Model-Independent Test of General Relativity:
An Extended post-Einsteinian Framework with Complete Polarization Content
We develop a model-independent test of General Relativity that allows for the constraint of the gravitational wave (GW) polarization content with GW detections of binary compact object inspirals. We first consider three modified gravity theories (Brans-Dicke theory, Rosen’s theory and Lightman-Lee theory) and calculate the response function of ground-based detectors to gravitational waves in the inspiral phase. This allows us to see how additional polarizations predicted in these theories modify the General Relativistic prediction of the response function. We then consider general power-law modifications to the Hamiltonian and radiation-reaction force and study how these modify the time-domain and Fourier response function when all polarizations are present. From these general arguments and specific modified gravity examples, we infer an improved parameterized post-Einsteinian template family with complete polarization content. This family enhances General Relativity templates through the inclusion of new theory parameters, reducing to the former when these parameters acquire certain values, and recovering modified gravity predictions for other values, including all polarizations. We conclude by discussing detection strategies to constrain these new, polarization theory parameters by constructing certain null channels through the combination of output from multiple detectors.
Gravitational waves (GWs) will soon be detected by ground-based detectors, such as Advanced LIGO Abramovici et al. (1992); Collaboration (2007); lig () and Advanced Virgo Acernese et al. (2007); vir (). These waves will provide invaluable information about the gravitational interaction in the so-called strong-field regime, where the non-linearity and strong dynamics of the Einstein equations play an important role. Although General Relativity (GR) has passed all Solar System and binary pulsar tests with flying colors, the strong-field regime remains mostly unexplored Will (1993, 2005). For example, observations have not yet been able to confirm GR’s no-hair theorems or the non-linear part of the Einstein equations in the GW generation.
One of the primary targets of these terrestrial detectors are waves generated in the late inspiral of compact objects. Detectors that can observe signals as low as will be able to follow the binary inspiral from astronomically small separations down to merger. For example, for a binary neutron star inspiral, ground-based detectors should be able to detect inspirals from initial separations of , where is the total mass of the binary (here and in what follows we use geometric units ). In this late inspiral regime, the non-linearities of the field equations and the strong-field nature of GR are essential, while one can still employ perturbation theory to model the associated GWs until right before the compact objects plunge into each other and merge.
One of the predictions of GR that one would wish to test is that the GW metric perturbation only possesses two propagating degrees of freedom. In a general theory of gravity, there are up to degrees of freedom allowed. In GR, however, due to the structure of the field equations, only of them are physical, with the remaining being gauge degrees of freedom. Although ground-based detectors are expected to observe GWs at low signal-to-noise ratio, the simultaneous detection of such waves by multiple detectors should allow us to constrain the existence of additional polarization modes (2 scalar and 2 vectorial). The main question, of course, is precisely how to carry out such tests and the degree to which they will allow us to rule out or confirm additional polarization modes.
Many modified gravity theories exist where all degrees of freedom are physical and must be accounted for when computing the GW response function. Perhaps the most well-known examples are scalar-tensor theories, where the presence of a scalar field leads to the existence of dipole radiation and a scalar (so-called breathing) polarization mode, in addition to the two standard transverse-traceless modes of GR Brans and Dicke (1961); Will (1993, 2005). Vector-tensor theories usually predict the existence of preferred directions and the excitation of vector modes Will (1993), while tensor-vector-scalar theories, such as TeVeS Bekenstein (2004); Bekenstein and Sanders (2005), predict the existence of all polarization modes and Einstein-Aether theories Jacobson and Mattingly (2004); Eling et al. (2004); Jacobson (2008) predict the existence of polarization modes . Moreover, bimetric and stratified theories also predict the existence of all polarization modes Rosen (1971, 1974); Lightman and Lee (1973); Will (1993).
Recently, a model-independent framework to test GR was developed: the parameterized post-Einsteinian (ppE) framework Yunes and Pretorius (2009); Cornish et al. (2011). In this scheme, one enhances the GR GW template family through the addition of new theory parameters ; when these ppE parameters acquire certain values one recovers GR, while for other values one recovers predictions from modified gravity theories. As a first step, Yunes and Pretorius Yunes and Pretorius (2009) proposed a simple ppE model that treated the inspiral phase in a post-Newtonian (PN) approximation, where the orbital velocity is much less than the speed of light , and that neglected the direct excitation of additional polarization modes in the GW response function. Such modes were only indirectly accounted for through possible modifications to the orbital binding energy and its rate of change, which were captured through ppE parameters.
