Fundamental Theoretical Bias in Gravitational Wave Astrophysics
and the Parameterized Post-Einsteinian Framework
We consider the concept of fundamental bias in gravitational wave astrophysics as the assumption that general relativity is the correct theory of gravity during the entire wave-generation and propagation regime. Such an assumption is valid in the weak field, as verified by precision experiments and observations, but it need not hold in the dynamical strong-field regime where tests are lacking. Fundamental bias can cause systematic errors in the detection and parameter estimation of signals, which can lead to a mischaracterization of the universe through incorrect inferences about source event rates and populations. We propose a remedy through the introduction of the parameterized post-Einsteinian framework, which consists of the enhancement of waveform templates via the inclusion of post-Einsteinian parameters. These parameters would ostensibly be designed to interpolate between templates constructed in general relativity and well-motivated alternative theories of gravity, and also include extrapolations that follow sound theoretical principles, such as consistency with conservation laws and symmetries. As an example, we construct parameterized post-Einsteinian templates for the binary coalescence of equal-mass, non-spinning compact objects in a quasi-circular inspiral. The parametrized post-Einsteinian framework should allow matched filtered data to select a specific set of post-Einsteinian parameters without a priori assuming the validity of the former, thus either verifying general relativity or pointing to possible dynamical strong-field deviations.
i.1 The concept of bias and
the validity of General Relativity
The problem of bias has had a negative impact on many scientific endeavors by causing scientists to draw incorrect inferences from valid data. As the next generation of gravitational wave (GW) detectors LIG (); VIR (); GEO () promises to open a new observational window onto the universe, one wonders what bias might be present in this new branch of astronomy that could lead to incorrect conclusions about the universe seen in GWs. This is a particularly pertinent question to address early on, as the first observations are expected to have quite low signal-to-noise ratio (SNR).
Bias is here to be understood as some set of a priori assumptions, prejudices or preconceptions that affect conclusions derived from properly collected data. We purposely exclude from this definition experimentally-induced systematic errors, as we wish to concentrate on errors introduced during post-processing and analysis. In GW astrophysics, a principal source of bias is the a priori assumption, often unstated, that general relativity (GR) is the correct theory that describes all gravitational phenomena at the scales of relevance to GW generation and propagation. This bias is ingrained in many of the detection and parameter estimation tools developed to mine GW data, and extract astrophysical information from them (see for example Fairhurst et al. (2009) and references therein).
The study of theoretical bias in GW astrophysics is not completely new. Recently, Cutler and Vallisneri Cutler and Vallisneri (2007) analyzed the issue of systematic error generated by the use of inaccurate template families. This issue can be broadly thought of as a modeling bias, where the preconception relates to physical assumptions to simplify the solutions considered (e.g. that all binaries have circularized prior to merger), or unverified assumptions about the accuracy of the solution used to model the given event. The study in Cutler and Vallisneri (2007) considered inaccuracies arising from errors in solutions to the Einstein equations due to the use of the post-Newtonian (PN) approximation scheme, and they found these inaccuracies could dominate the error budget. This paper differs from that analysis in that we are concerned with the validity of the Einstein equations themselves, an assumption that can be considered a fundamental bias.
Systematic errors created by fundamental bias may be as large, if not larger, than those induced by modeling bias, as waveforms could deviate from the GR prediction dramatically in the dynamical strong-field, if GR does not adequately describe the system in that region. This is particularly worrisome for template-based searches, as the event that will be ascribed to a detection will be the member of the template bank with the largest SNR. Given that GR is quite well tested in certain regimes, many sources cannot have deviations so far from GR as to prevent detection with GR templates (albeit with lower SNR). Thus, if templates are used based solely on GR models, although the corresponding events may be “heard”, any unexpected information the signals may contain about the nature of gravity will be filtered out.
But is it sensible to expect GR deviations at all? GR has been tested to high accuracy in the Solar System and with binary pulsars Will (2006, 1993). However, in the Solar System gravitational fields are weak and particle velocities are small relative to the speed of light, and all such tests only probe linear perturbations in the metric beyond a flat Minkowski background. Binary pulsar tests begin to probe the strong-field regime in terms of the compactness of the source (i.e. the ratio of the source mass to its radius), though known systems are still quite weak with regard to the strength of the dynamical gravitational fields (characterized by the ratio of the total mass of the system to the binary separation). In this sense then, the dynamical, strong-field region of GR has so far eluded direct observational tests Psaltis (2008).
The lack of experimental verification of GR in the dynamical strong-field could be remedied through GW observations. For example, a compact object such as a black hole (BH) or neutron star (NS) spiraling into a supermassive BH will emit waves that carry a map of the gravitational field of the central BH, a program of relevance to the planned Laser Interferometer Space Antenna mission (LISA) lis (), and sometimes referred to as bothrodesy Ryan (1997a); Collins and Hughes (2004); Hughes and Menou (2005); Schutz et al. (2009). Such maps would then allow for detailed tests of alternative theories of gravity. Additional work on constraining alternative theories have concentrated on using either the binary inspiral or the post-merger ringdown phase. With the former, studies have focused on the search for the graviton Compton wavelength Will (1998); Finn and Sutton (2002); Sutton and Finn (2002); Will and Yunes (2004); Berti et al. (2005a, b); Arun and Will (2009); Stavridis and Will (2009), the existence of a scalar component to the gravitational interaction Damour and Esposito-Farese (1998); Scharre and Will (2002); Will and Yunes (2004); Berti et al. (2005a, b); Yagi and Tanaka (2009); Cannella et al. (2009); Stavridis and Will (2009); Arun and Will (2009) and the existence of gravitational parity violation Alexander et al. (2008a); Yunes and Finn (2009). Studies of the ringdown phase have concentrated on violations of the GR no-hair theorem (that the mass and spin completely determine the gravitational field of a rotating BH) by proposing to verify certain consistency relations between quasinormal ringdown (QNR) modes Thorne (1997); Dreyer et al. (2004); Berti et al. (2006); Berti and Cardoso (2006). All of the above tests, however, have focused on specific alternative theories and have not investigated the issue of fundamental bias that we wish to address here.
i.2 Toward a framework for detecting fundamental bias in GW observations of compact object mergers
In this paper, we are not so much interested in specific tests of particular alternative theories, but rather we want to introduce a framework that may allow one to quantifiably investigate the consequences of fundamental bias in GW astronomy. We propose to do so via the introduction of the parameterized post-Einsteinian (ppE) framework, analogous to the parameterized post-Newtonian (ppN) Nordtvedt (1968); Will and Nordtvedt (1972); Will (1971); Nordtvedt and Will (1972); Will (1973, 2006) or the parameterized post-Keplerian (ppK) ones Damour and Taylor (1991, 1992); Will (2006), where model waveforms are enhanced in a systematic and well-motivated manner by parameters that can measure deviations from GR. The construction of such a ppE framework is a quixotic task in general, and so to make it more manageable we will concentrate on a subset of possible GW sources: the inspiral, merger and ringdown of binary BH-like compact objects.
