5 Compact Binary Tests
In this section, we discuss gravitational wave tests of GR with signals emitted by compact binary systems. We begin by explaining the difference between direct and generic tests. We then proceed to describe the many direct or topdown tests and generic or bottomup tests that have been proposed once gravitational waves are detected, including tests of the nohair theorems. We concentrate here only on binaries composed of compact objects, such as neutron stars, black holes or other compact exotica. We will not discuss tests one could carry out with electromagnetic information from binary (or double) pulsars, as these are already described in [438*]. We will also not review tests of GR with accretion disk observations, for which we refer the interested reader to [359*].
5.1 Direct and generic tests
Gravitationalwave tests of Einstein’s theory can be classed into two distinct subgroups: direct tests and generic tests. Direct tests employ a topdown approach, where one starts from a particular modified gravity theory with a known action, derives the modified field equations and solves them for a particular gravitational waveemitting system. On the other hand, generic tests adopt a bottomup approach, where one takes a particular feature of GR and asks what type of signature its absence would leave on the gravitationalwave observable; one then asks whether the data presents a statisticallysignificant anomaly pointing to that particular signature.
Direct tests have by far been the traditional approach to testing GR with gravitational waves. The prototypical examples here are tests of Jordan–Fierz–Brans–Dicke theory. As described in Section 2, one can solve the modified field equations for a binary system in the postNewtonian approximation to find a prediction for the gravitationalwave observable, as we will see in more detail later in this section. Other examples of direct tests include those concerning modified quadratic gravity models and noncommutative geometry theories.
The main advantage of such direct tests is also its main disadvantage: one has to pick a particular modified gravity theory. Because of this, one has a welldefined set of field equations that one can solve, but at the same time, one can only make predictions about that modified gravity model. Unfortunately, we currently lack a particular modified gravity theory that is particularly compelling; many modified gravity theories exist, but none possess all the criteria described in Section 2, except perhaps for the subclass of scalartensor theories with spontaneous scalarization. Lacking a clear alternative to GR, it is not obvious which theory one should pick. Given that the full development (from the action to the gravitational wave observable) of any particular theory can be incredibly difficult, time and computationally consuming, carrying out direct tests of all possible modified gravity models once gravitational waves are detected is clearly unfeasible.
Given this, one is led to generic tests of GR, where one asks how the absence of specific features contained in GR could impact the gravitational wave observable. For example, one can ask how such an observable would be modified if the graviton had a mass, if the gravitational interaction were Lorentz or parity violating, or if there existed large extra dimensions. From these general considerations, one can then construct a “meta”observable, i.e., one that does not belong to a particular theory, but that interpolates over all known possibilities in a welldefined way. This model has come to be known as the parameterized postEinsteinian framework, in analogy to the parameterized postNewtonian scheme used to test GR in the solar system [438*]. Given such a construction, one can then ask whether the data points to a statisticallysignificant deviation from GR.
The main advantage of generic tests is precisely that one does not have to specify a particular model, but instead one lets the data select whether it contains any statisticallysignificant deviations from our canonical beliefs. Such an approach is, of course, not new to physics, having most recently been successfully employed by the WMAP team [57]. The intrinsic disadvantage of this method is that, if a deviation is found, there is no onetoone mapping between it and a particular action, but instead one has to point to a class of possible models. Of course, such a disadvantage is not that limiting, since it would provide strong hints as to what type of symmetries or properties of GR would have to be violated in an ultraviolet completion of Einstein’s theory.
5.2 Direct tests
5.2.1 Scalartensor theories
Let us first concentrate on Jordan–Fierz–Brans–Dicke theory, where black holes and neutron stars have been shown to exist. In this theory, the gravitational mass depends on the value of the scalar field, as Newton’s constant is effectively promoted to a function, thus leading to violations of the weakequivalence principle [160, 434, 441*]. The usual prescription for the modeling of binary systems in this theory is due to Eardley [160].^{8} He showed that such a scalarfield effect can be captured by replacing the constant inertial mass by a function of the scalar field in the distributional stressenergy tensor and then Taylor expanding about the cosmological constant value of the scalar field at spatial infinity, i.e.,
where the subscript stands for different sources, while and the sensitivities and are defined by where we remind the reader that , the derivatives are to be taken with the baryon number held fixed and evaluated at . These sensitivities encode how the gravitational mass changes due to a nonconstant scalar field; one can think of them as measuring the gravitational binding energy per unit mass. The internal gravitational field of each body leads to a nontrivial variation of the scalar field, which then leads to modifications to the gravitational binding energies of the bodies. In carrying out this expansion, one assumes that the scalar field takes on a constant value at spatial infinity , disallowing any homogeneous, cosmological solution to the scalar field evolution equation [Eq. (19*)].With this at hand, one can solve the massless Jordan–Fierz–Brans–Dicke modified field equations [Eq. (19*)] for the nondynamical, nearzone field of compact objects to obtain [441*]
where runs from 1 to , we have defined the spatial field point distance , the parameterized postNewtonian quantity and we have chosen units in which . This solution is obtained in a postNewtonian expansion [75*], where the ellipses represent higherorder terms in and . From such an analysis, one can also show that compact objects follow geodesics of such a spacetime, to leading order in the postNewtonian approximation [160], except that Newton’s constant in the coupling between matter and gravity is replaced by , in geometric units.
As is clear from the above analysis, blackhole and neutronstar solutions in this theory generically depend on the quantities and . The former determines the strength of the correction, with the theory reducing to GR in the limit [164]. The latter depends on the compact object that is being studied. For neutron stars, this quantity can be computed as follows. First, neglecting scalar corrections to neutronstar structure and using the Tolman–Oppenheimer–Volkoff equation, one notes that the mass , for a fixed equation of state and central density, with the total baryon number. Thus, using Eq. (126*), one has that
where the derivative is to be taken holding fixed. In this way, given an equation of state and central density, one can compute the gravitational mass as a function of baryon number, and from this, obtain the neutron star sensitivities. Eardley [160], Will and Zaglauer [441], and Zaglauer [474*] have shown that these sensitivities are always in the range for a soft equation of state and for a stiff one, in both cases monotonically increasing with mass in . Recently, Gralla [202] has found a more general method to compute sensitivities is generic modified gravity theories.What is the sensitivity of black holes in generic scalartensor theories? Will and Zaglauer [474*] have argued that the nohair theorems require for all black holes, no matter what their mass or spin is. As already explained in Section 2, stationary black holes that are the byproduct of gravitational collapse (i.e., with matter that satisfies the energy conditions) in a general class of scalartensor theories are identical to their GR counterparts [224, 408, 159, 398].^{9} This is because the scalar field satisfies a free wave equation in vacuum, which forces the scalar field to be constant in the exterior of a stationary, asymptoticallyflat spacetime, provided one neglects a homogeneous, cosmological solution. If the scalar field is to be constant, then by Eq. (127), for a single blackhole spacetime.
Such an argument formally applies only to stationary scenarios, so one might wonder whether a similar argument holds for binary systems that are in a quasistationary arrangement. Will and Zaglauer [474] and Mirshekari and Will [315] extended this discussion to quasistationary spacetimes describing blackhole binaries to higher postNewtonian order. They argued that the only possible deviations from are due to tidal deformations of the horizon due to the companion, which are known to arise at very high order in postNewtonian theory, . Recently, Yunes et al. [465*] extended this argument further by showing that to all orders in postNewtonian theory, but in the extreme massratio limit, black holes cannot have scalar hair in generic scalartensor theories. Finally, Healy et al. [230] have carried out a full numerical simulation of the nonlinear field equations, confirming this argument in the full nonlinear regime.
The activation of dynamics in the scalar field for a vacuum spacetime requires either a nonconstant distribution of initial scalar field (violating the constant cosmological scalar field condition at spatial infinity) or a pure geometrical source to the scalar field evolution equation. The latter would lead to the quadratic modified gravity theories discussed in Section 2.3.3. As for the former, Horbatsch and Burgess [235] have argued that if, for example, one lets , which clearly satisfies in a Minkowski background,^{10} then a Schwarzschild black hole will acquire modifications that are proportional to . Alternatively, scalar hair could also be induced by spatial gradients in the scalar field [67], possibly anchored in matter at galactic scales. Such cosmological hair, however, is likely to be suppressed by a long time scale; in the example above must have units of inverse time, and if it is to be associated with the expansion of the universe, then it would be natural to assume , where is the Hubble parameter. Therefore, although such cosmological hair might have an effect on black holes in the early universe, it should not affect black hole observations at moderate to low redshifts.
Scalar field dynamics can be activated in nonvacuum spacetimes, even if initially the stars are not scalarized provided one considers a more general scalartensor theory, like the one introduced by Damour and EspositoFarèse [129*, 130*]. As discussed in Section 2.3.1, when the conformal factor takes on a particular functional form, nonlinear effects induced when the gravitational energy exceeds a certain threshold can spontaneously scalarize merging neutron stars, as demonstrated recently by Barausse, et al [51*]. Therefore, neutron stars in binaries are likely to have hair in generic scalartensor theories, even if they start their inspiral unscalarized.
What do gravitational waves look like in Jordan–Fierz–Brans–Dicke theory? As described in Section 2.3.1, both the scalar field perturbation and the new metric perturbation satisfy a sourced wave equation [Eq. (19*)], whose leadingorder solution for a twobody inspiral is [436*]
where is the distance to the detector, is a unit vector pointing toward the detector, is the magnitude of relative position vector , with the trajectory of body , is the reduced mass and is the total mass, is the relative velocity vector and we have defined the shorthands
We have also introduced multiindex notation here, such that . Such a solution is derived using the Lorenz gauge condition and in a postNewtonian expansion, where we have left out subleading terms of relative order or .
Given the new metric perturbation , one can reconstruct the gravitational wave metric perturbation, and from this, the response function, associated with the quasicircular inspiral of compact binaries. After using Kepler’s third law to simplify expressions [, where is the orbital angular frequency and is the total mass and is the orbital separation], one finds for a groundbased Lshaped detector [102*]:
where is the symmetric mass ratio, is the chirp mass, is the inclination angle, and where we have used the beampattern functions in Eq. (58*). In Eq. (136) and henceforth, we linearize all expressions in . Jordan–Fierz–Brans–Dicke theory predicts the generic excitation of three polarizations: the usual plus and cross polarizations, and a breathing, scalar mode. We see that the latter contributes to the response at two, one and zero times the orbital frequency. One should note that all of these corrections arise during the generation of gravitational waves, and not due to a propagation effect. In fact, gravitational waves travel at the speed of light (and the graviton remains massless) in standard Jordan–Fierz–Brans–Dicke theory.
The quantities and are the orbital phase and frequency respectively, which are to be found by solving the differential equation
where the ellipses stand for higherorder terms in the postNewtonian approximation. In this expression, and henceforth, we have kept only the leadingorder dipole term and all known postNewtonian, GR terms. If one wished to include higher postNewtonian–order BransDicke terms, one would have to include monopole contributions as well as postNewtonian corrections to the dipole term. The first term in Eq. (137*) corresponds to dipole radiation, which is activated by the scalar mode. That is, the scalar field carries energy away from the system modifying the energy balance law to [436*, 379*, 440*] where the ellipses stand again for higherorder terms in the postNewtonian approximation. Solving the frequency evolution equation perturbatively in , one findswhere we have defined . In deriving these equations, we have neglected the last term in Eq. (137*), as this is a constant that can be reabsorbed into the chirp mass. Notice that since the two definitions of chirp mass differ only by a term of , the first term of Eq. (137*) is not modified.
One of the main ingredients that goes into parameter estimation is the Fourier transform of the response function. This can be estimated in the stationaryphase approximation, for a simple, nonspinning, quasicircular inspiral. In this approximation, one assumes the phase is changing much more rapidly than the amplitude [56*, 125*, 153*, 457*]. One finds [102*]
where we have defined the amplitudes
and the Fourier phase
where the Brans–Dicke correction is kept only to leading order in and , while are postNewtonian GR coefficients (see, e.g., [265]). In writing the Fourier response in this way, we had to redefine the phase of coalescence via
where is the Kronecker delta and is the GR phase of coalescence (defined as an integration constant when the frequency diverges). Of course, in this calculation we have neglected amplitude corrections that arise purely in GR, if one were to carry out the postNewtonian approximation to higher order.
