Parameterized Post-Einsteinian Gravitational Waveforms in
Various Modified Theories of Gravity
Despite the tremendous success of general relativity so far, modified theories of gravity have received increased attention lately, motivated from both theoretical and observational aspects. Gravitational wave observations opened new possibilities for testing the viability of such theories in the strong-field regime. One could test each theory against observed data one at a time, though a more efficient approach would be to first probe gravity in a theory-agnostic way and map such information to that on specific theories afterward. One example of such model-independent tests with gravitational waves is the parameterized post-Einsteinian formalism, where one introduces generic parameters in the amplitude and phase that capture non-Einsteinian effects. In this paper, we derive gravitational waveforms from inspiraling compact binaries in various modified theories of gravity that violate at least one fundamental pillar in general relativity. We achieve this by deriving relations between corrections to the waveform amplitude/phase and those to the frequency evolution and Kepler’s third law, since the latter two have already been (or can easily be) derived in many theories. Such an analysis allows us to derive corrections to the waveform amplitude, which extends many of previous works that focused on phase corrections only. Moreover, we derive gravitational waveforms in theories with varying gravitational constant. We extend the previous work by introducing two gravitational constants (the conservative one entering in the binding energy and the dissipative one entering in the gravitational wave luminosity) and allowing masses of binary constituents to vary with time. We also correct some errors in previous literature. Our results can be used to improve current analyses of testing general relativity as well as to achieve new projected constraints on many modified theories of gravity.
General relativity (GR) is one of the cornerstones of modern physics, and so far the most successful theory of gravitation. Along with the elegant mathematical structure and solid conceptual foundation, GR has passed all the tests with high accuracy Will (2014). However, there are theoretical and observational motivations which lead to the demand of a modified theory of gravitation. Regarding the former, GR is a purely classical theory and incompatible with quantum mechanics. Strong gravitational fields at Planck scale where quantum effects cannot be ignored Adler (2010); Ng (2003), such as in the vicinity of black holes (BHs) and the very early universe, require a consistent theory of quantum gravity for their complete description. Regarding the latter, puzzling observations such as the accelerated expansion of the universe Abbott (1988); Copeland et al. (2006); Perlmutter et al. (1999); Riess et al. (1998, 2004); Weinberg (1989); van Albada et al. (1985); Weinberg et al. (2013) and anomalous kinematics of galaxies Bosma (1981a, b); Begeman et al. (1991); Rubin and Ford (1970); Rubin et al. (1980); Ostriker and Peebles (1973); Ostriker (1993) also suggest that one may need to go beyond GR to explain such cosmological phenomena if one does not wish to introduce dark energy or dark matter that are currently unknown.
Before gravitational waves (GWs) were directly detected by Advanced LIGO and Virgo, tests of gravity mainly focused on using solar system experiments and observations of radio pulsars and cosmology. Each of these cover different ranges of length scale and curvature strength. Solar system experiments constrain gravity in the weak-field and slow-motion environment. In terms of relativistic equations of motion, such experiments give access mostly to first order corrections to Newtonian dynamics Berti et al. (2015); Will (2014). Pulsar timing observations of neutron stars (NSs) offer us both weak-field and strong-field tests of gravity Freire et al. (2012); Kramer et al. (2006); Liu et al. (2012); Ransom et al. (2014); Stairs et al. (2002, 1998); Stairs (2003); Taylor et al. (1992); Wex (2014); Yunes and Siemens (2013). On one hand, binary components are widely separated and the relative motion of two stars in a binary is slow (and thus weak-field). On the other hand, binary pulsars consist of NSs which are compact and are strong-field sources of gravity. Cosmological observations constrain gravity in the weak field regime but at length scales which are many orders of magnitude larger compared to other tests Berti et al. (2015); Clifton et al. (2012); Jain and Khoury (2010); Joyce et al. (2015); Koyama (2016). Cosmological tests of gravity include observations of cosmic microwave background radiation Osborne et al. (2011); Ade et al. (2016); Salvatelli et al. (2016); Ade et al. (2016); Bennett et al. (2013); Hinshaw et al. (2013), studies of Big Bang Nucleosynthesis Clifton et al. (2005); Coc et al. (2006); Damour and Pichon (1999); Komatsu et al. (2011); Mathews et al. (2017); Olive et al. (2000); Santiago et al. (1997), weak gravitational lensing Bartelmann and Schneider (2001); Collett et al. (2018); Clowe et al. (2006); Huterer (2010); Lewis and Challinor (2006) and observations of galaxies Berti et al. (2015). Other tests include using the orbital motion of stars near the Galactic Center Ghez et al. (2005); Hees et al. (2017); Abuter et al. (2018).
Up until now, six GW sources have been discovered (five of them being consistent with binary BH mergers Abbott et al. (2016a, b, 2017a, 2017b, 2017c) while the remaining one being consistent with a binary NS merger Abbott et al. (2017d)), which opened completely new ways of testing GR. GWs provide the opportunity to probe gravity in the strong-field and highly dynamical regime. Binary BH merger events have been used to carry out a model-independent test of gravity by estimating the amount of residuals in the detected signals of GW150914 from the best-fit waveform Abbott et al. (2016c). GW150914 has also been used to perform a consistency test of GR between the inspiral and post-inspiral phases Abbott et al. (2016c). An addition of Virgo allowed one to look for non-tensorial polarization modes of GWs Abbott et al. (2017c). Meanwhile, the arrival time difference between gravitons and photons in the binary NS merger event GW170817 can be used to constrain the deviation in the propagation speed of the former from the latter to one part in , to place bounds on the violation of Lorentz invariance and to carry out a new test of the equivalence principle via the Shapiro time delay Abbott et al. (2017e). Such a constraint on the propagation speed of GWs has led one to rule out many of modified theories of gravity that can explain the current accelerating expansion of our universe without introducing dark energy Baker et al. (2017); Creminelli and Vernizzi (2017); Ezquiaga and Zumalacarregui (2017); Battye et al. (2018); Ezquiaga and Zumalacarregui (2018). So far, no evidence has been found that indicates non-GR effects.
One can carry out yet another type of tests of GR by directly measuring or constraining non-GR parameters in the waveform. One can derive modifications to GR waveforms by choosing specific modified theories of gravity, though perhaps a more efficient approach is to perform the test in a model-independent way. A pioneering work along this line has been carried out in Arun et al. (2006a, b); Mishra et al. (2010), where the authors treat each post-Newtonian (PN) term in the waveform independently and look for consistency among them. Based on this, a data analysis pipeline (TIGER) was developed Agathos et al. (2014); Meidam et al. (2014). One drawback of such a formalism is that one can only treat PN terms in non-GR theories that are also present in GR, which means that one cannot capture e.g. scalar dipole radiation effect entering at a negative PN order that is absent in GR. To overcome this, Yunes and Pretorius Yunes and Pretorius (2009) proposed a new framework called parameterized post-Einsteinian (PPE) formalism, where they introduced new parameters that can capture non-GR effects in waveforms in a generic way. The original work focused on tensorial polarizations for quasi-circular binaries and introduced only the leading PN non-GR corrections in Fourier domain. Such an analysis was later extended to include non-tensorial polarizations Chatziioannou et al. (2012) and multiple PN correction terms Sampson et al. (2013), and for time domain waveforms Huwyler et al. (2015), eccentric binaries Loutrel et al. (2014) and a sudden turn on of non-GR effects Sampson et al. (2014a, b). The LIGO Scientific Collaboration and Virgo Collaboration developed a generalized IMRPhenom model Abbott et al. (2017e) that is similar to the PPE formalism Yunes et al. (2016). Generic non-GR parameters in the waveform phase have been constrained in Abbott et al. (2017e, 2016d); Yunes et al. (2016); Abbott et al. (2017a) with the observed GW events.
In this paper, we derive PPE waveforms in various modified theories of gravity. Many of previous literature focused on deriving phase corrections since matched filtering is more sensitive to such phase corrections than to amplitude corrections. Having said this, there are situations where amplitude corrections are more useful to probe, such as amplitude birefringence in parity-violating theories of gravity Alexander et al. (2008); Yunes and Finn (2009); Yunes et al. (2010a); Yagi and Yang (2018) and testing GR with astrophysical stochastic GW backgrounds Maselli et al. (2016). We first derive PPE amplitude and phase corrections in terms of generic modifications to the frequency evolution and Kepler’s third law that determine the waveform in Fourier domain. For our purpose, this formalism is more useful than that in Chatziioannou et al. (2012), which derives the amplitude and phase corrections in terms of generic modifications to the binding energy of a binary and the GW luminosity. We follow the original PPE framework and focus on deriving leading PN corrections in tensorial modes only Yunes and Pretorius (2009); Cornish et al. (2011). Non-tensorial GW modes also typically exist in theories beyond GR, though at least in scalar-tensor theories, the amplitude of a scalar polarization is of higher PN order than amplitude corrections to tensor modes Chatziioannou et al. (2012); Arun (2012).
We also derive non-GR corrections in varying- theories, considering a PPE formalism with variable gravitational constants. Although there is only one gravitational constant in GR, many modified theories allow more than one gravitational constants that appear in different sectors. We consider two different gravitational constants, one entering in the GW luminosity and the other in Kepler’s third law or the binding energy. We also promote the binary masses and the specific angular momentum to vary with time via the sensitivities Nordtvedt (1990), which closely follow testing variation in with binary pulsars Wex (2014). Our work extends the previous work of Ref. Yunes et al. (2010b) where dissipative and conservative constants were taken to be the same and the masses of binary components were assumed to be constant. Furthermore, we correct the energy-balance law used in Yunes et al. (2010b) for varying- theories by taking into account the non-conservation of binding energy in the absence of gravitational radiation.
Non-GR corrections can enter in the gravitational waveform through activation of different theoretical mechanisms, which can be classified as generation mechanisms and propagation mechanisms Yunes et al. (2016). Generation mechanisms take place close to the source (binary), while propagation mechanisms occur in the far-zone and accumulate over distance as the waves propagate. In this paper, we focus on the former111PPE waveforms due to modifications in the propagation sector can be found in Mirshekari et al. (2012); Yunes et al. (2016); Nishizawa (2018), which have been used for GW150914, GW151226 Yunes et al. (2016) and GW170104 Abbott et al. (2017a) to constrain the mass of the graviton and Lorentz violation.. The PPE parameters in various modified theories of gravity are summarized in Tables 3 (phase corrections) and 2 (amplitude corrections). Some of the amplitude corrections were derived here for the first time. We also correct some errors in previous literature.
