Gravitational-wave polarizations in generic linear massive gravity and generic higher-curvature gravity
aa r X i v : . [ g r- q c ] F e b Gravitational-wave polarizations in generic linear massive gravity and generichigher-curvature gravity
Tomoya Tachinami, ∗ Shinpei Tonosaki, † and Yuuiti Sendouda ‡ Graduate School of Science and Technology, Hirosaki University, Hirosaki, Aomori 036-8561, Japan (Dated: February 11, 2021)We study the polarizations of gravitational waves (GWs) in two classes of extended gravity theo-ries. As a preparatory yet complete study, we formulate the polarizations in linear massive gravity(MG) with generic mass terms of non-Fierz–Pauli type by identifying all the independent variablesthat obey Klein–Gordon-type equations. The dynamical degrees of freedom (dofs) in the genericMG consist of spin-2 and spin-0 modes, the former breaking down into two tensor (helicity-2), twovector (helicity-1) and one scalar (helicity-0) components, while the latter just corresponding to ascalar. We find convenient ways of decomposing the two scalar modes of each spin into distinct linearcombinations of the transverse and longitudinal polarizations with coefficients directly expressed bythe mass parameters, thereby serving as a useful tool in measuring the masses of GWs. Then weanalyze the linear perturbations of generic higher-curvature gravity (HCG) whose Lagrangian is anarbitrary polynomial of the Riemann tensor. When expanded around a flat background, the lineardynamical dofs in this theory are identified as massless spin-2, massive spin-2 and massive spin-0modes. As its massive part encompasses the identical structure to the generic MG, GWs in thegeneric HCG provide six massive polarizations on top of the ordinary two massless modes. In par-allel to MG, we find convenient representations for the scalar-polarization modes directly connectedto the coupling constants of HCG. In the analysis of HCG, we employ two distinct methods; Onetakes full advantage of the partial equivalence between the generic HCG and MG at the linear level,whereas the other relies upon a gauge-invariant formalism. We confirm that the two results agree.We also discuss methods to determine the theory parameters by GW-polarization measurements.
I. INTRODUCTION
In the era of gravitational-wave astronomy opened by the historic event GW150914 [1], increasingly more attentionhas been attracted to the nature of the dynamical degrees of freedom (dofs) of gravity propagating as gravitationalwaves (GWs). Detection of any deviations from the predictions of general relativity (GR), exactly luminal propagationand two orthogonal modes, would immediately signal the presence of the gravitational theory beyond GR.A representative example of extended gravity theories beyond GR is the linear massive gravity (MG), in whichgravitational waves acquire masses. Adding a generic combination of possible two mass terms to GR breaks its fourlinear gauge symmetries and gives rise to six dofs in total. A special class of MG introduced by Fierz and Pauli (FP)[2] with a single mass parameter m preserves one gauge symmetry and avoids the appearance of the spin-0 mode thatwould have a negative kinetic term.If GWs are massive, their velocity c g deviates from the speed of light c due to the modification of the dispersionrelation. The multi-messenger analysis of the GW event GW170817 [3] has put a tight constraint on the deviation, | c g − c | /c . O (10 − ) [4], which can be interpreted as an upper limit on its mass. It was also pointed out that anon-zero mass, however small, of graviton in the FP theory would lead to a bending angle of light around a massivebody discontinuously different from GR [5–7], which could, however, be circumvented by certain non-linearity [8].It was argued in [9, 10] that GWs in a certain generic class of gravity theories can have maximally six polarizations.So far, there has been no contradiction with the hypothesis of only two polarizations in the GW experiments [11]including the observed orbital decay rate of a neutron star binary PSR B1913+16 [12]. However, in practice, a completedecomposition into possible six polarizations and determination of each amplitude in an observed gravitational wavecannot be done with the limited number of detectors that we currently have. Indeed, there have been only weakconstraints on the existence of non-GR polarizations; An example is the constraint on the vector-type polarizationsobtained by a method to detect scalar and vector polarizations with four interferometers of LIGO (Hanford andLivingston), Virgo and KAGRA developed in [13–15]. In this regards, there are still open and wide theoreticalpossibilities to explore.One of our purposes in this paper is to study gravitational-wave polarizations in gravity theories whose Lagrangiancan be written in a generic form L = f ( R µνρσ , g µν ) / κ , where g µν is the space-time metric, R µνρσ the Riemann ∗ tachinami(a)tap.st.hirosaki-u.ac.jp † tonosaki(a)tap.st.hirosaki-u.ac.jp ‡ sendouda(a)hirosaki-u.ac.jp tensor and κ the bare gravitational constant. The scalar function f is almost generic but we here conservativelyassume