Given that we want to liberate GW astronomy from the assumption that GR is correct in all regimes, what sense then does it make to begin with a binary BH merger, an event wholly within the realm of GR? First, there is strong observational evidence that highly compact objects exist, and in many cases the observations are consistent with the supposition that they are BHs Narayan (2005): the high luminosity of quasars and other active galactic nuclei (AGN) can be explained by gravitational binding energy released through gas accretion onto supermassive () BHs at the centers of the galaxies Rees (1984); Ferrarese and Ford (2004); several dozen X-ray binary systems discovered to date have compact members too massive to be NSs, and exhibit phenomena consistent with matter interactions originating in the strong-field regime of an inner accretion disk McClintock and Remillard (2003); the dynamical motions of stars and gas about the centers of nearby galaxies and our Milky Way Galaxy suggest the presence of very massive, compact objects, the most plausible explanation being supermassive BHs Gebhardt et al. (2000); Schodel et al. (2002); Ghez et al. (2005). Therefore, it is not too bold to assume that even if GR does not accurately describe compact objects, whichever theory does must nevertheless still permit BH-like solutions. Second, from studies of the orbital decay of the Hulse-Taylor Hulse and Taylor (1975) and more recently discovered binary pulsars Burgay et al. (2003); Kramer et al. (2006), it is again a rather conservative conclusion that the early evolution of binary compact objects in the universe is adequately governed by GR—namely, that binary systems are unstable to the emission of quadrupole gravitational radiation.
Based on the above considerations, starting with GR binary BH merger waveforms seems sound. We are then faced with the question of how to modify these waveform in a sensible manner. In theory, there are uncountably many conceivable modifications to GR that only manifest in the late stages of the merger. To make this question manageable, we shall guide our search for ppE expansions by looking to alternative theories that satisfy as many of the following criteria as possible:
Metric theories of gravity: theories where gravity is a manifestation of curved spacetime, described via a metric tensor, and which satisfies the weak-equivalence principle Misner et al. (1973).
Weak-field consistency: theories that reduce to GR sufficiently when gravitational fields are weak and velocities are small, i.e. to pass all precision, experimental and observational tests.
Strong-field inconsistency: theories that modify GR in the dynamical strong-field by a sufficient amount to observably affect binary merger waveforms.
Notice that the weak-field consistency criterion also requires the existence and stability of physical solutions, such as the Newtonian limit of the Schwarzschild metric to describe physics in the Solar System. One might also wish that other criteria be satisfied, such as well-posedness of the initial-value problem, the existence of a well-defined, relativistic action, and that the theory be well-motivated from fundamental physics (eg. string theory, loop quantum gravity, the Standard Model of elementary interactions, etc), but we shall not impose such additional requirements here.
Instead of concentrating on a particular theory that satisfies the above criteria, here we are more concerned with being able to measure generic deviations from GR predictions. In spite of the possibly uncountable number of modifications one can introduce to the GR action, parameterizations of generic deviations is easier if one concentrates on the waveform observable, the response function, which in turn can be described by a complex function in the frequency domain. Deviations from GR then translate into deviations in the waveform amplitude and phase. Such deviations could be induced, for example, by modifications to the GW emission formulae (e.g. the quadrupole formula), new polarization modes, new propagating degrees of freedom and new effective forces. In order to ease the flow of the paper, we will not further discuss details of how certain alternative theories modify the gravitational waveform here, referring the interested reader to Appendix A.
i.3 Two key questions related to fundamental bias
In view of the above considerations, we shall attempt to lay the foundations to answer the following two questions related to fundamental bias in GW astronomy, here focusing on binary compact object mergers:
Suppose gravity is described by a theory differing from GR in the dynamical, strong-field regime, but one observes a population of merger events filtered through a GR template bank. What kinds of systematic errors and incorrect conclusions might be drawn about the nature of the compact object population due to this fundamental bias?
Given a set of observations of merger events obtained with a GR template bank, can one quantify or constrain the level of consistency of these observations with GR as the underlying theory describing these events?
As we will discuss in the next subsection, the ppE framework will be able to answer question , at least for the class of deviations from GR considered here. As for question , we shall not specifically address it in this paper, though will here give a couple of brief examples to clarify the question. As a first example, suppose the “true” theory of gravity differs from GR in that scalar radiation is produced in the very late stages of a merger, and during the ringdown. This will cause the inspiral to happen more quickly compared to GR, resulting in a late-time dephasing of the waveform relative to a GR template. Also, less power may be radiated in GWs during the ringdown phase. In all then, these events may be detected by GR templates, though with systematically lower SNR, i.e. be seemingly more distant. Here then, the fundamental bias could make one incorrectly infer that merger events were more frequent in the past.