Many studies have been carried out to determine the level at which such corrections to the waveform could be measured or constrained once a gravitational wave has been detected. The first such study was carried out by Will [436*], who determined that given a LIGO detection at SNR of a blackhole/neutronstar nonspinning, quasicircular inspiral, one could constrain . Scharre and Will [379*] carried out a similar analysis but for a LISA detection with of a intermediatemass blackhole/neutronstar, nonspinning, quasicircular inspiral, and found that one could constrain . Such an analysis was then repeated by Will and Yunes [440*] but as a function of the classic LISA instrument. They found that the bound is independent of the LISA arm length, but inversely proportional to the LISA position noise error, if the position error noise dominates over laser shot noise. All such studies considered an angleaveraged signal that neglected the spin of either body, assumptions that were relaxed by Berti et al. [63*, 64]. They carried out MonteCarlo simulations over all signal sky positions that included spinorbit precession to find that the projected bound with LISA deteriorates to for the same system and SNR. This was confirmed and extended by Yagi et al. [450*], who in addition to spinorbit precession allowed for noncircular (eccentric) inspirals. In fact, when eccentricity is included, the bound deteriorates even further to . The same authors also found that similar gravitationalwave observations with the nextgeneration detector DECIGO could constrain . Similarly, for a nonspinning neutronstar/blackhole binary, the future groundbased detector, the Einstein Telescope (ET) [361], could place constraints about 5 times stronger than the Cassini bound, as shown in [38*].
All such projected constraints are to be compared with the current solar system bound of placed through the tracking of the Cassini spacecraft [73*]. Table 1 presents all such bounds for ease of comparison,^{11} normalized to an SNR of 10. As should be clear, it is unlikely that LIGO observations will be able to constrain better than current solar system bounds. In fact, even LISA would probably not be able to do better than the Cassini bound. Table 1 also shows that the inclusion of more complexity in the waveform seems to dilute the level at which can be constrained. This is because the inclusion of eccentricity and spin forces one to introduce more parameters in the waveform, without these modifications truly adding enough waveform complexity to break the induced degeneracies. One would then expect that the inclusion of amplitude modulation due to precession and higher harmonics should break such degeneracies, at least partially, as was found for massive blackhole binary [279, 280]. However, even then it seems reasonable to expect that only thirdgeneration detectors will be able to constrain beyond solarsystem levels.
Reference  Binary mass  Properties  
[73]  x  4  Solar system 
[436*]  0.1  LIGO, Fisher, Ang. Ave.  
circular, nonspinning  
[379]  24  LISA, Fisher, Ang. Ave.  
circular, nonspinning  
[440*]  20  LISA, Fisher, Ang. Ave.  
circular, nonspinning  
[63*]  0.7  LISA, Fisher, MonteCarlo  
circular, w/spinorbit  
[450*]  0.5  LISA, Fisher, MonteCarlo  
eccentric, spinorbit  
[451*]  160  DECIGO, Fisher, MonteCarlo  
eccentric, spinorbit  
[38]  10  ET, Fisher, Ang. Ave.  
circular, nonspinning  
The main reason that solarsystem constraints of Jordan–Fierz–Brans–Dicke theory cannot be beaten with gravitationalwave observations is that the former are particularly wellsuited to constrain weakfield deviations of GR. One might have thought that scalartensor theories constitute strongfield tests of Einstein’s theory, but this is not quite true, as argued in Section 2.3.1. One can see this clearly by noting that scalartensor theory predicts dipolar radiation, which dominates at low velocities over the GR prediction (precisely the opposite behavior that one would expect from a strongfield modification to Einstein’s theory).
However, one should note that all the above analysis considered only the inspiral phase of coalescence, usually truncating their study at the innermost stablecircular orbit. The merger and ringdown phases, where most of the gravitational wave power resides, have so far been mostly neglected. One might expect that an increase in power will be accompanied by an increase in SNR, thus allowing us to constrain further, as this scales with 1/SNR [262*]. Moreover, during merger and ringdown, dynamical strongfield gravity effects in scalartensor theories could affect neutron star parameters and their oscillations [395], as well as possibly induce spontaneous scalarization [51*]. All of these nonlinear effects could easily lead to a strengthening of projected bounds. However, to date no detailed analysis has attempted to determine how well one could constrain scalartensor theories using full information about the entire coalescence of a compact binary.
The subclass of scalartensor models described by Jordan–Fierz–Brans–Dicke theory is not the only type of model that can be constrained with gravitationalwave observations. In the extreme–massratio limit, for binaries consisting of a stellarmass compact object spiraling into a supermassive black hole, Yunes et al. [465*] have recently shown that generic scalartensor theories reduce to either massless or massive Jordan–Fierz–Brans–Dicke theory. Of course, in this case the sensitivities need to be calculated from the equations of structure within the full scalartensor theory. The inclusion of a scalar field mass leads to an interesting possibility: floating orbits [94*]. Such orbits arise when the small compact object experiences superradiance, leading to resonances in the scalar flux that can momentarily counteract the gravitationalwave flux, leading to a temporarilystalled orbit that greatly modifies the orbitalphase evolution. These authors showed that if an extreme massratio inspiral is detected with a template consistent with GR, this alone allows us to rule out a large region of phase space, where is the mass of the scalar (see Figure 1 in [465*]). This is because if such an inspiral had gone through a resonance, a GR template would be grossly different from the signal. Such bounds are dramatically stronger than the current most stringent bound and obtained from Cassini measurements of the Shapiro timedelay in the solar system [20*]. Even if resonances are not hit, Berti et al. [71] have estimated that secondgeneration groundbased interferometers could constrain the combination with the observation of gravitational waves from neutronstar/binary inspirals at an SNR of . These bounds can also be stronger than current constraints, especially for large scalar mass.
Lastly one should mention possible gravitationalwave constraints on other types of scalar tensor theories. Let us first consider Brans–Dicke type scalartensor theories, where the coupling constant is allowed to vary. Will [436] has argued that the constraints described in Table 1 go through, with the change
where . In the limit, this implies the replacement . Of course, this assumes that there is neither a potential nor a geometric source driving the evolution of the scalar field, and is not applicable for theories where spontaneous scalarization is present [129*].Another interesting scalartensor theory to consider is that studied by Damour and EspositoFarèse [129, 130]. As explained in Section 2.3.1, this theory is defined by the action of Eq. (14*) with the conformal factor . In standard Brans–Dicke theory, only mixed binaries composed of a black hole and a neutron star lead to large deviations from GR due to dipolar emission. This is because dipole emission is proportional to the difference in sensitivities of the binary components. For neutron–star binaries with similar masses, this difference is close to zero, while for black holes it is identically zero (see Eqs. (134) and (144)). However, in the theory considered by Damour and EspositoFarèse , when the gravitational energy is large enough, as in the very late inspiral, nonlinear effects can lead to drastic modifications from the GR expectation, such as spontaneous scalarization [51]. Unfortunately, most of this happens at rather high frequency, and thus, it is not clear whether such effects are observable with current groundbased detectors.
5.2.2 Modified quadratic gravity
Black holes exist in the classes of modified quadratic gravity that have so far been considered. In nondynamical theories (when and the scalarfields are constant, refer to Eq. (25)), Stein and Yunes [473*] have shown that all metrics that are Ricci tensor flat are also solutions to the modified field equations (see also [360*]). This is not so for dynamical theories, since then the field is sourced by curvature, leading to corrections to the field equations proportional to the Riemann tensor and its dual.
In dynamical Chern–Simons gravity, stationary and sphericallysymmetric spacetimes are still described by GR solutions, but stationary and axisymmetric spacetimes are not. Instead, they are represented by [466*, 272*]
with the scalar field where is the line element of the Kerr metric and we recall that in the notation of Section 2.3.3. These expressions are obtained in Boyer–Lindquist coordinates and in the smallrotation/smallcoupling limit to in [466*, 272] and to in [455]. The linearinspin corrections modify the framedragging effect and they are of 3.5 postNewtonian order. The quadraticinspin corrections modify the quadrupole moment, which induces 2 postNewtonianorder corrections to the binding energy. However, the stability of these black holes has not yet been demonstrated.In EinsteinDilatonGauss–Bonnet gravity, stationary and sphericallysymmetric spacetimes are described, in the smallcoupling approximation, by the line element [473*]
in Schwarzschild coordinates, where is the line element on the twosphere, is the Schwarzschild factor and we have definedwhile the corresponding scalar field is
This solution is not restricted just to EinsteinDilatonGauss–Bonnet gravity, but it is also the most general, stationary and sphericallysymmetric solution in quadratic gravity. This is because all terms proportional to are proportional to the Ricci tensor, which vanishes in vacuum GR, while the term does not contribute in spherical symmetry (see [473*] for more details). Linear slowrotation corrections to this solution have been found in [345]. Although the stability of these black holes has not yet been demonstrated, other dilatonic black hole solutions obtained numerically (equivalent to those in EinsteinDilatonGauss–Bonnet theory in the limit of small fields) [257] have been found to be stable under axial perturbations [258, 409, 343].Neutron stars also exist in quadratic modified gravity. In dynamical Chern–Simons gravity, the massradius relation remains unmodified to first order in the slowrotation expansion, but the moment of inertia changes to this order [469, 19*], while the quadrupole moment and the mass measured at spatial infinity change to quadratic order in spin [448*]. This is because the massradius relation, to first order in slowrotation, depends on the sphericallysymmetric part of the metric, which is unmodified in dynamical Chern–Simons gravity. In EinsteinDilatonGauss–Bonnet gravity, the massradius relation is modified [342*]. As in GR, these functions must be solved for numerically and they depend on the equation of state.
Gravitational waves are also modified in quadratic modified gravity. In dynamical Chern–Simons gravity, Garfinkle et al. [190] have shown that the propagation of such waves on a Minkowski background remains unaltered, and thus, all modifications arise during the generation stage. In EinsteinDilatonGauss–Bonnet theory, no such analysis of the propagation of gravitational waves has yet been carried out. Yagi et al. [447*] studied the generation mechanism in both theories during the quasicircular inspiral of comparablemass, spinning black holes in the postNewtonian and smallcoupling approximations. They found that a standard postNewtonian analysis fails for such theories because the assumption that black holes can be described by a distributional stressenergy tensor without any further structure fails. They also found that since black holes acquire scalar hair in these theories, and this scalar field is anchored to the curvature profiles, as black holes move, the scalar fields must follow the singularities, leading to dipole scalarfield emission.
During a quasicircular inspiral of spinning black holes in dynamical Chern–Simons gravity, the total gravitational wave energy flux carried out to spatial infinity (equal to minus the rate of change of a binary’s binding energy by the balance law) is modified from the GR expectation to leading order by [447*]
due to scalar field radiation and corrections to the metric perturbation that are of magnetictype, quadrupole form. In this equation, is the leadingorder GR prediction for the total energy flux, is the dimensionless Chern–Simons coupling parameter, is the magnitude of the relative velocity with unit vector , , where is the Kerr spin parameter of the th black hole and is the unit vector in the direction of the spin angular momentum, the unit vector points from body one to two, and the angle brackets stand for an average over several gravitational wave wavelengths. If the black holes are not spinning, then the correction to the scalar energy flux is greatly suppressed [447*] where we have defined the reduced mass difference . Notice that this is a 7 postNewtonian–order correction , instead of a 2 postNewtonian correction as in Eq. (153*). In the nonspinning limit, the dynamical Chern–Simons correction to the metric tensor induces a 6 postNewtonian–order correction to the gravitational energy flux [447*], which is consistent with the numerical results of [344].On the other hand, in EinsteinDilatonGauss–Bonnet gravity, the corrections to the energy flux are [447*]
which is a postNewtonian correction. This is because the scalar field behaves like a monopole (see Eq. (152*)), and when such a scalar monopole is dragged by the black hole, it emits electrictype, dipole scalar radiation. Any hairy black hole with monopole hair will thus emit dipolar radiation, leading to postNewtonian corrections in the energy flux carried to spatial infinity.Such modifications to the energy flux modify the rate of change of the binary’s binding energy through the balance law, , which in turn modify the rate of change of the gravitational wave frequency and phase, . For dynamical Chern–Simons gravity (when the spins are aligned with the orbital angular momentum) and for EinsteinDilatonGauss–Bonnet theory (in the nonspinning case), the Fourier transform of the gravitationalwave response function in the stationary phase approximation becomes [447*, 454*]
where is the Fourier transform of the response in GR, with the gravitational wave frequency and [447*, 454*]where we have defined the symmetric and antisymmetric spin combinations . We have here neglected any possible amplitude correction, but we have included both deformations to the binding energy and Kepler’s third law, in addition to changes in the energy flux, when computing the phase correction. However, in EinsteinDilatonGauss–Bonnet theory the binding energy is modified at higher postNewtonian order, and thus, corrections to the energy flux control the modifications to the gravitationalwave response function.
From the above analysis, it should be clear that the corrections to the gravitationalwave observable in quadratic modified gravity are always proportional to the quantity . Thus, any measurement that is consistent with GR will allow a constraint of the form , where is a number of order unity, and is the accuracy of the measurement. Solving for the coupling constants of the theory, such a measurement would lead to [390*]. Therefore, constraints on quadratic modified gravity will weaken for systems with larger characteristic mass. This can be understood by noticing that the corrections to the action scale with positive powers of the Riemann tensor, while this scales inversely with the mass of the object, i.e., the smaller a compact object is, the larger its curvature. Such an analysis then automatically predicts that LIGO will be able to place stronger constraints than LISAlike missions on such theories, because LIGO operates in the 100 Hz frequency band, allowing for the detection of stellarmass inspirals, while LISAlike missions operate in the mHz band, and are limited to supermassive blackholes inspirals.