that, when Taylor expanded around R µνρσ = 0, it only gives positive powers of the Riemann tensor so thatMinkowski space-time is a solution of the full theory. Einstein’s general relativity is defined by the linear function f = R , while presence of any higher-order terms characterizes how the theory differs from GR. Studies of such modelsdate back to Weyl [16], who suggested f = C µνρσ C µνρσ , where C µνρσ is the Weyl curvature tensor. Relativelymodern motivations also come from the developments in string theories, e.g., [17].Higher-curvature gravity (HCG) generically exhibits more dynamical dofs than GR does. For instance, the theorywith f = R + β R can be shown to be equivalent to a scalar–tensor theory, which can be generalized to the case ofa generic function f of the Ricci scalar, called f ( R ) gravity. As a cosmological application, the R + β R model wasutilized by Starobinsky to realize inflation [18]. A generic class of f ( R ) gravity has provided candidates for the darkenergy although such theories with negative powers of curvature is out of our scope in this paper. Also, it was shownby Stelle [19] that in the theory with f = R − α C µνρσ C µνρσ + β R , there arises another massive spin-2 particleon top of zero-mass graviton, which was utilized to render the quantum theory renormalizable [20]. Afterwards, ageneral Hamiltonian analysis of f (Riemann) gravity, keeping f undetermined, was done in [21].Our purpose is to give a complete classification of GWs in generic higher-curvature theories in terms of six types ofpolarizations. In the literature, some special cases have been studied. For instance, Bogdanos et al. [22] considereda generic theory whose Lagrangian consists of quadratic scalar invariants, showing that the GWs have full six polar-izations. More recently, an analysis of f ( R ) gravity using gauge-invariant variables was done by Moretti et al. [23].However, no complete studies have been conducted on the case of generic f .In the studies of linear perturbations of HCG, what is significant would be the equivalence between the quadraticcurvature gravity and massive bi-gravity at the linear-order level [19], which was recently extended to arbitrarybackground with an Einstein metric [24]. This means that the analyses of the extra massive dofs in HCG can be donein parallel to MG, which motivates us to start with studying MG in this paper. On the other hand, we show that adedicated analysis based on a gauge-invariant formalism is also useful, and that the two results agree.The organization of this paper is as follows. In Sec. II, we briefly introduce the basic notion of the polarizationsof gravitational waves on the basis of geodesic deviation. In Sec. III, we analyze the polarizations of GWs in severaltheories of gravity: Starting with reproducing the standard result in general relativity in Sec. III A, we study thegeneric linear MG and generic HCG in Secs. III B and III C, respectively. In Sec. III C 1, we introduce auxiliary fieldswith spin-2 and spin-0 to rewrite the action, count the number of dofs and reveal the origin of six polarizations. InSec. III C 2, as an another approach, a gauge-invariant formulation to analyze GWs in HCG is developed. We confirmthat the same result is obtained in the two different approaches. In Sec. IV, we briefly discuss possible methodsfor determining the theory parameters by GW-polarization observations using laser interferometers or pulsar timing.Finally, we conclude in Sec. V.Throughout the paper, we will work with natural units with c = 1. Greek indices of tensors such as µ, ν, · · · areof space-time while Latin ones such as i, j, · · · are spatial. We introduce background coordinates ( x , x , x , x ) =( t, x, y, z ) in which the Minkowski metric is η µν = diag( − , , , ∂ µ denotes partial differentiation ∂∂x µ . (cid:3) ≡ η µν ∂ µ ∂ ν is the d’Alembertian and △ ≡ δ ij ∂ i ∂ j the Laplacian. The Riemann tensor is defined as R µνρσ = ∂ ρ Γ µνσ − · · · . Parentheses around tensor indices denote symmetrization such as T ( µν ) ≡ ( T µν + T νµ ), whilesquare brackets denote anti-symmetrization such as T [ µν ] ≡ ( T µν − T νµ ). II. POLARIZATIONS OF GRAVITATIONAL WAVES
In this section, we introduce the notion of polarizations of GWs leaving gravity theory unspecified. On the micro-scopic side, the graviton treated as a massless particle with spin 2 is irreducibly decomposed into the helicity statescalled polarizations . On the macroscopic side where classical GWs are considered, a way to define polarization of GWsis the geodesic deviation in space-times. To follow the latter perspective, we shall take Minkowskian background η µν and define the variable as the deviation of the space-time metric g µν from the background, h µν ≡ g µν − η µν . (1)Geometric quantities such as curvature is expanded in h µν . The space-time (space) indices of linear quantities suchas the perturbation itself h µν are raised and lowered by the background metric η µν ( δ ij ). Hereafter, the backgroundLaplacian △ ≡ δ ij ∂ i ∂ j acting on a perturbative quantity is assumed to be invertible. A. Geodesic deviation and gauge-invariant perturbations