The rest of the paper is organized as follows: In Sec. II, we revisit the standard PPE formalism. In Sec. III, we derive the PPE parameters in some example theories following the formalism in Sec. II. In Sec. IV, we derive the PPE parameters in varying- theories. We summarize our work and discuss possible future prospects in Sec. V. Appendix A discusses the original PPE formalism. In App. B, we derive the frequency evolution in varying- theories from the energy-balance law. We use the geometric units throughout this paper except for varying- theories.
|Theories||PPE Phase Parameters||Binary Type|
|Magnitude ()||Exp. ()|
|Scalar-Tensor Scharre and Will (2002); Berti et al. (2005)||Any|
|EdGB Yagi et al. (2012a)||Any|
|DCS Yagi et al. (2012b); Yunes et al. (2016)||BH/BH|
|Einstein-Æther Hansen et al. (2015)||Any|
|Khronometric Hansen et al. (2015)||Any|
|Noncommutative Kobakhidze et al. (2016)||BH/BH|
|Varying- Yunes et al. (2010b)||Any|
|Theories||PPE Amplitude Parameters|
|Magnitude ()||Exponent ()|
|Scalar-Tensor Arun (2012); Chatziioannou et al. (2012); Liu et al. (2018)|
|Einstein-Æther Hansen et al. (2015)|
|Khronometric Hansen et al. (2015)|
|Varying- Yunes et al. (2010b)|
Ii PPE Waveform
We begin by reviewing the PPE formalism. The original formalism (that we explain in detail in App. A) was developed by considering non-GR corrections to the binding energy and GW luminosity Yunes and Pretorius (2009); Chatziioannou et al. (2012). The former (latter) correspond to conservative (dissipative) corrections. Here, we take a slightly different approach and consider corrections to the GW frequency evolution and the Kepler’s law , where is the orbital separation while is the GW frequency. This is because these two quantities directly determine the amplitude and phase corrections away from GR, and hence, the final expressions are simpler than the original ones. Moreover, non-GR corrections to and have already been derived in previous literature for many modified theories of gravity.
PPE gravitational waveform for a compact binary inspiral in Fourier domain is given by Yunes and Pretorius (2009)
where is the gravitational waveform in GR. corresponds to the non-GR correction to the GW amplitude while is that to the GW phase with
is the chirp mass with component masses and . is proportional to the relative velocity of the binary components. represents the overall magnitude of the amplitude correction while gives the velocity dependence of the correction term. In a similar manner, one can rewrite the phase correction as
, , , and are called the PPE parameters. When , Eq. (1) reduces to the waveform in GR.
One can count the PN order of non-GR corrections in the waveform as follows. A correction term is said to be of PN relative to GR if the relative correction is proportional . Thus, the amplitude correction in Eq. (1) is of PN order. On the other hand, given that the leading GR phase is proportional to (see Eq. (78)), the phase correction in Eq. (3) is of PN order.
As we mentioned earlier, the PPE modifications in Eq. (1) enter through corrections to the orbital separation and the frequency evolution. We parameterize the former as
where and are non-GR parameters which show the deviation of the orbital separation away from the GR contribution . To leading PN order, is simply given by the Newtonian Kepler’s law as . Here is the total mass of the binary while is the orbital angular frequency. The above correction to the orbital separation arises purely from conservative corrections (namely corrections to the binding energy).
Similarly, we parameterize the GW frequency evolution with non-GR parameters and as
Unlike the correction to the orbital separation, the one to the frequency evolution originates corrections from both the conservative and dissipative sectors.
Below, we will derive how the PPE parameters are given in terms of and . We will also show how the amplitude PPE parameters can be related to the phase PPE ones in certain cases. We will assume that non-GR corrections are always smaller than the GR contribution and keep only to leading order in such corrections at the leading PN order.
ii.1 Amplitude Corrections
Let us first look at corrections to the waveform amplitude. Within the stationary phase approximation Damour et al. (2000); Yunes et al. (2009), the waveform amplitude for the dominant quadrupolar radiation in Fourier domain is given by
Here is the waveform amplitude in the time domain while represents time at the stationary point. can be obtained by using the quadrupole formula for the metric perturbation in the transverse-traceless gauge given by Blanchet (2002)
Here is the source’s luminosity distance and is the source’s quadruple moment tensor.
For a quasi-circular compact binary, in Eq. (7) then becomes
where is the amplitude of the Fourier waveform in GR. Notice that this expression is much simpler than that in the original formalism in Eq. (73).
Let us now show the expressions for the PPE parameters and for three different cases using Eq. (10):
When conservative corrections dominate, and there is an explicit relation between and . Though finding such a relation is quite involved and one needs to go back to the original PPE formalism as explained in App. A. Non-GR corrections to the GW amplitude in such a formalism is shown in Eq. (80). Setting the dissipative correction to zero, one finds
Comparable Dissipative and Conservative Case
If dissipative and conservative corrections enter at the same PN order, we can set in Eq. (10). Since there is no generic relation between and in this case, one simply finds
ii.2 Phase Corrections
Next, let us study corrections to the GW phase. The phase in Fourier domain is related to the frequency evolution as Tichy et al. (2000)
which can be rewritten as
Substituting Eq. (5) to the right hand side of the above equation and keeping only to leading non-GR correction, we find
We are now ready to derive and extract the PPE parameters and . Using , we can integrate Eq. (18) twice to find
for and . Here we only keep to leading non-GR correction and is the GR contribution given in Eq. (78) to leading PN order. Similar to the amplitude case, the above expression is much simpler than that in the original formalism in Eq. (A). Comparing this with Eqs. (1) and (3), we find
The above relation is valid for all three types of corrections considered for the GW amplitude case.
ii.3 Relations among ppE Parameters
which holds in all three cases considered previously. Let us consider such three cases in turn below to derive relations between and .