How well can these modifications be measured with gravitationalwave observations? Yagi et al. [447*] predicted, based on the results of Cornish et al. [124*], that a skyaveraged LIGO gravitationalwave observation with SNR of 10 of the quasicircular inspiral of nonspinning black holes with masses would allow a constraint of , where we recall that . A similar skyaveraged, eLISA observation of a quasicircular, spinaligned blackhole inspiral with masses would constrain [447]. The loss in constraining power comes from the fact that the constraint on will scale with the total mass of the binary, which is six orders of magnitude larger for spaceborne sources. These constraints are not stronger than current bounds from the existence of compact objects [342] () and from the change in the orbital period of the lowmass xray binary A0620–00 () [444], but they are independent of the nature of the object and sample the theory in a different energy scale. In dynamical Chern–Simons gravity, one expects similar projected gravitationalwave constraints on , namely , where is the total mass of the binary system in kilometers. Therefore, for binaries detectable with groundbased interferometers, one expects constraints of order . In this case, such a constraint would be roughly six orders of magnitude stronger than current LAGEOS bounds [19]. Dynamical Chern–Simons gravity cannot be constrained with binary pulsar observations, since the theory’s corrections to the postKeplerian observables are too high postNewtonian order, given the current observational uncertainties [448]. However, the gravitational wave constraint is more difficult to achieve in the dynamical Chern–Simons case, because the correction to the gravitational wave phase is degenerate with spin. However, Yagi et al. [454*] argued that precession should break this degeneracy, and if a signal with sufficiently high SNR is observed, such bounds would be possible. One must be careful, of course, to check that the smallcoupling approximation is still satisfied when saturating such a constraint [454].
5.2.3 Noncommutative geometry
Black holes exist in noncommutative geometry theories, as discussed in Section 2.3.5. What is more, the usual Schwarzschild and Kerr solutions of GR persist in these theories. This is not because such solutions have vanishing Weyl tensor, but because the quantity happens to vanish for such metrics. Similarly, one would expect that the twobody, postNewtonian metric that describes a blackhole–binary system should also satisfy the noncommutative geometry field equations, although this has not been proven explicitly. Similarly, although neutronstar spacetimes have not yet been considered in noncommutative geometries, it is likely that if such spacetimes are stationary and satisfy the Einstein equations, they will also satisfy the modified field equations. Much more work on this is still needed to establish all of these concepts on a firmer basis.
Gravitational waves exist in noncommutative gravity. Their generation for a compact binary system in a circular orbit was analyzed by Nelson et al., in [326*, 325*]. They began by showing that a transversetraceless gauge exists in this theory, although the transversetraceless operator is slightly different from that in GR. They then proceeded to solve the modified field equations for the metric perturbation [Eq. (42*)] via a Green’s function approach:
where recall that acts like a mass term, the integral is taken over the entire past light cone, is the Bessel function of the first kind, is the distance from the source to the observer and the quadrupole moment is defined as usual: where is the timetime component of the matter stressenergy tensor. Of course, this is only the first term in an infinite multipole expansion.Although the integral in Eq. (159*) has not yet been solved in the postNewtonian approximation, Nelson et al. [326*, 325*] did solve for its time derivative to find
where is the orbital angular frequency and we have defined
and one has assumed that the binary is in the  plane and the observer is on the axis. However, if one expands these expressions about , one recovers the GR solution to leading order, plus corrections that decay faster than . This then automatically implies that such modifications to the generation mechanism will be difficult to observe for sources at astronomical distances.
Given such a solution, one can compute the flux of energy carried by gravitational waves to spatial infinity. Stein and Yunes [400*] have shown that in quadratic gravity theories, this flux is still given by
where is the tracereversed metric perturbation, the integral is taken over a 2sphere at spatial infinity, and we recall that the angle brackets stand for an average over several wavelengths. Given the solution in Eq. (161), one finds that the energy flux is The asymptotic expansion of the term in between square brackets about is which then leads to an energy flux identical to that in GR, as any subdominant term goes to zero when the 2sphere of integration is taken to spatial infinity. In that case, there are no modifications to the rate of change of the orbital frequency. Of course, if one were not to expand about , then the energy flux would lead to certain resonances at , but the energy flux is only welldefined at future null infinity.The above analysis was used by Nelson et al. [326*, 325*] to compute the rate of change of the orbital period of binary pulsars, in the hopes of using this to constrain . Using data from the binary pulsar, they stipulated an orderofmagnitude constraint of . However, such an analysis could be revisited to relax a few assumptions used in [326*, 325*]. First, binary pulsar constraints on modified gravity theories require the use of at least three observables. These observables can be, for example, the rate of change of the period , the line of nodes and the perihelion shift . Any one observable depends on the parameters in GR or in noncommutative geometries, where are the component masses. Therefore, each observable corresponds to a surface of codimension one, i.e., a twodimensional surface or sheet in the threedimensional space . If the binary pulsar observations are consistent with Einstein’s theory, then all sheets will intersect at some point, within a certain uncertainty volume given by the observational error. The simultaneous fitting of all these observables is what allows one to place a bound on . The analysis of [326*, 325*] assumed that all binary pulsar observables were known, except for , but degeneracies between could potentially dilute constraints on these quantities. Moreover, this analysis should be generalized to eccentric and inclined binaries, since binary pulsars are known to not be on exactly circular orbits.
But perhaps the most important modification that ought to be made has to do with the calculation of the energy flux itself. The expression for in Eq. (164*) in terms of derivatives of the metric perturbation derives from the effective gravitationalwave stressenergy tensor, obtained by perturbatively expanding the action or the field equations and averaging over several wavelengths (the Isaacson procedure [241, 242]). In modified gravity theories, the definition of the effective stressenergy tensor in terms of the metric perturbation is usually modified, as found for example in [400*]. In the case of noncommutative geometries, Stein and Yunes [400] showed that Eq. (164*) still holds, provided one considers fluxes at spatial infinity. However, the analysis of [326*, 325*] evaluated this energy flux at a fixed distance, instead of taking the limit.
The balance law relates the rate of change of a binary’s binding energy with the gravitational wave flux emitted by the binary, but for it to hold, one must require the following: (i) that the binary be isolated and possess a welldefined binding energy; (ii) the total stressenergy of the spacetime satisfies a local covariant conservation law. If (ii) holds, one can use this conservation law to relate the rate of change of the volume integral of the energy density, i.e., the energy flux, to the volume integral of the current density, which can be rewritten as an integral over the boundary of the volume through Stokes’ theorem. Since in principle one can choose any integration volume, any physicallymeaningful result should be independent of the surface of that volume. This is indeed the case in GR, provided one takes the integration sphere to spatial infinity. Presumably, if one included all the relevant terms in , without taking the limit to , one would still find a result that is independent of the surface of this twosphere. However, this has not yet been verified. Therefore, the analysis of [326, 325] should be taken as an interesting first step toward understanding possible changes in the gravitationalwave metric perturbation in noncommutative geometries.
Not much beyond this has been done regarding noncommutative geometries and gravitational waves. In particular, one lacks a study of what the final response function would be if the gravitationalwave propagation were modified, which of course depends on the timeevolution of all propagating gravitationalwave degrees of freedom, and whether there are only the two usual dynamical degrees of freedom in the metric perturbation.
5.3 Generic tests
5.3.1 Massive graviton theories and Lorentz violation
Several massive graviton theories have been proposed to later be discarded due to ghosts, nonlinear or radiative instabilities. Thus, little work has gone into studying whether black holes and neutron stars in these theories persist and are stable, and how the generation of gravitational waves is modified. Such questions will depend on the specific massive gravity model considered, and of course, if a Vainshtein mechanism is employed, then there will not be any modifications.
However, a few generic properties of such theories can still be stated. One of them is that the nondynamical (nearzone) gravitational field will be corrected, leading to Yukawalike modifications to the gravitational potential [437*]
where is the distance from the source to a field point. For example, the latter parameterization arises in gravitational theories with compactified extra dimensions [261]. Such corrections lead to a fifth force, which then in turn allows us to place constraints on through solar system observations [404*]. Nobody has yet considered how such modifications to the nearzone metric could affect the binding energy of compact binaries and their associated gravitational waves.Another generic consequence of a graviton mass is the appearance of additional propagating degrees of freedom in the gravitational wave metric perturbation. In particular, one expects scalar, longitudinal modes to be excited (see, e.g., [148*]). This is, for example, the case if the action is of Pauli–Fierz type [169*, 148]. Such longitudinal modes arise due to the nonvanishing of the and Newman–Penrose scalars, and can be associated with the presence of spin0 particles, if the theory is of Type N in the classification [438*]. The specific form of the scalar mode will depend on the structure of the modified field equations, and thus, it is not possible to generically predict its associated contribution to the response function.
A robust prediction of massive graviton theories relates to how the propagation of gravitational waves is affected. If the graviton has a mass, its velocity of propagation will differ from the speed of light, as given for example in Eq. (23*). Will [437*] showed that such a modification in the dispersion relation leads to a correction in the relation between the difference in time of emission and arrival of two gravitons:
where is the redshift, is the graviton’s Compton wavelength, and are the emission frequencies of the two gravitons and is the distance measure where is the present value of the Hubble parameter, is the matter energy density and is the vacuum energy density (for a zero spatialcurvature universe).Even if the gravitational wave at the source is unmodified, the graviton time delay will leave an imprint on the Fourier transform of the response function by the time it reaches the detector [437*]. This is because the Fourier phase is proportional to
where is now not a constant but a function of frequency as given by Eq. (168*). Carrying out the integration, one finds that the Fourier transform of the response function becomes where is the Fourier transform of the response function in GR, we recall that and we have defined Such a correction is of 1 postNewtonian order relative to the leadingorder, Newtonian term in the Fourier phase. Notice also that there are no modifications to the amplitude at all.Numerous studies have considered possible bounds on . The most stringent solar system constraint is and it comes from observations of Kepler’s third law (mainly Mars’ orbit), which if the graviton had a mass would be modified by the Yukawa factor in Eq. (167*). Observations of the rate of decay of the period in binary pulsars [174*, 53] can also be used to place the more stringent constraint . Similarly, studies of the stability of Kerr black holes in Pauli–Fierz theory [169*] have yielded constraints of [88*]. Gravitationalwave observations of binary systems could also be used to constrain the mass of the graviton once gravitational waves are detected. One possible test is to compare the times of arrival of coincident gravitational wave and electromagnetic signals, for example in whitedwarf binary systems. Larson and Hiscock [281*] and Cutler et al. [126*] estimated that one could constrain with classic LISA. Will [437*] was the first to consider constraints on from gravitationalwave observations only. He considered skyaveraged, quasicircular inspirals and found that LIGO observations of equalmass black holes would lead to a constraint of with a Fisher analysis. Such constraints are improved to with classic LISA observations of , equalmass black holes. This increase comes about because the massive graviton correction accumulates with distance traveled (see Eq. (171*)). Since classic LISA would have been able to observe sources at Gpc scales with high SNR, its constraints on would have been similarly stronger than what one would achieve with LIGO observations. Will’s study was later generalized by Will and Yunes [440*], who considered how the detector characteristics affected the possible bounds on . They found that this bound scales with the squareroot of the LISA arm length and inversely with the square root of the LISA acceleration noise. The initial study of Will was then expanded by Berti et al. [63*], Yagi and Tanaka [450*], Arun and Will [39*], Stavridis and Will [399*] and Berti et al. [70*] to allow for non–skyaveraged responses, spinorbit and spinspin coupling, higher harmonics in the gravitational wave amplitude, eccentricity and multiple detections. Although the bound deteriorates on average for sources that are not optimally oriented relative to the detector, the bound improves when one includes spin couplings, higher harmonics, eccentricity, and multiple detections as the additional information and power encoded in the waveform increases, helping to break parameter degeneracies. However, all of these studies neglected the merger and ringdown phases of the coalescence, an assumption that was relaxed by Keppel and Ajith [262*], leading to the strongest projected bounds . Moreover, all studies until then had computed bounds using a Fisher analysis prescription, an assumption relaxed by del Pozzo et al. [142*], who found that a Bayesian analysis with priors consistent with solar system experiments leads to bounds stronger than Fisher ones by roughly a factor of two. All of these results are summarized in Table 2, normalizing everything to an SNR of 10. In summary, projected constraints on are generically stronger than current solar system or binary pulsar constraints by several orders of magnitude, given a LISA observation of massive blackhole mergers. Even an aLIGO observation would do better than current solar system constraints by a factor between a few [142*] to an order of magnitude [262*], depending on the source.