The principle of detecting gravitational waves with the interferometers like LIGO, Virgo and KAGRA or the pulsartiming arrays (PTAs) is to measure the spatial separation of a nearby, as compared with the wavelength of GWs, pairof test bodies at rest, ζ i , whose motion is governed by the geodesic deviation equation¨ ζ i = − (1) R i j ζ j , (2)where the dot denotes derivative with respect to x = t and (1) R i j is the linear Riemann tensor given in terms ofthe metric perturbation as (1) R i j = −
12 ¨ h ij + ∂ ( i ˙ h j )0 − ∂ i ∂ j h . (3)Thus, the separation of two test bodies would fluctuate in response to the gravitational waves embodied by h µν , andconversely, by tracking their movements, one could decode the dynamical contents of the gravitational theory.Generally, complications arise in the analysis of gravitational perturbations from the intrinsic degrees of freedomof choosing the background coordinates. The metric perturbations are transformed by a small coordinate change x µ → x µ + ξ µ ( x ) with an arbitrary four vector ξ µ as h µν → h µν − ∂ µ ξ ν − ∂ ν ξ µ . (4)An effective way to isolate the physical degrees of freedom is the gauge-invariant formalisms originally developed inthe cosmological context [25–27]. In this formalism, we begin with introducing scalar, vector and tensor-type variablesto decompose each component of the metric perturbation as h = − A , h i = − ∂ i B − B i , h ij = 2 δ ij C + 2 ∂ i ∂ j E + 2 ∂ ( i E j ) + 2 H ij , (5)where the vector and tensor variables satisfy ∂ i B i = ∂ i E i = 0 , H ii = 0 , ∂ j H ij = 0 . (6)Gauge transformations of each variable are summarized in Appendix A. We can find a reduced number of variables thatare invariant under the transformation (4): The tensor variable H ij is invariant; For the vector part, a combinationof the variables B i and E i which is invariant is Σ i ≡ B i + ˙ E i ; (7)For the scalar part, a useful set of invariant combinations isΨ ≡ A − ˙ B − ¨ E , Φ ≡ C . (8)At this point, we have found six gauge-invariant variables out of the original ten in h µν . A remarkable propertyof the linear Riemann tensor (3) is its invariance under (4). Indeed, the Riemann tensor can be written in terms ofthese invariant variables as (1) R i j = − ¨ H ij − ∂ ( i ˙Σ j ) + ∂ i ∂ j Ψ − δ ij ¨Φ . (9)It is worth stressing that, in spite of their apparent advantage, each gauge-invariant variable does not necessarily asingle dynamical degree of freedom in a given gravity theory. Instead, we generally expect that the gauge-invariantvariables become linear combinations of independent dofs. In particular, in the scalar part a mixture of different spinsoccurs as we will see in the specific cases of massive gravity and higher-curvature gravity. B. Polarization basis
The connection between the irreducible decomposition of the Riemann tensor (9) and observables in gravitational-wave experiments is made explicit by considering a wave solution propagating in a fixed direction. As we alreadynoted, we should keep in mind that each part can contain multiple degrees of freedom.Suppose a tensor variable T ij ( z − c T t ) propagating in the z direction with velocity c T . This would constitutethe tensor gauge-invariant variable H ij . A conventional orthogonal basis that is compatible with the conditions T ii = ∂ i T ij = 0 is the + and × polarizations e + ij ≡ − , e × ij ≡ . (10)Using these basis tensors, we can decompose the tensor variable as T ij = T + e + ij + T × e × ij , (11)where the polarization components are defined as T λ ≡ e ijλ T ij (12)for λ = + , × . Explicitly, T + = T xx = − T yy and T × = T xy = T yx .For a vector variable propagating in the z direction with velocity c V , V i ( z − c V t ), a polarization basis compatiblewith the divergence-free condition ∂ i V i = 0 is e xij ≡ , e yij ≡ . (13)Using these, we can decompose the vector-part of the symmetric tensor as ∂ ( i V j ) = 12 V ′ x e xij + 12 V ′ y e yij , (14)where the prime denotes derivative with respect to z .As for the scalar-type polarizations, natural transverse (“breathing”) and longitudinal polarization bases are e B ij ≡ , e L ij ≡ √ . (15)For a scalar variable propagating in the z direction with velocity c S , S ( z − c S t ),( ∂ i ∂ j − △ δ ij ) S = e B ij S ′′ , ∂ i ∂ j S = e L ij S ′′ . (16)Sometimes it is also useful to introduce other scalar basis instead of B and L, such as e T ij ≡ r e B ij + 1 √ e L ij = r , e T ij ≡ √ e B ij − r e L ij = 1 √ − . (17)These basis tensors satisfy an orthonormal condition e αij e βij = 2 δ αβ for α, β = + , × , x, y, B , L. Seen as a spatialsymmetric tensor, the Riemann tensor (1) R i j can be decomposed with the above introduced polarization basis. Itis understood from (2) that the amplitude of small oscillation of the distance of a pair of two bodies δζ i due to the α polarization component of GW is proportional to A α ≡ A e ijα (1) R i j , where A is a constant independent of the typeof polarization. III. GRAVITATIONAL-WAVE POLARIZATIONS IN GENERIC THEORIES
In this section, we treat concrete examples of gravitational theories: general relativity (GR), generic massive gravity(MG) and generic higher curvature gravity (HCG).
