Testing local Lorentz invariance with gravitational waves

The effects of local Lorentz violation on dispersion and birefringence of gravitational waves are investigated. The covariant dispersion relation for gravitational waves involving gauge-invariant Lorentz-violating operators of arbitrary mass dimension is constructed. The chirp signal from the gravitational-wave event GW150914 is used to place numerous first constraints on gravitational Lorentz violation.

The recent discovery of gravitational waves [1], a century after their prediction by Einstein [2], opens the door to a new class of experimental tests of General Relativity (GR).
While GR is an impressively successful classical field theory of gravity, incorporating a consistent description of quantum effects is widely expected to involve changes to its underlying principles. An essential foundation of GR is the Einstein equivalence principle, which combines the requirements of local Lorentz invariance with local position invariance and the weak equivalence principle. In this work, we demonstrate that the observation of gravitational waves from coalescing black holes at cosmological distances presents an opportunity for clean tests of local Lorentz invariance in the pure-gravity sector. We use the chirp data from the gravitational-wave event GW150914 to place first constraints on certain types of local Lorentz violation involving the gravitational field.
Experimental studies of local Lorentz invariance, which includes symmetry under local rotations and boosts, have enjoyed a resurgence in popularity in recent decades [3,4], triggered by the demonstration that minuscule Lorentz violation could naturally emerge in quantum-gravity theories such as strings [5]. A general and model-independent approach to describing the effects of Lorentz violation in quantum gravity is provided by effective field theory [6]. We are interested here in the corresponding action for pure gravity, which is formed as the sum of the usual Einstein-Hilbert action with cosmological constant together with all possible terms involving operators formed from gravitational fields. This theory is a piece of the general effective field theory for gravity and matter, the gravitational Standard-Model Extension (SME). A Lorentz-violating term in the action is an observer-independent quantity containing a Lorentz-violating operator contracted with a coefficient governing the size of its effects. Using natural units, each operator can be assigned a mass dimension d, and the associated coefficient then has mass dimension 4 − d. Under the plausible assumption that Lorentz violation is suppressed by powers of the Planck mass, which is the natural mass scale associated with the Newton gravitational coupling, operators of higher d can be expected to induce smaller effects. Laboratory [7][8][9][10][11][12][13][14] and analytical [15][16][17][18][19][20][21][22] investigations using this pure-gravity effective field theory have largely concentrated on minimal Lorentz-violating operators, which have dimension d = 4. Theoretical aspects of nonminimal operators with dimensions d = 5, 6 have been studied [23], and constraints on some nonrelativistic combinations of operators with d = 6 have been obtained via laboratory tests of short-range gravity [24][25][26]. The tighest constraints to date on local Lorentz violation in the gravity sector have been deduced from the absence of gravitationalČerenkov radiation by cosmic rays [30][31][32][33][34][35][36] with a large class of effects for d = 4, 6, and 8 now being excluded at sharp levels [35]. Reviews can be found, for example, in Refs. [3,4,[27][28][29]. Here, we construct the general quadratic Lagrange density for gravitational waves in the presence of Lorentz-violating operators of arbitrary d, and we extract the covariant dispersion relation involving gauge-invariant effects. We show that observations of gravitational waves provide sensitivities to nonminimal Lorentz violation independent of matter-sector effects, and we use the gravitational-wave event GW150914 to place numerous first limits on gravity-sector operators with d ≥ 5.
The effective field theory for gravitational Lorentz violation [6] can be linearized in a flat-spacetime background with Minkowski metric, g µν = η µν + h µν . Our first goal is to construct the general Lagrange density quadratic in the dimensionless metric perturbation h µν , allowing for both Lorentz-invariant and Lorentz-violating terms. A generic term of this type takes the form where is an operator of mass dimension d ≥ 2 and the coefficients K To construct explicitly the operators K (d)µνρσ , we perform a decomposition into irreducible pieces and examine the properties of each. This reveals that 14 independent classes of operators can control the behavior of gravitational waves. However, many violate the usual gauge symmetry of GR under the transformaton h µν → h µν + ∂ µ ξ ν + ∂ ν ξ µ . Performing this transformation on the term (2) shows that the condition for gauge invariance is Only three of the 14 classes of irreducible operators obey this condition, and they are therefore of particular interest. Their existence can be understood as following from spontaneous breaking of the diffeomorphism and Lorentz invariance [37], which hides symmetry rather than explicitly violating it and is therefore automatically compatible with the Bianchi identities [6]. The gauge invariance maintains the standard counting of degrees of freedom in h µν , insuring that the three classes of operators induce perturbative modifications to the two usual propagating modes in a gravitational wave. Note that in principle the higher derivatives occuring in the term (2) introduce additional modes, but these are nonperturbative in Lorentz violation and occur only at high energies outside the domain of validity of the effective field theory.
