that $\zeta_{i}$ is the proper deformation parameter Yunes and Pretorius (2009a).

III Expansion of the field equations

Let us decompose the GR metric tensor into a flat background plus a metric perturbation:

[TABLE]

We emphasize here that throughout this paper, $h_{\mu\nu}$ denotes the metric perturbation in GR while $\mathfrak{h}_{\mu\nu}$ is the metric deformation away from GR.

In expanding the modified field equations, we will also find it useful to define the standard trace-reversed metric perturbation in GR as

[TABLE]

In particular, notice that when the background is flat ${\bar{h}}_{\mu\nu}={h}_{\mu\nu}-\frac{1}{2}{h}\eta_{\mu\nu}$ and ${h}_{\mu\nu}={\bar{h}}_{\mu\nu}-\frac{1}{2}{\bar{h}}\eta_{\mu\nu}$ to linear order in GR. We also define the deformed trace reversed metric perturbation as

[TABLE]

The harmonic gauge condition reduces to $\bar{h}^{\mu\nu}{}_{,\nu}=0$ and $\bar{\mathfrak{h}}^{\mu\nu}{}_{,\nu}=0$. Throughout this paper, we only study the GR deformation up to $\mathcal{O}(\alpha_{i}/\beta)$ for $\vartheta$ and $\mathcal{O}(\zeta_{i})$ for $\mathfrak{h}_{\mu\nu}$.

III.1 Scalar field

The evolution equation for the scalar field at leading order in the metric perturbation becomes

[TABLE]

with relative remainders of ${\mathcal{O}}(h)$. Here, $\epsilon^{\mu\nu\delta\sigma}$ is the Levi-Civita symbol with convention $\epsilon^{0123}=+1$ in an orthonormal, positively oriented frame, and we have used the harmonic gauge condition.

III.2 Metric perturbation

Let us now perturb the metric field equations [Eq. (5)] about $\varsigma=0$. The deformed metric wave equation at linear order in $\mathfrak{h}_{\mu\nu}$ becomes

[TABLE]

where the tensors on the right-hand side are given by

[TABLE]

where $\square_{\eta}$ is the d’Alembertian of flat spacetime, $h=h_{\mu}{}^{\mu}$, and $T_{\mu\nu}^{(\vartheta)}$ is given as

[TABLE]

The quantity $\delta T_{\mu\nu}^{{\mbox{\tiny mat}}}$ stands for the perturbation to the energy-momentum tensor for matter. Even when dealing with BHs, $\delta T_{\mu\nu}^{{\mbox{\tiny mat}}}\neq 0$ because we treat BHs as distributional point particles and their trajectories are generically modified at ${\mathcal{O}}(\varsigma)$. However, in this paper we concentrate on the dissipative sector of the theory only, and not on modifications to the shape of the orbits (conservative dynamics). The latter does modify the GW phase evolution Yunes and Stein (2011); Yunes and Pretorius (2009a), as we discuss in Sec. VIII.

The evolution equation for the metric perturbation takes on the same form (a sourced wave equation) as that for the scalar field. The source terms in both of these equations depend on the GR metric perturbation, which we here assume to be that of a compact binary quasi-circular inspiral in the PN approximation, i.e. moving at small velocities relative to the speed of light and producing weak gravitational fields. We provide explicit expressions for the GR metric perturbation in the subsequent subsection.

III.3 Post-Newtonian metric and trajectories

In this subsection, we provide explicit expressions for the linear metric perturbation in GR that we use to evaluate all source terms. We are here interested in a binary system, composed of two compact objects with masses $m_{1}$ and $m_{2}$ and initially separated by a distance $r_{12}\equiv b$. The objects’ trajectories can be parameterized via

[TABLE]

where $m\equiv m_{1}+m_{2}$ is the total mass and where we have assumed they are located on the $x$–$y$ plane. Throughout this paper, vectors are sometimes denoted with a boldface. We also define

[TABLE]

where we follow the conventions of Blanchet (2006), with

[TABLE]

We further assume these objects are on a quasi-circular orbit with leading-order angular velocity $\omega=(1/b)(m/b)^{1/2}$ and orbital velocity $v=(m/b)^{1/2}$. The orbital separation $b$ is assumed constant, as its time-evolution is driven by GW emission at high-order in $v/c$.

The GR spacetime metric for such a binary is expanded as in Eq. (13). In the near zone, the metric perturbation is given by

[TABLE]

where ${\mathcal{O}}(v^{A})$ stands for an $(A/2)$PN remainder, i.e. a term of ${\cal{O}}((v/c)^{A})$, and the notation $+(1\leftrightarrow 2)$ means that one should add the same terms with the labels $1$ and $2$ interchanged. The potentials $U_{A}$ and $V_{Ai}$ with $A=(1,2)$ are defined as

[TABLE]

where $\rho_{A}$ and $v_{A}^{i}\equiv\dot{x}_{A}^{i}$ are the density and the center of mass velocities of the respective objects, with the overhead dot standing for time differentiation. Field variables associated with a prime, e.g. $\rho^{\prime}_{A}$, are to be evaluated at $\bm{x}^{\prime}$. In the point-particle limit, the metric becomes

[TABLE]

with remainders of relative $\mathcal{O}(v^{2})$. We have kept the PN leading terms in the metric that are proportional to $m_{A}$ only, but higher-order terms can be found in Blanchet et al. (1998), while terms proportional to the spin of each BH can be found in Tagoshi et al. (2001).

IV Scalar field evolution

In this section, we solve the evolution equation for the scalar field both for field points in the far and near-zones, as defined in Sec. IV.1. The former will allow us to evaluate the energy flux carried by the scalar field at infinity, while the latter will be essential to find effective source terms that reproduce the known strong field solutions and to solve the evolution equations for the metric deformation.

IV.1 Zones

As shown in Fig. 2, let us decompose the geometry into three regions: an inner zone (IZ), a near zone (NZ) and a far zone (FZ); see e.g. Alvi (2000); Yunes et al. (2005); Johnson-McDaniel et al. (2009) for further details. The IZs are centered at each object with radii $\mathcal{R}_{\mathrm{IZ}}$. These radii are defined as the boundary inside which either $T_{\mu\nu}^{{\mbox{\tiny mat}}}\neq 0$ or the usual PN approximation breaks down due to strong-gravity effects. We here take them to be sufficiently larger than $m_{A}$ and much less than $b$. The NZ is centered at the binary’s center of mass with radius $\mathcal{R}_{\mathrm{NZ}}$ and excluding the IZs. This radius is defined as the boundary outside which time-derivatives cannot be assumed to be small compared with spatial derivatives due to the wave-like nature of the metric perturbation. We here take this boundary to be roughly equal to $\lambda_{\mathrm{GW}}$, where $\lambda_{\mathrm{GW}}$ denotes the GW wavelength. The FZ is also centered at the binary’s center of mass, but it extends outside $\mathcal{R}_{\mathrm{NZ}}$.

One can only apply the PN formalism when the gravitational field is weak and velocities are small. When we deal with strong field sources like BHs and NSs, therefore, one can use the PN scheme in the NZ and FZ only. In the IZs, one may not be able to use PN theory, since the gravitational field may be too strong. In this case, we have to asymptotically match our PN solution in the NZ with the strong field solutions valid in the IZs, inside some buffer regions that overlap both NZ and each IZ (see Refs. D’Eath (1975); Thorne and Hartle (1984); Damour (1983) for a description of how to carry this out in GR). The strong field solution for BHs was found in Refs. Yunes and Stein (2011) and Yunes and Pretorius (2009a) in the class of theories considered here.

IV.2 Near zone solutions

Since the NZ is in the weak field regime, we can apply the PN formalism to compact binary systems. Let us consider the even and odd-parity sectors separately.

IV.2.1 Even-parity sector

The evolution equation for the even-parity sector is

[TABLE]

with $\rho\equiv\rho_{1}+\rho_{2}$ and remainders of ${\mathcal{O}}(h^{3})$.

First, let us consider weakly-gravitating objects, i.e. not BHs or NSs, in which case the PN expansion is valid also in the IZ. By substituting the GR PN metric of Eqs. (32)-(34), the NZ solution to the above wave equation at leading PN order becomes

[TABLE]

again with remainders of ${\mathcal{O}}(h^{3})$, with $U\equiv U_{1}+U_{2}$ and $\mathcal{M}$ denoting the constant-time, NZ+IZ hypersurface. We can safely neglect the contribution from the FZ, since the fall-off of the source term is sufficiently fast.

The solution in Eq. (40) can be simplified by integrating by parts several times and using that $\square U=-4\pi\rho$ and $\square{|\bm{x}-\bm{x}^{\prime}|}^{-1}=-4\pi\delta^{(3)}(\bm{x}-\bm{x}^{\prime})$ to obtain

[TABLE]

Expanding this solution in terms of particles $1$ and $2$, we arrive at

[TABLE]

with

[TABLE]

and

[TABLE]

$\vartheta_{{\mbox{\tiny self}}}$ is the part of $\vartheta$ that can be evaluated by considering a single object only, while $\vartheta_{{\mbox{\tiny cross}}}$ is the part that depends on the fields of both bodies.

The integrals that define both $\vartheta_{{\mbox{\tiny self}}}$ and $\vartheta_{{\mbox{\tiny cross}}}$ have support in the IZs only, and thus, the NZ integral operator is homogeneous (source-free). When we discuss the NZ behavior of fields associated with compact objects, such as BHs or NSs, we cannot directly evaluate such IZ integrals. These are derived under the assumption that the PN expansion is valid everywhere, which fails for compact objects in the IZs. Instead, we need to determine these homogeneous solutions through asymptotic matching. Before doing so, it is helpful to study the meaning of each term for weakly-gravitating objects.