Reference  Binary mass  Properties  
[404]  x  0.0028  Solarsystem dynamics 
[174]  x  Binary pulsar orbital period  
in Visser’s theory [424]  
[88]  x  0.024  Stability of black holes 
in Pauli–Fierz theory [169]  
[437*]  0.006  LIGO, Fisher, Ang. Ave.  
circular, nonspinning  
[437]  69  LISA, Fisher, Ang. Ave.  
circular, nonspinning  
[281, 126]  0.03  LISA, WDWD, coincident  
with electromagnetic signal  
[440]  50  LISA, Fisher, Ang. Ave.  
circular, nonspinning  
[63]  10  LISA, Fisher, MonteCarlo  
circular, w/spinorbit  
[39]  10  LISA, Fisher, Ang. Ave.  
higherharmonics, circular, nonspinning  
[450]  22  LISA, Fisher, MonteCarlo  
eccentric, spinorbit  
[451]  2.4  DECIGO, Fisher, MonteCarlo  
eccentric, spinorbit  
[399]  50  LISA, Fisher, MonteCarlo  
circular, w/spin modulations  
[262]  400  LISA, Fisher, Ang. Ave.  
circular, nonspinning, w/merger  
[142]  0.006 – 0.014  LIGO, Bayesian, Ang. Ave.  
circular, nonspinning  
[70]  30  eLISA, Fisher, MonteCarlo  
multiple detections, circular, nonspinning  
Before proceeding, we should note that the correction to the propagation of gravitational waves due to a nonzero graviton mass are not exclusive to binary systems. In fact, any gravitational wave that propagates a significant distance from the source will suffer from the time delays described in this section. Binary inspirals are particularly useful as probes of this effect because one knows the functional form of the waveform, and thus, one can employ matched filtering to obtain a strong constraint. But, in principle, one could use gravitationalwave bursts from supernovae or other sources.
We have so far concentrated on massive graviton theories, but, as discussed in Section 2.3.2, there is a strong connection between such theories and Lorentz violation. Modifications to the dispersion relation are usually a result of a modification of the Lorentz group or its action in real or momentum space. For this reason, it is interesting to consider generic Lorentzviolatinginspired, modified dispersion relations of the form of Eq. (24*), or more precisely [316*]
where controls the structure of the modification and its amplitude. When and one recovers the standard modified dispersion relation of Eq. (23*). Eq. (173*) introduces a generalized time delay between subsequent gravitons of the form [316*] where we have defined , with Planck’s constant, and the generalized distance measure [316*] Such a modification then leads to the following correction to the Fourier transform of the response function [316*] where is the Fourier transform of the response function in GR and we have defined [316*] The case is special leading to the Fourier phase correction [316*] The reason for this is that when the Fourier phase is proportional to the integral of , which then leads to a natural logarithm.Different limits deserve further discussion here. Of course, when , one recovers the standard massive graviton result with the mapping . When , the dispersion relation is identical to that in Eq. (23*), but with a redefinition of the speed of light, and should thus be unobservable. Indeed, in this limit the correction to the Fourier phase in Eq. (176*) becomes linear in frequency, and this is 100% degenerate with the time of coalescence parameter in the standard GR Fourier phase. Finally, relative to the standard GR terms that arise in the postNewtonian expansion of the Fourier phase, the new corrections are of postNewtonian order. Then, if LIGO gravitationalwave observations were incapable of discerning between a 4 postNewtonian and a 5 postNewtonian waveform, then such observations would not be able to see the modified dispersion effect if . Mirshekari et al. [316] confirmed this expectation with a Fisher analysis of nonspinning, comparablemass quasicircular inspirals. They found that for , one can place very weak bounds on , namely with a LIGO observation of a neutron star inspiral, with an enhancedLISA or NGO observation of a blackhole inspiral, assuming a SNR of 10 and 100 respectively. A word of caution is due here, though, as these analyses neglect any Lorentzviolating correction to the generation of gravitational waves, including the excitation of additional polarization modes. One would expect that the inclusion of such effects would only strengthen the bounds one could place on Lorentzviolating theories, but this must be done on a theory by theory basis.
5.3.2 Variable G theories and large extra dimensions
The lack of a particular Lagrangian associated with variable theories, excluding scalartensor theories, or extra dimensions, makes it difficult to ascertain whether blackhole or neutronstar binaries exist in such theories. Whether this is so will depend on the particular variable model considered. In spite of this, if such binaries do exist, the gravitational waves emitted by such systems will carry some generic modifications relative to the GR expectation.
Most current tests of the variability of Newton’s gravitational constant rely on electromagnetic observations of massive bodies, such as neutron stars. As discussed in Section 2.3.4, scalartensor theories can be interpreted as variable theories, where the variability of is really a variation in the coupling between gravity and matter. However, Newton’s constant serves the more fundamental role of defining the relationship between geometry or length and energy, and such a relationship is not altered in most scalartensor theories, unless the scalar fields are allowed to vary on a cosmological scale (background, homogeneous scalar solution).
For this reason, one might wish to consider a possible temporal variation of Newton’s constant in pure vacuum spacetimes, such as in blackhole–binary inspirals. Such temporal variation would encode at the time and location of the merger event. Thus, once a sufficiently large number of gravitational wave events has been observed and found consistent with GR, one could reconstruct a constraint map that bounds along our past light cone (as a function of redshift and sky position). Since our pastlight cone with gravitational waves would have extended to roughly redshift with classic LISA (limited by the existence of merger events at such high redshifts), such a constraint map would have been much more complete than what one can achieve with current tests at redshift almost zero. Big Bang nucleosynthesis constraints also allow us to bound a linear drift in from to zero, but these become degenerate with limits on the number of relativistic species. Moreover, these bounds exploit the huge leverarm provided by integrating over cosmic time, but they are insensitive to local, oscillatory variations of with periods much less than the cosmic observation time. Thus, gravitationalwave constraint maps would test one of the pillars of GR: local position invariance. This principle (encoded in the equivalence principle) states that the laws of physics (and thus the fundamental constants of nature) are the same everywhere in the universe.
Let us then promote to a function of time of the form [468*]
where and are constants, and the subindex means that these quantities are evaluated at coalescence. Clearly, this is a Taylor expansion to first order in time and position about the coalescence event , which is valid provided the spatial variation of is much smaller than its temporal variation, i.e., , and the characteristic period of the temporal variation is longer than the observation window (at most, years for classic LISA), so that . Similar parameterization of have been used to study deviations from Newton’s second law in the solar system [149, 430, 427, 411]. Thus, one can think of this modification as the consequence of some effective theory that could represent the predictions of several different alternative theories.The promotion of Newton’s constant to a function of time changes the rate of change of the orbital frequency, which then directly impacts the gravitationalwave phase evolution. To leading order, Yunes et al. [468*] find
where is the rate of change of the orbital frequency in GR, due to the emission of gravitational waves and . Such a modification to the orbital frequency evolution leads to the following modification [468*] to the Fourier transform of the response function in the stationaryphase approximation [56, 125, 153, 457] where we recall again that and have defined the constant parameters [468*] to leading order in the postNewtonian approximation. We note that this corresponds to a correction of postNewtonian order in the phase, relative to the leadingorder term, and that the corrections are independent of the symmetric mass ratio, scaling only with the redshifted chirp mass . Due to this, one expects the strongest effects to be seen in lowfrequency gravitational waves, such as those one could detect with LISA or DECIGO/BBO.Given such corrections to the gravitationalwave response function, one can investigate the level to which a gravitationalwave observation consistent with GR would allow us to constrain . Yunes et al. [468*] carried out such a study and found that for comparablemass blackhole inspirals of total redshifted mass with LISA, one could constrain or better to redshift 10 (assuming an SNR of . Similar constraints are possible with observations of extreme massratio inspirals. The constraint is strengthened when one considers intermediatemass blackhole inspirals, where one would be able to achieve a bound of . Although this is not as stringent as the strongest constraints from other observations (see Section 2.3.4), we recall that gravitationalwave constraints would measure local variations at the source, as opposed to local variations at zero redshift or integrated variations from the very early universe.
The effect of promoting Newton’s constant to a function of time is degenerate with several different effects. One such effect is a temporal variability of the black hole masses, i.e., if . Such timevariation could be induced by gravitational leakage into the bulk in certain braneworld scenarios [255], as explained in Section 2.3.4. For a black hole of mass , the rate of black hole evaporation is given by
where is the size of the large extra dimension. As expected, such a modification to a blackhole–binary inspiral will lead to a correction to the Fourier transform of the response function that is identical in structure to that of Eq. (181*), but the parameters change to [449*] A similar expression is found for a neutronstar/blackhole inspiral, except that the dependent factor in between parenthesis is corrected.Given a gravitationalwave detection consistent with GR, one could then, in principle, place an upper bound on . Yagi et al. [449*] carried out a Fisher analysis and found that a 1year LISA detection would constrain with a binary inspiral at an SNR of 100. This constraint is roughly two orders of magnitude weaker than current tabletop experiment constraints [7]. Moreover, the constraint weakens somewhat for more generic inspirals, due to degeneracies between and eccentricity and spin. However, a similar observation with the third generation detector DECIGO/BBO should be able to beat current constraints by roughly one order of magnitude. Such a constraint could be strengthened by roughly one order of magnitude further, if one included the statistical enhancement in parameter estimation due to detection of order sources by DECIGO/BBO.
Another way to place a constraint on is to consider the effect of mass loss in the orbital dynamics [308*]. When a system loses mass, the evolution of its semimajor axis will acquire a correction of the form , due to conservation of specific orbital angular momentum. There is then a critical semimajor axis at which this correction balances the semimajor decay rate due to gravitational wave emission. McWilliams [308] argues that systems with are then gravitationalwave dominated and will thus inspiral, while systems with will be massloss dominated and will thus outspiral. If a gravitational wave arising from an inspiraling binary is detected at a given semimajor axis, then is automatically constrained to about . Yagi et al. [449] extended this analysis to find that such a constraint is weaker than what one could achieve via matched filtering with a waveform in the form of Eq. (181*), using the DECIGO detector.
The correction to the gravitationalwave phase evolution is also degenerate with cosmological acceleration. That is, if a gravitational wave is generated at highredshift, its phase will be affected by the acceleration of the universe. To zerothorder, the correction is a simple redshift of all physical scales. However, if one allows the redshift to be a function of time
then the observed waveform at the detector becomes structurally identical to Eq. (181*) but with the parameters However, using the measured values of the cosmological parameters from the WMAP analysis [271, 156], one finds that this effect is roughly times smaller than that of a possible correction at the level of the possible bounds quoted above [468]. Of course, if one could in the future constrain better by 3 orders of magnitude, possible degeneracies with would become an issue.A final possible degeneracy arises with the effect of a third body [463*], accretion disk migration [267*, 462*] and the interaction of a binary with a circumbinary accretion disk [229*]. All of these effects introduce corrections to the gravitationalwave phase of negative PN order, just like the effect of a variable gravitational constant. However, degeneracies of this type are only expected to affect a small subset of blackhole–binary observations, namely those with a third body sufficiently close to the binary, or a sufficiently massive accretion disk.
5.3.3 Parity violation
As discussed in Section 2.3.6 the simplest action to model parity violation in the gravitational interaction is given in Eq. (45*). Black holes and neutron stars exist in this theory, albeit nonrotating. A generic feature of this theory is that parity violation imprints onto the propagation of gravitational waves, an effect that has been dubbed amplitude birefringence. Such birefringence is not to be confused with optical or electromagnetic birefringence, in which the gauge boson interacts with a medium and is doublyrefracted into two separate rays. In amplitude birefringence, right (left)circularly polarized gravitational waves are enhanced or suppressed (suppressed or enhanced) relative to the GR expectation as they propagate [245, 295, 11*, 460*, 17, 464*].
One can understand amplitude birefringence in gravitational wave propagation due to a possible noncommutativity of the parity operator and the Hamiltonian. The Hamiltonian is the generator of time evolution, and thus, one can write [464*]
where is the gravitationalwave angular frequency, is time, and are the gravitational wave Fourier components with wavenumber . The quantity models possible background curvature effects, with for propagation on a Minkowski metric, and proportional to redshift for propagation on a Friedman–Robertson–Walker metric [277]. The quantity models possible parityviolating effects, with in GR. One can rewrite the above equation in terms of right and leftcircular polarizations, to find Amplitude birefringence has the effect of modifying the eigenvalues of the diagonal propagator matrix for right and leftpolarized waves, with right modes amplified or suppressed and left modes suppressed or enhanced relative to GR, depending on the sign of . In addition to these parityviolating propagation effects, parity violation should also leave an imprint in the generation of gravitational waves. However, such effects need to be analyzed on a theory by theory basis. Moreover, the propagationdistance–independent nature of generation effects should make them easily distinguishable from the propagation effects we consider here.The degree of parity violation, , can be expressed entirely in terms of the waveform observables via [464*]