The three classes of gauge-invariant operators are determined by their symmetries, which are given by the Young tableaux in Table I In what follows, it is also useful to define quantities that are the sums over d of each of these sets of operators, The operator q µρνσ is antisymmetric in the first pair of indices and symmetric in the second, while the operator s µρνσ is antisymmetric in both the first and second pairs of indices, and k µνρσ is totally symmetric. Any contraction of these operators with a derivative vanishes.
The complete gauge-invariant quadratic Lagrange density, including all Lorentz-violating and Lorentz-invariant terms of arbitrary mass dimension d, then takes the form where L 0 is the quadratic approximation to the Einstein-Hilbert action and is a subset of the d = 4 component of s µρνσ . We remark in passing that the introduction of a dual operator via s µρνσ = −ǫ µρακ ǫ ανσβλ s κλ ∂ α ∂ β reveals that s µρνσ contributes as a momentum-dependent metric perturbation, η κλ → η κλ − s κλ . The effects of s κλ on gravitationalČerenkov radiation is the subject of Ref. [35]. Also, a treatment of terms linear in h µν is provided elsewhere, along with a discussion of the other 11 classes of gauge-violating operators, which can describe additional gravitational modes beyond the usual helicity-two states of GR [38].
Following methods developed for the study of Lorentz invariance in the photon sector of the SME [39], the covariant dispersion relation for propagation of a gravitational wave of 4-momentum p µ = (ω, p) can be derived from the Lagrange density (4). Some algebra reveals that the leading-order dispersion relation takes the form with where now the derivative factors ∂ µ in the coefficients (3)  , so thatn has spherical polar angles θ ≃ 160 • , φ ≃ 120 • in the Sun-centered celestial-equatorial frame canonically used to report results of searches for Lorentz violation [41]. For practical applications, it is therefore useful to perform a decomposition of the dispersion relation (5) in spherical harmonics. Since the metric perturbation h µν has helicity-2 components, and since the Lagrange density (4) is quadratic in h µν , the decomposition involves spin-weighted spherical harmonics [42] of spin weight s ≤ 4, which we denote as s Y jm (n). A summary of the properties of these harmonics can be found in Appendix A of Ref. [39]. Note in particular that 0 Y jm (n) ≡ Y jm (n), the usual scalar spherical harmonics.
Investigation shows that ς 0 , ς 3 are rotation scalars while ς 1 , ς 2 are helicity-4 tensors. The decomposition then can be written as where |s| ≤ j ≤ d − 2. The CPT-odd operators in the Lagrange density are controlled by the coefficients k Armed with the above tools, we can use the gravitational-wave event GR150914 to test local Lorentz invariance in gravity. Since operators with larger d are expected to be more suppressed, it might seem natural to study first effects involving the coefficients k (4) (I)jm . However, the corresponding operators are nondispersive and nonbirefringent, so detecting their effects is more challenging and typically would require a comparison to light or neutrinos propagating from the same source [35,44]. Instead, we consider in turn the d = 5 coefficients k (5) (V )jm and the d = 6 cofficients k (6) (E)jm , k (6) (B)jm . Since polarimetric information for GW150914 and its source is unavailable, we focus here on dispersive effects.