In this paper, we relax this assumption and improve the ppE framework to allow for the direct presence of additional polarization modes in the response function. In order to achieve this, we first look at three modified gravity theories that predict the existence of additional polarization modes, namely Brans-Dicke theory Brans and Dicke (1961), Rosen’s bimetric theory Rosen (1971, 1974) and Lightman-Lee theory Lightman and Lee (1973)111We choose these theories as toy models because substantial work has already been done to analyze their properties Will (1977); Will and Zaglauer (1989). Once similar work is done for other theories, the analysis of this paper can be extended to include them.. For each of these theories, we calculate the GW metric perturbation, we extract from this the polarization modes, construct the GW response function and Fourier transform the latter in the stationary-phase approximation (SPA) Bender and Orszag (1999); Droz et al. (1999); Yunes et al. (2009).
We then use physical arguments to predict the general functional form of the time-domain response function, assuming all polarization states are present. We begin by noting that the existence of additional modes leads to new terms in the time-domain response function that are proportional to the harmonics of the orbital phase. When this is Fourier transformed, however, the harmonic does not possess a stationary point, and thus, it is subdominant relative to the modes. When Fourier transforming, we allow for a parametric deformation of the Hamiltonian and the radiation-reaction force through the addition of relative power-law corrections. As Yunes and Pretorius found Yunes and Pretorius (2009), power-law modifications to the binding energy and energy flux introduce power-law modifications to the Fourier amplitude and phase. The inclusion of additional polarizations introduces a new term that is proportional to the harmonic of the orbital phase and that was not included in the original ppE scheme.
With these general arguments and solutions from modified gravity theories at hand, we generalize the ppE framework to allow for the remaining four polarization modes in the response function. Given a single detector, we find that to parameterize only the harmonic one requires ppE parameters for the most general case, in agreement with Yunes and Pretorius Yunes and Pretorius (2009); if one wishes to include all polarization modes, one needs a total of ppE parameters instead. However, if one restricts attention only to power-law modifications to the Hamiltonian and radiation-reaction force in the generation of GWs, then for single-detectors one needs only ppE parameters to parameterize the harmonic and ppE parameters to parameterize all polarizations.
Part of our work is similar to a paper by Arun that recently appeared in the literature Arun (2012). In that paper, he first defines the dipolar mode of the GW as the harmonic component of the wave. Then, assuming a general functional form for this component and a GR quadrupole plus dipole frequency evolution, he calculates the Fourier transform of the response function. He finds that the Fourier response can be parametrized by 2 parameters: one that measures the amplitude of the dipole mode relative to the leading quadrupole mode; and one that measures the relative strength of dipole emission. The reason that Arun manages to parameterize the waveform with a smaller number of parameter is that he neglects corrections to the conservative part of the dynamics and he restricts attention to dipole emission only. Our paper generalizes his results, allowing for generic deformations in the Hamiltonian and radiation-reaction force, while simultaneously accounting for all polarization modes.
Using a single spherical Forward (1971) or truncated icosohedral Johnson and Merkowitz (1993) acoustic detector it is possible to extract and separate Bianchi et al. (1996) the various polarization modes. In contrast, a standard two-arm interferometric detector only provides a single projection of the polarization state, and it takes multiple independent projections from an array of detectors to fully constrain the polarization content. We construct a ppE model that could be used with multiple interferometers by restoring the beam-pattern dependencies of the ppE amplitude. This forces us to break-up the overall amplitude into several terms, each with new ppE theory parameters, thus enlarging the parameter space to theory parameters in the general case and for power-law corrections. With this at hand, we then discuss a new multiple detector strategy to extract the additional polarization modes through the construction of null streams, ie. combinations that isolate specific polarization modes. One such null stream has the property that if GR were right, the stream would be free of gravitational wave energy for any designated sky location. If each detector output is thought of as a vector in signal space, then the construction of null channels reduces to the continuous projection of this signal in directions orthogonal to the GR polarization modes. In general, we find that at least detectors are necessary to constrain additional modes, detectors are necessary to constrain the vector modes and with detectors it is possible to construct streams that are null in every theory of gravity.
The rest of this paper is organized as follows: Section II describes how the gravitational response function is computed in a generic modified gravity theory and summarizes the SPA. Section III discusses Brans-Dicke theory, Rosen’s theory and Lightman-Lee’s theory, computing the response function and its Fourier transform in the SPA for each of them. Section B calculates the complete waveform for generic deviations in the systems binding energy and balance law. Section V uses the results from the previous sections to construct a generalized ppE framework that allows for the existence of additional polarizations modes. Section VI discusses the construction of null streams for generic waveforms with polarization modes. Section VII concludes and discusses possible future research directions.