where is the GR expectation for a right or leftpolarized gravitational wave. In the last equality we have also introduced the notation , where is the GR gravitationalwave phase and where is a constant factor, is the conformal wave number and are conformal coordinates for propagation in a Friedmann–Robertson–Walker universe. The precise form of will depend on the particular theory under consideration. For example, in nondynamical Chern–Simons gravity with a field , and in an expansion about , one finds [464*] where is the Chern–Simons scalar field at the detector, with the Chern–Simons coupling constant [see, e.g., Eq. (45*)], is redshift, is the comoving distance and is the value of the Hubble parameter today and is the observed gravitationalwave frequency. When considering propagation on a Minkowski background, one obtains the above equation in the limit as , so the second term dominates, where is the scale factor. To leadingorder in a curvature expansion, the parityviolating coefficient will always be linear in frequency, as shown in Eq. (191*). For more general parity violation and flatspacetime propagation, will be proportional to , where is a coupling constant of the theory (or a certain derivative of a coupling field) with units of (in the previous case, , so the correction was simply proportional to , where ).How does such parity violation affect the waveform? By using Eq. (188*) one can easily show that the Fourier transform of the response function becomes [11*, 460*, 464*]
Of course, one can rewrite this in terms of a real amplitude correction and a real phase correction. Expanding in to leading order, we find [464*] where is the Fourier transform of the response function in GR and we have definedWe see then that amplitude birefringence modifies both the amplitude and the phase of the response function. Using the nondynamical Chern–Simons expression for in Eq. (191*), we can rewrite Eq. (193*) as [464*]
where we have defined the coefficientswhere we recall that . The phase correction corresponds to a term of 5.5 postNewtonian order relative to the Newtonian contribution, and it scales quadratically with the Chern–Simons coupling field , which is why it was left out in [464*]. The amplitude correction, on the other hand, is of 1.5 postNewtonian order relative to the Newtonian contribution. Since both of these appear as positiveorder, postNewtonian corrections, there is a possibility of degeneracy between them and standard waveform template parameters.
Given such a modification to the response function, one can ask whether such parity violation is observable with current detectors. Alexander et al. [11*, 460*] argued that a gravitational wave observation with LISA would be able to constrain an integrated measure of , because LISA can observe massive–blackhole mergers to cosmological distances, while amplitude birefringence accumulates with distance traveled. For such an analysis, one cannot Taylor expand about its present value, and instead, one finds that
where we have definedWe can solve the above equation to find
where in the second equality we have linearized about and . Alexander et al. [11*, 460*] realized that this induces a timedependent change in the inclination angle (i.e., the apparent orientation of the binary’s orbital angular momentum with respect to the observer’s lineofsight), since the latter can be defined by the ratio . They then carried out a simplified Fisher analysis and found that a LISA observation of the inspiral of two massive black holes with component masses at redshift would allow us to constrain the integrated dimensionless measure to . One might worry that such an effect would be degenerate with other standard GR processes that induce similar timedependencies, such as spinorbit coupling. However, this timedependence is very different from that of the parityviolating effect, and thus, Alexander et al. [11, 460] argued that these effects would be weakly correlated.Another test of parity violation was proposed by Yunes et al. [464*], who considered the coincident detection of a gravitational wave and a gammaray burst with the SWIFT [193] and GLAST/Fermi [97] gammaray satellites, and the groundbased LIGO [2] and Virgo [6] gravitational wave detectors. If the progenitor of the gammaray burst is a neutronstar/neutronstar merger, the gammaray jet is expected to be highly collimated. Therefore, an electromagnetic observation of such an event implies that the binary’s orbital angular momentum at merger must be pointing along the line of sight to Earth, leading to a strongly–circularlypolarized gravitationalwave signal and to maximal parity violation. If the gammaray burst observation were to provide an accurate sky location, one would be able to obtain an accurate distance measurement from the gravitational wave signal alone. Moreover, since GLAST/Fermi observations of gammaray bursts occur at low redshift, one would also possess a purely electromagnetic measurement of the distance to the source. Amplitude birefringence would manifest itself as a discrepancy between these two distance measurements. Therefore, if no discrepancy is found, the error ellipse on the distance measurement would allow us to place an upper limit on any possible gravitational parity violation. Because of the nature of such a test, one is constraining generic parity violation over distances of hundreds of Mpc, along the light cone on which the gravitational waves propagate.
The coincident gammaray burst/gravitationalwave test compares favorably to the pure LISA test, with the sensitivity to parity violation being about 2 – 3 orders of magnitude better in the former case. This is because, although the fractional error in the gravitationalwave distance measurement is much smaller for LISA than for LIGO, since it is inversely proportional to the SNR, the parity violating effect also depends on the gravitationalwave frequency, which is much larger for neutronstar inspirals than massive blackhole coalescences. Mathematically, the simplest models of gravitational parity violation will lead to a signature in the response function that is proportional to the gravitationalwave wavelength^{12} . Although the coincident test requires small distances and low SNRs (by roughly 1 – 2 orders of magnitude), the frequency is also larger by a factor of 5 – 6 orders of magnitude for the LIGOVirgo network.
The coincident gammaray burst/gravitationalwave test also compares favorably to current solar system constraints. Using the motion of the LAGEOS satellites, Smith et al. [388] have placed the bound assuming . A similar assumption leads to a bound of with a coincident gammaray burst/gravitationalwave observation. Moreover, the latter test also allows us to constrain the second timederivative of the scalar field. Finally, a LISA observation would constrain the integrated history of along the past light cone on which the gravitational wave propagated. However, these tests are not as stringent as the recently proposed test by Dyda et al. [158], , assuming the effective theory cutoff scale is less than 10 eV and obtained by demanding that the energy density in photons created by vacuum decay over the lifetime of the universe not violate observational bounds.
The coincident test is somewhat idealistic in that there are certain astrophysical uncertainties that could hamper the degree to which we could constrain parity violation. One of the most important uncertainties relates to our knowledge of the inclination angle, as gammaray burst jets are not necessarily perfectly aligned with the line of sight. If the inclination angle is not known a priori, it will become degenerate with the distance in the waveform template, decreasing the accuracy to which the luminosity could be extracted from a pure gravitational wave observation by at least a factor of two. Even after taking such uncertainties into account, Yunes et al. [464] found that could be constrained much better with gravitational waves than with current solar system observations.
5.3.4 Parameterized postEinsteinian framework
One of the biggest disadvantages of a topdown or direct approach toward testing GR is that one must pick a particular theory from the beginning of the analysis. However, given the large number of possible modifications to Einstein’s theory and the lack of a particularly compelling alternative, it is entirely possible that none of these will represent the correct gravitational theory in the strong field. Thus, if one carries out a topdown approach, one will be forced to make the assumption that we, as theorists, know which modifications of gravity are possible and which are not [467*]. The parameterized postEinsteinian (ppE) approach is a framework developed specifically to alleviate such a bias by allowing the data to select the correct theory of nature through the systematic study of statistically significant anomalies.
For detection purposes, one usually expects to use match filters that are consistent with GR. But if GR happened to be wrong in the strong field, it is possible that a GR template would still extract the signal, but with the wrong parameters. That is, the best fit parameters obtained from a matched filtering analysis with GR templates will be biased by the assumption that GR is sufficiently accurate to model the entire coalescence. This fundamental bias could lead to a highly distorted image of the gravitationalwave universe. In fact, recent work by Vallisneri and Yunes [417*] indicates that such fundamental bias could indeed be present in observations of neutron star inspirals, if GR is not quite the right theory in the strongfield.
One of the primary motivations for the development of the ppE scheme was to alleviate fundamental bias, and one of its most dangerous incarnations: stealthbias [124*]. If GR is not the right theory of nature, yet all our future detections are of low SNR, we may estimate the wrong parameters from a matchedfiltering analysis, yet without being able to identify that there is a nonGR anomaly in the data. Thus, stealth bias is nothing but fundamental bias hidden by our limited SNR observations. Vallisneri and Yunes [417] have found that such stealthbias is indeed possible in a certain sector of parameter space, inducing errors in parameter estimation that could be larger than statistical ones, without us being able to identify the presence of a nonGR anomaly.
5.3.4.1 Historical development
The ppE scheme was designed in close analogy with the parameterized postNewtonian (ppN) framework, developed in the 1970s to test GR with solar system observations (see, e.g., [438] for a review). In the solar system, all direct observables depend on a single quantity, the metric, which can be obtained by a smallvelocity/weakfield postNewtonian expansion of the field equations of whatever theory one is considering. Thus, Will and Nordtvedt [331, 432, 439, 332, 433] proposed the generalization of the solar system metric into a metametric that could effectively interpolate between the predictions of many different alternative theories. This metametric depends on the product of certain Green function potentials and ppN parameters. For example, the spatialspatial components of the metametric take the form where is the Kronecker delta, is the Newtonian potential and is one of the ppN parameters, which acquires different values in different theories: in GR, in Jordan–Fierz–Brans–Dicke theory, etc. Therefore, any solar system observable could then be written in terms of system parameters, such as the masses of the planets, and the ppN parameters. An observation consistent with GR allows for a bound on these parameters, thus simultaneously constraining a large class of modified gravity theories.
The idea behind the ppE framework was to develop a formalism that allowed for similar generic tests but with gravitational waves instead of solar system observations. The first such attempt was by Arun et al. [37, 317*], who considered the quasicircular inspiral of compact objects. They suggested the waveform template family
This waveform depends on the standard system parameters that are always present in GR waveforms, plus one theory parameter that is to be constrained. The quantity is a number chosen by the data analyst and is restricted to be equal to one of the postNewtonian predictions for the phase frequency exponents, i.e., .The template family in Eq. (204*) allows for postNewtonian tests of GR, i.e., consistency checks of the signal with the postNewtonian expansion. For example, let us imagine that a gravitational wave has been detected with sufficient SNR that the chirp mass and mass ratio have been measured from the Newtonian and 1 postNewtonian terms in the waveform phase. One can then ask whether the 1.5 postNewtonian term in the phase is consistent with these values of chirp mass and mass ratio. Put another way, each term in the phase can be thought of as a curve in space. If GR is correct, all these curves should intersect inside some uncertainty box, just like when one tests GR with binary pulsar data. From that standpoint, these tests can be thought of as nulltests of GR and one can ask: given an event, is the data consistent with the hypothesis for the restricted set of frequency exponents ?
A Fisher and a Bayesian data analysis study of how well could be constrained given a certain was carried out in [317*, 240*, 290*]. Mishra et al. [317] considered the quasicircular inspiral of nonspinning compact objects and showed that aLIGO observations would allow one to constrain to 6% up to the 1.5 postNewtonian order correction (). Thirdgeneration detectors, such as ET, should allow for better constraints on all postNewtonian coefficients to roughly 2%. Clearly, the higher the value of , the worse the bound on because the power contained in higher frequency exponent terms decreases, i.e., the number of useful additional cycles induced by the term decreases as increases. Huwyler et al. [240] repeated this analysis but for LISA observations of the quasicircular inspiral of black hole binaries with spin precession. They found that the inclusion of precessing spins forces one to introduce more parameters into the waveform, which dilutes information and weakens constraints on by as much as a factor of 5. Li et al. [290*] carried out a Bayesian analysis of the oddsratio between GR and restricted ppE templates given a nonspinning, quasicircular compact binary inspiral observation with aLIGO and adVirgo. They calculated the odds ratio for each value of listed above and then combined all of this into a single probability measure that allows one to quantify how likely the data is to be consistent with GR.
5.3.4.2 The simplest ppE model
One of the main disadvantages of the postNewtonian template family in Eq. (204*) is that it is not rooted on a theoretical understanding of modified gravity theories. To alleviate this problem, Yunes and Pretorius [467*] reconsidered the quasicircular inspiral of compact objects. They proposed a more general ppE template family through generic deformations of the harmonic of the response function in Fourier space : where now are all free parameters to be fitted by the data, in addition to the usual system parameters. This waveform family reproduces all predictions from known modified gravity theories: when , the waveform reduces exactly to GR, while for other parameters one reproduces the modified gravity predictions of Table 3.
Theory 