For a second hypothetical example, consider an extreme mass ratio merger, where a small compact object spirals into a supermassive BH. Suppose that a Chern-Simons (CS)-like correction is present, altering the near-horizon geometry of the BH as described in Yunes and Pretorius (2009); Konno et al. (2009). To leading order, the CS correction reduces the effective gravitomagnetic force exerted by the BH on the compact object; in other words, the GW emission would be similar to a compact object spiraling into a GR Kerr BH, but with smaller spin parameter . Suppose further that near-extremal () BHs are common (how rapidly astrophysical BHs can spin is an interesting and open question). Observation of a population of CS-modified Kerr BHs using GR templates would systematically underestimate the BH spin, leading to the erroneous conclusion that near-extremal BHs are uncommon, which could further lead to incorrect inferences about astrophysical BH formation and growth mechanisms.
i.4 Toward a ppE Construction
The concept of a ppE framework is in close analogy to the ppN one, proposed by Nordtvedt and Will Nordtvedt (1968); Will and Nordtvedt (1972); Will (1971); Nordtvedt and Will (1972); Will (1973, 2006) to deal with an outbreak of alternative theories of gravity in the 1970’s. Such a framework allows for model-independent tests of GR in the Solar System, through the introduction of ppN parameters in the weak-field expansion of the metric tensor. When these parameters take on a specific set of values, the metric tensor becomes identical to that predicted by GR, while when it takes on different values, the gravitational field becomes that predicted by certain alternative theories. In the same way, we introduce ppE parameters to interpolate between different theories, but we parameterize the GW response function instead of the metric tensor, as the former is the observable in GW astrophysics.
We shall follow the ppN route in the construction of a ppE framework: we will explore a set of alternative theories and their effects on GWs, and from these, we will phenomenologically infer and engineer a ppE template family that not only captures the waveforms from known theories, but also other phenomenological corrections. As we shall see, our approach reproduces corrections to the GW response functions predicted by a large class of alternative theories of gravity proposed to date, including Brans-Dicke theory, massive graviton theories and non-dynamical CS modified gravity.
As a first step, we shall not consider here the effect of scalar, vectorial or all six tensorial polarizations on the response function, nor will we attempt to classify alternative theories based on the different perturbative modes that can be excited. In fact, such a classification already exists, the scheme Eardley et al. (1973); Will (1993), which organizes different alternative theories according to whether they excite certain contractions of the Weyl tensor as characterized by the Newman-Penrose (NP) scalars. Here we will concentrate on the two “plus” and “cross” tensorial degrees of freedom, and how these change in alternative theories. Additional excited modes will only be considered insofar as they affect the source dynamics, and thus indirectly the structure of the plus and cross modes.
In the future it would be useful to extend the ppE templates to include additional GW polarizations and modes. However that would be a non-trivial effort, the main reason being that currently there is a lack of alternative theories that simultaneously satisfy the criteria discussed above and allow for the excitation of additional modes. For example, Brans-Dicke theory does possess a breathing mode, though it is strongly suppressed by constraints on its coupling constant. Similarly, Einstein-Aether theory generically has 5 propagating degrees of freedom, but these decouple and radiation can be shown to remain quadrupolar Jacobson (2008). Thus, unlike with the other ppE extensions considered here, there is little theoretical guidance available on how GR waveforms would be modified if additional GW polarizations are excited. Without such guidance then, it will be almost impossible to argue that any proposed extensions are well-motivated.
A separate issue is whether the effect of additional polarization states can be observed with contemporary or planned detectors. For direct measurement one requires arms to be sensitive to polarization states Cornish (), for example using multiple ground-based detectors, Doppler tracking of multiple spacecraft, or pulsar timing arrays Hellings (1978). Alternatively, a space-borne detector such as LISA can be sensitive to multiple polarization states for events that last a sizable fraction of the spacecraft’s orbit, as the motion of detector then effectively samples the waveform with multiple arm orientations. For ground-based detectors, binary merger events happen too quickly for a similar strategy to be effective. However, other sources, for example the waves that could be emitted by a “mountain” on a pulsar, will produce coherent GWs over many rotation periods of the Earth, and thus could also contain information on multiple polarization states. We will not consider these interesting issues here.
i.5 A ppE template family for the inspiral, merger and ringdown of black hole-like compact objects
For this initial study, we construct ppE templates describing only the quasi-circular coalescence of non-spinning and equal-mass compact objects. Even within this restricted class of events, the ppE construction is non-unique, and certainly more refined versions could be developed. We introduce several ppE template families that vary in the number of ppE parameters, and thus, in the amount of fundamental bias each family assumes in its construction. The frequency-domain (overhead tilde) ppE templates with the least number of ppE parameters that we derive is the following (more general representations are provided in Sec. V.3):
where the subscript and stand for inspiral-merger and merger-ringdown. In the inspiral phase (), the GW is described by a “chirping” complex exponential, consisting of the GR component corrected by ppE amplitude and phase functions with parameters . Here is the inspiral reduced frequency, is the chirp mass with symmetric mass ratio and total mass (though again, we will only focus on the equal mass case). The merger phase () is treated as an interpolating region between inspiral and ringdown, where the merger parameters are set by continuity, and the merger ppE parameters are . In the ringdown phase (), the GW is described by a single-mode generalized Lorentzian, with real and imaginary dominant frequencies and , ringdown parameter also set by continuity, and ppE ringdown parameters . The transition frequencies can either be treated as ppE parameters, or (for example) set to the GR light-ring frequency and the fundamental ringdown frequency, respectively. In a later section, we shall present more general ppE waveforms with inherently less fundamental bias, some of which are reminiscent to the work of Arun et al. (2006a, b); Cannella et al. (2009), except that here we are interested in tests of GR through the determination of ppE parameters, instead of the measurement of PN ones.
The ppE template presented above is non-unique; in fact, the ppE framework as a whole is inherently non-unique, as are the ppN or ppK ones, because a finite parameterization cannot represent an infinite space of alternative theory templates. However, this ppE family is minimal within the class of templates considered here, as it employs the smallest number of ppE parameters necessary to reproduce corrections to the GW response function from well-known alternative theories of gravity in the inspiral phase:
GR is reproduced with , and ;
Jordan-Brans-Dicke-Fierz, or simply Brans-Dicke theory (BD), with , and related to the Brans-Dicke coupling parameter (see Eq. (58));
Massive graviton (MG) theories with , and related to the graviton Compton wavelength (see Eq. (62));
CS modified gravity with , and related to the CS coupling parameter (see eg. Eq. (70)).