Comparable Dissipative and Conservative Case
When dissipative and conservative corrections enter at the same PN order, there is no explicit relation between and . This is because depends both on and (see Eq. (14)) while depends only on the latter (see Eq. (19)), and there is no relation between the former and the latter. Thus, one can rewrite in terms of and substitute into Eq. (14) but cannot eliminate from the expression for .
Iii Example Theories
In this section, we consider several modified theories of gravity where non-GR corrections arise from generation mechanisms. We briefly discuss each theory, describing differences from GR and its importance. We derive the PPE parameters for each theory following the formalism in Sec. II. Among the various example theories we present here, dissipative corrections dominate in scalar-tensor theories, EdGB gravity, Einstein-Æther theory, and khronometric gravity. On the other hand, dissipative and conservative corrections enter at the same PN order in dCS gravity, noncommutative gravity, and varying- theories. We do not consider any theories where conservative corrections dominate dissipative ones, though such a situation can be realized for e.g. equal-mass and equal-spin binaries in dCS gravity, where the scalar quadrupolar radiation is suppressed and dominant corrections arise from the scalar dipole interaction and quadrupole moment corrections in the conservative sector.
iii.1 Scalar-Tensor Theories
Scalar-tensor theories are one of the most well-established modified theories of gravity where at least one scalar field is introduced through a non-minimal coupling to gravity Berti et al. (2015); Chiba et al. (1997); Harrison (1972). Such theories arise naturally from the dimensional reduction of higher dimensional theories, such Kaluza-Klein theory Fujii and Maeda (2007); Overduin and Wesson (1997) and string theories Polchinski (1998a, b). Scalar-tensor theories have implications to cosmology as well since they are viable candidates for accelerating expansion of our universe Brax et al. (2004); Kainulainen and Sunhede (2006); Baccigalupi et al. (2000); Riazuelo and Uzan (2002); Schimd et al. (2005), structure formation Brax et al. (2006), inflation Burd and Coley (1991); Barrow and Maeda (1990); Clifton et al. (2012), and primordial nucleosynthesis Coc et al. (2006); Damour and Pichon (1999); Larena et al. (2007); Torres (1995). Such theories also offer simple ways to self-consistently model possible variations in Newton’s constant Clifton et al. (2012) (as we discuss in Sec. IV). One of the simplest scalar-tensor theories is Brans-Dicke (BD) theory, where a non-canonical scalar field is non-minimally coupled to the metric with an effective strength inversely proportional to the coupling parameter Brans and Dicke (1961); Scharre and Will (2002). So far the most stringent bound on the theory has been placed by the Cassini-Huygens satellite mission via Shapiro time delay measurement, which gives Bertotti et al. (2003). Another class of scalar-tensor theories that has been studied extensively is Damour-Esposito-Farèse (DEF) gravity (or sometimes called quasi Brans-Dicke theory), which has two coupling constants . This theory reduces to BD theory when is set to 0 and is directly related to . This theory predicts nonperturbative spontaneous or dynamical scalarization phenomena for NSs Damour and Esposito-Farèse (1993); Barausse et al. (2013).
When scalarized NSs form compact binaries, these systems emit scalar dipole radiation that changes the orbital evolution from that in GR. Such an effect can be used to place bounds on scalar-tensor theories. For example, combining observational orbital decay results from multiple binary pulsars, the strongest upper bound on that controls the magnitude of scalarization in DEF gravity has been obtained as at confidence level Shao et al. (2017). More recently, observations of a hierarchical stellar triple system PSR J0337+1715 placed strong bounds on the Strong Equivalence Principle (SEP) violation parameter444SEP violation parameter is defined as , where and are respectively the gravitational and inertial mass of a pulsar Archibald et al. (2018). as at confidence level Archibald et al. (2018). This bound stringently constrained the parameter space of DEF gravity Damour and Esposito-Farèse (1993); Nordtvedt (1970); Bergmann (1968); Horbatsch and Burgess (2011); Wagoner (1970).
Can BHs also possess scalar hair like NSs in scalar-tensor theories? BH no-hair theorem can be applied to many of scalar-tensor theories that prevents BHs to acquire scalar charges Hawking (1972); Bekenstein (1995); Sotiriou and Faraoni (2012); Hui and Nicolis (2013); Maselli et al. (2015a) including BD and DEF gravity, though exceptions exist, such as EdGB gravity Yunes and Stein (2011); Sotiriou and Zhou (2014a, b); Silva et al. (2018); Doneva and Yazadjiev (2018) that we explain in more detail in the next subsection. On the other hand, if the scalar field cosmologically evolves as a function of time, BHs can acquire scalar charges, known as the BH miracle hair growth Jacobson (1999); Horbatsch and Burgess (2012) (see also Healy et al. (2012); Berti et al. (2013) for related works).