Neglecting the size of the weakly-gravitating objects, the first term in Eq. (43) in the NZ is evaluated as

[TABLE]

with remainders of relative ${\mathcal{O}}(m/r)$, while the second term becomes

[TABLE]

The last equality can be shown by exchanging the integration variables333In fact, this integral vanishes to all orders in $x$. This is because $(\rho_{1}U_{1,i})_{,i}$ is spherically symmetric, and thus, when it acts as a source to a wave equation, the solution should either scale as $1/r$ or it should vanish identically. We have here shown that there is no $1/r$ part.. Thus, one can approximate $\vartheta_{{\mbox{\tiny self}}}$ as

[TABLE]

with the scalar monopole charge defined by

[TABLE]

with $A=(1,2)$. Here we put “IZ” to the integral to emphasize that the integration can be restricted to both IZs because the integrand is localized.

The first term in Eq. (48) represents the monopole field around object 1. These monopole fields give the leading PN contribution in the NZ unless both monopole charges $q_{1}$ and $q_{2}$ vanish. This is indeed the case in EDGB theory, where $(\alpha_{1},\alpha_{2},\alpha_{3})=(1,-4,1)\alpha_{\mathrm{EDGB}}$. We will later show that this cancellation does really survive even if we consider NSs. If this cancellation occurs, the higher order terms of ${\mathcal{O}}(m^{2}/r^{2})$ in the expansion of Eq. (45) become the dominant contribution to $\vartheta$. The second term in Eq. (48) is much higher PN order compared with the first term and hence sub-dominant in the NZ.

Let us now consider $\vartheta_{{\mbox{\tiny self}}}$ for compact objects, where the IZ integrals must be treated carefully. Since the PN expansion is no longer valid in the IZ, one cannot use the simple extrapolation of the above result. In Sec. IV.3, we match the NZ solution to the one obtained for isolated BHs in the strong-field Yunes and Stein (2011); Yunes and Pretorius (2009a). We will not discuss the matching for NSs in this paper, but the order of magnitude estimate

[TABLE]

should still be valid, where $R_{A}$ is the radius of the $A$th NS. When $\alpha_{1}+\alpha_{2}+3\alpha_{3}=0$, the cancellation observed in the weakly gravitating objects may still persist even for NSs. However, the cancellation will not in general be exact, except for the EDGB subcase. In EDGB theory, the NS scalar monopole charge vanishes independently of the equation of state. Mathematically speaking, this is because the monopole charge is given by the integral of the Gauss-Bonnet invariant $R_{\mathrm{GB}}\equiv R^{2}-4R_{\mu\nu}R^{\mu\nu}+R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}$, which vanishes for any simply-connected, asymptotically flat geometry. A more explicit proof is given in Appendix A.

Let us now return to the $\vartheta_{\mbox{\tiny cross}}$ contribution and consider first weakly-gravitating objects. To evaluate Eq. (44), one can use point-particle expansions of the potentials and the density, i.e. $\rho_{A}=m_{A}\delta^{(3)}(\bm{x}-\bm{x}_{A})$ and $U_{A}=m_{A}/r_{A}$. Simple substitution leads to

[TABLE]

The first term in parentheses comes from the term $U_{1,i}U_{2,i}$ in Eq. (44). The remaining two terms come from the integral in Eq. (44). The second and third terms in parentheses look like scalar dipole moments for bodies 2 and 1 respectively. However, a Taylor expansion about the center of mass of each body, shows that the $1/r_{A}^{2}$ piece of $\vartheta_{{\mbox{\tiny cross}}}$ cancels, which implies that there is no scalar dipole.

Let us now consider $\vartheta_{{\mbox{\tiny cross}}}$ for compact objects. As discussed in the previous paragraph, one might expect a scalar dipole charge induced by the acceleration of object 1 due to the gravitational field of object 2 (\propto U_{2,i}(\mbox{\boldmathx}_{1})). In GR, however, acceleration is understood as geodesic motion in a perturbed geometry. The deviation of the local geometry from the unperturbed isolated geometry originates due to tides, and this is a relative $4$PN effect. This is much smaller than the scalar monopole charge contribution from $\vartheta_{{\mbox{\tiny self}}}$.

To summarize, the dominant contribution to $\vartheta$ comes from the monopole charge associated with each object, which depends on its internal structure.

IV.2.2 Odd-parity sector

In the odd-parity case, the scalar field evolution equation is

[TABLE]

plus terms of ${\mathcal{O}}(h^{3})$. Again, we first consider weakly gravitating objects. At leading PN order, the above equation becomes

[TABLE]

with remainders of relative ${\mathcal{O}}(v^{2})$. As in the even-parity case, we write the solution to this wave equation as

[TABLE]

where

[TABLE]

and

[TABLE]

Let us first consider self-interaction terms $\vartheta_{{\mbox{\tiny self}}}$. Integrating by parts several times, we find

[TABLE]

where we have used the relations $\square U_{1}=-4\pi\rho_{1}$, $\square V_{1}^{k}=-4\pi\rho_{1}v_{1}^{k}$ and $\square{|\bm{x}-\bm{x}^{\prime}|}^{-1}=-4\pi\delta^{(3)}(\bm{x}-\bm{x}^{\prime})$. The third term vanishes when we take the point-particle limit444 The vanishing of this term is a general consequence of the symmetry of the system. The source term contains an $\epsilon^{ijk}$ symbol, which must be contracted with other vectors to produce a scalar. We here have only two possible vectors to contract with, i.e. the velocity $v_{1}^{i}$ and the unit vector $n_{1}^{i}$ from object 1. Hence, any contraction with the Levi-Civita symbol should vanish., i.e. $\rho_{A}=m_{A}\delta^{(3)}(\bm{x}-\bm{x}_{A})$, $U_{A}=m_{A}/r_{A}$ and $V_{Ai}=m_{A}v_{Ai}/r_{A}$.

Let us evaluate the first and the second terms in the NZ. Keeping only the leading PN term in the NZ, we find

[TABLE]

where we have defined

[TABLE]

This leading-order PN term in $\vartheta_{{\mbox{\tiny self}}}$ represents a magnetic-type dipole.

As in the even-parity case, to extend this result to compact objects we have to determine the value of $\mu_{i}^{(A)}$ by matching the NZ solution in Eq. (58) to a strong field solution. This will be carried out in Sec. IV.3 for the BH case. For NSs, we just present an order of magnitude estimate based on a simple extrapolation of weakly-gravitating results:

[TABLE]

where $S_{A}^{i}$ is the spin angular momentum of the object. Following the procedure in Appendix A, we can show that NSs cannot have scalar monopole charge in the dynamical CS case.

Next, we consider the cross term $\vartheta_{{\mbox{\tiny cross}}}$ in the weakly-gravitating case. Integrating by parts several times, we find

[TABLE]

One can take the point-particle limit of this expression without any trouble to obtain

[TABLE]

Since $\mu_{i}^{(A)}$ in Eq. (58) do not depend on $v$, these terms are of relative $\mathcal{O}(v^{5})$ compared to the leading-order term of $\vartheta_{{\mbox{\tiny self}}}$.

As for compact objects, the results found in the even-parity case also apply here. Terms proportional to $1/r_{A}^{2}$ in the above expression suggest that each object has a dipole component induced by the companion. When we expand this expression around $r_{A}\ll b$, however, the terms proportional to $1/(r_{A}^{2}b^{2})$ cancel each other, as in the even-parity case, leading to no induced dipole moment. Even if this were not the case, however, the corrections to the dipole moment would be higher order than the contributions from $\vartheta_{{\mbox{\tiny self}}}$.

To conclude, the dominant contribution to $\vartheta$ is clearly that of $\vartheta_{{\mbox{\tiny self}}}$ given in Eq. (58), which again depends on the structure of the source and thus violates the effacement principle.

IV.3 Matching near zone and strong-field solutions and finding the effective source terms

In alternative theories of gravity, the point-particle limit is not always valid and the multipole moments of compact objects may depend on the internal structure of the source. In the previous subsections, we found that the dominant contributions to the scalar field come from self-interaction terms, which in turn depend on certain structure constants. In this subsection, we determine these constants by matching the $\vartheta$ solution to that of an isolated BH.

IV.3.1 Even-parity sector

In the even-parity case, the monopole charges $q_{1}$ and $q_{2}$ in Eq. (48) must be determined by matching to a BH solution. An isolated BH sources a scalar field Yunes and Stein (2011), whose leading PN behavior is

[TABLE]

Matching this solution to the NZ solution of Eq. (48) we obtain

[TABLE]

Notice that this monopole charge does not depend on $(\alpha_{1},\alpha_{2})$, as for pure BH spacetimes, these coupling constants appear in combination with the Ricci scalar and tensor, which vanishes. This is to be contrasted with the NS case, in which $q_{A}$ depends on $\alpha_{1}$ and $\alpha_{2}$ as well as $\alpha_{3}$ and vanishes in EDGB theory. Interestingly, BHs do not have scalar hair in more traditional (Brans-Dicke type) scalar-tensor theories, while NSs do possess them. This situation is reversed in EDGB theory.