We follow here the conventions of Misner, Thorne and Wheeler Misner et al. (1973): Greek letters stand for spacetime time indices, while Latin letters in the middle of the alphabet stand for spatial indices only; commas in index lists stand for partial derivatives and semi-colons for covariant derivatives; parentheses and square brackets in index lists stand for symmetrization and anti-symmetrization respectively such as and ; the metric is denoted via with signature ; the Einstein summation convention and geometric units with is assumed, unless otherwise specified.
Ii Fourier Transform of a Generic Response Function
In this section, we construct the response function for a generic modified gravity theory, given the GW metric perturbation. We then briefly explain how the SPA to the Fourier transform of this response can be calculated. We conclude the section by showing how the algorithm works in the standard GR limit.
ii.1 Polarizations from the Metric Perturbation
where are angular pattern functions and are waveform polarizations. The former are given by Poisson and Will ()
The waveform polarizations can be computed from the contraction of certain basis vectors (see e.g. Kidder (1995); Poisson and Will () noting that in Poisson and Will () , ie. the basis vectors orthogonal to , the unit vector pointing from the source to the detector, are denoted as ) with the waveform amplitudes , namely
In these equations, is the amplitude of the scalar breathing mode, is the amplitude of the scalar longitudinal mode, are the amplitudes of the vectorial modes and are the amplitudes of the transverse-traceless modes.
The usual way to find the waveform amplitudes is to compute the linearized Riemann tensor evaluated with the trace-reversed metric perturbation Misner et al. (1973); Poisson and Will (). A more straightforward way to accomplish the same result, however, is to construct operators that act on the trace-reversed metric perturbation directly and return the waveform amplitudes. In terms of these, the amplitudes are given by
where is a projection operator orthogonal to , a unit vector pointing from the source to the detector, while is the trace-reversed metric perturbation and is the Kronecker delta.
One might wonder whether we can reconstruct the full metric perturbation from the GW polarization modes in Eq. (9). Notice, though, that Eq. (9) contains only 6 degrees of freedom (1 in , 1 in , 2 in because it is transverse and 2 in because it is transverse and traceless), while the full metric perturbation generically contains 10 degrees of freedom. Thus, for the inversion to be unique one must make a gauge choice, such as a pure traceless (yet not fully transverse) gauge. Doing so, the metric perturbation can be written as
Of course, such a metric reconstruction is unnecessary for our purposes because the observable is the response function and we can project out the relevant degrees of freedom (those in Eq. (9)) without making any gauge choice.
ii.2 Stationary Phase Approximation
In GW data analysis, one often works with the Fourier transform of the response function, which can be obtained analytically in the SPA Bender and Orszag (1999); Droz et al. (1999); Yunes et al. (2009). We here briefly review this method. The goal of the SPA is to compute the generalized Fourier integral
assuming that the response function is composed of a slowly varying amplitude and a rapidly varying phase , namely
with the harmonic number and the orbital phase.
The first term in square brackets does not have a stationary point, ie. a value of for which the derivative of the argument of the exponential vanishes, . Terms without a stationary point contribute subdominantly to the generalized Fourier integral, and thus, they can be neglected by the Riemann-Lebesgue Lemma Bender and Orszag (1999). The second term in Eq. (13) does have a stationary point, which after Taylor expanding occurs when the derivative of the argument in the exponential vanishes, ie. , where is the orbital angular frequency.
With this at hand, the SPA of is Yunes et al. (2009)
where the phase GW is given by
In the rest of this paper, we will use these expressions to find an analytic representation of the Fourier transform of the response function in different modified gravity theories.
As we will see in Sec. III, the time-domain response function in a generic modified gravity theory will contain terms proportional to the harmonic of the orbital phase, as shown in Eq. (12). Terms in the Fourier transform of the response function proportional to the harmonic are of the form
where the power depends on how the frequency evolves. Such an integral vanishes for when the limits of integration are , because, as the complex exponential oscillates, contributions from subsequent intervals cancel out. Indeed, to first order we keep only the leading quadrupole emission for which . Therefore, we see that the harmonic of the orbital phase in the response function will contribute subdominantly to the SPA of the Fourier response.
ii.3 General Relativity Limit
As an illustrative example, let us apply the above formalism to the trace-reversed metric perturbation in GR. The trace-reversed metric perturbation for a two-body, quasi-circular orbit is given, to leading-order in the PN expansion, by (see e.g. Blanchet (2006))
where the quadrupole moment is
is the reduced mass, is the total mass, is the distance to the source, is the orbital separation and are orbital trajectory and orbital velocity unit vectors.