A. General relativity
We first reproduce the standard result in GR. The action of GR expanded around a flat background to the quadraticorder is S GR [ h µν ] = − κ Z d x (1) G µν h µν , (18)where (1) G µν is the linearized Einstein tensor (1) G µν ≡ − (cid:3) h µν + ∂ ( µ ∂ λ h ν ) λ − ∂ µ ∂ ν h + 12 η µν ( (cid:3) h − ∂ ρ ∂ σ h ρσ ) . (19)The equation of motion (eom) for h µν in vacuum is obtained as (1) G µν = 0 . (20)The GR action (18) and eom (20) are invariant under the gauge transformation (4). We can choose the transverse-traceless (TT) gauge h µν → h TT µν such that h TT00 = 0 , h
TT0 i = 0 , δ ij h TT ij = 0 , ∂ i h TT ij = 0 . (21)Then the eom (20) reduces to a massless Klein–Gordon-type equation for h TT ij : (cid:3) h TT ij = 0 . (22)We can take a plane-wave solution h TT ij ∝ e i ω ( z − t ) and obtain the linear Riemann tensor (3) as (1) R i j = −
12 ¨ h TT ij = 12 ω (cid:0) h TT+ e + ij + h TT × e × ij (cid:1) . (23)The same conclusion can be drawn in the gauge-invariant formulation, in which the GR action (18) is rewritten interms of the gauge-invariant variables as S GR [ H ij , Σ i , Φ , Ψ] = 12 κ Z d x (cid:20) H ij (cid:3) H ij + 12 ∂ j Σ i ∂ j Σ i − (cid:3) Φ − △ (Ψ − Φ) (cid:21) . (24)The equations of motion in vacuum are (cid:3) H ij = 0 , △ Σ i = 0 , − △ Ψ − △ Φ = 0 , △ Φ = 0 . (25)These imply Σ i = 0 and Φ = Ψ = 0 thanks to the assumed invertibility of Laplacian. Therefore, for a plane-wavesolution H ij ∝ e i ω ( z − t ) , the Riemann tensor (9) is (1) R i j = − ¨ H ij = ω (cid:0) H + e + ij + H × e × ij (cid:1) . (26) B. Generic linear massive gravity
Next, we consider linear massive gravity (MG) specified by the action S MG [ h µν ] = S GR [ h µν ] − m κ Z d x (cid:2) h µν h µν − (1 − ǫ ) h (cid:3) = 14 κ Z d x (cid:20) − (1) G µν h µν − m h µν h µν − (1 − ǫ ) h ) (cid:21) , (27)where S GR is the GR action (18), (1) G µν is the linear Einstein tensor (19), m corresponds to the mass of spin-2graviton and ǫ is a non-dimensional parameter. This is the most generic extension of linear general relativity thatincorporates Lorentz-invariant mass terms. For ǫ = 0, the action reduces to that of the Fierz–Pauli theory [2], wherethe graviton is pure spin-2, otherwise a spin-0 “ghost” graviton emerges, as we shall confirm below.In order to treat the dynamical dofs efficiently, we use the tensor (T), vector (V) and scalar (S) variables definedas (5). The action (27) is decomposed as S MG [ h µν ] = S (T)MG [ H ij ] + S (V)MG [ B i , E i ] + S (S)MG [ A, B, C, E ] (28)with S (T)MG [ H ij ] = 12 κ Z d x (cid:2) H ij (cid:3) H ij − m H ij H ij (cid:3) , (29) S (V)MG [ B i , E i ] = 12 κ Z d x (cid:20) −
12 Σ i △ Σ i + m (cid:0) B i B i + E i △ E i (cid:1)(cid:21) , (30) S (S)MG [ A, B, C, E ] = 12 κ Z d x (cid:20) − (cid:3) Φ − △ (Ψ − Φ) − m (cid:0) ǫ A + B △ B + 6 (3 ǫ − C + 2 ǫ E △ E + 4 (3 ǫ − C △ E + 4 ( ǫ − A △ E + 12 ( ǫ − A C (cid:1)(cid:21) . (31)Contrary to GR, these vector and scalar actions cannot be solely expressed with the gauge-invariant variables dueto the lack of the gauge symmetries. Thus, in order to calculate the Riemann tensor (9), we have to manipulate theoriginal variables.The eom for the tensor variable H ij is obtained from (29) as (cid:3) H ij − m H ij = 0 . (32)The eom for the gauge-invariant variable Σ i does not directly derive, but the eoms for B i and E i from (30) are − △ ( B i + ˙ E i ) + m B i = 0 , ˙ B i + ¨ E i + m E i = 0 . (33)Combining these, we find that Σ i = B i + ˙ E i obeys (cid:3) Σ i − m Σ i = 0 . (34)The most complicated is to find the governing equations for the gauge-invariant scalars Ψ and Φ. The scalar eomsfrom variations of (31) with respect to A, B, C, E are, respectively,2 △ C + m ( ǫA + ( ǫ − △ E + 3 ( ǫ − C ) = 0 , △ h C + m B i = 0 , (cid:3) C + 2 △ ( A − ˙ B − ¨ E ) − △ C + m (3 (3 ǫ − C + (3 ǫ − △ E + 3 ( ǫ − A ) = 0 , △ h − C + m ( ǫ △ E + (3 ǫ − C + ( ǫ − A ) i = 0 . (35)Gathering these equations, it is found that the following two combinations W ≡ A − ˙ B − ¨ E − C , h ≡ A + 6 C + 2 △ E (36)satisfy Klein–Gordon equations (cid:3) W − m W = 0 , ǫ (cid:3) h − − ǫ m h = 0 . (37)Observe that W = Ψ − Φ and h = η µν h µν . The second equation implies that, if the Fierz–Pauli tuning ǫ = 0 isrealized, then the four-dimensional trace h is constrained to vanish. We assume ǫ = 0 and define m ≡ − ǫ ǫ m as themass of h . These W and h are the only dynamical dofs. Indeed, the set of equations (35) can be solved for A, B, C, E in terms of W and h as A = 23 m △ W + (cid:18) − ǫ − ǫ m △ (cid:19) h ,B = 43 m △ ˙ W − ǫ m ˙ h ,C = − m △ W + ǫ h ,E = 1 m W − m △ W + ǫ m h . (38)Moreover, the gauge-invariant variables are expressed asΨ = A − ˙ B − ¨ E = W − m △ W + ǫ h , Φ = C = − m △ W + ǫ h , (39)where we have used (37) to eliminate ¨ W and ¨ h . Using these relationships, the scalar part of the linear Riemann tensoris written as (1) R (S) i j ≡ ∂ i ∂ j Ψ − δ ij ¨Φ= 3 ∂ i ∂ j W − m △ (cid:16) ∂ i ∂ j W − δ ij ¨ W (cid:17) + ǫ (cid:16) ∂ i ∂ j h − δ ij ¨ h (cid:17) . (40)Having found that the variables H ij , Σ i , W and h obey Klein–Gordon-type equations, we are allowed to considerplane-wave solutions propagating along the z direction, H ij ∝ e i ( k H z − ω H t ) , Σ i ∝ e i ( k Σ z − ω Σ t ) , W ∝ e i ( k W z − ω W t ) , h ∝ e i ( k h z − ω h t ) (41)with k I ≡ q ω I − m ( I = H, Σ , W ) , k h ≡ q ω h − m . (42)Then the Riemann tensor is calculated as (1) R i j = ω H ( H + e + ij + H × e × ij ) − q ω − m ω Σ (cid:0) Σ x e xij + Σ y e yij (cid:1) − m △ W (cid:16) ω W e B ij − √ m e L ij (cid:17) + ǫ h (cid:18) ω h e B ij + m √ e L ij (cid:19) . (43)This expression tells us that the separated six variables provide different polarizations. In particular, the informationcarried by the two scalar variables is distinctive. W , the helicity-0 mode of the spin-2 graviton, can be split into thetransverse, or “breathing” (B), and longitudinal (L) polarizations based on its different dependences on the frequency:the former is proportional to ω W while the latter is m . Similarly, the spin-0 graviton h , which only exists if ǫ = 0,can be decomposed into the transverse and longitudinal polarizations. Thus, if the amplitudes of each polarization areseparately measured in future gravitational-wave experiments, the longitudinal modes will provide a direct measureof the masses of the spin-2 and spin-0 gravitons. We will come back to this issue in Sec. IV.Finally, let us mention the effectively massless case m ≪ ω I . Resembling the discontinuity in the bending angleof light [5–7], taking the continuous limit does not recover the set of polarizations expected in GR as the vector andthe transverse scalar polarizations would remain. C. Generic higher-curvature gravity
Next we consider a class of extended theories of gravity whose full action is of the form S = 12 κ Z d x √− g f ( R µνρσ , g µν ) , (44)where f contains terms non-linear in the Riemann curvature. We begin with discussing perturbative degrees offreedom in this higher-curvature gravity (HCG) in the Minkowski background. If the Lagrangian f consists only ofterms that have smooth behavior around R µνρσ = 0, its expansion in curvature tensors up to the quadratic order canbe arranged as f = χ R − α C µνρσ C µνρσ + β R + γ (cid:0) R µνρσ R µνρσ − R µν R µν + R (cid:1) + O ( R µνρσ ) , (45)where χ , α , β and γ are constants and C µνρσ is the Weyl curvature tensor. The combination in the parentheses,so-called Gauss–Bonnet invariant, is topological in four dimensions and can be discarded in the action integral. Thenwe find that the generic higher-curvature action expanded up to the second order in the metric perturbation h µν S HCG [ h µν ] = 12 κ Z d x (cid:16) − χ (1) G µν h µν − α (1) C µνρσ (1) C µνρσ + β (1) R (cid:17) , (46)where (1) C µνρσ and (1) R are the linear perturbation of the Weyl tensor and Ricci scalar, respectively, whose expressionsare presented in Appendix B. When χ = 0, the theory cannot be seen as GR with corrections and, moreover, as wewill see in Appendix D, there arise instabilities in the tensor and scalar parts. So, hereafter we assume χ = 0.Below, we take two different approaches to analyze the GWs in generic HCG.