Let us now derive the PPE parameters in scalar tensor theories. Gravitational waveforms are modified from that in GR through the scalar dipole radiation. Using the orbital decay rate of compact binaries in scalar-tensor theories in Freire et al. (2012); Wex (2014), one can read off the non-GR corrections to as
with . Given that the leading correction to the waveform is the dissipative one in scalar-tensor theories, one can use Eq. (20) to derive the PPE phase correction as
with . Here represents the scalar charge of the th binary component. Using further Eq. (22), one finds the amplitude correction as
with . These corrections enter at PN order relative to GR.
The scalar charges depend on specific theories and compact objects. For example, in situations where the BH no-hair theorem Hawking (1972); Bekenstein (1995); Sotiriou and Faraoni (2012) applies, . On the other hand, if the scalar field is evolving cosmologically, BHs undergo miracle hair growth Jacobson (1999) and acquire scalar charges given by Horbatsch and Burgess (2012)
where is the growth rate of the scalar field while and are the mass and the magnitude of the dimensionless spin angular momentum of the body respectively. The PPE phase parameter for binary BHs in such a situation was derived in Yunes et al. (2016). Another well-studied example is Brans-Dicke theory, where one can replace in Eqs. (26) and (27) as Freire et al. (2012). Here is the sensitivity of the th body and roughly equals to its compactness (0.5 for BHs and for NSs). The PPE parameters in this theory has been found in Chatziioannou et al. (2012). Scalar charges and the PPE parameters in generic screened modified gravity have recently been derived in Zhang et al. (2017a); Liu et al. (2018).
The phase correction in Eq. (26) has been used to derive current and future projected bounds with GW interferometers. Regarding the former, GW150914 and GW151226 do not place any meaningful bounds on Yunes et al. (2016). On the other hand, by detecting GWs from BH-NS binaries, aLIGO and Virgo with their design sensitivities can place bounds that are stronger than the above binary pulsar bounds from dynamical scalarization for certain equations of state and NS mass range Shibata et al. (2014); Taniguchi et al. (2015); Sampson et al. (2014b); Shao et al. (2017)555One needs to multiply Eq. (26) by a step-like function to capture the effect of dynamical scalarization.. Einstein Telescope, a third generation ground-based detector, can yield constraints on BD theory from BH-NS binaries that are 100 times stronger than the current bound Zhang et al. (2017b). Projected bounds with future space-borne interferometers, such as DECIGO, can be as large as four orders of magnitude stronger than current bounds Yagi and Tanaka (2010a), while those with LISA may not be as strong as the current bound Berti et al. (2005); Yagi and Tanaka (2010b).
Up until now, we have focused on theories with a massless scalar field, but let us end this subsection by commenting on how the above expressions for the PPE parameters change if one considers a massive scalar field instead. In such a case, the scalar dipole radiation is present only when the mass of the scalar field is smaller than the orbital angular frequency . Then, if the Yukawa-type correction to the binding energy is subdominant, Eqs. (26) and (27) simply acquire an additional factor of , where is the Heaviside function. For example, the gravitational waveform phase in massive BD theory is derived in Berti et al. (2012). The situation is similar if massive pseudo-scalars are present, such as axions Huang et al. (2018).
iii.2 Einstein-dilaton Gauss-Bonnet Gravity
EdGB gravity is a well-known extension of GR, which emerges naturally in the framework of low-energy effective string theories and gives one of the simplest viable high-energy modifications to GR Moura and Schiappa (2007); Pani and Cardoso (2009). It also arises as a special case of Horndeski gravity Zhang et al. (2017c); Berti et al. (2015), which is the most generic scalar-tensor theory with at most second-order derivatives in the field equations. One obtains the EdGB action by adding a quadratic-curvature term to the Einstein-Hilbert action, where the scalar field (dilaton) is non-minimally coupled to the Gauss-Bonnet term with a coupling constant Kanti et al. (1996)666We use barred quantities for coupling constants so that one can easily distinguish them from the PPE parameters.. A stringent upper bound on such a coupling constant has been placed using the orbital decay measurement of a BH low-mass X-ray binary (LMXB) as cm Yagi (2012). A similar upper bound has been placed from the existence of BHs Pani and Cardoso (2009). Equation-of-state-dependent bounds from the maximum mass of NSs have also been derived in Pani et al. (2011a).
BHs in EdGB gravity are of particular interest since they are fundamentally different from their GR counterparts. Perturbative but analytic solutions are available for static Mignemi and Stewart (1993); Mignemi (1995); Yunes and Stein (2011); Sotiriou and Zhou (2014b) and slowly rotating EdGB BHs Pani et al. (2011b); Ayzenberg and Yunes (2014); Maselli et al. (2015b) while numerical solutions have been found for static Kanti et al. (1996); Torii et al. (1997); Alexeev and Pomazanov (1997) and rotating Pani and Cardoso (2009); Kleihaus et al. (2011, 2014) BHs. One of the important reasons for considering BHs in EdGB is that BHs acquire scalar monopole charges Yagi et al. (2012a); Sotiriou and Zhou (2014b); Berti et al. (2018); Prabhu and Stein (2018) while ordinary stars such as NSs do not if the scalar field is coupled linearly to the Gauss-Bonnet term in the action Yagi et al. (2012a, 2016). This means that binary pulsars are inefficient to constrain the theory, and one needs systems such as BH-LMXBs Yagi (2012) or BH/pulsar binaries Yagi et al. (2016) to have better probes on the theory.