Jordan–Fierz–Brans–Dicke 




Dissipative EinsteinDilatonGauss–Bonnet Gravity 




Massive Graviton 




Lorentz Violation 




Theory 




Extra Dimensions 




NonDynamical Chern–Simons Gravity 




Dynamical Chern–Simons Gravity 




In Table 3, recall that is the difference in the square of the sensitivities and is the Brans–Dicke coupling parameter (see Section 5.2.1; we have here neglected the scalar mode), is the coupling parameter in EinsteinDilatonGauss–Bonnet theory (see Section 5.2.2), where we have here included both the dissipative and the conservative corrections, is a certain distance measure and is the Compton wavelength of the graviton (see Section 5.3.1), is a distance scale at which Lorentzviolation becomes important and is the graviton momentum exponent in the deformation of the dispersion relation (see Section 5.3.1), is the value of the time derivative of Newton’s constant at coalescence and is the mass loss due to enhanced Hawking radiation in extradimensional scenarios (see Section 5.3.2), is given in Eq. (157) and are given in Eqs. (198) and (199) of Section 5.3.3.
Although there are only a few modified gravity theories where the leadingorder postNewtonian correction to the Fourier transform of the response function can be parameterized by postNewtonian waveforms of Eq. (204*), all such predictions can be modeled with the ppE templates of Eq. (205*). In fact, only massive graviton theories, certain classes of Lorentzviolating theories and dynamical Chern–Simons gravity lead to waveform corrections that can be parameterized via Eq. (204*). For example, the lack of amplitude corrections in Eq. (204*) does not allow for tests of gravitational parity violation or nondynamical Chern–Simons gravity.
However, this does not imply that Eq. (205*) can parameterize all possible deformations of GR. First, Eq. (205*) can be understood as a singleparameter deformation away from Einstein’s theory. If the correct theory of nature happens to be a deformation of GR with several parameters (e.g., several coupling constants, mass terms, potentials, etc.), then Eq. (205*) will only be able to parameterize the one that leads to the most useful cycles. This was recently verified by Sampson et al. [376*]. Second, Eq. (205*) assumes that the modification can be represented as a power series in velocity, with possibly noninteger values. Such an assumption does not allow for possible logarithmic terms, which are known to arise due to nonlinear memory interactions at sufficientlyhigh postNewtonian order. It also does not allow for interactions that are screened, e.g., in theories with massive degrees of freedom. Nonetheless, the parameterization in Eq. (205*) will still be able to signal that the detection is not a pure Einstein event, at the cost of biasing their true value.
The inspiral ppE model of Eq. (205*) is motivated not only from examples of modified gravity predictions, but from generic modifications to the physical quantities that drive the inspiral: the binding energy or Hamiltonian and the radiationreaction force or the fluxes of the constants of the motion. Yunes and Pretorius [467*] and Chatziioannou et al. [102*] considered generic modifications of the form
where , since otherwise one would lose analyticity in the limit of zero velocities for circular inspirals, and where are parameters that depend on the modified gravity theory and, in principle, could depend on dimensionless quantities like the symmetric mass ratio. Such modifications lead to the following corrections to the SPA Fourier transform of the timedomain response function for a quasicircular binary inspiral template (to leading order in the deformations and in postNewtonian theory)
Of course, usually one of these two modifications dominates over the other, depending on whether or . In Jordan–Fierz–Brans–Dicke theory, for example, the radiationreaction correction dominates as . If, in addition to these modifications in the generation of gravitational waves, one also allows for modifications in the propagation, one is then led to the following template family [102*]
Here and are ppE parameters induced by modifications to the generation and propagation of gravitational waves respectively, where still , while is fully determined by the former set via
if the modifications to the binding energy dominate, if the modifications to the energy flux dominate, or if both corrections enter at the same postNewtonian order. Noticing again that if only a single term in the phase correction dominates in the postNewtonian approximation (or both will enter at the same postNewtonian order), one can map Eq. (207) to Eq. (205*) by a suitable redefinition of constants.
5.3.4.3 More complex ppE models
Of course, one can introduce more ppE parameters to increase the complexity of the waveform family, and thus, Eq. (205*) should be thought of as a minimal choice. In fact, one expects any modified theory of gravity to introduce not just a single parametric modification to the amplitude and the phase of the signal, but two new functional degrees of freedom: where these functions will depend on the frequency , as well as on system parameters and theory parameters . In a postNewtonian expansion, one expects these functions to reduce to leadingorder on the lefthand sides of Eq.s (213*), but also to acquire postNewtonian corrections of the form
where here the structure of the series is assumed to be of the form with . Such a model, also suggested by Yunes and Pretorius [467*], would introduce too many new parameters that would dilute the information content of the waveform model. Recently, Sampson et al. [376*] demonstrated that the simplest ppE model of Eq. (205*) suffices to signal a deviation from GR, even if the injection contains three terms in the phase.
In fact, this is precisely one of the most important differences between the ppE and ppN frameworks. In ppN, it does not matter how many ppN parameters are introduced, because the observations are of very high SNR, and thus, templates are not needed to extract the signal from the noise. On the other hand, in gravitational wave astrophysics, templates are essential to make detections and do parameter estimation. Spurious parameters in these templates that are not needed to match the signal will deteriorate the accuracy to which all parameters can be measured because of an Occam penalty. Thus, in gravitational wave astrophysics and data analysis one wishes to minimize the number of theory parameters when testing GR [124*, 376*]. One must then find a balance between the number of additional theory parameters to introduce and the amount of bias contained in the templates.
At this junction, one must emphasize that frequency exponents in the amplitude and phase correction were above assumed to be integers, i.e., . This must be the case if these corrections arise due to modifications that can be represented as integer powers of the momenta or velocity. We are not aware of any theory that predicts corrections proportional to fractional powers of the velocity for circular inspirals. Moreover, one can show that theories that introduce noninteger powers of the velocity into the equations of motion will lead to issues with analyticity at zero velocity and a breakdown of uniqueness of solutions [102*]. In spite of this, modified theories can introduce logarithmic terms, that for example enter at high postNewtonian order in GR due to nonlinear propagation effects (see, e.g., [75] and references therein). Moreover, certain modified gravity theories introduce screened modifications that become “active” only above a certain frequency. Such effects would be modeled through a Heaviside function, for example needed when dealing with massive Brans–Dicke gravity [147, 94, 20, 465]. However, even these nonpolynomial injections would be detectable with the simplest ppE model. In essence, one finds similar results as if one were trying to fit a 3parameter injection with the simplest 1parameter ppE model [376].
Of course, one can also generalize the inspiral ppE waveform families to more general orbits, for example through the inclusion of spins aligned or counteraligned with the orbital angular momentum. More general inspirals would still lead to waveform families of the form of Eq. (205*) or (209), but where the parameters would now depend on the mass ratio, mass difference, and the spin parameters of the black holes. With a single detection, one cannot break the degeneracy in the ppE parameters and separately fit for its system parameter dependencies. However, given multiple detections one should be able to break such a degeneracy, at least to a certain degree [124*]. Such breaking of degeneracies begins to become possible when the number of detections exceeds the number of additional parameters required to capture the physical parameter dependencies of .
PpE waveforms can be extended to account for the merger and ringdown phases of coalescence. Yunes and Pretorius have suggested the following template family to account for this as well [467*]
where the subscripts IM and MRD stand for inspiral merger and merger ringdown, respectively. The merger phase () is modeled here as an interpolating region between the inspiral and ringdown, where the merger parameters are set by continuity and differentiability, and the ppE merger parameters should be fit for. In the ringdown phase (), the response function is modeled as a singlemode generalized Lorentzian, with real and imaginary dominant frequencies and , ringdown parameter also set by continuity and differentiability, and the ppE ringdown parameters are to be fit for. The transition frequencies can either be treated as ppE parameters or set via some physical criteria, such as at lightring frequency and the fundamental ringdown frequency, respectively.Recently, there has been effort to generalize the ppE templates to allow for the excitation of nonGR gravitationalwave polarizations. Modifications to only the two GR polarizations map to corrections to terms in the timedomain Fourier transform that are proportional to the harmonic of the orbital phase. However, Arun suggested that if additional polarizations are present, other terms proportional to the and harmonic will also arise [36*]. Chatziioannou, Yunes and Cornish [102*] have found that the presence of such harmonics can be captured through the more complete singledetector template family
where we have defined .
The ppE theory parameters are now . Of course, one may ignore altogether, if one wishes to ignore propagation effects. Such a parameterization recovers the predictions of Jordan–Fierz–Brans–Dicke theory for a singledetector response function [102*], as well as Arun’s analysis for generic dipole radiation [36].
One might worry that the corrections introduced by the harmonic, i.e., terms proportional to in Eq. (217), will be degenerate with postNewtonian corrections to the amplitude of the mode (not displayed in Eq. (217)). However, this is clearly not the case, as the latter scale as with an integer greater than 0, while the mode is proportional to , which would correspond to a postNewtonian order correction, i.e., . On the other hand, the ppE amplitude corrections to the mode, i.e., terms proportional to in the amplitude of Eq. (217), can be degenerate with such postNewtonian corrections when is an integer greater than .
5.3.4.4 Applications of the ppE formalism
The two models in Eq. (205*) and (209) answer different questions. The latter contains a stronger prior (that ppE frequency exponents be integers), and thus, it is ideal for fitting a particular set of theoretical models. On the other hand, Eq. (205*) with continuous ppE frequency exponents allows one to search for generic deviations that are statistically significant, without imposing such theoretical priors. That is, if a deviation from GR is present, then Eq. (205*) is more likely to be able to fit it, than Eq. (209). If one prioritizes the introduction of the least number of new parameters, Eq. (205*) with can still recover deviations from GR, even if the latter cannot be represented as a correction proportional to an integer power of velocity.
Given these ppE waveforms, how should they be used in a data analysis pipeline? The main idea behind the ppE framework is to match filter or perform Bayesian statistics with ppE enhanced template banks to allow the data to select the bestfit values of . As discussed in [467, 124*] and then later in [290*], one might wish to first run detection searches with GR template banks, and then, once a signal has been found, do a Bayesian model selection analysis with ppE templates. The first such Bayesian analysis was carried out by Cornish et al. [124], who concluded that an aLIGO detection at SNR of 20 for a quasicircular, nonspinning blackhole inspiral would allow us to constrain and much better than existent constraints for sufficiently strongfield corrections, e.g., . This is because for lower values of the frequency exponents, the corrections to the waveform are weakfield and better constrained with binary pulsar observations [461]. The large statistical study of Li et al. [290] uses a reduced set of ppE waveforms and investigates our ability to detect deviations of GR when considering a catalogue of aLIGO/adVirgo detections. Of course, the disadvantage of such a pipeline is that it requires a first detection, and if the gravitational interaction is too different from GR’s prediction, it is possible that a search with GR templates might miss the signal all together; we deem this possibility to be less likely.
A builtin problem with the ppE and the ppN formalisms is that if a nonzero ppE or ppN parameter is detected, then one cannot necessarily map it back to a particular modified gravity action. On the contrary, as suggested in Table 3, there can be more than one theory that predicts structurallysimilar corrections to the Fourier transform of the response function. For example, both Jordan–Fierz–Brans–Dicke theory and the dissipative sector of EinsteinDilatonGauss–Bonnet theory predict the same type of leadingorder correction to the waveform phase. However, if a given ppE parameter is measured to be nonzero, this could provide very useful information as to the type of correction that should be investigated further at the level of the action. The information that could be extracted is presented in Table 4, which is derived from knowledge of the type of corrections that lead to Table 3.