The allowed range of the ppE parameters is not completely free, as Solar System and binary pulsar experiments have already constrained some of them. The subset of inspiral ppE templates that reproduce well-known theories after fixing and is then constrained to some range in . More precisely, since the Brans-Dicke coupling parameter has been constrained to by the Cassini spacecraft Bertotti et al. (2003), this automatically forces when , where are the sensitivities of the binary components. For example, for a binary NS system, when . Similarly, since the Compton wavelength of a massive graviton has already been constrained to from pulsar timing observations Baskaran et al. (2008), this implies when , where is a distance measure to the source. For example, for a binary NS system at a redshift of , one finds the prior when . Finally, the non-dynamical CS coupling parameter has been constrained by binary pulsar observations to be such that for an aligned binary (zero inclination angle), when , which for a binary NS at redshift translates to .
One can also consider placing priors on other ppE parameters introduced in Eq. (1) without fixing the waveform to represent a particular well-known theory. However, since the merger and the ringdown have not been constrained at all by observation, one cannot really restrict . On physical grounds, one can only require that and real and also real, such that the waveform is well behaved at large frequencies. Since the quadrupole formula has been verified to leading order with binary pulsar observations, one also expects that for low-frequencies , which for example implies that if then , while if then is essentially unconstrained. If one does not expect gravitational radiation to be sourced below dipolar order one must have ; similarly, if one expects strong-field, source generation deviations to only arise beyond quadrupolar order then . Of course, the stronger the priors imposed, the stronger the bias the ppE model inherits.
i.6 The Possible Role of the ppE Framework in Data Analysis
We do not propose here to employ the ppE templates for direct detection, but rather for post-detection analysis. Following the detection of a GW by a pure GR template, that segment of data could then be reanalyzed with the ppE templates. In principle, one should search over all system and ppE parameters, but in practice, since the ppE waveforms are deformations of GR templates, one can restrict attention to a neighborhood of the system parameter space centered around the best-fit ones obtained through filtering with a GR waveform.
The templates presented above do contain some modeling error, i.e., the subset of the ppE template family with all parameters set to the GR values—the ppE-GR templates—are not the “exact” solution to the event in GR (for which there is no known closed form solution). What is more important though is that the ppE-GR templates have a high overlap or fitting factor with the correct waveform, which is why the ppE templates have been built as generalizations of GR approximations with that particular property Ajith et al. (2008). The modeling error in the ppE-GR templates could be determined by computing fitting factors between them and the corresponding GR templates, folding in the detector noise curve, or following the formalism laid out in Cutler and Vallisneri (2007). This would then provide a measure of the efficacy of the ppE templates in detecting deviations from GR, or constraining alternative models, with the given event.
Although the practical implementation of an efficient post-detection analysis pipeline will be relegated to future work, one should not necessarily restrict such analysis to a frequentist approach. An alternative would be to perform a Bayesian model selection study Umstatter and Tinto (2008); Cornish and Littenberg (2007) to determine whether the evidence points toward GR or a GR deviation (through the non-vanishing of ppE parameters). The idea is that more parameters in a template bank will generically improve the fit to the data, even if there is no signal present. Thus, any search should be guided by careful consideration of priors (such as those discussed earlier) or through the computation of evidence associated with a given model, for example as in Feroz et al. (2009). Such considerations are particularly important for low SNR events, such as those expected for ground-based detector sources.
A clear path then presents itself for possible strategies that one might pursue in the study of fundamental bias:
Given a GR signal and a ppE template, how well can the latter extract the former? how much modeling error is intrinsic in the ppE template family?
Given a non-GR signal and a GR template, how much fundamental bias-induced systematic error is generated in the estimation of parameters? Can the signal even be extracted?
Given a non-GR signal and a ppE template, how well can the latter extract the former? How well can intrinsic and ppE parameters be estimated?
These questions constitute a starting point for future data analysis studies to investigate the issue of fundamental bias in GW astronomy and refinements of the ppE framework.
i.7 Organization of this paper
An outline of the remainder of this paper is as follows: Sec. II discusses the anatomy of binary BH coalescence in GR; Sec. III derives some model-independent modifications to the inspiral GW amplitude and phase, while Sec. IV does the same for the merger and ringdown phase; Sec. V uses the results from the preceding sections to construct an particularly simple example of a ppE template bank; Sec. VI concludes and points to future research. Appendix A presents a brief summary of well-known alternative theories and their effect on the GW observable, and Appendix B describes suggested “exotic” alternatives to BHs.
We follow here the conventions and notation of Misner, Thorne and Wheeler Misner et al. (1973). In particular, Greek letters stand for spacetime time indices, while Latin letters stand for spatial indices only. Commas in index lists stand for partial differentiation, while semi-colons stand for covariant differentiation. The Einstein summation convention is assumed unless otherwise specified. The metric signature is and we use geometric units where . In the Appendices, we reinstate the powers of G and c, as one must distinguish the GR coupling constants from those in alternative theories of gravity.
Ii Anatomy of a Compact Object Binary Coalescence in GR
Let us now consider the quasi-circular binary coalescence of non-spinning BH-like compact objects—namely, events that exhibit inspiral, merger and ringdown phases akin to binary BH mergers in GR. Certain classes of “exotic” horizon-less compact objects, such as boson stars (see Appendix B), may exhibit similar merger waveforms, and insofar as they are viable candidates to explain compact objects in the universe that today are classified as BHs, they are also within the scope of the ppE templates we will construct here. Neutron star mergers, for example, are not to be consider part of this family, as these are a separate class of GW sources that would require a different ppE expansion (though ostensibly the inspiral phase of a NS/NS merger ppE template would be the same as the BH/BH one, and if the merger regime for the former happens outside the detector’s sensitivity window, as is expected to be the case with LIGO/GEO/Virgo, then the binary BH ppE templates would suffice).
We divide the coalescence into three stages: (i) the inspiral; (ii) the plunge and merger; (iii) the ringdown. In the first stage, the objects start widely separated and slowly spiral in via GW radiation-reaction. In the second stage, the objects rapidly plunge and merge, roughly when the object’s separation is somewhere around the location of the light-ring. In the third stage, after a common apparent horizon has formed, the remnant rings down and settles to an equilibrium configuration. Note that this classification is somewhat fuzzy, as studies of numerical simulation results have shown that a sharp transition does not exist between them in GR Buonanno et al. (2007); Sperhake et al. (2008); Berti et al. (2007a). In the remainder of this section, we concentrate on GR coalescences and leave discussions about possible modifications to the next section.