The matching carried out above dealt with the monopole part of $\vartheta$. That is, we have ignored any tidal deformation of either BH induced by its binary companion. In BH perturbation theory, one can calculate the deformation of the isolated BH metric to find that it depends on the sum of electric and magnetic tidal tensors, leading to a metric deformation that scales as $(r_{1}/b)^{2}(m_{2}/b)$ for $r_{1}\ll b$ Alvi (2000); Yunes et al. (2005); Martel and Poisson (2005); Poisson (2005); Yunes and Tichy (2006); Johnson-McDaniel et al. (2009). Thus, in the IZ of object 1, tidal deformations lead to corrections of ${\mathcal{O}}(m^{3}/b^{3})$, which are much smaller than the effects considered here. Therefore, it suffices in this section to consider an isolated BH when matching the scalar fields.

With this at hand, we can now treat BHs in even-parity, quadratic modified gravity as delta function sources of matter energy density, and with effective scalar charge density

[TABLE]

In the PN expansion such sources reproduce the BH solution found by Yunes and Stein Yunes and Stein (2011) at leading order.

Let us make a few observations about the effective source term approach. First, notice that the scalar field diverges as $m_{A}\to 0$, which violates the small-coupling approximation. This is related to the fact that as one shrinks a BH, the radius of curvature at the horizon also goes to zero, probing increasingly shorter length scales. When the small-coupling approximation is violated, one can no longer neglect the scalar field’s stress-energy tensor and the $({\mathcal{H}}_{\mu\nu},{\mathcal{I}}_{\mu\nu},{\mathcal{J}}_{\mu\nu},{\mathcal{K}}_{\mu\nu})$ tensors that would dominate over the Einstein tensor. Of course, one cannot take this limit seriously, as we are considering here a low-energy effective theory, which is missing higher-curvature terms that would need to be included. Notice also that this is different from the behavior of scalar fields in traditional scalar-tensor theories, where the scalar field vanishes in the $m_{A}\to 0$ limit.

IV.3.2 Odd-parity sector

In the odd-parity case, the dipole charges of the respective objects in Eq. (58) are to be determined by matching against the appropriate BH solutions. An isolated non-spinning BH in the odd-parity case does not support a scalar field. By contrast, a spinning BH does, and in the slow-rotation limit, neglecting higher order PN corrections, it is given by  Yunes and Pretorius (2009a)

[TABLE]

where $\chi_{A}\equiv S_{A}^{i}/m_{A}^{2}$ is the normalized spin angular momentum vector of the $A$th BH. Matching this solution to the NZ $\vartheta_{\mbox{\tiny self}}$ in Eq. (58), we obtain

[TABLE]

With this at hand, we can now treat BHs in odd-parity, quadratic modified gravity as distributional sources of matter energy density and effective scalar charge density

[TABLE]

In the PN expansion, such sources reproduce the BH solution found by Yunes and Pretorius Yunes and Pretorius (2009a) at leading order.

Let us make a few observations about this solution. First, notice that the pseudo-scalar dipole charge is well behaved in the limit $m_{A}\to 0$, because there is a maximum BH spin $|\chi_{A}^{i}|<1$. Second, notice that in the $|\chi_{A}^{i}|\to 0$ limit, this dipole charge vanishes, which is a consequence of Birkhoff’s theorem holding in CS gravity Yunes and Sopuerta (2008); Grumiller and Yunes (2008); Alexander and Yunes (2009). Namely, non-spinning BHs in CS theory are the same as BHs in GR (i.e. Schwarzschild BHs). Therefore, in this case the point-particle limit is well-justified and the metric deformation or the scalar field does not depend on the internal structures of non-spinning sources.

IV.4 Far-zone field point solutions

Let us assume that we have the wave equation

[TABLE]

where $\tau$ denotes the source term. The far-zone field point solution to this wave equation is given as Will and Wiseman (1996); Pati and Will (2000)

[TABLE]

with $u\equiv t-r$. By using $u_{,i}=-n_{i}$ and by keeping only terms proportional to $1/r$, the above solution reduces to

[TABLE]

Here, the region $\mathcal{M}$ denotes the hypersurface of $t-r=\mathrm{const}$. In the following, we apply these formulas to the even and odd parity cases separately.

IV.4.1 Even-parity sector

Following the discussion in Sec. IV.2, the evolution equation for the scalar field is dominantly

[TABLE]

From Eq. (72), this wave equation can be solved as

[TABLE]

The $m=0$ term gives

[TABLE]

where we have defined the total scalar monopole charge $q\equiv q_{1}+q_{2}$. Recall that this monopole charge $q$ refers to the scalar field, and not to an electromagnetic one. For a BH binary or a NS binary in a quasi circular orbit, $q$ only changes during merger, as mass is carried away in radiation. Thus, monopole radiation is inefficient and suppressed.

For the $m=1$ case, we find

[TABLE]

where we have defined the total scalar dipole moment as

[TABLE]

When we evaluate this for circular orbits, we find

[TABLE]

where we have defined the relative velocity $v^{k}_{12}\equiv v_{1}^{k}-v_{2}^{k}$.

The $m=1$ term clearly leads to dipole radiation in the FZ, which is less relativistic than GR quadrupole radiation, becoming stronger at smaller velocities. Of course, this term is proportional to the coupling constants of the theory, which are assumed much smaller than one. . Reference Yunes and Stein (2011) failed to recognize such dipolar emission because they considered the motion of test particles that had no scalar charge. We cannot think of any mechanism that would suppress such dipolar radiation.

IV.4.2 Odd-parity sector: spinning bodies

As in the previous Section, the evolution equation for the scalar field is dominantly

[TABLE]

By using Eq. (71), the far-zone field point solution is obtained as

[TABLE]

When $m=0$ there is obviously no contribution to the scalar field. When $m=1$,

[TABLE]

and thus

[TABLE]

with $\mu_{i}\equiv\mu_{1i}+\mu_{2i}$. Notice that we recover the solution of Yunes and Pretorius Yunes and Pretorius (2009a) for the first term of the above equation with $\mu_{A}^{i}$ given as in Eq. (68). These terms will not strongly radiate because $\dot{\mu_{i}}$ is non-vanishing only for spin-precessing systems. Even then, such radiation would be suppressed by the ratio of the orbital timescale to the precession timescale.

The $m=2$ contribution, by contrast, depends on the much shorter orbital timescale. We look for terms of $\mathcal{O}(r^{-1})$ since they are the only ones that contribute to the energy flux at infinity. Keeping in mind that the function being differentiated depends on retarded time, we can rewrite Eq. (80) as

[TABLE]

When $m=2$, we have that

[TABLE]

where the pseudo-tensor quadrupole moment (not to be confused with $\mu^{i}\mu^{j}$) is defined as

[TABLE]

The $m=2$ contribution becomes

[TABLE]

where the final equality is evaluated on a circular orbit. Notice that such a scalar field will strongly radiate because $\mu^{ij}$ depends on the orbital timescale.

IV.4.3 Odd-parity sector: non-spinning bodies

When both objects are non-spinning, the self-interaction terms produced by the effective source identically vanish. One is then left with the source term constructed from the product of the gravitational fields of objects 1 and 2. These terms will be proportional to $m_{1}m_{2}$. As we will see, there are many contributions that turn out to vanish upon NZ integration. For pedagogical reasons, we will show here explicitly how this happens and eventually arrive at contributions that do not vanish.

The evolution equation for the scalar field to leading PN order is

[TABLE]

where we substituted the NZ metric components in the point-particle approximation. The leading order term of the solution to this differential equation, i.e. the $m=0$ term in the sum of Eq. (72), is evaluated as

[TABLE]

Here we integrated over the NZ+IZ hypersurface ${\mathcal{M}}$ without taking any care of the strong gravity region in the IZs. One can easily show that the contribution from the IZs is not large in the present case. In the second line, we replaced partial derivatives with respect to $x^{i}$ acting on $1/r_{A}$ with (minus the) particle derivatives with respect to $x_{A}^{i}$:

[TABLE]

with $A=(1,2)$. We commuted these particle derivatives with the integral, and finally obtained a typical NZ integral, discussed in Appendix B. From Eq. (156), we know that $Y=b$, and by taking all particle derivatives, the last equality is established.

We could have inferred that the $m=0$ term in the sum does not contribute for non-spinning BHs without any explicit calculations. The argument here is similar to that in footnote 4. Possible vectors to contract with the Levi-Civita symbol include the velocities $v_{A}^{i}$ and the unit vectors $n_{A}^{i}$, but not spin vectors $S_{A}^{i}$, as we here consider non-spinning BHs. In particular, for the $m=0$ case, there cannot be any FZ vectors $n^{i}$ present. Thus, all vectors that can be contracted onto the Levi-Civita symbol must lie in the same orbital plane and this obviously vanishes. This argument should be true at all PN orders555One may think that one can construct a vector that does not lie in the orbital plane by taking the cross product of two vectors that lie on this plane, e.g. $\bm{n}_{12}\times\bm{v}_{12}$. However, since GR is parity even, such a vector cannot be present in the PN metric..

Let us then consider the next-order term. This will arise from the leading-order source term [right-hand side of Eq. (87)] with $m=1$ in the NZ sum:

[TABLE]

where we have used Eq. (155), which defines $Y_{p}$. By direct evaluation, one can show that this term also identically vanishes. The first non-vanishing contribution coming from an $m=1$ term must then be ${\mathcal{O}}(v^{3})$ smaller than the ordering of the $m=0$ term.

Finally, let us consider the (next)2-order term. This can arise only from the leading-order source term with $m=2$ in the NZ sum:

[TABLE]

which simplifies to

[TABLE]

where we have defined the mass difference $\delta m\equiv m_{1}-m_{2}$ and the symmetric mass ratio $\eta\equiv m_{1}m_{2}/m^{2}$. We have here used Kepler’s law and expanded the STF tensors. This is the dominant FZ behavior of the scalar field, which as we see is much suppressed relative to the odd-parity solution we found for spinning BHs.