Interpretation 


Parity violation 


Anomalous acceleration, Extra dimensions, Violation of position invariance 


Dipole gravitational radiation, Electric dipole scalar radiation 


Massive graviton propagation 
spin 

Magnetic dipole scalar radiation, Quadrupole moment correction, Scalar dipole force 
Moreover, if a followup search is done with the ppE model in Eq. (209), one could infer whether the correction is one due to modifications to the generation or the propagation of gravitational waves. In this way, a nonzero ppE detection could inform theories of what type of GR modification is preferred by nature.
5.3.4.5 Degeneracies
However, much care must be taken to avoid confusing a ppE theory modification with some other systematic, such as an astrophysical, a mismodeling or an instrumental effect. Instrumental effects can be easily remedied by requiring that several instruments, with presumably unrelated instrumental systematics, independently derive a posterior probability for that peaks away from zero. Astrophysical uncertainties can also be alleviated by requiring that different events lead to the same posteriors for ppE parameters (after breaking degeneracies with system parameters). However, astrophysically there are a limited number of scenarios that could lead to corrections in the waveforms that are large enough to interfere with these tests. For comparablemass–ratio inspirals, this is usually not a problem as the inertia of each binary component is too large for any astrophysical environment to affect the orbital trajectory [229]. Magnetohydrodynamic effects could affect the merger of neutronstar binaries, but this usually occurs outside of the sensitivity band of groundbased interferometers. However, in extreme–massratio inspirals the small compact object can be easily nudged away by astrophysical effects, such as the presence of an accretion disk [462*, 267*] or a third supermassive black hole [463]. However, these astrophysical effects present the interesting feature that they correct the waveform in a form similar to Eq. (205*) but with . This is because the larger the orbital separation, the stronger the perturbations of the astrophysical environment, either because the compact object gets closer to the third body or because it leaves the inner edge of the accretion disk and the disk density increases with separation. Such effects, however, are not likely to be present in all sources observed, as few extreme–massratio inspirals are expected to be embedded in an accretion disk or sufficiently close to a third body ( 0.1 pc) for the latter to have an effect on the waveform.
Perhaps the most dangerous systematic is mismodeling, which is due to the use of approximation schemes when constructing waveform templates. For example, in the inspiral one uses the postNewtonian approximation series, expanding and truncating the waveform at a given power of orbital velocity. Moreover, neutron stars are usually modeled as testparticles (with a Dirac distributional density profile), when in reality they have a finite radius, which will depend on its equation of state. Such finitesize effects enter at 5 postNewtonian order (the effacement principle [227, 128]), but with a postNewtonian coefficient that can be rather large [320, 72, 175]. Ignorance of the postNewtonian series beyond 3 postNewtonian order can lead to systematics in the determination of physical parameters and possibly also to confusion when carrying out ppElike tests. Much more work is needed to determine the systems and SNRs for which such systematics are truly a problem.
5.3.5 Searching for nontensorial gravitationalwave polarizations
Another way to search for generic deviations from GR is to ask whether any gravitationalwave signal detected contains more than the two traditional polarizations expected in GR. A general approach to answer this question is through null streams, as discussed in Section 4.3. This concept was first studied by Gürsel and Tinto [212] and later by Chatterji et al. [101] with the aim to separate falsealarm events from real detections. Chatziioannou et al. [102*] proposed the extension of the idea of null streams to develop null tests of GR, which was proposed using stochastic gravitational wave backgrounds in [329, 330] and recently implemented in [228] to reconstruct the independent polarization modes in timeseries data of a groundbased detector network.
Given a gravitationalwave detection, one can ask whether the data is consistent with two polarizations by constructing a null stream through the combination of data streams from 3 or more detectors. As explained in Section 4.3, such a null stream should be consistent with noise in GR, while it would present a systematic deviation from noise if the gravitational wave metric perturbation possessed more than two polarizations. Notice that such a test would not require a template; if one were parametrically constructed, such as in [102], more powerful null tests could be applied to such a null steam. In the future, we expect several gravitational wave detectors to be online: the two aLIGO ones in the United States, adVIRGO in Italy, LIGOIndia in India, and KAGRA in Japan. Given a gravitationalwave observation that is detected by all five detectors, one can then construct three enhanced GR null streams, each with power in a signal null direction.
5.3.6 ILoveQ tests
Neutron stars in the slowrotation limit can be characterized by their mass and radius (to zerothorder in spin), by their moment of inertia (to firstorder in spin), and by their quadrupole moment and Love numbers (to secondorder in spin). One may expect these quantities to be quite sensitive to the neutron star’s internal structure, which can be parameterized by its equation of state, i.e., the relation between its internal pressure and its internal energy density. Since the equation of state cannot be wellconstrained at supernuclear densities in the laboratory, one is left with a variety of possibilities that predict different neutronstar massradius relations.
Recently, however, Yagi and Yunes [453*, 452*] have demonstrated that there are relations between the moment of inertia (), the Love numbers (, and the quadrupole moment (), the ILoveQ relations that are essentially insensitive to the equation of state. Figure 5* shows two of these relations (the normalized ILove and QLove relations – see caption) for a variety of equations of state, including APR [10], SLy [150, 385], Lattimer–Swesty with nuclear incompressibility of 220 MeV (LS220) [283, 335*], Shen [382, 383, 335], the latter two with temperature of 0.01 MeV and an electron fraction of 30%, and polytropic equations of state with indices of , and .^{13} The bottom panels show the difference between the numerical results and the analytical, fitting curve. Observe that all equations of state lead to the same ILove and QLove relations, with discrepancies smaller than 1% for realistic neutronstar masses. These results have recently been verified in [304] through the postNewtonianAffine approach [168, 305], which proves the ILoveQ relations hold not only during the inspiral, but also close to plunge and merger.
Given the independent measurement of any two members of the ILoveQ trio, one could carry out a (null) modelindependent and equationofstateindependent test of GR [453*, 452*]. For example, assume that electromagnetic observations of the binary pulsar J0737–3039 have measured the moment of inertia to 10% accuracy [282, 273, 274]. The slowrotation approximation is perfectly valid for this binary pulsar, due to its relatively long spin period. Assume further that a gravitationalwave observation of a neutronstar–binary inspiral, with individual masses similar to that of the primary in J0737–3039, manages to measure the neutron star tidal Love number to 60% accuracy [453*, 452*]. These observations then lead to an error box in the ILove plane, which must contain the curve in the leftpanel of Figure 5*.
A similar test could be carried out by using data from only binary pulsar observations or only gravitational wave detections. In the case of the latter, one would have to simultaneously measure or constrain the value of the quadrupole moment and the Love number, since the moment of inertia is not measurable with gravitational wave observations. In the case of the former, one would have to extract the moment of inertia and the quadrupole moment, the latter of which will be difficult to measure. Therefore, the combination of electromagnetic and gravitational wave observations would be the ideal way to carry out such tests.
Such a test of GR, of course, is powerful only as long as modified gravity theories predict ILoveQ relations that are not degenerated with the general relativistic ones. Yagi and Yunes [453*, 452*] investigated such a relation in dynamical Chern–Simons gravity to find that such degeneracy is only present in the limit . That is, for any finite value of , the dynamical Chern–Simons ILoveQ relation differs from that of GR, with the distance to the GR expectation increasing for larger . Yagi and Yunes [453*, 452*] predicted that a test similar to the one described above could constrain dynamical Chern–Simons gravity to roughly , where recall that .
The test described above, of course, only holds provided the ILoveQ relations are valid, which in turn depends on the assumptions made in deriving them. In particular, Yagi and Yunes [453, 452] assumed that the neutron stars are uniformly and slowly rotating, as well as only slightly tidally deformed by their rotational velocity or companion. These assumptions would not be valid for newlyborn neutron stars, which are probably differentially rotating and doing so quickly. However, the gravitational waves emitted by neutronstar inspirals are expected to have binary components that are old and not rapidly spinning by the time they enter the detector sensitivity band [74]. Some shortperiod, millisecond pulsars may spin at a nonnegligible rate, for which the normalized moment of inertia, quadrupole moment and Love number would not be independent of the rotational angular velocity. However, if then the above tests should still be possible, since binary pulsar observations would also automatically determine the rotational angular velocity, for which a unique ILoveQ relation should exist in GR.
5.4 Tests of the nohair theorems
Another important class of generic tests of GR are those that concern the nohair theorems. Since much work has been done on this area, we have decided to separate this topic from the main generic tests section (5.3). In what follows, we describe what these theorems are and the possible tests one could carry out with gravitationalwave observations emitted by blackhole–binary systems.
5.4.1 The nohair theorems
The nohair theorems state that the only stationary, vacuum solution to the Einstein equations that is nonsingular outside the event horizon is completely characterized by three quantities: its mass , its spin and its charge . This conclusion is arrived at by combining several different theorems. First, Hawking [223*, 222*] proved that a stationary black hole must have an event horizon with a spherical topology and that it must be either static or axially symmetric. Israel [243, 244] then proved that the exterior gravitational field of such static black holes is uniquely determined by and and it must be given by the Schwarzschild or the Reissner–Nordström metrics. Carter [98] constructed a similar proof for uncharged, stationary, axiallysymmetric black holes, where this time black holes fall into disjoint families, not deformable into each other and with an exterior gravitational field uniquely determined by and . Robinson [363] and Mazur [306] later proved that such black holes must be described by either the Kerr or the Kerr–Newman metric. See also [318, 352] for more details.
The nohair theorems apply under a restrictive set of conditions. First, the theorems only apply in stationary situations. Blackhole horizons can be tidally deformed in dynamical situations, and if so, Hawking’s theorems [223, 222] about spherical horizon topologies do not apply. This then implies that all other theorems described above also do not apply, and thus, dynamical black holes will generically have hair. Second, the theorems only apply in vacuum. Consider, for example, an axiallysymmetric black hole in the presence of a nonsymmetrical matter distribution outside the event horizon. One might naively think that this would tidally distort the event horizon, leading to a rotating, stationary black hole that is not axisymmetric. However, Hawking and Hartle [226] showed that in such a case the matter distribution torques the black hole forcing it to spin down, thus leading to a nonstationary scenario. If the black hole is nonstationary, then again the nohair theorems do not apply by the arguments described at the beginning of this paragraph, and thus nonisolated black holes can have hair. Third, the theorems only apply within GR, i.e., through the use of the Einstein equations. Therefore, it is plausible that black holes in modified gravity theories or in GR with singularities outside any event horizons (naked singularities) will have hair.
The nohair theorems imply that the exterior gravitational field of isolated, stationary, uncharged and vacuum black holes (in GR and provided the spacetime is regular outside all event horizons) can be written as an infinite sum of mass and current multipole moments, where only two of them are independent: the mass monopole moment and the current dipole moment . One can extend these relations to include charge, but astrophysical black holes are expected to be essentially neutral due to charge accretion. If the nohair theorems hold, all other multipole moments can be determined from [195, 194, 213]
where and are the th mass and current multipole moments. Even if the blackhole progenitor was not stationary or axisymmetric, the nohair theorems guarantee that any excess multipole moments will be shedoff during gravitational collapse [356, 357]. Eventually, after the black hole has settled down and reached an equilibrium configuration, it will be described purely in terms of and , where is the Kerr spin parameter.An astrophysical observation of a hairy black hole would not imply that the nohair theorems are wrong, but rather that one of the assumptions made in deriving these theorems is not appropriate to describe nature. As described above, the three main assumptions are stationarity, vacuum and that GR and the regularity condition hold. Astrophysical black holes will generically be hairy due to a violation of the first two assumptions, since they will neither be perfectly stationary, nor exist in a perfect vacuum. Astrophysical black holes will always suffer small perturbations by other stars, electromagnetic fields, other forms of matter, like dust, plasma or dark matter, etc, which will induce nonzero deviations from Eq. (219*) and thus evade the nohair theorems. However, in all cases of interest such perturbations are expected to be too small to be observable, which is why one argues that even astrophysical black holes should obey the nohair theorems if GR holds. Put another way, an observation of the violation of the nohair theorems would be more likely to indicate a failure of GR in the strongfield, than an unreasonably large amount of astrophysical hair.
Tests of the nohair theorems come in two flavors: through electromagnetic observations [250, 251, 253, 254] and through gravitational wave observations [370*, 371*, 112*, 196*, 44*, 50, 289, 390*, 471, 422*, 421*, 184*, 423*, 364]. The former rely on radiation emitted by accelerating particles in an accretion disk around black holes. However, such tests are not clean as they require the modeling of complicated astrophysics, with matter and electromagnetic fields. Gravitational wave tests are clean in that respect, but unlike electromagnetic tests, they cannot be carried out yet due to lack of data. Other electromagnetic tests of the nohair theorems exist, for example through the observation of close stellar orbits around Sgr A* [312, 313, 373] and pulsar–blackhole binaries [431], but these cannot yet probe the nearhorizon, strongfield regime, since electromagnetic observations cannot yet resolve horizon scales. See [359] for reviews on this topic.
5.4.2 Extreme massratio tests of the nohair theorem
Gravitational wave tests of the nohair theorems require the detection of either extreme massratio inspirals or the ringdown of comparablemass blackhole mergers with future spaceborne gravitationalwave detectors [25, 24]. Extreme massratio inspirals consist of a stellarmass compact object spiraling into a supermassive black hole in a generic orbit within astronomical units from the event horizon of the supermassive object [23]. These events outlive the observation time of future detectors, emitting millions of gravitational wave cycles, with the stellarmass compact object essentially acting as a tracer of the supermassive black hole spacetime [397]. Ringdown gravitational waves are always emitted after black holes merge and the remnant settles down into its final configuration. During the ringdown, the highlydistorted remnant radiates all excess degrees of freedom and this radiation carries a signature of whether the nohair theorems hold in its quasinormal mode spectrum (see, e.g., [68*] for a recent review).
Both electromagnetic and gravitational wave tests need a metric with which to model accretion disks, quasiperiodic oscillations, or extreme massratio inspirals. One can classify these metrics as direct or generic, paralleling the discussion in Section 5.2. Direct metrics are exact solutions to a specific set of field equations, with which one can derive observables. Examples of such metrics are the Manko–Novikov metric [302] and the slowlyspinning blackhole metric in dynamical Chern–Simons gravity [466*]. When computing observables with these metrics, one usually assumes that all radiative and dynamical process (e.g., the radiationreaction force) are as predicted in GR. Generic metrics are those that parametrically modify the Kerr spacetime, such that for certain parameter choices one recovers identically the Kerr metric, while for others, one has a deformation of Kerr. Generic metrics can be further classified into two subclasses, Ricciflat versus nonRicciflat, depending on whether they satisfy .
Let us first consider direct metric tests of the nohair theorem. The most studied direct metric is the Manko–Novikov one, which although an exact, stationary and axisymmetric solution to the vacuum Einstein equations, does not represent a black hole, as the event horizon is broken along the equator by a ring singularity [302]. Just like the Kerr metric, the Manko–Novikov metric possesses an ergoregion, but unlike the former, it also possesses regions of closed timelike curves that overlap the ergoregion. Nonetheless, an appealing property of this metric is that it deviates continuously from the Kerr metric through certain parameters that characterize the higher multiple moments of the solution.
The first geodesic study of Manko–Novikov spacetimes was carried out by Gair et al. [182*]. They found that there are two ringlike regions of bound orbits: an outer one where orbits look regular and integrable, as there exist four isolating integrals of the motion; and an inner one where orbits are chaotic and thus ergodic. Gair et al. [182*] suggested that orbits that transition from the integrable to the chaotic region would leave a clear observable signature in the frequency spectrum of the emitted gravitational waves. However, they also noted that chaotic regions exist only very close to the central body and are probably not astrophysically accessible. The study of Gair et al. [182] was recently confirmed and followed up by Contopoulos et al. [116]. They studied a wide range of geodesics and found that, in addition to an inner chaotic region and an outer regular region, there are also certain Birkhoff islands of stability. When an extreme massratio inspiral traverses such a region, the ratio of resonant fundamental frequencies would remain constant in time, instead of increasing monotonically. Such a feature would impact the gravitational waves emitted by such a system, and it would signal that the orbit equations are nonintegrable and the central object is not a Kerr black hole.
The study of chaotic motion in geodesics of nonKerr spacetimes is by no means new. Chaos has also been found in geodesics of Zipoy–Voorhees–Weyl and Curzon spacetimes with multiple singularities [391, 392] and in general for Zipoy–Voorhees spacetimes in [296], of perturbed Schwarzschild spacetimes [287], of Schwarzschild spacetimes with a dipolar halo [286, 288, 209] of Erez–Rosen spacetimes [210], and of deformed generalizations of the Tomimatsy–Sato spacetime [154]. One might worry that such chaotic orbits will depend on the particular spacetime considered, but recently Apostolatos et al. [31*] and Lukes–Gerakopoulos et al. [297*] have argued that the Birkhoff islands of stability are a general feature. Although the Kolmogorov, Arnold, and Moser theorem [270, 35, 321] states that phase orbit tori of an integrable system are only deformed if the Hamiltonian is perturbed, the Poincare–Birkhoff theorem [292] states that resonant tori of integrable systems actually disintegrate, leaving behind a chain of Birkhoff islands. These islands are only characterized by the ratio of winding frequencies that equals a rational number, and thus, they constitute a distinct and generic feature of nonintegrable systems [31, 297]. Given an extreme massratio gravitationalwave detection, one can monitor the ratio of fundamental frequencies and search for plateaus in their evolution, which would signal nonintegrability. Of course, whether detectors can resolve such plateaus depends on the initial conditions of the orbits and the physical system under consideration (these determine the thickness of the islands), as well as the mass ratio (this determines the radiationreaction timescale) and the distance and mass of the central black hole (this determines the SNR).