In describing the waveform, we shall concentrate on the Fourier transform, , of the GW response function, , as the former is more directly applicable to GW data analysis (see eg. Finn (1992); Jaranowski and Krolak (2005) for a data analysis review) . The response function of an interferometer, , is a time-series defined via the projection , where are beam-pattern functions that describe the intrinsic detector response to a GW, while are the projections of the GW metric perturbation onto a plus/cross polarization basis. We will split the Fourier transform of the response function into three components, , and , corresponding to the inspiral, merger and ringdown phases respectively. We will further decompose each of these complex functions into their real amplitude and phase components:
all of which will be modified in the ppE framework.
The inspiral phase can be modeled very well by the restricted PN approximation (see eg. Yunes et al. (2009)). In this approximation, the amplitude of the time-series is assumed to vary slowly relative to the phase, allowing its Fourier transform to be calculated via the stationary-phase approximation Bender and Orszag (1999); Droz et al. (1999); Yunes et al. (2009). Improving on the restricted approximation leads to sub-leading amplitude corrections that introduce higher harmonics in the Fourier transform, though we shall not consider that here Droz et al. (1999); Arun et al. (2007). However, we should note that if considering extreme-mass ratio inspirals (EMRIs) or eccentric orbits, then higher harmonics will play a significant role and should be accounted for (see eg. Yunes et al. (2009)).
In GR, within the restricted PN approximation the Fourier transform of the response function can be written as
where we recall that is the reduced frequency parameter, is the chirp mass, is the symmetric mass ratio, is the total mass and is twice the Keplerian orbital frequency. The coefficient depends on the chirp mass, the luminosity distance and the orientation of the binary relative to the detector (parameterized through the inclination and polarization angle ) via111When modeling the Fourier transform of the response function for LISA, one must multiply by a factor of to account for the geometry of the experiment. See eg. Eq. and of Yunes et al. (2009). Moreover, the beam pattern functions further depend on the motion of the detector around the Sun, which can be modeled as in eg. Cutler (1998); Finn and Chernoff (1993):
where is a function of the beam-pattern functions , the inclination and the polarization angle .222Note that the function as defined here differs from that defined in Yunes et al. (2009) by a factor of . The quantities and are constants, related to the time and phase of coalescence, while are constant PN phase parameters that depend on the mass ratio. To PN order, these coefficients are given by
and expressions to higher order are also known Blanchet et al. (2002).
The merger phase cannot be modeled analytically by any controlled perturbation scheme. We shall thus treat this phase as a transition region that interpolates between the inspiral and the ringdown. During this phase, numerical simulations have shown that the amplitude and the phase can be fit by Buonanno et al. (2007); Ajith et al. (2008)
where is the frequency of transition between inspiral and merger and where and are new constants, and can be related to and by demanding continuity of the waveform. The frequency dependence of the amplitude in the merger phase, ,is not universal, but specific to the test system (equal mass, non-spinning) studied here333Though note that in Baumgarte et al. (2008) it was suggested that the exponent might be closer to ..
The ringdown phase can be modeled very well through BH perturbation theory techniques (see eg. Berti et al. (2009) for a recent review). In this approximation, the GW response function can be written as a sum of exponentially damped sinusoids, the Fourier transform of each being a Lorentzian function. Furthermore, for equal mass mergers, a single (least damped) QNR mode dominates the waveform, and so to good approximation we can write Ajith et al. (2008)
where is the width of the Lorentzian that essentially describes the mean duration of the ringdown phase (the damping time). Notice that during the plunge and the ringdown the phase remains the same, modulo non-linear effects, which conveniently guarantees continuity of the phase across this boundary.
The phenomenological parameters can be fit to numerical simulations to find Ajith et al. (2008)
where the arrow stands for the approximate value for equal mass binaries. The quantities denote the boundaries of the inspiral-merger region and merger-ringdown region, at which the piece-wise function has been constructed to be continuous. Notice that the merger and ringdown phases are both rather short compared to the inspiral phase, lasting approximately Hz and Hz respectively. Due to the short duration of these phases, they can be modeled via the simple linear relation in the phase shown above, and in fact, as the merger phase is fairly constant, it could be absorbed in the overall amplitude.
The parameterization presented here is purposely similar to that proposed in Ajith et al. (2008). In that study, however, the phase was modeled in all stages of coalescence by the same function, namely Eq. (4), where the parameters are not fixed to the PN value, but instead fitted to numerical relativity waveforms (see eg. Table in Ajith et al. (2008)). If one prefers, one could model the GW phase in the merger and ringdown with the fitted parameters presented in Table of Ajith et al. (2008). Alternatively, one could also follow the prescription of Berti et al. (2006) and construct the Fourier transform of the response function for an infinite sum of modes, absorbing the phase into a complex amplitude and paying more careful attention to the response function, but we leave such details for future work.
Iii Model-Independent Modifications to the Inspiral Phase
We shall now consider leading-order deviations from GR in the inspiral phase. In the restricted PN and stationary phase approximations, the Fourier transform of the response function can be written as (see eg. Eq. in Yunes et al. (2009))
where is the time-domain amplitude evaluated at the stationary point, defined via , and overhead dots stand for partial differentiation with respect to time. is the orbital frequency, and is its rate of change evaluated at the stationary point ; recall that is the GW frequency. The quantity is the GW phase, which is given by (see eg. Eq. in Yunes et al. (2009))
The orbital frequency and its rate of change completely determine the Fourier transform of the response function in the stationary phase approximation. The rate of change of the orbital frequency is generically computed through the product rule , where is the orbital binding energy. The quantity is calculated by invoking the so-called balance law. This law states that the amount of energy carried away in GWs is equal to the amount of orbital binding energy lost by the binary system, i.e. , where is the GW luminosity. The luminosity can be computed perturbatively in terms of a multipolar decomposition of the effective energy-momentum tensor of the system Thorne (1980), which to leading order yields , where is the reduced quadrupole moment tensor of the binary.