1. Massive-bigravity approach
As first shown by Stelle [19], there is an equivalence of the action (46) to GR “minus” massive gravity, S [ φ µν , ˜ φ µν ] = χ S GR [ φ µν ] − χ S MG [ ˜ φ µν ]= χ κ Z d x (cid:20) − (1) G µν [ φ ] φ µν + (1) G µν [ ˜ φ ] ˜ φ µν + m (cid:16) ˜ φ µν ˜ φ µν − (1 − ǫ ) ˜ φ (cid:17)(cid:21) , (47)where we introduced m ≡ χ/ (2 α ) and ǫ = 9 β/ (2 α + 12 β ), see Appendix C for the derivation extended to Einsteinmanifolds [24]. Clearly, we need to assume α = 0. The case with α = 0 can be treated in a similar manner by meansof a conformal transformation, after which the theory takes a form of a scalar–tensor theory; See [23] for example.In this formalism, the original metric perturbation is given by h µν = φ µν + ˜ φ µν , (48)and, as seen from the structure of the action (47), the dynamical contents in this theory are φ µν as a massless spin-2field and ˜ φ µν as a mixture of a massive spin-2 and a spin-0 fields. It is worth mentioning that, while φ µν field has thesame gauge symmetry as GR, ˜ φ µν is not subject to gauge transformations.For the massless spin-2, we can choose the TT gauge as in GR treated in Sec. III A. The only dynamical dofs are φ TT ij = 2 H ij with its eom being (cid:3) H ij = 0. For the massive spin-2, the analysis is completely parallel to the case ofmassive gravity treated in Sec. III B. We decompose ˜ φ µν as˜ φ = − A , ˜ φ i = − ∂ i ˜ B − ˜ B i , ˜ φ ij = 2 ˜ C δ ij + 2 ∂ i ∂ j ˜ E + 2 ∂ ( i ˜ E j ) + 2 ˜ H ij (49)and define two scalar variables ˜ W ≡ ˜ A − ˙˜ B − ¨˜ E − ˜ C , ˜ φ ≡ ˜ φ µµ = 2 ˜ A + 6 ˜ C + 2 △ ˜ E . (50)Then we find the equations of motion for the dynamical variables (cid:0) (cid:3) − m (cid:1) ˜ H ij = 0 , (cid:0) (cid:3) − m (cid:1) ˜Σ i = 0 , (cid:0) (cid:3) − m (cid:1) ˜ W = 0 , (cid:18) βχ (cid:3) − (cid:19) ˜ φ = 0 . (51)In this case, the Fierz–Pauli tuning ǫ = 0 is realized and the spin-0 mode ˜ φ is required to vanish when β = 0, otherwiseit acquires a finite mass m ≡ χ/ (6 β ). Considering plane wave solutions H ij ∝ e i ω H ( z − t ) , ˜ H ij ∝ e i ( k ˜ H z − ω ˜ H t ) , ˜Σ i ∝ e i ( k ˜Σ z − ω ˜Σ t ) , ˜ W ∝ e i ( k ˜ W z − ω ˜ W t ) , ˜ φ ∝ e i ( k ˜ φ z − ω ˜ φ t ) (52)with dispersion relations k I = (p ω I − m ( I = ˜ H, ˜Σ , ˜ W ) p ω I − m ( I = ˜ φ ) , (53)we get the following expression for the Riemann tensor: (1) R i j = X λ =+ , × h ω H H λ + ω H ˜ H λ i e λij − q ω − m ω ˜Σ X p = x,y ˜Σ p e pij − m △ ˜ W (cid:16) ω W e B ij − √ m e L ij (cid:17) + ǫ φ (cid:18) ω φ e B ij + m √ e L ij (cid:19) . (54)To summarize, we have established that the gravitational-wave polarizations in generic HCG with α = 0 is a sum ofthose in GR and in MG, with a difference that the mass parameters m and m in this case are given by the expansioncoefficients χ , α and β inherent in the non-linear Lagrangian f .
2. Gauge-invariant approach
Next, we take a distinct approach starting with introduction of the gauge-invariant variables. We first assume that χ = 0 as in the previous section, but α = 0 is not mandatory. We will discuss some special cases in the main text andin Appendix D.Using the result presented in Appendix B, the action (46) can be rewritten in terms of the gauge-invariant variablesas S HCG [ h µν ] = S (T)HCG [ H ij ] + S (V)HCG [Σ i ] + S (S)HCG [Φ , Ψ] , (55)where each part is S (T)HCG [ H ij ] = 12¯ κ Z d x (cid:2) H ij (cid:3) H ij − α (cid:3) H ij (cid:3) H ij (cid:3) , (56) S (V)HCG [Σ i ] = 12¯ κ Z d x (cid:20) ∂ j Σ i ∂ j Σ i − ¯ α (cid:16) ∂ j ˙Σ i ∂ j ˙Σ i − △ Σ i △ Σ i (cid:17)(cid:21) , (57) S (S)HCG [Φ , Ψ] = 12¯ κ Z d x (cid:20) − (cid:3) Φ − △ (Ψ − Φ) −
43 ¯ α ( △ Ψ − △ Φ) + ¯ β (1) R (cid:21) (58)with (1) R = − (cid:3) Φ − △ (Ψ − Φ), ¯ κ ≡ κ/χ , ¯ α ≡ α/χ and ¯ β ≡ β/χ . As seen in the above expressions, when α = 0, thetensor and vector parts reduce to the ones in GR. Conversely, the tensor and vector parts are modified in comparisonto GR by the presence of the Weyl-squared term but unaffected by the Ricci-squared term in the expansion of theLagrangian (45). On the other hand, the scalar part is affected by both the Weyl-squared and Ricci-squared terms.In the following, we proceed to the analyses for each part.First, the tensor part is only affected by the presence of the Weyl term as seen in (56). The tensor eom is (cid:3) H ij − α (cid:3) H ij = 0 . (59)When α = 0, the tensor eom reduces to that of GR, and the same result for