IV.5 Summary of this section

Let us summarize the results found so far for later use. In the even-parity case, generically at least one of the binary component objects will have a scalar monopole charge. Since the scalar field excitation due to the induced monopole is dominant, we neglect all the other less important contributions. Weakly gravitating objects need not have a scalar monopole charge if $\alpha_{1}+\alpha_{2}+3\alpha_{3}=0$, and BHs have no scalar monopole charge if $\alpha_{3}=0$. In EDGB theory, NSs have no scalar monopole charge. In the odd-parity case, the dominant contribution is the magnetic-type scalar dipole moment induced by spins. Generically, astrophysical objects will possess spin, but we will continue to include non-spinning results to compare with previous work.

In the NZ, we can parametrize the leading PN terms of the scalar field as

[TABLE]

where $(A,B,a,b,c,d)$ are given in Table 1 and for compactness of the Table we define

[TABLE]

In the FZ, we can parametrize the scalar field as

[TABLE]

where $C$ is also given in Table 1 and we define

[TABLE]

V Metric evolution

In this section, we solve the evolution equations for the metric deformation in the FZ, so that we can calculate the gravitational energy flux at infinity. Note that throughout, we use the Newtonian relationship $v^{2}=m/b$ (and similarly for the acceleration). This relationship must be corrected at higher PN order or at $\mathcal{O}(\varsigma)$. As we mentioned earlier, here we do not take into account the corrections to the orbital motion due to the conservative force at $\mathcal{O}(\varsigma)$. These conservative effects do not interfere at $\mathcal{O}(\varsigma)$ with the radiative effects that we are concerned with in this paper. Therefore the corrections to the GW waveform become a simple summation of these two different types of effects.

For the FZ field points, the solution to the metric deformation equation of motion [Eq. (17)] can be read from Eq. (72):

[TABLE]

where we have defined the source term as

[TABLE]

Notice that this corresponds to an IZ+NZ integration for FZ field points, where we have neglected the FZ integration because it is subdominant.

The integrals presented above have to be carried out also in the IZ, where the PN expansion is not valid anymore. In GR, however, such divergences can be ignored, using a regularization scheme. Since both the true solution and an appropriately regularized solution satisfy the field equations in the NZ, their difference due to the IZ contribution is only through a homogeneous solution. Such homogeneous solutions are regular in the NZ and FZ, but can be divergent in the IZ. They are characterized by the multipole moments of the respective objects, which can be determined by studying tidal perturbations around a strongly gravitating object. One can then perform matching of the metric solution, as for the scalar solution, but the metric matching is beyond the scope of this paper. In what follows, we only consider the regularized contribution, following Hadamard partie finie (FP) regularization Blanchet and Faye (2000). We comment more on the divergent contribution at the end of this Section.

V.1 Even-parity sector

Let us focus on the metric perturbation in the even-parity sector first. The leading order term both in the PN and $1/r$ expansion at infinity is formally given by

[TABLE]

where we have defined $\tilde{\cal{J}}_{ij}\equiv\vartheta\tilde{\mathcal{J}}^{(0)}_{ij}+\tilde{\mathcal{J}}^{(1)}_{ij}$. The source terms $\tilde{\cal{H}}_{\mu\nu}$ and $\tilde{\cal{I}}_{\mu\nu}$ do not contribute to this expression since they identically vanish in the NZ where $R_{\mu\nu}=0$.

We can estimate the order of magnitude of both $\mathfrak{h}_{ij}^{{\cal{J}}}$ and $\mathfrak{h}_{ij}^{T}$ as follows:

[TABLE]

Here we factored out $v^{2}$ in the final expressions, since the GR leading quadrupolar field is also proportional to $v^{2}$. Clearly, the dominant contribution comes from Eq. (102).

Let us now make this computation more precise. The stress-energy tensor will contain self-interactions of the form $\vartheta_{A,i}\vartheta_{A,j}$ and cross terms of the form $\vartheta_{1,i}\vartheta_{2,j}$. The former case leads to divergent integrals, which must be determined by strong-field matching, so we do not consider them here. Let us concentrate on the latter, which take the form

[TABLE]

which sources the metric perturbation

[TABLE]

where we used an integration formula for the triangle potential given in Appendix B. We can see that this correction is 0PN relative to the radiative metric perturbation in GR, just as we predicted in Eq. (102). However, this correction turns out to be still smaller in the energy flux than the dipole scalar radiation, which gives a -1PN correction.

V.2 Odd-parity sector

We now focus on the odd-parity sector, for which the solution is given by the term proportional to $\alpha_{4}$ in Eq. (97), namely

[TABLE]

The stress-energy contribution $\mathfrak{h}_{ij}^{T}$ is the same as in Eq. (104).

The $\mathcal{K}$ contribution to Eq. (107) is more involved. The leading-order behavior of the $\mathcal{K}$ tensor is

[TABLE]

Other terms are of higher PN order. By applying the Lorenz or harmonic gauge condition $h^{\mu\nu}{}_{,\nu}=0$, substituting $h_{ij}=h_{00}\delta_{ij}$ into Eq. (109), and using $\epsilon_{jkl}h^{\mu\nu,kl}=\epsilon_{jkl}\vartheta_{,kl}=0$, we get

[TABLE]

The $\tilde{\cal{K}}_{ij}$ term in Eq. (107) is then a sum of four terms, namely

[TABLE]

where we have defined

[TABLE]

When we substitute the PN metric into the above terms, the right-hand sides depend on the velocity vectors $v_{A}^{i}$ (which depend on time only). The field $\vartheta$ is given in Eq. (93) and its derivative can be computed simply from that equation. Since this field is a NZ one, it depends on time through the positions of the objects, which implies that its time derivative can be converted into a spatial derivative via $\partial_{t}f(r_{1})=-v_{1}^{i}\partial_{i}f(r_{1})$.

Let us begin by making a simple order of magnitude estimate of how large the regularized contribution is. For this, it suffices to look at Eqs. (102) and (112):

[TABLE]

The $\vartheta$ field here is that of the NZ, and hence

[TABLE]

where $\chi$ stands for the magnitude of $\chi_{1}^{i}$ and $\chi_{2}^{i}$. From this analysis, $\mathfrak{h}_{ij}^{T}$ is clearly larger for rapidly spinning objects, leading to a 2PN effect.

For the non-spinning case, one might expect the ${\cal{K}}$ contribution to lead to a 6PN effect, but as we explain in Appendix C, these leading-order effects actually vanish. This cancellation can also rather easily be seen by integrating by parts in Eqs. (112)-(115). After discarding boundary terms (taking into account the boundary term is equivalent to adding homogeneous solutions, corresponding to deformed multipole moments of compact objects), we obtain expressions of the form $\epsilon_{jkl}\,\vartheta\,h_{00,kl\ldots}$, which obviously vanishes by the antisymmetry of the Levi-Civita tensor. We carry out a more careful analysis in Appendix C, where we explicitly show that the leading and first sub-leading order terms vanish666In Appendix C, we only show this for non-spinning BHs, but a similar calculation can be performed for spinning BHs to $\mathcal{O}(\chi)$.. The first non-vanishing term is then of ${\mathcal{O}}(v^{2})$ smaller than the order of magnitude estimates in Eqs. (118) and (119), leading to 7PN and $4.5$PN contributions at $\mathcal{O}(\chi^{0})$ and $\mathcal{O}(\chi^{1})$, respectively.

Since the largest contribution seems to arise for spinning BHs from the $\mathfrak{h}_{ij}^{T}$ term, let us consider this in more detail. Two possible contributions are generated here: one that depends only on self-interaction terms, and one that depends on the cross-interaction. The former leads to divergent integrals, which need to be matched from strong-field solutions, and we do not consider these here. The latter leads to the metric deformation

[TABLE]

which is clearly of the order predicted in Eq. (118), i.e. 2PN order relative to GR. This is of the same order as the energy flux correction carried by the pseudo-scalar radiation.

V.3 Multipole moments

In this Subsection, we discuss the additional contribution from the IZs, which enter as additional homogeneous solutions in the NZ and FZ, These contributions are homogeneous in the sense that they arise from sources that have support only in the IZs, and thus they vanish in the NZ and FZ (see e.g. the discussion prior to Eq. (45)). The homogeneous solutions are characterized by the mass and current multipole moments of the strong-field bodies, which must be determined by matching to strong gravity solutions in the IZ. When we solve the non-linear equations of motion iteratively, the source terms in general can be classified into two pieces: a self-interaction part and a cross-interaction part, as in the case of $\vartheta$ in Sec. IV. The cross-interaction part is sourced by the companion, while the self-interaction part is not.

The self-interaction part is rather easy to handle because matching involves only a single isolated object. As described in Sec. IV.2, these self-interaction terms can be thought of as homogeneous solutions that have support only in the IZ. As such, in the small-coupling approximation, they satisfy homogenous field equations that take Einstein form. If the spin of the object is neglected, the only possible linear perturbation to such a homogeneous solution that is compatible with asymptotic flatness is a shift of the body’s mass (in the $1/r$ piece of the $(t,t)$ and diagonal parts of the metric). In essence, this is a consequence of Birkhoff’s theorem, which holds for homogeneous solutions. Such a shift is consistent with the strong-field, non-spinning BH solution in EDGB theory found in Yunes and Stein (2011). In that case, the mass shift is simply $m_{A}\to(49/80)\zeta_{3}m_{A}$.