We now show the expressions of the PPE parameters for EdGB gravity. The scalar monopole charge of EdGB BHs generates scalar dipole radiation, which leads to an earlier coalescence of BH binaries compared to GR. Such scalar radiation modifies the GW phase with the PPE parameters given by Yunes et al. (2016); Yagi et al. (2012a)
and . Here, is the dimensionless EdGB coupling parameter and are the spin-dependent factors of the BH scalar charges given by Berti et al. (2018); Prabhu and Stein (2018)777 are zero for ordinary stars like NSs Yagi et al. (2012a, 2016).. In EdGB gravity, the leading order correction to the phase enters through the correction of the GW energy flux, and hence the theory corresponds to a dissipative-dominated case. We can then use Eq. (22) to calculate the amplitude PPE parameters as
and . These corrections enter at PN order.
One can use the phase correction in Eq. (29) to derive bounds on EdGB gravity with current Yunes et al. (2016) and future Yagi (2012) GW observations. Similar to the scalar-tensor theory case, current binary BH GW events do not allow us to place any meaningful bounds on the theory. Future second- and third-generation ground-based detectors and LISA can place bounds that are comparable to current bounds from LMXBs Yagi (2012). On the other hand, DECIGO has the potential to go beyond the current bounds by three orders of magnitude.
iii.3 Dynamical Chern-Simons Gravity
DCS gravity is described by Einstein-Hilbert action with a dynamical (pseudo-)scalar field which is non-minimally coupled to the Pontryagin density with a coupling constant Alexander and Yunes (2009); Jackiw and Pi (2003). Similar to EdGB gravity, dCS gravity arises as an effective field theory from the compactification of heterotic string theory Green and Schwarz (1984); McNees et al. (2016). Such a theory is also important in the context of particle physics Alexander and Yunes (2009); Mariz et al. (2004, 2008); Gomes et al. (2008), loop quantum gravity Mercuri and Taveras (2009); Taveras and Yunes (2008), and inflationary cosmology Weinberg (2008). Demanding that the critical length scale (below which higher curvature corrections beyond quadratic order cannot be neglected in the action) has to be smaller than the scale probed by table-top experiments, one finds Yagi et al. (2012c). Similar constraints have been placed from measurements of the frame-dragging effect by Gravity Probe B and LAGEOS satellites Ali-Haimoud and Chen (2011).
We now derive the expressions of the PPE parameters for dCS gravity. While BHs in EdGB gravity possess scalar monopole charges, BHs in dCS gravity possess scalar dipole charges which induce scalar quadrupolar emission Yagi et al. (2012a). On the other hand, scalar dipole charges induce a scalar interaction force between two BHs. Each BH also acquires a modification to the quadrupole moment away from the Kerr value. All of these modifications result in both dissipative and conservative corrections entering at the same order in gravitational waveforms. For spin-aligned binaries888See recent works Loutrel et al. (2018a, b) for precession equations in dCS gravity., corrections to Kepler’s law and frequency evolution in dCS gravity are given in Yagi et al. (2012b) within the slow-rotation approximation for BHs, from which we can derive
with , and
with , and
with . Here are the symmetric and antisymmetric combinations of dimensionless spin parameters and is the fractional difference in masses relative to the total mass. The above corrections enter at 2 PN order.
Can GW observations place stronger bounds on the theory? Current GW observations do not allow us to put any meaningful bounds on dCS gravity Yunes et al. (2016) (see also Yagi and Yang (2018)). However, future observations have potential to place bounds on the theory that are six to seven orders of magnitude stronger than current bounds Yagi et al. (2012b). Such stronger bounds can be realized due to relatively strong gravitational field and large spins that source the pseudo-scalar field. Measuring GWs from extreme mass ratio inspirals with LISA can also place bounds that are three orders of magnitude stronger than current bounds Canizares et al. (2012).
iii.4 Einstein-Æther and Khronometric Theory
In this section, we study two example theories that break Lorentz invariance in the gravity sector, namely Einstein-Æther and khronometric theory. Lorentz-violating theories of gravity are candidates for low-energy descriptions of quantum gravity Blas and Lim (2015); Horava (2009). Lorentz-violation in the gravity sector has not been as stringently constrained as that in the matter sector Mattingly (2005); Jacobson et al. (2006); Liberati (2013) and several mechanisms exist that prevents percolation of the latter to the former Liberati (2013); Pospelov and Shang (2012).
Einstein-Æther theory is a vector-tensor theory of gravity, where along with the metric, a spacetime is endowed with a dynamical timelike unit vector (Æther) field Jacobson and Mattingly (2001); Jacobson (2007). Such a vector field specifies a particular rest frame at each point in spacetime, and hence breaks the local Lorentz symmetry. The amount of Lorentz violation is controlled by four coupling parameters . Einstein-Æther theory preserves diffeomorphism invariance and hence is a Lorentz-violating theory without abandoning the framework of GR Jacobson (2007). Along with the spin-2 gravitational perturbation of GR, the theory predicts the existence of the spin-1 and spin-0 perturbations Foster (2006); Jacobson and Mattingly (2004); Foster (2007). Such perturbation modes propagate at speeds that are functions of the coupling parameters , and in general differ from the speed of light Jacobson and Mattingly (2004).