the polarization is obtained. When α = 0,we find that the above eom admits two independent solutions H ij = φ ij and H ij = ˜ φ ij which respectively satisfy (cid:3) φ ij = 0 , (cid:3) ˜ φ ij − m ˜ φ ij = 0 , (60)where, as before, m = 1 / (2 ¯ α ). Hence the general solution for H ij is H ij = φ ij + ˜ φ ij . (61)Considering plane-wave solutions propagating in the z direction, φ ij ∝ e i ω φ ( z − t ) , ˜ φ ij ∝ e i ( k ˜ φ z − ω ˜ φ t ) , (62)where k ˜ φ = q ω φ − m , and substituting these into (9), we obtain the tensor part of the Riemann tensor as (1) R (T) i j = ω φ (cid:0) φ + e + ij + φ × e × ij (cid:1) + ω φ (cid:16) ˜ φ + e + ij + ˜ φ × e × ij (cid:17) . (63)Identifying φ λ with H λ and ˜ φ λ with ˜ H λ , we reproduce the tensor part of (54).Next, as for the vector part, the equation of motion from the action (57) is(1 − α (cid:3) ) △ Σ i = 0 . (64)When α = 0, Σ i = 0 as expected. When α = 0, the eom admits a plane-wave solutionΣ i ∝ e i ( k Σ z − ω Σ t ) (65)with k Σ = p ω − m . For this, the vector part of the Riemann tensor (9) is (1) R (V) i j = − q ω − m ω Σ (cid:0) Σ x e xij + Σ y e yij (cid:1) , (66)0which is identical with the vector part of (54).Finally, we analyze the scalar part. As for scalar, counting of the number of dofs is a non-trivial task since theaction (58) contains second-order time derivatives non-linearly in the Ricci-squared term. Let us first introduce avariable Θ = △ (Φ − Ψ) to simplify the scalar action as S (S)HCG [Φ , Θ] = 12¯ κ Z d x h − (cid:3) Φ + 6Φ Θ − α Θ + ¯ β (1) R i . (67)To have an equivalent action with only derivatives lower than or equal to second order, we replace the Ricci scalar (1) R = − (cid:3) Φ + 3Θ with an auxiliary variable Ξ introducing a Lagrange multiplier λ as S (S)HCG [Φ , Θ , Ξ , λ ] = 12¯ κ Z d x h − (cid:3) Φ + 6Φ Θ − α Θ + ¯ β Ξ + λ ( (1) R − Ξ) i . (68)The variation of the above action with respect to Ξ gives a constraint λ = 2 ¯ β Ξ, which can be used to eliminate λ as S (S)HCG [Φ , Θ , Ξ] = 12¯ κ Z d x (cid:2) − (cid:3) Φ + 6Φ Θ − α Θ − ¯ β Ξ −
12 ¯ β Ξ (cid:3) Φ + 6 ¯ β Ξ Θ (cid:3) . (69)Now, after some manipulation, we obtain three independent equations from the variations of the above action withrespect to each variable, Θ − α (cid:3) Θ = 0 , (70)¯ β Ξ − β (cid:3) Ξ = 0 , (71)Φ = ¯ α Θ − ¯ β Ξ . (72)The eoms (70) and (71) imply that Θ and Ξ are independent dofs with masses m = 1 / (2 ¯ α ) and m = 1 / (6 ¯ β ),respectively. The algebraic constraint (72) indicates that when ¯ α = 0 ( ¯ β = 0), Θ (Ξ) can be eliminated. Whenboth ¯ α and ¯ β are non-zero, we can solve (70) and (71) for Θ and Ξ and obtain Φ via (72). We then have the othergauge-invariant variable Ψ = ¯ α Θ − △ − Θ − ¯ β Ξ . (73)Assuming plane-wave solutions Θ ∝ e i ( k Θ z − ω Θ t ) , Ξ ∝ e i ( k Ξ z − ω Ξ t ) (74)with k Θ = p ω − m and k Ξ = p ω − m , we arrive at the expression for the scalar part of the linear Riemanntensor (9) (1) R (S) i j = ¯ α Θ (cid:16) ω e B ij − √ m e L ij (cid:17) − ¯ β Ξ (cid:18) ω e B ij + m √ e L ij (cid:19) . (75)This is identical with the scalar part of (54) if we identify as¯ α Θ = − m △ ˜ W , ¯ β Ξ = − ǫ φ . (76)Our final task here is to investigate special cases with α = 0 or β = 0. When α = 0, the eom (70) reduces to aconstraint Θ = 0, which implies Ψ = Φ. Equation (72) reduces to Φ = − ¯ β Ξ, so the variable Φ represents the spin-0dof and obeys the eom (cid:3) Φ − m Φ = 0 . (77)Therefore, recalling Ψ = Φ, we find the scalar-type polarization in (9) in this case as (1) R (S) i j = Φ (cid:16) ω e B ij + √ m e L ij (cid:17) . (78)Also, the vector and tensor perturbations give the same result as in GR. This agrees with the result of Moretti etal. [23].1When β = 0, the constraint (72) reduces to Φ = ¯ α Θ , which implies thatΨ = Φ −
32 ¯ α △ − Φ (79)and the eom for Φ is (cid:3) Φ − m Φ = 0 . (80)It is clear that Φ in this case is the helicity-0 component of the massive spin-2. Recalling the relation to Ψ as givenby (79), the scalar-type polarization in (9) is calculated as (1) R (S) i j = Φ (cid:18) ω e B ij − m √ e L ij (cid:19) . (81) IV. DETERMINING THEORY PARAMETERS BY OBSERVATIONS