For spinning objects, one expects there to be higher multipole moments in the strong-field solution. However, one should be able to absorb current dipole moment modifications by a redefinition of the spin parameter, while the mass dipole moment will be absorbed by the redefinition of the position of the center of mass. Therefore, the leading-order corrections that survive are the mass quadrupole moment, which produces a metric perturbation in the NZ proportional to $1/r^{3}$. As we will see, when we consider FZ solution, there is an additional factor of $v^{2}$ that enters.

Therefore, contributions to the energy flux from the quadrupole or higher multipole moments are at least 3PN order relative to that from the GR quadrupole formula. We will later find that corrections to the energy flux due to scalar radiation appear at -1PN and 2PN relative order for the even and odd-parity cases, respectively. Hence, the contributions from the multipole moments that we discussed here are definitely smaller than those introduced by scalar radiation in the even-parity case, and at most, the same order in the odd-parity case.

Let us take a look at spinning BHs in the odd-parity sector in more detail. At ${\mathcal{O}}(\chi)$ there is freedom in adding a homogeneous solution proportional to $1/r^{2}$ in the $h_{0i}$ component. This corresponds to a freedom in shifting the Kerr parameter measured at infinity. Reference Yunes and Pretorius (2009a) set this homogeneous solution to zero so that there is no shift in the Kerr parameter. At $\mathcal{O}(\chi^{2})$, there should be corrections proportional to $1/r^{3}$ in $h_{ij}$ which shifts the quadrupole moment. Since there is no parameter in the Kerr geometry that can absorb this correction in the quadrupole moment, this $1/r^{3}$ correction cannot be eliminated.

The effective source term that reproduces this correction should look like

[TABLE]

where $Q_{A}=\mathcal{O}(\zeta_{4}m_{A}a_{A}^{2})$ and $\hat{S}_{A,k}\equiv S^{i}_{A}/m_{A}^{2}$ is a unit spin angular momentum vector. The solution of this wave equation at $\mathcal{O}(1/r)$ is given by

[TABLE]

The leading-order contributions at $m=0$ (2PN) and $m=1$ ($2.5$PN) vanish, leading to the first non-zero contribution at $m=2$

[TABLE]

which is 3PN relative to GR. Therefore, the self-interacting correction in the metric at $\mathcal{O}(\chi^{2})$ is smaller compared to the corrections in the energy flux carried by the scalar field and the metric field with regularized modification.

The cross-interaction part is more complicated. In this case, we have to consider the induced multipole moments due to the presence of the secondary object. Thus, even if we consider non-spinning objects, higher multipole moments might be induced. Another important difference is that neither the mass monopole nor the spin dipole can be simply absorbed by a redefinition of the mass and spin of each object. This is because the shifts of these multipole moments depend on the orbital parameters, such as separation $b$. Notice, however, that the effects of the secondary object propagate only through the scalar field or the gravitational tidal force.

The order of magnitude of the former scalar field effect is more complicated to estimate and it depends on the situation. In the even-parity case, $\vartheta$ sourced by the secondary body at the position of the primary body is proportional to $1/b$. In EDGB theory, since $\vartheta$ has shift symmetry within the context of the classical theory, the effects are suppressed by the gradient of the field, i.e. they are proportional to $1/b^{2}$. In the odd-parity case, there is again shift symmetry and the monopole scalar charge is absent. Because of these two reasons, the suppression is proportional to $1/b^{3}$ in CS theory. These suppressions will be sufficient to conclude that the effects are relatively at least 1PN and 3PN in the even and odd-parity cases respectively, which is smaller than the effects induced by scalar radiation.

In the odd-parity non-spinning case, the latter gravitational tidal force dominates over the scalar propagation effect. To calculate this tidal force properly requires asymptotic matching between the IZ solution and a strong-field, perturbed Schwarzschild solution in CS gravity. Perturbations of the Schwarzschild spacetime can be decomposed as a sum over electric and magnetic tidal tensors (see e.g. Poisson (2005)). The former scale as $1/b^{3}(1+v+v^{2}+\ldots)$, while the latter scales as $v/b^{3}(1+v+v^{2}+\ldots)$ Johnson-McDaniel et al. (2009). Such tidal deformations will induce gravitational waves that will scale as the second-time derivatives of the electric and magnetic quadrupole deformations, i.e. they will scale as $\omega^{2}/b^{3}(1+v+v^{2}+\ldots)$ and $\omega^{2}v/b^{3}(1+v+v^{2}+\ldots)$. In GR, the leading order effect is induced by the electric quadrupole moment and it scales as $\omega^{2}/b^{3}$, a 5PN order effect. In CS, we expect the magnetic quadrupole moment to provide the leading-order deformation, and at the level of energy flux, this couples to the GR metric perturbation produced by either the mass octupole or current dipole moment, leading to 6PN correction. This interpretation seems to be consistent with the results of Pani et al. Pani et al. (2011a).

VI Energy flux

The inspiral of a compact binary system is controlled by the system’s change in binding energy and angular momentum. The binding energy changes according to the dissipation of energy carried by all dynamical fields, which here includes the metric perturbation and the scalar field. The stress-energy tensor (SET) associated with each field quantifies the density and flux of energy and momentum. The energy loss is calculated as the integral of the energy flux through a 2-sphere of radius $r$ in the limit $r\to\infty$ and in the direction of the sphere’s outward unit normal $n^{i}$. That is, for some field $\varphi$ (be it $h_{ij}$, $\mathfrak{h}_{ij}$, or $\vartheta$) with SET $T^{(\varphi)}_{\mu\nu}$,

[TABLE]

where the angle brackets with subscript $\omega$ stand for orbit averaging.

The total energy flux can be ordered in powers of $\varsigma$ as

[TABLE]

The GR energy flux $\dot{E}^{{\mbox{\tiny GR}}}$ is given by the GR metric perturbation only, without any contributions from the scalar field at ${\cal{O}}(\varsigma^{0})$, as there is no scalar field in GR. For circular orbits, this is

[TABLE]

The $\mathcal{O}(\varsigma)$ correction, $\delta\dot{E}$, can be decomposed into

[TABLE]

where the first term is the scalar field contribution and the second term is the contribution of the deformed metric perturbation.

The scalar field contribution is calculated with the SET given by Eq. (7):

[TABLE]

Since we are taking the $r\to\infty$ limit, $\vartheta$ must be that valid in the FZ.

The metric deformation contribution to the energy flux is slightly more subtle. This modification to the GR flux can have three distinct sources: (i) the effective SET in terms of $h_{ij}$ and $\mathfrak{h}_{ij}$ may be functionally different, but as shown in Stein and Yunes (2011), this is not so for the class of theories we consider here777Reference Stein and Yunes (2011) showed that the TT gauge exists in quadratic gravity as $r\to\infty$. Any non-TT propagating mode that is sourced in the NZ vanishes in the FZ at all orders. This is in contrast to scalar-tensor theories in the Jordan frame, where the scalar “breathing” mode is present in the metric. This difference comes from the way the metric deformation and the scalar field couple in the field equations. In the quadratic gravity case, $\vartheta$ does not multiply $G_{\mu\nu}$ in the field equations (the Einstein-Hilbert sector of the action is unmodified), while the opposite is true in scalar-tensor theories in the Jordan frame. Therefore, in the former $\mathfrak{h}_{\mu\nu}$ and $\vartheta$ decouple in the $r\rightarrow\infty$ limit and there is no breathing mode. In contrast, in the latter the coupling between $\mathfrak{h}_{\mu\nu}$ and $\vartheta$ remains in the limit $r\rightarrow\infty$, leading to a non-vanishing breathing mode and a modification to the effective SET.; (ii) The orbital equations of motion, and the associated relations $m/b=v^{2}$ and $\omega=v^{3}/m$, might be modified at $\mathcal{O}(\varsigma)$, as was partially calculated in Yunes and Stein (2011); (iii) The generation mechanism of the FZ metric perturbation is modified, i.e. the radiative part of the metric perturbation is deformed. We consider here only the dissipative modifications introduced by (iii), as (ii) would require an analysis of the equations of motion, which is beyond the scope of this paper888 The distinction between (ii) and (iii) can be ambiguous at higher PN order, because how the orbital parameters are modified depends on the gauge choice. However, as long as we impose the harmonic gauge condition on both GR and the deformed metric perturbations, we do not have to worry about this gauge issue at least up to next-to-leading PN order. .

Letting $H_{\alpha\beta}=h_{\alpha\beta}+\varsigma\mathfrak{h}_{\alpha\beta}+{\cal{O}}(\varsigma^{2})$, the effective SET of GWs is given by Stein and Yunes (2011)

[TABLE]

where the angle brackets with a subscript $\lambda$ stand for a quasi-local average over several wavelengths and TT stands for the transverse-traceless projection

[TABLE]

with $P_{ij}=\delta_{ij}-n_{ij}$ the projector onto the plane perpendicular to the line from the source to a FZ field point. Expanding this SET in orders of $\varsigma$, the ${\cal{O}}(\varsigma^{0})$ part leads to $\dot{E}_{{\mbox{\tiny GR}}}$, while the ${\cal{O}}(\varsigma)$ part is

[TABLE]

which leads to

[TABLE]

As before, the $h_{\alpha\beta}$ and $\mathfrak{h}_{\alpha\beta}$ are those valid in the FZ.