We will consider GR modifications of the Fourier transform of the response function during the inspiral arising either from changes to the conserved binary binding energy (the Hamiltonian, see eg. Blanchet (2006) for a more precise definition), or changes to the energy balance law. The former are essentially modifications to the conservative dynamics, while the latter modifies the dissipative dynamics. We shall tackle each of these corrections next.
iii.1 Modifications to the Binary Hamiltonian
The Hamiltonian for a binary system, or equivalently, the binary’s center of mass binding energy, can usually be expressed as a linear combination of kinetic energy and potential energy : , where and stand for generalized coordinates and conjugate momenta. For a binary system, one can extract the Newtonian contribution and rewrite such a generic Hamiltonian as
where is the relative velocity, is the reduced mass, is the orbital separation, and and are as before the total mass and symmetric mass ratio. The quantity is some additional contribution to the Hamiltonian that can be due either to relativistic effects, or to non-GR contributions, for example from additional scalar or dipolar degrees of freedom. Of course, not all functions are allowed, since binary pulsar observations have already verified GR in a quasi-weak field, which translates into the requirement when .
From such a Hamiltonian, one can derive a modified version of Kepler’s third law (or equivalently Newton’s second law), namely
where is the binary’s orbital angular frequency and . We see then that modifications to the Hamiltonian and to Kepler’s law are not independent, as expected.
Kepler’s law must be inverted to obtain the separation as a function of angular frequency, which can then be inserted into the Hamiltonian to obtain . Although the unspecified form of does not allow an explicit inversion, we can always write the inversion implicitly as
where is some function of determined from the inversion of Eq. (15). As for , the quantity must also satisfy the condition in the weak field.
One possible parameterization of that satisfies the above condition is
where is a PN parameter, while and are dimensionless, real numbers. The parameters and determine roughly where the non-Newtonian correction to Kepler’s law begins to become important, and they must be real since is real. Furthermore, the condition that when implies that in the inspiral phase. The choice of parameterization of in Eq. (17) is equivalent to choosing a parameterization for the potential in Eq. (14); with our choice one finds that
to first order in .
We have here chosen a parameterization for that allows an inversion analytically in terms of known functions. A more general parameterization would be to add terms with exponents . However, the higher the exponent, the smaller its contribution, since the inspiral phase must terminate outside the light-ring where in GR (for simplicity, we here work with one exponent only). In fact, in GR one finds that perturbatively
when from Eq. of Blanchet (2006), where here is a gauge parameter associated with harmonic coordinates. From the GR expression, we also infer that the constant could depend on the system parameters, such as the mass ratio. Also, must depend on some non-GR coupling intrinsic to the alternative theory, such that in the limit as this parameter vanishes, one recovers the GR prediction.
The binary’s center-of-mass energy can now be expressed entirely in terms of the orbital frequency. Requiring the circular orbit condition , we find
In principle, we should not expand in , since we are interested in corrections that could arise in the late inspiral (though we know that even close to merger and in GR ). To simplify the following analysis, we shall perform this expansion regardless, as it only serves to motivate the final ppE parameterizations, where can take large values in the late inspiral. Performing the expansion, we find
and differentiating with respect to we obtain
The above considerations provide motivation to extrapolate a model-independent, non-GR parameterization for :
where we recall that for circular orbits. We have simplified the above parameterization by introducing and . Such an expression resembles the PN expansion of GR, which one can derive from Eq. of Blanchet (2006):
Similarly, the parameter could depend on but it must also be proportional to a positive power of the alternative theory couplings, such that when these vanish, one recovers the PN expansion of the GR expectation.
iii.2 Modifications to the Energy Balance Law
The energy flux at infinity is usually calculated in GR by assuming that the only source of radiation are GWs, so that , where we recall that is the GW luminosity. In alternative theories of gravity, however, there could be additional sources of radiation from other propagating degrees of freedom. The balance law then becomes .
The calculation of the GW luminosity is generically performed in two steps. First, one employs perturbation theory to determine the effective (Isaacson) GW stress-energy tensor Isaacson (1968a, b), the time-time component of which provides an expression for in terms of time derivatives of the metric perturbation . In GR, this expression is schematically . Second, one performs a multipolar decomposition of the metric into mass and current multipoles Thorne (1980), which are given by integrals over the sources and for point sources are simply proportional to their trajectories and velocities. In GR, one then schematically finds , where is the reduced quadrupole moment and is the distance to the observer. Combining both steps into one single procedure, one finds that the GW luminosity in GR is proportional to third time-derivatives of the reduced quadrupole moment .
In general, one can decompose the GW luminosity in a superposition of mass and current multipole moments Thorne (1980)
where is the monopole of the system, is its dipole moment and are proportionality constant for the monopole, dipole and quadrupole contributions respectively. The quantity has the effect of re-normalizing Newton’s gravitational constant , and thus, can be bundled together with . On the other hand, the quantity is related to the self-binding energy of the gravitating bodies, and it produces an independent contribution to the GW luminosity. In GR, these two contributions (monopole and dipole) identically vanish due to center-of-mass and linear-momentum conservation. Evaluating Eq. (25) with standard definitions of multipole moments for a binary system Thorne (1980), one finds to leading order
where are the so-called Peters-Mathews parameters, whose numerical value depends on the alternative theory under consideration, while is a dipole self-gravitational energy contribution, with the difference in self-gravitational binding energy per unit mass (see eg. Eq. in Will (1993)). For example, in GR , while in Brans-Dicke theory without a cosmological constant Will (1993), where is the BD coupling parameter.
Using the modified Kepler’s law from the previous subsection, we can re-express the power as a function of frequency only, namely
where we have set , corresponding to a circular orbit. The dipolar term dominates over GR’s quadrupolar emission during the early inspiral, which explains why Brans-Dicke theory, whose dominant signature is precisely dipolar radiation (see Appendix A.1), has been constrained so well by binary pulsar observations. On the other hand, the modification to Kepler’s law becomes important as the binary approaches the end of the inspiral phase, as expected since it is meant to model dynamical strong-field corrections to GR.