In order to explicitly calculate the waveform amplitudes via Eq. (9) we must first express in the source system
Choosing the coordinate system so that the vector from the source to the observer is on the - plane the vectors and the polarization tensors are
where we recall that is the orbital phase and is the inclination angle. Applying the operators in Eq. (9), we obtain
Once we have the amplitudes, we can compute the waveform polarization modes from Eq. (8) to obtain
and all other modes vanish. From Eq. (1), the response is then simply
is the GW phase and is the chirp mass, with the symmetric mass ratio. We have here used Kepler’s third law to simplify the final result, neglecting subdominant terms in the PN approximation.
The SPA of the Fourier transform of this response function can be computed following Sec. II.2. Using the balance law to relate the rate of change of binding energy to the GW luminosity, we can calculate the frequency evolution, which to leading-order in the PN expansion is given by
With this in hand, we can now compute the well-known (restricted, ie. leading-order in the amplitude) Fourier transform of the response function in the SPA, namely
where we have also introduced for future convenience
although only the harmonic enters the GR waveform. Here, are known PN coefficients that can be read for example from Eq. in Buonanno et al. (2009), and we have defined the reduced -harmonic frequency
such that , with the GW frequency.
Up until now, we have concentrate on the restricted PN approximation, but later on it will be important to determine whether modified gravity corrections to the Fourier amplitude are degenerate with PN amplitude corrections in GR. Amplitude corrections arise because the PN waveform contains an infinite number of higher harmonics, as one can see e.g. in Eq. of Blanchet (2006). Therefore, the Fourier transform of such a waveform leads also to a sum of harmonic terms. The dominant one is the mode, which was already described above in Eq. (32). The next order terms are the and harmonics, which scale as Van Den Broeck and Sengupta (2007)
where are amplitude factors that depend on different combinations of the inclination and polarization angles. The key point here is that these terms enter at 1PN order higher in the amplitude relative to the dominant mode, as can be established by looking at its frequency dependence.
Iii Modified Gravity Theories
iii.1 Brans-Dicke Theory
Brans Dicke theory Brans and Dicke (1961) is defined by the gravitational action (in Jordan frame)
where is the determinant of the metric , is the Ricci scalar, is a dynamical scalar field, is a potential for the scalar field and is a coupling constant. Usually, one sets the potential to zero, unless one is considering massive Brans-Dicke theory Alsing et al. (2011). Such a theory is a subset of scalar-tensor theories, where the coupling constant . Variation of this action with respect to the metric and the scalar field leads to the modified field equations of the theory. Linearizing these field equations about a flat background , one can obtain evolution equations for . We refer the interested reader to Will and Zaglauer (1989); Yunes et al. (2011) for further details.
In Brans-Dicke theory, it is convenient to define the trace-reversed metric perturbation in terms of two other fields, a covariantly conserved tensor and the scalar field , to obtain
where is the asymptotic value of the scalar field at spatial infinity. The linearized field equations prescribe the evolution of both and , whose solution in the PN approximation is
where we have defined
In these equations, we have also defined for , , , where is the sensitivity of the th object, as defined in Brans Dicke theory, and . Clearly, Brans-Dicke theory reduces to GR in the (or ) limit.
With this at hand, we can now compute the polarization modes as in Sec. II.1. Using the Lorenz gauge condition from Will (1977) 222The evolution equation for is , where is a complicated extension of the Landau-Lifshitz pseudo-tensor. One can easily verify that this differential equation preserves the Lorenz gauge condition., the waveform amplitudes are
where the longitudinal and vectorial modes vanish. The polarization modes are then
where again the longitudinal and vectorial modes vanish.
Putting all pieces together and simplifying expressions through the leading PN order expression for Kepler’s third law
where stands for the complex conjugate and where we have kept terms only linear in . In these equations, we have also defined
The Fourier integral of Eq. (47) can be easily calculated with the SPA, but this requires use of the orbital frequency evolution. The rate of change of the binding energy is determined by both dipole and quadrupole radiation, and to first order in and to leading-order in the PN approximation, it is calculated in Will (1977)
When this is combined with the binding energy and Eq. (46), one obtains the orbital frequency evolution
where recall that as given below Eq. (III.1).