VI.1 Scalar field correction to the energy flux

VI.1.1 Even-parity sector

In the even-parity case, $\vartheta^{{\mbox{\tiny FZ}}}$ is dominated by the dipole component [Eq. (76)], which we repeat here for convenience: $\vartheta^{{\mbox{\tiny FZ}}}=\dot{D}_{i}n^{i}/r$, where $D_{i}$ is the NZ dipole given in Eq. (77). This is inserted into the energy loss formula, Eq. (128). Since the FZ scalar field depends on retarded time, both time and spatial derivatives can be written as time derivatives of the NZ moments. This gives

[TABLE]

which for circular orbits gives

[TABLE]

Note that here, as before, the $m\to 0$ limit diverges, because the effective theory breaks down on short length scales and $\varsigma\ll 1$ is violated.

When the compact bodies are BHs, their scalar monopole charges are given by Eq. (65), $q_{A}=2\alpha_{3}/(\beta m_{A})$, which then leads to

[TABLE]

Comparing this with the GR energy flux, we find

[TABLE]

a relative -1PN effect. That is, the energy lost to the scalar field due to dipole radiation would enter as a lower-order in $v$ effect than the energy loss in GR. If one takes the limit $m_{2}\to\infty$ while keeping $(m_{1},v)$ fixed, then the above ratio scales as $m_{1}^{-4}$; i.e. the energy flux ratio is sensitive to the smallest horizon scale of the system. The effect is of a similar size for comparable stellar-mass binary and EMRI system. A SMBH-SMBH binary experiences the smallest effect.

VI.1.2 Odd-parity sector: spinning bodies

The scalar field $\vartheta^{{\mbox{\tiny FZ}}}$ is here dominated by the quadrupole component [Eq. (86)], which we repeat here for convenience $\vartheta^{{\mbox{\tiny FZ}}}=\ddot{\mu}_{ij}n^{ij}/r=-\omega^{2}\mu_{ij}n^{ij}/r$, where the quadrupole tensor $\mu_{ij}$ is defined in Eq. (85). Inserting this into the energy loss formula [Eq. (128)] gives

[TABLE]

Let us evaluate this for quasi-circular orbits with non-precessing spins. The third time derivative of the quadrupole tensor $\mu_{ij}$ becomes

[TABLE]

and the total energy flux is

[TABLE]

where $\hat{v}_{12}$ is the unit vector in the direction of the relative velocity and the dimensionless quantity $\bar{\Delta}$ is defined as

[TABLE]

Notice that $\delta\dot{E}^{(\vartheta)}$ in Eq. (139) is finite in the EMRI limit. Note also that when both spins are perpendicular to the orbital plane, $\bar{\Delta}$ is as well, and the second term of $\delta\dot{E}^{(\vartheta)}$ vanishes. Comparing Eq. (139) with GR,

[TABLE]

hence scalar radiation in the odd-parity sector is clearly a relative 2PN effect. This effect was not included in the work of Pani et al. Pani et al. (2011a), who found a $7$PN correction, since their simulations did not include spins. If one takes the limit $m_{2}\to\infty$ while keeping $(m_{1},v)$ fixed, then the above ratio scales as $m_{1}^{-2}m_{2}^{-2}$; i.e. the energy flux ratio is sensitive to the geometric mean of the two horizon scales in the system. This implies that the effect is greatest for comparable stellar-mass binaries.

VI.1.3 Odd-parity sector: non-spinning bodies

The odd-parity $\vartheta_{{\mbox{\tiny FZ}}}$ in Eq. (92) can be used to evaluate the energy loss in Eq. (128):

[TABLE]

Compared to the GW radiation in GR [Eq. (126)], this scalar radiation becomes

[TABLE]

which shows that this is a relative 7PN effect. In contrast with the cases of even-parity and odd-parity with spins, this effect is dominantly controlled by the total mass, rather than the mass ratio. The effect is greatest for a system of stellar-mass BHs.

The above result can be compared to numerical calculations recently performed by Pani et al. Pani et al. (2011a). They estimated the effect of scalar radiation in dynamical CS gravity Alexander and Yunes (2009) for non-spinning, circular EMRIs. They numerically solved the master perturbation equations on a Schwarzschild background to obtain the time evolution of the scalar field and the metric perturbation, caused by a non-spinning point particle. Figure 3 compares their results to ours, found in Eq. (143). Observe that the numerical results of Pani et al. are in excellent agreement with our post-Newtonian calculation, which extends it to comparable mass-ratios (notice the factor of $\delta m/m$).

VI.2 Metric deformation correction to the energy flux

For the even-parity case, the correction to the energy flux that arises from the deformation to the gravitational metric perturbation is at least of 0PN order relative to GR. This is higher PN order compared to the scalar dipole radiation found in Sec. VI.1.1, and thus, we will not consider it further.

For the odd-parity case with spinning BHs, one of the leading contribution comes from the metric correction sourced by $T_{ij}^{(\vartheta)}$, which is given in Eq. (LABEL:h-odd-a2). Inserting this metric perturbation into Eq. (132), the energy flux correction relative to GR becomes

[TABLE]

which is of relative 2PN order, just as the contribution due to scalar radiation in Eq. (141). Notice that both the metric deformation and scalar field corrections to the energy flux are of $\mathcal{O}(\chi^{2})$, but the latter is larger by a factor of ${\cal{O}}(\eta^{-1})$.

We expect $\mathcal{O}(\chi)$ corrections to the energy flux due to the metric deformation to be higher PN order. For very slowly spinning binaries, however, they may give larger corrections compared to the $\mathcal{O}(\chi^{2})$ 2PN ones presented here.

In the odd-parity sector with non-spinning objects, the regularized contributions to the metric deformation can only provide energy flux corrections of at least 7PN order. However, as explained in Sec. V.3, we expect that matching strong-field solutions to the non-regular NZ ones may generate 6PN corrections in the energy flux, similar to those found by Pani et al. Pani et al. (2011a).

VII Impact on gravitational wave phase

How do all these modifications to the energy flux affect the GW observable? To answer this question, we compute the Fourier transform of the phase of the GW response function in the stationary phase approximation (SPA), where we assume the GW phase changes much more rapidly than the GW amplitude Droz et al. (1999).

We begin by parameterizing all the corrections to the energy flux that we have studied so far via the following power law:

[TABLE]

where $(A,a)$ are summarized in Table 2 for the four different sectors considered.

With the generic energy flux parameterization, the orbital phase for a quasi-circular inspiral becomes

[TABLE]

where $F$ and $\omega=2\pi F$ are the linear and angular orbital frequency, $\phi_{\mathrm{GR}}=-1/(32\eta)(2\pi mF)^{-5/3}$ is the GR orbital phase and $E(\omega)=-(\mu/2)(m\omega)^{2/3}$ is the binary’s binding energy to Newtonian order. Recall here that $m=m_{1}+m_{2}$ is the total mass of the binary, while $\mu=m_{1}m_{2}/m$ is the reduced mass and $\eta=\mu/m$ is the symmetric mass ratio. Equation (146) is not valid when $a=5$ (a 2.5PN correction), as then the integrand becomes proportional to $\omega^{-1}$, which leads to a log term.

Before we compute the Fourier phase, we must first define $t_{0}$, the time at which the stationary phase condition is satisfied $F(t_{0})=f/2$, where $f$ is the GW frequency. This condition can be solved to yield

[TABLE]

where $t_{0,\mathrm{GR}}$ is the GR $t_{0}$. Again, this expression is not valid at $a=8$, because once more the correction to $t_{0}(f)$ would be a log term.

With this at hand, we can now compute the Fourier phase in the SPA:

[TABLE]

where $\Psi_{\mathrm{GR}}\equiv(3/128)(\pi\mathcal{M}f)^{-5/3}$, and where $\mathcal{M}=\eta^{3/5}m$ is the chirp mass. Again, these expressions are not valid when $a=5$ or $a=8$, for the reasons described above.

The corrections to the GW phase found here map directly to the parameterized post-Einsteinian (ppE) framework Yunes and Pretorius (2009b). In that framework, one postulates that modified gravity theories affect the Fourier phase of the GW response function in the SPA via

[TABLE]

where $(\beta_{\mbox{\tiny ppE}},b_{\mbox{\tiny ppE}})$ are ppE parameters. We see that this is identical to the corrections introduced by a change in the energy flux, with the mapping

[TABLE]

This is not surprising, as the ppE framework was in part motivated by studying power-law (in velocity) modifications to the energy flux and the binding energy Yunes and Pretorius (2009b).

We have then found that a large number of energy flux corrections associated with extra gravitational and scalar field emissions can be mapped to the ppE framework. In the even parity case, the leading-order frequency exponent $b_{{\mbox{\tiny ppE}}}=-7/3$, while in the odd-parity case $b_{{\mbox{\tiny ppE}}}=-1/3$, unless the binary is non-spinning in which case $b_{{\mbox{\tiny ppE}}}=+3$.

The results found in this paper could help in the generalization of the ppE framework to more generic quasi-circular inspirals. The original framework considered only non-spinning, equal mass inspirals, while recently Cornish et al. Cornish et al. (2011) generalized it to non-spinning, unequal mass systems through $A\to A\eta^{c}$. In this paper we have found that $A$ does not only depend on a simple power law of $\eta$, but also on the mass difference $\delta m/m=\sqrt{1-4\eta}$ and on combinations of the spins. For single detections, however, such a generalization is not needed as one only measures a single number, $\beta_{\mbox{\tiny ppE}}$, and one cannot extract the dependencies on $\eta$, $\delta m/m$, and the spins.