Another example of a direct metric test of the nohair theorem is through the use of the slowlyrotating dynamical Chern–Simons black hole metric [466]. Unlike the Manko–Novikov metric, the dynamical Chern–Simons one does represent a black hole, i.e., it possesses an event horizon, but it evades the nohair theorems because it is not a solution to the Einstein equations. Sopuerta and Yunes [390] carried out the first extreme massratio inspiral analysis when the background supermassive black hole object is taken to be such a Chern–Simons black hole. They used a semirelativistic model [368] to evolve extreme massratio inspirals and found that the leadingorder modification comes from a modification to the geodesic trajectories, induced by the nonKerr modifications of the background. Because the latter correspond to a strongfield modification to GR, modifications in the trajectories are most prominent for zoomwhirl orbits, as the small compact object zooms around the supermassive black hole in a region of unstable orbits, close to the event horizon. These modifications were then found to propagate into the gravitational waves emitted, leading to a dephasing that could be observed or ruled out with future gravitationalwave observations to roughly the horizon scale of the supermassive black hole, as has been recently confirmed by Canizares et al. [93]. However, these studies may be underestimates, given that they treat the black hole background in dynamical Chern–Simons gravity only to firstorder in spin.
A final example of a direct metric test of the nohair theorems is to consider black holes that are not in vacuum. Barausse et al. [52] studied extreme–massratio inspirals in a Kerr–blackhole background that is perturbed by a selfgravitating, homogeneous torus that is compact, massive and close to the Kerr black hole. They found that the presence of this torus impacts the gravitational waves emitted during such inspirals, but only weakly, making it difficult to distinguish the presence of matter. Yunes et al. [462] and Kocsis et al. [267] carried out a similar study, where this time they considered a small compact object inspiraling completely within a geometrically thin, radiationpressure dominated accretion disk. They found that diskinduced migration can modify the radiationreaction force sufficiently so as to leave observable signatures in the waveform, provided the accretion disk is sufficiently dense in the radiationdominated regime and a gap opens up. However, these tests of the nohair theorem will be rather difficult as most extreme–massratio inspirals are not expected to be in an accretion disk.
Let us now consider generic metric tests of the nohair theorem. Generic Ricciflat deformed metrics will lead to Laplacetype equations for the deformation functions in the farfield since they must satisfy to linear order in the perturbations. The solution to such an equation can be expanded in a sum of mass and current multipole moments, when expressed in asymptotically Cartesian and masscentered coordinates [407]. These multipoles can be expressed via [112*, 422*, 421*]
where and are mass and current multipole deformations. Ryan [370, 371] showed that the measurement of three or more multipole moments would allow for a test of the nohair theorem. Generic nonRicci flat metrics, on the other hand, will not necessarily lead to Laplacetype equations for the deformation functions in the far field, and thus, the farfield solution and Eq. (220*) will depend on a sum of and multipole moments.The first attempt to construct a generic, Ricciflat metric was by Collins and Hughes [112*]: the bumpy blackhole metric. In this approach, the metric is assumed to be of the form
where is a bookkeeping parameter that enforces that is a perturbation of the Kerr background. This metric is then required to satisfy the Einstein equations linearized in , which then leads to differential equations for the metric deformation. Collins and Hughes [112*] assumed a nonspinning, stationary spacetime, and thus only possessed two degrees of freedom, both of which were functions of radius only: , which must be a harmonic function and which changes the Newtonian part of the gravitational field at spatial infinity; and which is completely determined through the linearized Einstein equations once is specified. One then has the freedom to choose how to prescribe and Collins and Hughes investigate [112] two choices that correspond physically to pointlike and ringlike naked singularities, thus violating cosmic censorship [347]. Vigeland and Hughes [422] and Vigeland [421] then extend this analysis to stationary, axisymmetric spacetimes via the Newman–Janis method [327*, 151*], showing how such metric deformations modify Eq. (220*), and computing how these bumps imprint themselves onto the orbital frequencies and thus the gravitational waves emitted during an extreme–massratio inspiral.That the bumps represent unphysical matter should not be a surprise, since by the nohair theorems, if the bumps are to satisfy the vacuum Einstein equations they must either break stationarity or violate the regularity condition. Naked singularities are an example of the latter. A Lorentzviolating massive field coupled to the Einstein tensor is another example [155]. Gravitational wave tests with bumpy black holes must then be understood as null tests: one assumes the default hypothesis that GR is correct and then sets out to test whether the data rejects or fails to reject this hypothesis (a null hypothesis can never be proven). Unfortunately, however, bumpy black hole metrics cannot parameterize spacetimes in modified gravity theories that lead to corrections in the field equations that are not proportional to the Ricci tensor, such as for example in dynamical Chern–Simons or in EinsteinDilatonGauss–Bonnet modified gravity.
Other bumpy black hole metrics have also been recently proposed. Glampedakis and Babak [196*] proposed a different type of stationary and axisymmetric bumpy black hole through the Hartle–Thorne metric [218], with modifications to the quadrupole moment. They then constructed a “kludge” extreme massratio inspiral waveform and estimated how well the quadrupole deformation could be measured [44*]. However, this metric is valid only when the supermassive black hole is slowlyrotating, as it derives from the Hartle–Thorne ansatz. Recently, Johansen and Psaltis [252*] proposed yet another metric to represent bumpy stationary and sphericallysymmetric spacetimes. This metric introduces one new degree of freedom, which is a function of radius only and assumed to be a series in . Johansen and Psaltis then rotated this metric via the Newman–Janis method [327, 151] to obtain a new bumpy metric for axiallysymmetric spacetimes. However, such a metric possesses a naked ring singularity on the equator, and naked singularities on the poles. As before, none of these bumpy metrics can be mapped to known modified gravity black hole solutions, in the Glampedakis and Babak case [196] because the Einstein equations are assumed to hold to leading order in the spin, while in the Johansen and Psaltis case [252] because a single degree of freedom is not sufficient to model the three degrees of freedom contained in stationary and axisymmetric spacetimes [401, 423*].
The only generic nonRicciflat bumpy blackhole metric so far is that of Vigeland, Yunes and Stein [423*]. They allowed generic deformations in the metric tensor, only requiring that the new metric perturbatively retained the Killing symmetries of the Kerr spacetime: the existence of two Killing vectors associated with stationarity and axisymmetry, as well as the perturbative existence of a Killing tensor (and thus a Carterlike constant), at least to leading order in the metric deformation. Such requirements imply that the geodesic equations in this new background are fully integrable, at least perturbatively in the metric deformation, which then allows one to solve for the orbital motion of extreme–massratio inspirals by adapting previously existing tools. Brink [83, 84, 85, 86, 87] studied the existence of such a secondorder Killing tensor in generic, vacuum, stationary and axisymmetric spacetimes in Einstein’s theory and found that these are difficult to construct exactly. By relaxing this exact requirement, Vigeland, Yunes and Stein [423] found that the existence of a perturbative Killing tensor poses simple differential conditions on the metric perturbation that can be analytically solved. Moreover, they also showed how this new bumpy metric can reproduce all known modified gravity black hole solutions in the appropriate limits, provided these have an at least approximate Killing tensor; thus, these metrics are still vacuum solutions even though , since they satisfy a set of modified field equations. Although unclear at this junction, it seems that the imposition that the spacetime retains the Kerr Killing symmetries leads to a bumpy metric that is wellbehaved everywhere outside the event horizon (no singularities, no closedtimelike curves, no loss of Lorentz signature). Recently, Gair and Yunes [184] studied how the geodesic equations are modified for a testparticle in a generic orbit in such a spacetime and showed that the bumps are indeed encoded in the orbital motion, and thus, in the gravitational waves emitted during an extreme–massratio inspiral.
One might be concerned that such nohair tests of GR cannot constrain modified gravity theories, because Kerr black holes can also be solutions in the latter [360]. This is indeed true provided the modified field equations depend only on the Ricci tensor or scalar. In EinsteinDilatonGauss–Bonnet or dynamical Chern–Simons gravity, the modified field equations depend on the Riemann tensor, and thus, Ricciflat metric need not solve these modified set [473]. Moreover, just because the metric background is identically Kerr does not imply that inspiral gravitational waves will be identical to those predicted in GR. All studies carried out to date, be it direct metric tests or generic metric tests, assume that the only quantity that is modified is the metric tensor, or equivalently, the Hamiltonian or binding energy. Inspiral motion, of course, does not depend just on this quantity, but also on the radiationreaction force that pushes the small object from geodesic to geodesic. Moreover, the gravitational waves generated during such an inspiral depend on the field equations of the theory considered. Therefore, all metric tests discussed above should be considered as partial tests. In general, strongfield modified gravity theories will modify the Hamiltonian, the radiationreaction force and the wave generation.
5.4.3 Ringdown tests of the nohair theorem
Let us now consider tests of the nohair theorems with gravitational waves emitted by comparablemass binaries during the ringdown phase. Gravitational waves emitted during ringdown can be described by a superposition of exponentiallydamped sinusoids [69*]:
where is the distance from the source to the detector, the asterisk stands for complex conjugation, the real mode amplitudes and and the real phases and depend on the initial conditions, are spheroidal functions evaluated at the complex quasinormal ringdown frequencies , and the real physical frequency and the real damping times are both functions of the mass and the Kerr spin parameter only, provided the nohair theorems hold. These frequencies and damping times can be computed numerically or semianalytically, given a particular blackhole metric (see [68] for a recent review). The Fourier transform of a given mode is [69*]where we have defined as well as the Lorentzian functions
Ringdown gravitational waves will all be of the form of Eq. (222*) provided that the characteristic nature of the differential equation that controls the evolution of ringdown modes is not modified, i.e., provided that one only modifies the potential in the Teukolsky equation or other subdominant terms, which in turn depend on the modified field equations.Tests of the nohair theorems through the observation of blackhole ringdown date back to Detweiler [146], and it was recently worked out in detail by Dreyer et al. [152*]. Let us first imagine that a single complex mode is detected and one measures separately its real and imaginary parts. Of course, from such a measurement, one cannot extract the measured harmonic triplet , but instead one only measures the complex frequency . This information is not sufficient to extract the mass and spin angular momentum of the black hole because different quintuplets can lead to the same complex frequency . The best way to think of this is graphically: a given observation of traces a line in the complex plane; a given triplet defines a complex frequency that also traces a curve in the complex plane; each intersection of the measured line with defines a possible doublet ; since different triplets lead to different curves and thus different intersections, one ends up with a set of doublets , out of which only one represents the correct blackhole parameters. We thus conclude that a single mode observation of ringdown gravitational waves is not sufficient to test the nohair theorem [152*, 69*].
Let us then imagine that one has detected two complex modes, and . Each detection leads to a separate line and in the complex plane. As before, each triplet leads to separate curves which will intersect with both and in the complex plane. Each intersection between and leads to a set of doublets , while each intersection between and leads to another set of doublets . However, if the nohair theorems hold sets and must have at least one element in common. Therefore, a twomode detection allows for tests of the nohair theorem [152, 69*]. However, when dealing with a quasicircular blackhole–binary inspiral within GR one knows that the dominant mode is . In such a case, the observation of this complex mode by itself allows one to extract the mass and spin angular momentum of the black hole. Then, the detection of the real frequency in an additional mode can be used to test the nohair theorem [69*, 65*].
Although the logic behind these tests is clear, one must study them carefully to determine whether all systematic and statistical errors are sufficiently under control so that they are feasible. Berti et al. [69*, 65*] investigated such tests carefully through a frequentist approach. First, they found that a matchedfiltering type analysis with twomode ringdown templates would increase the volume of the template manifold by roughly three orders of magnitude. A better strategy then is perhaps to carry out a Bayesian analysis, like that of Gossan et al. [256, 201]; through such a study one can determine whether a given detection is consistent with a twomode or a onemode hypothesis. Berti et al. [69, 65] also calculated that a SNR of would be sufficient to detect the presence of two modes in the ringdown signal and to resolve their frequencies, so that nohair tests would be possible. Strong signals are necessary because one must be able to distinguish at least two modes in the signal. Unfortunately, however, whether the ringdown leads to such strong SNRs and whether the subdominant ringdown modes are of a sufficiently large amplitude depends on a plethora of conditions: the location of the source in the sky, the mass of the final black hole, which depends on the rest mass fraction that is converted into ringdown gravitational waves (the ringdown efficiency), the mass ratio of the progenitor, the magnitude and direction of the spin angular momentum of the final remnant and probably also of the progenitor and the initial conditions that lead to ringdown. Thus, although such tests are possible, one would have to be quite fortunate to detect a signal with the right properties so that a twomode extraction and a test of the nohair theorems is feasible.
5.4.4 The hairy search for exotica
Another way to test GR is to modify the matter sector of the theory through the introduction of matter corrections to the Einstein–Hilbert action that violate the assumptions made in the nohair theorems. More precisely, one can study whether gravitational waves emitted by binaries composed of strange stars, like quark stars, or horizonless objects, such as boson stars or gravastars, are different from waves emitted by more traditional neutronstar or blackhole binaries. In what follows, we will describe such hairy tests of the existence of compact exotica.
Boson stars are a classic example of a compact object that is essentially indistinguishable from a black hole in the weak field, but which differs drastically from one in the strong field due to its lack of an event horizon. A boson star is a coherent scalarfield configuration supported against gravitational collapse by its selfinteraction. One can construct several Lagrangian densities that would allow for the existence of such an object, including miniboson stars [178, 179], axiallysymmetric solitons [372], and nonsolitonic stars supported by a noncanonical scalar potential energy [113]. Boson stars are wellmotivated from fundamental theory, since they are the gravitationallycoupled limit of qballs [108, 276], a coherent scalar condensate that can be described classically as a nontopological soliton and that arises unavoidably in viable supersymmetric extensions of the standard model [275]. In all studies carried out to date, boson stars have been studied within GR, but they are also allowed in scalartensor theories [46].
At this junction, one should point out that the choice of a boson star is by no means special; the key point here is to select a strawman to determine whether gravitational waves emitted during the coalescence of compact binaries are sensitive to the presence of an event horizon or the evasion of the nohair theorems induced by a nonvacuum spacetime. Of course, depending on the specific model chosen, it is possible that the exotic object will be unstable to evolution or even to its own rotation. For example, in the case of an extreme massratio inspiral, one could imagine that as the small compact object enters the boson star’s surface, it will accrete the scalar field, forcing the boson star to collapse into a black hole. Alternatively, one can imagine that as two supermassive boson stars merge, the remnant might collapse into a black hole, emitting the usual GR quasinormal modes. What is worse, even when such objects are in isolation, they are unstable under small perturbations if their angular momentum is large, possibly leading to gravitational collapse into a black hole or possibly a scalar explosion [95, 96]. Since most astrophysical black hole candidates are believed to have high spins, such instabilities somewhat limit the interest of horizonless objects. Even then, however, the existence of slowly spinning or non spinning horizonless compact objects cannot be currently ruled out by observation.
Boson stars evade the nohair theorems within GR because they are not vacuum spacetimes, and thus, their metric and quasinormal mode spectrum cannot be described by just their mass and spin angular momentum; one also requires other quantities intrinsic to the scalarfield energy momentum tensor, scalar hair. Therefore, as before, two types of gravitational wave tests for scalar hair have been proposed: extreme–massratio inspiral tests and ringdown tests. As for the former, several studies have been carried out that considered a supermassive boson star background. Kesden et al. [263*] showed that stable circular orbits exist both outside and inside of the surface of the boson star, provided the small compact object interacts with the background only gravitationally. This is because the effective potential for geodesic motion in such a bosonstar background lacks the Schwarzschildlike singular behavior at small radius, instead turning over and allowing for a new minimum. Gravitational waves emitted in such a system would then stably continue beyond what one would expect if the background had been a supermassive black hole; in the latter case the small compact object would simply disappear into the horizon. Kesden et al. [263] found that orbits inside the boson star exhibit strong precession, exciting high frequency harmonics in the waveform, and thus allowing one to easily distinguish between such boson stars from blackhole backgrounds.
Just as the inspiral phase is modified by the presence of a boson star, the merger phase is also greatly altered, but this must be treated fully numerically. A few studies have found that the merger of boson stars leads to a spinning bar configuration that either fragments or collapses into a Kerr black hole [339, 338]. Of course, the gravitational waves emitted during such a merger will be drastically different from those produced when black holes merge. Unfortunately, the complexity of such simulations makes predictions difficult for any one given example, and the generalization to other more complicated scenarios, such as theories with modified field equations, is currently not feasible.
Recently, Pani et al. [340*, 341*] revisited this problem, but instead of considering a supermassive boson star, they considered a gravastar. This object consists of a Schwarzschild exterior and a de Sitter interior, separated by an infinitely thin shell with finite tension [307, 100]. Pani et al. [341*] calculated the gravitational waves emitted by a stellarmass compact object in a quasicircular orbit around such a gravastar background. In addition to considering a different background, Pani et al. used a radiativeadiabatic waveform generation model to describe the gravitational waves [351, 238, 239, 458, 456, 459], instead of the kludge scheme used by Kesden et al. [49, 44, 456]. Pani el al. [341] concluded that the waves emitted during such inspirals are sufficiently different that they could be used to discern between a Kerr black hole and a gravastar.
On the ringdown side of nohair tests, several studies have been carried out. Berti and Cardoso [66] calculated the quasinormal mode spectrum of boson stars. Chirenti and Rezzolla [105] studied the nonradial, axial perturbations of gravastars, and Pani et al. [340] the nonradial, axial and polar oscillations of gravastars. Medved et al. [309, 310] considered the quasinormal ringdown spectrum of skyrmion black holes [386]. In all cases, it was found that the quasinormal mode spectrum of such objects could be used to discern between them and Kerr black holes. Of course, such tests still require the detection of ringdown gravitational waves with the right properties, such that more than one mode can be discerned and extracted from the signal (see Section 5.4.3).