The total Fourier-transformed waveform in Brans-Dicke theory is then simply
where the Fourier transform of the first term in Eq. (47) is
and that of the second term is
We have here assumed that the second and third terms in Eq. (III.1) are much smaller than the first term, ie. that Brans-Dicke theories introduces a small deformation away from GR. We have also here defined the -harmonic Brans-Dicke Fourier phase
with given in Eq. (33) (see also Eq. (V.2.1)) and given by Eq. (34). Notice that the second term in Eq. (III.1) is of PN order relative to the first one, a typical signature of dipole radiation. Such an amplitude correction is usually neglected, because GW interferometers are much more sensitive to the phase evolution. Of course, not all corrections to the amplitude and the phase will be measurable, and some of them can be degenerate with other system parameters, such as the luminosity distance or the chirp mass. Such issues will be discussed further in Sec. V.
iii.2 Rosen’s Theory
Rosen’s is an example of a bimetric theory Rosen (1971, 1974): a theory with a dynamical tensor gravitational field and a flat, non-dynamical metric, or prior geometry. Rosen’s theory is defined by the gravitational action Rosen (1971, 1974); Will (1993)
where is the determinant of the flat, non-dynamical metric and the operator stands for a covariant derivative with respect to . Although the field equations in this theory are quite different from Einstein’s, it has a standard parametrized post-Newtonian (ppN) limit, with only the ppN parameter different from its GR value. This parameter, however, has been constrained to be less than through observations of solar alignment with the ecliptic plane Will (1993, 2005).
Rosen’s theory is of class in the classification Will (1993, 2005), and thus, not all six polarization modes are observer-independent. In theories of this class, all observers agree on the magnitude of the longitudinal mode, but they disagree on the presence or absence of all other modes. This, however, does not mean that the other modes are not real or that they do not carry energy. It only implies that a spin-decomposition of the GWs is not invariant. Such frame dependence is irrelevant for our purposes, since the detector will be in a given frame, and thus, it will measure a certain number of polarization modes. See Will (1993, 2005) for more details on this theory.
The lack of definite helicity in Rosen’s theory is connected to the lack of positive definiteness in the sign of the emitted energy Will (1977). That is, for certain systems, Rosen’s theory predicts dipolar radiation that pumps energy into the system, leading to a total energy flux that is positive, instead of negative as in GR. In turn, this for example leads to an increase in the orbital period of binary pulsars with time Will (1977); Will and Eardley (1977); Weisberg and Taylor (1981). Since binary pulsar observations are consistent with GR, Rosen’s theory is today less appealing than in the 1970s. We here study it, not because we think of it as a particularly good candidate to replace Einstein’s theory, but as a toy model to determine how the GW response function is modified in theories that allow for the existence of all gravitational polarization modes Will (1993, 2005).
The variation of the action and the linearization of the resulting field equations give the evolution of the trace-reversed metric perturbation Will (1977); Will and Eardley (1977), which can be solved to find
where is the difference in the self-gravitational binding energy per unit mass of the binary components: . Due to the specific characteristics of the theory, one cannot find a particular gauge to simplify the above equations.
With this in hand, we can now compute the polarizations modes and the response function. Following the steps laid out in Sec. II.1, we find
Using the modified Kepler’s law
where , the time-domain response function is
where we recall that stands for complex conjugate and where are functions of the angles:
Before we can calculate the Fourier transform of the response function in the SPA for Rosen’s theory, we must first calculate the orbital frequency evolution. The energy evolution of the binary orbit due to GW emission is (to leading-order in the PN approximation and in ) given in Will (1977)
and this leads to the orbital frequency evolution
where the binding energy is not modified.
We can now calculate the Fourier transform in the SPA:
where the transform of the first term in Eq. (63) is
and that of the second term is
We have here assumed that and kept terms only to leading-order both in and in the PN approximation. We have also here defined the -harmonic Rosen Fourier phase
iii.3 Lightman-Lee Theory
Lightman-Lee theory Lightman and Lee (1973) is a bimetric theory of gravity, similar to Rosen’s. This theory is controlled by the metric , a dynamical gravitational tensor that is connected to and a flat, non-dynamical background metric Will (1993). The theory is defined by the gravitational action
where is the trace of the background metric (not to be confused with the symmetric mass ratio introduced earlier). The spacetime metric is connected to the tensor via
which can be expanded for weak gravitational fields as with