Although we currently lack any GW detections, we can still estimate the projected constraints that such detections would place on quadratic gravity. According to Table 2, the even-parity sector leads to the strongest deviations from GR, since $a$ is the most negative. Therefore, we consider EDGB theory, $(\alpha_{1},\alpha_{2},\alpha_{3},\beta)=(1,-4,1,\alpha_{\rm{EDGB}}^{-1})\alpha_{\mathrm{EDGB}}$, as a simple sub-case of the even-parity sector. Let us first imagine that we have detected a GW with Ad. LIGO and signal-to-noise ratio (SNR) of 20 that is consistent with GR and that originates from a non-spinning BH binary with masses $(m_{1},m_{2})=(6,12)M_{\odot}$. Given such a detection, Cornish et al. Cornish et al. (2011) estimated the projected bound $|\beta_{\mathrm{ppE}}|\lesssim 5\times 10^{-4}$ for $b_{\mathrm{ppE}}=-\frac{7}{3}$, which implies $|\alpha_{\mathrm{EDGB}}|^{1/2}\lesssim 4\times 10^{5}\;{\mathrm{cm}}$. Let us now assume that we have detected a GW with LISA classic with and SNR of $879$ and still consistent with GR, but that originates from a non-spinning BH binary with masses $(m_{1},m_{2})=(10^{6},3\times 10^{6})M_{\odot}$ at $z=1$. Given such a detection, Cornish et al. Cornish et al. (2011) estimated a bound on $|\beta_{\mathrm{ppE}}|\lesssim 10^{-6}$ for the same value of $b_{\mathrm{ppE}}$ as before, which leads to $\alpha_{\rm{EDGB}}^{1/2}\lesssim 10^{10}\;{{\rm cm}}$. In both cases, notice that these projected bounds are consistent with the small-coupling requirement $\zeta_{i}\ll 1$; i.e. saturating the projected Ad. LIGO and LISA constraints we have $\zeta_{\rm{Ad.~{}LIGO}}\sim 3\times 10^{-2}$ and $\zeta_{\rm{LISA}}\sim 10^{-5}$ for those particular binary systems, which is clearly much less than unity.

Comparing these results with the current constraint obtained by the Cassini satellite, $|\alpha_{\mathrm{EDGB}}|^{1/2}<1.3\times 10^{12}$cm Amendola et al. (2007), we see that Ad. LIGO and LISA could constrain $\alpha_{\mathrm{EDGB}}$ much more strongly. Unfortunately, it seems difficult to put constraints on EDGB with binary pulsar observations, since NSs have no scalar monopole charge in this theory. We emphasize again that this is opposite to the expectation from scalar-tensor theories, in which NSs have scalar monopole charges while BHs do not. Finally, one cannot estimate the bounds one could place on dynamical CS gravity, since one would have to properly account for modifications to the conservative equations of motion, which we have not calculated here.

VIII Conclusions and discussions

We have studied the binary inspiral problem in a wide class of quadratic gravity theories in the slow-motion, weak-gravity regime. The structure of a compact object in such theories affects the exterior scalar field sourced by the object. Despite this, we can model a compact object by an effective scalar field source characterized by its scalar monopole and dipole moments. The scalar monopole charge is enhanced inversely proportional to the mass of the object, while the dipole charge is independent of the mass for a fixed dimensionless spin parameter. With this effective source, we then derived and solved the modified field equations for the scalar field and metric deformation.

We find that the scalar field generically emits dipole radiation in the even-parity sector, and quadrupole radiation in the odd-parity sector. Such radiation affects the rate of change of the binary energy at relative $-1$PN order in the even-parity case and relative $2$PN order in the odd-parity case. The quadrupole contribution depends quadratically on the BH spins, and thus it is suppressed for non-spinning binaries. In that case, the odd-parity contribution becomes of relative $7$PN order, as found numerically in Pani et al. (2011a). We have found excellent agreement between their numerical results and our analytical calculations.

We have also calculated the metric perturbation in the FZ and its associated energy flux. In the even-parity sector, the dominant metric contribution leads to a 0PN relative correction in the energy flux, which is smaller than the -1PN correction induced by scalar dipolar radiation. In the odd-parity sector and for spinning BHs, the metric perturbation leads to a 2PN modification to the energy flux, which is of the same order as that induced by quadrupolar scalar radiation. In the odd-parity sector and for non-spinning BHs, we expect the energy flux correction due to the metric deformation is suppressed to at least of 6PN order, as found by Pani et al. Pani et al. (2011a).

Whether these corrections can be measured or constrained depends on whether they are degenerate with GR terms in the physical observable, i.e. the waveform. A $-1$PN effect cannot be degenerate, as there are no such terms predicted in GR. A $2$PN effect, however, could be degenerate with a spin-spin interaction for quasi-circular inspirals with aligned or counter-aligned spin components. That is, a renormalization of the spin magnitudes of both bodies can eliminate this $2$PN effect, assuming one truncates the waveform at that order. If higher-order PN waveforms are used, or if the orbit is more generic (i.e. if there is precession or eccentricity), then this degeneracy can be broken.

We also calculated the effects of such energy flux modifications on the gravitational waveform. The waveform phase depends sensitively on the rate of change of the orbital frequency, which in turn is governed by the rate of change of energy. We calculated the corrections that would be induced in the waveform and mapped them to the ppE framework. We then used a recent ppE study Cornish et al. (2011) to estimate the constraints that Ad. LIGO and LISA could potentially place on quadratic gravity theories. Given a GW detection, we found that the magnitude of the new length scale introduced by quadratic gravity theories (associated with a ratio of their coupling constants) could constrain at a level controlled by the smallest length-scale probed in the inspirals, i.e. the size of the smallest compact object’s event horizon or surface. The best projected bounds achievable with Ad. LIGO will thus come from stellar-mass BH or NS inspirals, while LISA will benefit the most from EMRIs. Since NSs have no scalar monopole charge in EDGB theory, this theory cannot be constrained from binary pulsar observations. This property is diametrically opposite to scalar-tensor theories where BHs have no hair.

There are several possible avenues for future work. Since we here mainly considered corrections due to the dissipative sector of the theory, one possibility is to calculate the non-dissipative corrections that would modify the binding energy (here, we mean both gravitational and scalar binding energy) and the equations of motion. There are two effects that should be accounted for: new scalar-scalar forces and metric deformations. Let us consider the former first. In the even parity case, compact objects have an associated scalar monopole charge, and thus, there is an additional scalar force with a $1/r$ potential that should lead to a relative 0PN non-dissipative correction. Similarly, in the odd-parity case, a spinning compact body possesses a current dipole charge, and hence, dipole-dipole interactions should arise. Since the dipole potential is proportional to $1/r^{2}$, while the dipole charge couples to the first derivative of the potential, the binding energy and the equations of motion should be corrected at relative 2PN order.

Another non-dissipative modification is induced by deformations of the background metric tensor. In the even-parity sector, such corrections enter at relative 0PN order, as found by Yunes and Stein Yunes and Stein (2011). In the odd-parity sector, there is no metric deformation for isolated non-spinning BHs, but for spinning ones there is a correction proportional to $r^{-4}$ to the $(t,i)$ components Yunes and Pretorius (2009a), which then leads to a $4.5$PN correction in the equations of motion when we consider boosted BHs. This then implies the following: (i) in the even-parity case, the conservative corrections to the equations of motion do not affect the leading-order modification to the waveforms, since this is dominated by the $-1$PN scalar radiation effect; (ii) in the odd-parity case, the conservative corrections from the metric deformation can be neglected, but those due to the scalar-scalar force will contribute at the same order as the effect calculated here. A complete analysis of the waveform observable would thus require the calculation of such a scalar-scalar, conservative effect.

Another possibility could be to study modified quadratic gravity in the context of BH perturbation theory. This would be a tremendous effort that would have to be split into separate parts. First, one would have to find an analytic, strong-field solution for arbitrarily-fast rotating BHs in quadratic gravity. This has only been found in the slow-rotation limit both in the even-parity Yunes and Stein (2011) and odd-parity sectors Yunes and Pretorius (2009a). Once this is accomplished, one would have to study the evolution of metric perturbations away from this solution. Such evolution equations would have to be decoupled in terms of some master function to derive Teukolsky-like master equations. Finally, with these equations at hand, one would have to solve them numerically, when the perturbations are sourced by a small object in a tight orbit. Such an analysis would be interesting because one would be able to derive not only the corrections to the energy flux carried out to infinity, but also that which is absorbed by the BH horizons and which we ignored in this paper.

A final follow-up would be to study how NS solutions are modified in quadratic gravity Pani et al. (2011b); Ali-Haimoud and Chen (2011) and how the energy flux from NS binaries is modified. This could then lead to direct constraints on quadratic gravity theories from double binary pulsar observations. Such constraints could be stronger , relative to current Solar System constraints, as they could potentially provide constraints of roughly the order of magnitude of the NS radius. Of course, in the case of EDGB theory or dynamical CS gravity, these constraints might not be stronger as NSs have no scalar monopole charge in such theories.

Acknowledgements.