The quantity serves to parameterize both additional contributions from scalar or vectorial degrees of freedom, as well as high-order curvature corrections to the GW effective stress-energy tensor. In the case of the former, one could for example have , where is some scalar field that is activated when the binary approaches the merger phase. In fact, such is the case in dynamical CS modified gravity (see Appendix A.3), where the CS field becomes strongly sourced during the late inspiral, plunge and merger, as the Riemann curvature grows. Here, one could for example have , an interaction that is absent in GR.
iii.3 Modified GW Phase
We can now combine the results from the previous two sections to find :
where we have rescaled , , , and where the new constant parameter can be related to , though the expression is not particularly illuminating. We have also here dropped second-order terms, such as the influence of the modified Keplerian relation on dipolar emission. The parameter has the effect of re-normalizing the value of or , which would not be directly observable in a GW detection, as it would be directly reabsorbed by physical quantities, like the chirp mass (unless one possesses an electromagnetic counterpart). The parameter parameterizes dipolar deviations and depends on the sensitivities of the bodies. The parameters and parameterize deviations from Kepler’s law, while describes possible additional sinks of energy.
We can now integrate Eq. (13) to obtain the modified GW phase:
where and are new parameters that again can be related to the old ones, though these relations are unilluminating. As is customary, we have here linearized in the GR corrections to be able to perform the integral analytically, an approximation that is valid in the inspiral, although not necessarily so in the plunge and merger. Note that the term proportional to mimics the Brans-Dicke correction to the GW phase, which is known to scale as (see discussion around Eq. (58)), while mimics the effects of a massive graviton on the propagation of GWs, which is known to scale as (see discussion around Eq. (62)).
iii.4 Modified GW Amplitude
The GW amplitude depends both on the amplitude of the time-domain waveform evaluated at the stationary point and the rate of change of the frequency . The latter has already been parameterized in Eq. (28). The former, however, requires further study. In GR, the quadrupole formula requires that
which leads to the following waveform amplitude to leading order for a binary system
Combining this expression with Eq. (28), we find that is equal to
where is another constant and we now kept only the leading-order corrections. Such an expression agrees identically with the GR expectation when and all other modified parameters vanish.
The above results hinge on the assumption that the quadrupole formula in Eq. (30) holds to leading order in the GR modification. In well-known alternative theories of gravity, including scalar-tensor ones, Eq. (30) is indeed valid to leading order, acquiring subdominant corrections. These corrections can be modeled by the undetermined function , in the same fashion as in Sec. III.2. Thus, the generalization of Eq. (30) leads to redundancies in the parameterization that we are after. In Sec. V we shall use the insight gained from the analysis of this section to infer a generalized parameterization for a model-independent modification of the inspiral gravitational waveform.
Iv Model-Independent Modifications to the Merger and Ringdown Phase
iv.1 Merger Phase
Let us now consider leading-order deviations from GR first during the merger and then the ringdown phase. The merger phase is the least understood regime of a binary coalescence, because no controlled approximation scheme can be employed to solve the field equations. In GR, numerical simulations of the merger of two, non-spinning, equal-mass BHs has produced the phenomenological fit
where the phase is a linear function . Comparing to the results of Sec. II we find that and . Linearity in the phase can be assumed due to the short duration of the merger phase (usually less than 2 GW cycles for BHs). This functional form is an approximate fit and found only a posteriori, i.e. after a full numerical solution has been obtained.
Lack of formally derivable analytic approximations of the merger waveform is not exclusive to GR. Furthermore, due to the dearth of numerical studies of this regime in alternative theories (but see Salgado et al. (2008) for promising plans to explore mergers in scalar-tensor theories), little guidance can be found on how the merger regime may differ from GR. How do we then parameterize model-independent corrections of the merger waveform when we possess no analytic or numerical guidance? For lack of a better prescription, we will here assume that the waveform produced during the merger can adequately be described as an interpolation between the inspiral and the ringdown waveforms. This is a good approximation to GR waveforms, and hence is a rather conservative approach to a ppE extrapolation. Specifically then, we propose
where and are an infinite set of undetermined parameters. We shall here keep only one term in the polynomial, setting , and all other amplitude parameters to zero, while we shall leave undetermined in the phase. The parameters are set be requiring continuity between the inspiral-merger interface at : and . The choice of a single term in the power-series representation of the Fourier transform of the merger response function forces us to consider coalescences with prompt mergers. Delayed mergers can occur, for example, with exotic compact objects such as boson stars Palenzuela et al. (2008), in a process akin to delayed-collapse scenarios in NS mergers. Of course, additional terms could be added to the series of Eq. (35) to account for such scenarios, however for simplicity here we will restrict attention to prompt mergers.
iv.2 Ringdown Phase
Let us now consider the leading-order modification to GWs that could be produced during the post-merger phase due to alternative theory corrections. GWs emitted in this stage are produced by oscillations in the remnant spacetime as it settles down to its final equilibrium configuration. Simulations show that in GR the ringdown very quickly becomes linear, and we will assume the same here for the ppE expansion. In GR, the metric perturbation in general is a sum of quasi-normal ringdown (QNR) modes, which are exponentially damped sinusoids, and so-called tail terms, which are constant phase solutions that decay via a temporal power law. The QNR modes dominate the early-time post-merger waveform, which is why we label this phase the ringdown phase.
We can write the ringdown time-domain response function as
where the phase is purely real and the amplitude is a decaying function of time. Due to the assumed linearity, we can then decompose the amplitude and phase into a sum of harmonics , where labels a spheroidal harmonic mode, and is an integer describing the overtone of the given mode. For the QNR component of each mode, the amplitude is a decaying exponential
where is the decay constant of the mode, is the Heaviside function that guarantees the waveform is well defined at negative infinite time, and is some constant amplitude determined by continuity with the merger phase, which as such must depend on the beam-pattern functions. The phase of each QNR mode is
where is the real quasinormal frequency of the mode.
In GR, the Fourier transform of the QNR component of the ringdown waveform is thus a sum of Lorentzian-like functions:
This result can be obtained by rewriting the cosine in Eq. (36) as a sum of complex exponentials using Euler’s formula, and then following the doubling prescription of Flanagan and Hughes (1998), where one essentially replaces in the Fourier integral and then multiplies the result by a correction factor that accounts for the doubling Berti et al. (2006).