Consider first the generic situation involving a source producing gravitational waves, such as the merger and ringdown of a black-hole binary. We can reasonably assume the observed wave is generated in a superposition of the two propagating modes. For example, the eigenmodes in the presence of d = 5 Lorentz violation are circularly polarized, so special physical circumstances would be required for a source to produce only one eigenmode. The dispersion between the two modes evinced in the dispersion relation (5) can then be used to constrain Lorentz violation by comparing their arrival times. The difference in their velocities generically depends on both the frequency ω of the wave and the locationn of its source. Both the source and the detector can be taken as comoving objects. The coordinate interval between them is therefore dl c = (1 + z)dl p = −v z dz/H z , where v z is the velocity of the source at redshift z, and H z = H 0 (Ω r ζ 4 + Ω m ζ 3 + Ω k ζ 2 + Ω Λ ) 1/2 , ζ ≡ 1 + z, is the Hubble expansion rate at z expressed in terms of the Hubble constant H 0 ≃ 67.3 km/s/Mpc radiation density Ω r ≃ 0, matter density Ω m ≃ 0.315, vacuum density Ω Λ ≃ 0.685, and curvature density Ω k = 1 − Ω r − Ω m − Ω Λ . Although the coordinate distance is identical for both modes, their velocities and hence their travel times differ in the presence of Lorentz violation. For example, the arrival-time difference between the two modes for coefficients k (d) (V )jm with fixed d is given by For GW150914, the linear combinations of k (B)jm appearing in the corresponding expression for ∆t are determined by spin-weighted spherical harmonics with approximate angular arguments θ ≃ 160 • , φ ≃ 120 • in the Sun-centered frame. The source is With the above values, we obtain the constraints The result (9) represents the first constraint on pure-gravity Lorentz-violating operators with d = 5. It also represents the first limit on CPT violation in gravitational waves, which here corresponds to a difference in propagation speed between the two circularly polarized CPT-conjugate eigenmodes. Note that sensitivity to d = 5 effects is empirically unavailable in the nonrelativistic limit and hence to typical laboratory experiments on Newton gravity, because the presence of d = 5 Lorentz-violating operators in the action leaves unaffected Newton's law [23]. This fact underscores the added value of the discovery of gravitational waves in the context of studies of the foundations of relativistic gravity. The result (10) represents the first bound on all birefringent coefficients at d = 6 and is competitive with existing laboratory bounds [24][25][26].
Some insight into the implications of these bounds can be gained by deriving from them the constraint on each individual coefficient in turn, under the assumption that the other components vanish. The resulting estimated bounds on the modulus of each component of k (E)jm , and k (B)jm are displayed in Table II. Note that each entry thereby also represents constraints on the moduli of the real and imaginary parts of each component.  This standard practice [3] is useful in comparing limits across different experiments and in constraining specific models. For example, models with rotation-invariant gravitational Lorentz violation [45,46] can involve at most the spherical coefficients k with jm = 00, for which it is convenient to definek The rotation-invariant limit of the dispersion relation (5) then takes the form (I) |p| ±k and the first row of Table II constrainsk (5) (V ) . Note that the ± signs reflect the presence of birefringence and CPT violation for even powers of ω, both of which are required to describe physics associated with an effective field theory [47].
In principle, methods related to the one adopted here could be used to obtain estimated constraints on other coefficients with d > 5, which are all associated with dispersive operators. The approach used above can be applied directly to k for d = 6, 8 would be significantly weaker than ones already deduced from the absence of gravitationalČerenkov radiation in cosmic rays [35]. Note, however, that dispersion limits of the type discussed here are particularly clean because they involve comparing the properties of two gravitational modes and hence lie entirely within the pure-gravity sector, whereas bounds from gravitationalČerenkov radiation involve comparative tests between the gravity and matter sectors. Indeed, gravitationalČerenkov radiation may even be forbidden for certain relative sizes of the coefficients for Lorentz violation for gravity and matter, which would obviate any bounds obtained via this technique. We also note in passing that the results in Ref. [35] are presented as limits on components of s µν µν , but the analysis effectively bounds ς 0 and hence k (d) (I)jm , which contains pieces of both s µν µν and k µν µν . The future detection of additional gravitational-wave events will yield direct improvements in the constraints obtained in this work. Moreover, the use of dispersion information from multiple astrophysical sources at different sky locations permits extraction of independent constraints on different coefficients, as has already been demonstrated for noniminal coefficients in the photon and neutrino sectors of the SME [48,49]. Improved sensitivities can also be expected for gravitational waves of higher frequency, as might be emitted in a supernova core collapse. The prospects are evidently bright for future studies of foundational physical principles via measurements of gravitational-wave properties.  [2] A. Einstein, Sitzungsber. K. Preuss. Akad. Wiss. 1, 688 (1916); Sitzungsber. K. Preuss. Akad.