We would like to thank Cliff Will, Eric Poisson and Luc Blanchet for invaluable teachings regarding the PN approximation. We also acknowledge Richard O’Shaughnessy, Yanbei Chen, Yacine Ali-Haïmoud, and Misao Sasaki for useful comments on the manuscript, and Paolo Pani, Vitor Cardoso and Leonardo Gualtieri for making some of their data accessible to us. NY thanks the Yukawa Institute for Theoretical Physics for their hospitality. KY is supported by the Japan Society for the Promotion of Science (JSPS) grant No. $22\cdot 900$. LCS acknowledges support from NSF Grant PHY-0449884 and from MIT’s Solomon Buchsbaum fund. NY acknowledges support from NSF grant PHY-1114374, as well as support provided by the National Aeronautics and Space Administration through Einstein Postdoctoral Fellowship Award Number PF0-110080, issued by the Chandra X-ray Observatory Center, which is operated by the Smithsonian Astrophysical Observatory for and on behalf of the National Aeronautics Space Administration under contract NAS8-03060. NY also acknowledges support from NASA grant NNX11AI49G, under sub-award 00001944. TT is supported by the Grant-in-Aid for Scientific Research (Nos. 21244033, 21111006 and 22111507). This work is also supported in part by the Grant-in-Aid for the Global COE Program “The Next Generation of Physics, Spun from Universality and Emergence” from the Ministry of Education, Culture, Sports, Science and Technology (MEXT) of Japan.

Appendix A The Balding of Neutron Stars in EDGB Gravity

In this appendix, we consider the scalar field equation in EDGB gravity for isolated NSs. Integrating the evolution equation, we find

[TABLE]

where we have defined the Gauss-Bonnet invariant ${\cal{R}}_{\rm GB}^{2}\equiv R^{2}-4R_{\mu\nu}R^{\mu\nu}+R_{\mu\nu\delta\sigma}R^{\mu\nu\delta\sigma}$. Since the Gauss-Bonnet combination is a topological invariant, the right-hand side identically vanishes for any simply-connected, asymptotically flat spacetime. Moreover, since we are considering isolated NSs, these must be stationary, and so the time integration can be removed.

With all of this and using Stokes’ theorem, Eq. (151) becomes

[TABLE]

where $n^{i}$ is the radial unit vector and the integral is performed over the 2-sphere at spatial infinity. Notice that $\sqrt{-g}\sim r^{2}$, while the scalar field must decay at infinity for it to have a finite energy.

Equation (152) does not vanish at spatial infinity for all scalar field solutions, i.e.  if we model $\vartheta=\vartheta_{n}/r$ with $\vartheta_{n}$ a constant, then Eq. (152) leads to the unique solution $\vartheta_{n}=0$. This is a physicists’s proof that the EDGB scalar field cannot have scalar monopole charge for a spherically symmetric NS. Similarly, one can show that NSs cannot have scalar monopole charge in dynamical CS gravity; the proof laid out above carries through with the replacement ${\cal{R}}_{\rm GB}^{2}\to{\,{}^{\ast}\!}R\,R$, since ${\,{}^{\ast}\!}R\,R$ is also a topological invariant.

Appendix B Integration techniques

In this appendix, we provide some useful integration techniques. When computing near-zone integrals, we are faced many times with integrals of the form

[TABLE]

When the point-particle approximation is valid, such near-zone integrals can be Hadamard regularized by keeping only the finite part. Let us then define Blanchet et al. (2002)

[TABLE]

to be evaluated in the near-zone and where ${\mathop{\mathrm{FP}}_{B=0}}$ stands for the finite part operator (in the limit $B\to 0$) and $|\tilde{\bm{x}}|$ is an analytic continuation factor Blanchet et al. (2002). The solution to this integral is

[TABLE]

The first few $Y_{\langle L\rangle}$ are simply

[TABLE]

The solution to the $Y_{\langle L\rangle}$ integral can also be derived by using certain Poisson integral identities Pati and Will (2000):

[TABLE]

where we have defined

[TABLE]

and the boundary term is

[TABLE]

As usual, we retain only those terms that are independent of the boundary $\mathcal{R}$.

Finally, there is yet another type of integral that commonly appears in near-zone integration:

[TABLE]

Let us then define the so-called triangle potential Pati and Will (2002)

[TABLE]

It is a bit of a miracle that the above integral has the closed-form solution $\mathcal{G}(\bm{x}_{A},\bm{x}_{B},\bm{x}_{C})=1-\ln\Delta(ABC)$, with $\Delta(ABC)\equiv|\bm{x}_{A}-\bm{x}_{B}|+|\bm{x}_{B}-\bm{x}_{C}|+|\bm{x}_{C}-\bm{x}_{A}|$.

One can show that the triangle potential satisfies a set of relations, including Pati and Will (2002)

[TABLE]

and more generally

[TABLE]

where $\mathcal{G}(ABC)\equiv\mathcal{G}(\bm{x}_{A},\bm{x}_{B},\bm{x}_{C})$.

Appendix C Odd-Parity, Non-Spinning,

Regularized Contribution in the Metric Correction

We consider here the odd-parity sector for non-spinning binaries, where, for the scalar field, the magnetic-type dipole moment vanishes, $\mu_{A}^{i}=0$, since $\chi_{A}=0$. For the regularized contribution, we only need to consider the cross-interaction terms since the isolated non-spinning BH solution in the odd-parity case is simply the Schwarzschild metric. The $\tilde{K}^{(1)}_{ij}$ source term gives the largest contribution and one is then left only with the pseudo-scalar generated by interaction terms, as given in Eq. (63).

The metric deformation is given by Eq. (97), the $m=0$ piece of which can be split as in Eqs. (112)-(115). Before tackling each of these terms separately, let us point out that many of them identically vanish. For example, one of the contribution in Eq. (112) is proportional to

[TABLE]

It is critical in this calculation and in the calculations that follow to replace the $x^{i}$ derivatives by particles derivatives, i.e. derivatives with respect to $x_{1}^{i}$ and $x_{2}^{i}$.

Let us then tackle the first contribution to the dissipative metric deformation. Equations (112)-(115) can then be rewritten as

[TABLE]

where we have defined

[TABLE]

and

[TABLE]

with $A,B,C$ denoting the multi-index lists. We provide a more detailed discussion of $J$ tensors in Appendix D. One can then show through explicit computation that the two terms combine to give $I_{1ij}+I_{2ij}=0$, $I_{3ijn}+I_{4ijn}=0$, $I_{5ijn}+I_{6ijn}=0$, and $I_{7ijn}+I_{8ijn}=0$. Therefore $\mathfrak{h}_{ij}^{(1\cdots 4)}=0$ at leading order.

Let us now look at contributions that are smaller by $\mathcal{O}(v)$. Such a correction can arise from two different terms: (i) the $\mathcal{O}(v)$ correction to the source term with $m=0$ in the sum of Eq. (97), or (ii) the $\mathcal{O}(v^{0})$ correction to the source term with $m=1$ in the sum of Eq. (97). For case (i), the next-order terms consist of two time derivatives and one factor of $h_{0i}$ (or three time derivatives and one factor of $h_{ij}$), which when combined are $\mathcal{O}(v^{2})$ smaller than the $\mathcal{O}(v^{0})$ contribution shown to vanish previously. Also, the next-order terms in the PN metric appears at $\mathcal{O}(v^{2})$ higher relative to the leading-order terms. Finally, $\vartheta^{{\mbox{\tiny NZ}}}$ in Eq. (56) expanded as in Eq. $(2.27)$ of Will and Wiseman (1996) with $m=1$ in the sum, gives an $\mathcal{O}(v)$ relative contribution to $\partial_{k}{\vartheta}$, but explicit calculation shows that

[TABLE]

For case (ii), the resulting $\dot{\mathfrak{h}}_{ij}$ contains one $n^{i}$ vector. The correction to the energy flux consists of $\dot{\mathfrak{h}}_{ij}$ multiplied by $h^{{\mbox{\tiny TT}}}_{ij}$ and averaged over a 2-sphere. However, since the leading contribution in $h^{{\mbox{\tiny TT}}}_{ij}$ contains even numbers of $n^{i}$ vectors, the correction only contains angular integrals of odd numbers of $n^{i}$’s which vanish exactly upon integration.

Since there is no $\mathcal{O}(G^{3},v)$ relative contribution to $\partial_{t}{\mathfrak{h}}_{ij}$, the first, non-vanishing contribution must be at least $\mathcal{O}(v^{2})$ smaller than what we computed in Eqs. (167)-(170), which amounts to a 7PN correction to the energy flux carried by the metric deformation, in the odd-parity, non-spinning case.

Appendix D Evaluating $J$ tensors

Recall that the definition of the $J$ tensors is

[TABLE]

The limit $3\to p$ which appears must be taken with care. There may be terms proportional to

[TABLE]

which have no finite part. In the evaluation of the $J$ tensors, only the finite part of the limit is kept. That is, a function can be expanded as a Laurent series about these points, and the finite part scales as $(r_{p3})^{0}$ in the limit as $3\to p$.

Another type of problematic limit is

[TABLE]

which does not formally exist, since it depends on the path taken as we describe below. Parameterize the path that particle 3 takes to the location of particle $p$ by the continuously differentiable path $\gamma(\lambda)$, with $\lambda$ a parameter of path length and $\lambda=0$ the location of particle $p$. There are an infinite number of paths one could choose, and each can be parameterized in two senses. Taking the limit along this path “from below” (i.e. from smaller values of $\lambda$ to larger values) yields

[TABLE]

where $\hat{v}_{\gamma}$ is the tangent vector to the curve $\gamma$. Taking the limit from above, we find

[TABLE]

The limit depends on the path’s tangent at the point of particle $p$, and the direction in which the limit is taken. Clearly, the final answer must be unique, which implies the limit must vanish.

A unique prescription to this problem is formalized as Hadamard regularization Blanchet and Iyer (2005). This can be summarized as follows. All possible paths are considered, with tangent vectors $\hat{v}_{\gamma}$. The average is then taken by integrating, e.g.

[TABLE]

The first few such limits, for example, are

[TABLE]