Tail terms in the post-merger waveform are induced either by linear interactions between the GWs and the background, or non-linear behavior right after merger. Linear tails have a long history in GR Price (1972a, b); Gundlach et al. (1994) and they are controlled by the asymptotic form of an effective potential far from the source. Such tails depend on the harmonic one is looking at but generically for a Schwarzschild background they can be parameterized by
Recently in Okuzumi et al. (2008) it was suggested that “non-linear tails” could be produced during merger. Such tails are structurally similar to the linear ones, except that one replaces the power , where is some positive integer. In this case, the Fourier transform is given by
for positive frequencies (the Fourier transform for a Schwarzschild linear tail can be easily inferred from the above equation). In GR then, one expects the complete ringdown GW to be a superposition of QNR terms and tail terms, leading to a Fourier transform that is the linear combination of Eq. (39) and (41).
What generic type of modifications could we expect from alternative theories of gravity? One possibility would be to modify the power in the exponential damping part of the ringdown, i.e. in Eq. (39) , with a positive real number. However, does not seem to be very “natural”, and would significantly complicate the analysis. For this study we simply keep . Another possible modification is to correct the tail behavior of the ringdown waves. Linear tails, however, are predominantly controlled by the dominant behavior of an effective potential in the far-field field, which in turn depends on the behavior of the background metric tensor. If we require that all stable post-merger metric tensors approach the Schwarzschild solution at late times in the far-field (by Solar System tests), it is unclear how much linear tails would be modified by non-linear terms in the gravitational field. On the other hand, non-linear tails could be modified in alternative theories of gravity, as suggested in Ching et al. (1995a, b). Nonetheless, alternative theory corrections to either linear or non-linear tail terms can both be parameterized by some undetermined power , leading to the Fourier transform shown in Eq. (41).
For simplicity then, we will propose a modified ringdown waveform with a similar functional form as the GR result given in Eq. (39), though allow the decay constants, frequencies and powers to take on non-GR values.
V Parameterized post-Einsteinian Framework
In this section we collate the results from the previous sections to construct ppE extensions of the inspiral, merger and ringdown waveform segments of a binary merger. In each of these phases, we shall divide this task into the construction of a ppE phase and a ppE amplitude, such that the waveform is given by:
where for the inspiral, merger and ringdown phases respectively. Then, we shall put these waveforms together to construct a ppE waveform for the entire coalescence. We conclude this section with a discussion of the data analysis cost incurred due to the addition of ppE parameters to the waveform.
Let us employ the insight gained by the analysis of Sec. III to propose inspiral ppE waveforms. Consider then the ppE GW phase, a generic form of which can be extrapolated from Eq. (III.3). One simple parameterization for the test system is the following:
where are PN phase coefficients, some of which are given in Eq. (6), while and are ppE parameters. Such a parameterization allows one to easily map between BD GW modifications and massive graviton ones, through the parameter choices and respectively.
One generalization would be to allow more than one alternative theory to be modeled simultaneously by the ppE waveform. With the test system in mind, such an idea suggests
The benefits of this parameterization is that it explicitly incorporates the GW phases predicted by alternative theories, as well as the PN prediction, while allowing, through the term, for new alternative theory modifications that have not yet been developed. Note however that now we have introduced a degeneracy into the waveforms; i.e. when , waveforms along the line are identical. In a sense then this is a non-optimal expansion.
A better suggestion might then be to postulate
The controlling factor in the phase expansion has been modified to instead of to allow for dipole radiation. The phase parameters can take on any values; the ppE-GR templates will be those where . Continuing in this vain then, the most general expansion we propose here for the phase is the following:
where now and are each a set of ppE parameters. The ppE-GR templates are those with , and . Again, this family can also capture the inspiral phase of several well known alternative theories (see Appendix A) with appropriate choices of the parameters. The parameterizations in are a generalization of the studies in Arun et al. (2006a, b); Cannella et al. (2009), except that we here allow for non-GR exponents in the u-dependence of the phase that could model dynamical strong-field corrections to GR.
Now let us concentrate on amplitude parameterizations. A generic form for the amplitude that can be extrapolated from Eq. (33) is the following
The quantities and are a ppE function and parameter respectively, where the former could depend on the GR parameter vector . Such a non-GR deformation of the GW amplitude is not degenerate with high-order PN corrections, since these scale with higher-harmonics of the GW signal, instead of the fundamental mode. Moreover, such a parameterization reduces to well-known alternative theory predictions, such as CS modified gravity when and reduces to the second term in Eq. (70). Of course, one could in principle introduce terms to the GW amplitude with different frequency exponents, though for simplicity we will not do so here; furthermore, GW detectors are most sensitive to phase, so there might be less to gain by introducing expansive amplitude modifications.
v.2 Merger and Ringdown
Here we employ the considerations discussed in Sec. IV to construct a ppE waveform that models the merger and ringdown phase in a non-GR, model-independent fashion. For the merger phase, we have (see Eq. (35) with Eq. (42))
The ppE parameters during the merger are then the real numbers , while the parameters are set by requiring continuity with the inspiral phase. When the ppE parameters are one recovers the GR prediction for the test system.
A simple proposal for a ppE parameterization of the ringdown waveform is
with , and where and are the dominant quasi-normal (QN) frequency and decay time, is a real ppE function of the final mass of the remnant, is a real ppE parameter, the quantity is determined by continuity with the merger phase, and where we have chosen the transition to occur at the fundamental ringdown mode . This models the ringdown with a single decaying mode, although even in GR there are an infinite number of such modes excited during ringdown (a possibility we analyze below).
A more general ringdown waveform is as follows Ferrari et al. (2001)
where and are the frequency and damping time of the -th mode, while the ppE function and parameter are assumed mode-independent. One could in principle postulate different ppE parameters for different modes, but this would introduce an immense number of such parameters, not all of which would necessarily be independent.
Traditionally, suggested tests of alternative theories of gravity using ringdown GWs are based on the “no-hair” theorem of GR. This theorem states that given the mass and angular momentum of a perturbed BH, all frequencies and damping times of the waveform are determined. For example, in GR the mode is given by Echeverria (1989); Berti et al. (2009)
where the total mass of the remnant is , the spin Kerr parameter of the remnant is , and we have defined the quality factor . The standard no-hair test assumes that two modes have been detected, the first of which can be used to determine and , while the second one can be used to check for consistency through equations similar to those above for additional modes. The violation of the two-hair theorem is common in alternative models for compact objects Berti and Cardoso (2006). With the parameterization of Eq. (