Khronometric theory is a variant of Einstein-Æther theory, where the ther field is restricted to be hypersurface-orthogonal. Such a theory arises as a low-energy limit of Hořava gravity, a power-counting renormalizable quantum gravity model with only spatial diffeomorphism invariance Blas et al. (2010); Berti et al. (2015); Horava (2009); Nishioka (2009); Visser (2009). The amount of Lorentz violation in the theory is controlled by three parameters, . Unlike Einstein-Æther theory, the spin-1 propagating modes are absent in khronometric theory.
Most of parameter space in Einstein-Æther and khronometric theory have been constrained stringently from current observations and theoretical requirements. Using the measurement of the arrival time difference between GWs and electromagnetic waves in GW170817, the difference in the propagation speed of GWs away from the speed of light has been constrained to be less than Abbott et al. (2017d, e). Such a bound can be mapped to bounds on Lorentz-violating gravity as Oost et al. (2018) and Emir Gumrukcuoglu et al. (2018)999Such bounds are consistent with the prediction in Hansen et al. (2015) based on Nishizawa and Nakamura (2014).. Imposing further constraints from solar system experiments Bailey and Kostelecky (2006); Foster and Jacobson (2006); Will (2006a), Big Bang nucleosynthesis Audren et al. (2013) and theoretical constraints such as the stability of propagating modes, positivity of their energy density Eling (2006) and the absence of gravitational Cherenkov radiationElliott et al. (2005), allowed regions in the remaining parameter space have been derived for Einstein-Æther Oost et al. (2018) and khronometric Emir Gumrukcuoglu et al. (2018) theory. Binary pulsar bounds on these theories were studied in Yagi et al. (2014a, b) before the discovery of GW170817, within a parameter space that is different from the allowed regions in Oost et al. (2018); Emir Gumrukcuoglu et al. (2018).
Let us now derive the PPE parameters in Einstein-Æther and khronometric theories. Propagation of the scalar and vector modes is responsible for dipole radiation and loss of angular momentum in binary systems, which increase the amount of orbital decay rate. Regarding Einstein-Æther theory, the PPE phase correction is given by Hansen et al. (2015)
with . Here is the propagation speed of the spin- modes in Einstein-Æther theory given by Jacobson (2007)
in Eq. (III.4) is the sensitivity of the -th body and has been calculated only for NSs Yagi et al. (2014a, b). Given that the leading order correction in Einstein-Æther theory arises from the dissipative sector Hansen et al. (2015), we can use Eq. (22) to find the PPE amplitude correction as101010Eqs. (III.4) and (III.4) correct errors in Hansen et al. (2015).
with . Similar to Einstein-Æther theory, the PPE parameters in khronometric theory is given by Hansen et al. (2015)
with , and
with . These corrections enter at PN order.
Above corrections to the gravitational waveform can be used to compute current and projected future bounds on the theories with GW observations, provided one knows what the sensitivities are for compact objects in binaries. Unfortunately, such sensitivities have not been calculated for BHs, and hence, one cannot derive bounds on the theories from recent binary BH merger events. Instead, Ref. Yunes et al. (2016) used the next-to-leading 0 PN correction that is independent of the sensitivities and derived bounds from GW150914 and GW151226, though such bounds are weaker than those from binary pulsar observations Yagi et al. (2014a, b). On the other hand, Ref. Hansen et al. (2015) includes both the leading and next-to-leading corrections to the waveform and estimate projected future bounds with GWs from binary NSs. The authors found that bounds from second-generation ground-based detectors are less stringent than existing bounds even with their design sensitivities. However, third-generation ground-based ones and space-borne interferometers can place constraints that are comparable, and in some cases, two orders of magnitude stronger compared to the current bounds Chamberlain and Yunes (2017); Hansen et al. (2015).
iii.5 Noncommutative Gravity
Although the concept of nontrivial commutation relations of spacetime coordinates is rather old Snyder (1947a, b), the idea has revived recently with the development of noncommutative geometry Connes (1985, 1995); Woronowicz (1987a); Landi (1997); Woronowicz (1987b), and the emergence of noncommutative structure of spacetime in a specific limit of string theory Witten (1986); Seiberg and Witten (1999). Quantum field theories on noncommutative spacetime have been studied extensively as well Douglas and Nekrasov (2001); Rivelles (2003); Szabo (2003). In the simplest model of noncommutative gravity, spacetime coordinates are promoted to operators, which satisfy a canonical commutation relation:
where is a real constant antisymmetric tensor. In ordinary quantum mechanics, Planck’s constant measures the quantum fuzziness of phase space coordinates. In a similar manner, introduces a new fundamental scale which measures the quantum fuzziness of spacetime coordinates Kobakhidze et al. (2016).
In order to obtain stringent constraints on the scale of noncommutativity, low-energy experiments are advantageous over high-energy ones Carroll et al. (2001); Mocioiu et al. (2000). Low-energy precision measurements such as clock-comparison experiments with nuclear-spin-polarized ions Prestage et al. (1985) give a constraint on noncommutative scale as TeV Carroll et al. (2001), where refers to the magnitude of the spatial-spatial components of 111111The corresponding bound on the time-spatial components of is roughly six orders magnitude weaker than that on the spatial-spatial components.. A similar bound has been obtained from the measurement of the Lamb shift Chaichian et al. (2001). Another speculative bound is derived from the analysis of atomic experiments which is 10 orders of magnitude stronger Berglund et al. (1995); Mocioiu et al. (2000). Study of inflationary observables using cosmic microwave background data from Planck gives the lower bound on the energy scale of noncommutativity as 19 TeV Calmet and Fritz (2015); P. K. et al. (2015).
Let us now review how the binary evolution is modified from that in GR in this theory. Several formulations of noncommutative gravity exist Aschieri et al. (2005, 2006); Calmet and Kobakhidze (2005); Chamseddine (2001); Kobakhidze (2008); Szabo (2006), though the first order noncommutative correction vanishes in all of them Calmet and Kobakhidze (2006); Mukherjee and Saha (2006) and the leading order correction enters at second order. On the other hand, first order corrections may arise from gravity-matter interactions Kobakhidze (2009); Mukherjee and Saha (2006). Thus one can neglect corrections to the pure gravity sector and focus on corrections to the matter sector (i.e., energy-momentum tensor) Kobakhidze et al. (2016). Making corrections to classical matter source and following an effective field theory approach, expressions of energy and GW luminosity for quasi-circular BH binaries have been derived in Ref. Kobakhidze et al. (2016), which give the correction to the frequency evolution in Eq. (5) as
We are now ready to derive the PPE parameters in noncommutative gravity. Given that the dissipative and conservative leading corrections enter at the same PN order, one can use Eqs. (44) and (45) in Eq. (14) to find the PPE amplitude correction as
with . can also be read off from the phase correction derived in Kobakhidze et al. (2016). The above corrections enter at 2 PN order.
The above phase correction has already been used to derive bounds on noncommutative gravity from GW150914 as Kobakhidze et al. (2016), which means that the energy scale of noncommutativity has been constrained to be the order of the Planck scale. Such a bound, so far, is the most stringent constraint on noncommutative scale and is 15 orders of magnitude stronger compared to the bounds coming from particle physics and low-energy precision measurements121212Notice that the GW bound is on the time-spatial components of , while most of particle physics and low-energy precision experiments place bounds on its spatial-spatial components..
Iv Varying- Theories
Many of the modified theories of gravity that violate the strong equivalence principle Di Casola et al. (2015); Will (2014); Bertotti and Grishchuk (1990) predict that locally measured gravitational constant () may vary with time Uzan (2011). Since the gravitational self-energy of a body is a function of the gravitational constant, in a theory where is time-dependent, masses of compact bodies are also time-dependent Nordtvedt (1990). The rate at which the mass of an object varies with time is proportional to the rate of change of the gravitational coupling constant Nordtvedt (1990). Such a variation of mass, together with the conservation of linear momentum, causes compact bodies to experience anomalous acceleration, which results in a time-evolution of the specific angular momentum Nordtvedt (1990). Existing experiments that search for variations in at present time (i.e., at very small redshift) include lunar laser ranging observations Williams et al. (2004), pulsar timing observations Deller et al. (2008); Kaspi et al. (1994), radar observations of planets and spacecraft Pitjeva (2005), and surface temperature observations of PSR J0437-4715 Jofre et al. (2006). Another class of constraints on a long-term variation of comes from Big Bang nucleosynthesis Bambi et al. (2005); Copi et al. (2004) and helioseismology Guenther et al. (1998). The most stringent bound on is of the order Genova et al. (2018).
More than one gravitational constants can appear in different areas of a gravitational theory. Here we introduce two different kinds of gravitational constant, one that arises in the dissipative sector and another in the conservative sector. The constant which enters in the GW luminosity through Einstein equations, i.e. the constant in Eq. (8), is the one we refer to as dissipative gravitational constant (), while that enters in Kepler’s law or binding energy of the binary is what we refer as the conservative one (). These two constants are the same in GR, but they can be different in some modified theories of gravity. An example of such a theory is Brans-Dicke theory with a cosmologically evolving scalar field Will (2006b).
The PPE parameters for varying- theories have previously been derived in Yunes et al. (2010b) for . Here, we improve the analysis by considering the two different types of gravitational constant and including variations in masses, which are inevitable for strongly self-gravitating objects when varies Nordtvedt (1990). We also correct small errors in Yunes et al. (2010b). We follow the analysis of Yagi et al. (2011) that derives gravitational waveforms from BH binary inspirals with varying mass effects from the specific angular momentum. We also present another derivation in App. B using the energy balance argument in Yunes et al. (2010b).
The formalism presented in Sec. II assumes that and the masses to be constant, and hence are not applicable to varying- theories. Thus, we will derive the PPE parameters in varying- theories by promoting the PPE formalism to admit time variation in the gravitational constants and masses as
where is the time of coalescence. Here we assumed that spatial variations of and are small compared to variations in time. gives the fractional difference between the rates at which and vary with time, and could be a function of parameters in a theory. The subscript denotes that the quantity is measured at the time . Other time variations to consider are those in the specific angular momentum and the total mass :
and can be written in terms of binary masses and sensitivities defined by
as Nordtvedt (1990)
Next, we explain how the binary evolution is affected by the variation of the above parameters. First, GW emission makes the orbital separation decay with the rate given by Cutler and Flanagan (1994)
Second, time variation of the total mass, (conservative) gravitational constant and specific angular momentum changes at a rate of
which is derived by taking a time derivative of the specific angular momentum . Having the evolution of at hand, one can derive the evolution of the orbital angular frequency using Kepler’s third law as