In this section, we provide a brief discussion of how one could determine the theory parameters, namely m or ǫ in MG and α and β in HCG. For brevity, below we collectively call the theory parameters “masses”. We considerinterferometers and pulsar timing arrays (PTAs) as GW measurement instruments.To discuss the ability of GW detectors for determining the masses via polarization measurements, it is necessaryto take into account detector’s response to each polarization in a GW propagating from the direction ( θ, φ ) withthe polarization angle ψ , which is represented by the antenna pattern functions F α ( θ, φ, ψ ) ( α = + , × , x, y, B , L)summarized in Appendix E. The whole signal takes the form S ( t ) = X α F α ( θ, φ, ψ ) A α ( t ) , (82)where A α is the waveform of each polarization being proportional to the coefficient of the polarization basis e αij appearing in the Riemann tensor.We note that different spin components have different velocity, so it would be plausible to treat each spin componentseparately if one is interested in the case of a short-duration source like a burst from a black-hole merger. Thus wefirst concentrate on the spin-0 GW alone, denoting its velocity as v . A spin-0 GW represented as a monochromaticplane-wave with frequency ω gives a signal of the form S ( t ) = F B A B ( t ) + F L A L ( t ) . (83)From (43) or (54) we see that there is a relationship between the waveforms A L ( t ) A B ( t ) = m ω . (84)Suppose that two detectors ( D = 1 ,
2) respond to a single spin-0 gravitational wave. In this case, the signals to bedetected by the two detectors are ( S ( t ) = F A B ( t ) + F A L ( t ) ,S ( t + ∆ t ) = F A B ( t ) + F A L ( t ) , (85)where ∆ t = L/v is the time delay between the arrivals at the two detectors separated by L along the propagationdirection. If the coefficient matrix F ≡ (cid:18) F F F F (cid:19) (86)is invertible, it is possible to solve for A B and A L as (cid:18) A B ( t ) A L ( t ) (cid:19) = F − (cid:18) S ( t ) S ( t + ∆ t ) (cid:19) . (87)2From this, the ratio of the waveforms R ≡ A L /A B is obtained in terms of observable signals and, therefore, the spin-0mass can be determined as m = ω R . (88)The above argument relies upon the invertibility of the matrix F . Unfortunately, LIGO-like interferometers havedegenerate antenna pattern functions F B = −
12 sin θ cos 2 φ , F L = 1 √ θ cos 2 φ , (89)so it is not possible to use the above method. On the other hand, PTAs have antenna functions [28, 29] F B = 12 sin θ v cos θ , F L = 1 √ θ v cos θ , (90)which are non-degenerate if the two detectors (pulsars) have different orientations with respect to the GW, θ = θ ,so they would allow the determination of the spin-0 mass. A straightforward extension to a setup with more detectors(pulsars) enables us to decompose a spin-2 GW into polarization modes and, in principle, determine the spin-2 mass m as well. Such multi-detector measurement would also be needed to analyze GW backgrounds.In an ideal case in which we know, or can predict, the spectral form of the spin-0 GWs A α ( ω, t ), i.e., its frequencydependence, we might be able to determine the mass even with a single detector. Consider two measurements of aGW at two different frequencies ω and ω (cid:26) S ( ω ) = F B A B ( ω ) + F L A L ( ω ) ,S ( ω ) = F B A B ( ω ) + F L A L ( ω ) , (91)where we omitted t . Substituting the relationship A L ( ω ) /A B ( ω ) = m /ω and solving for m , we obtain m = F B F L A B ( ω ) /A B ( ω ) − S ( ω ) /S ( ω ) ω − S ( ω ) /S ( ω ) − ω − A B ( ω ) /A B ( ω ) (92)Since the ratio A B ( ω ) /A B ( ω ) is assumed to be known, m can be measured. This method can also be extended tothe determination of the spin-2 mass m . V. CONCLUSION
In this paper, we studied gravitational-wave polarizations in generic linear massive gravity and generic higher-curvature gravity in the Minkowski background. We defined and analyzed the GW polarizations in terms of thecomponents of the Riemann tensor governing the geodesic deviation.In Sec. III B, we formulated the polarizations in linear MG with generic, non-Fierz–Pauli-type masses. We identifiedall the independent variables that obey Klein–Gordon-type equations. The dynamical dofs in the generic MG consistof spin-2 and spin-0 modes; the former breaks down into two tensor (helicity-2), two vector (helicity-1) and one scalar(helicity-0) polarizations, while the latter just corresponds to a scalar polarization. We found convenient ways ofdecomposing the two scalar modes of each spin into distinct linear combinations of the transverse and longitudinalpolarizations as in (43). This expression contains the graviton masses as the coefficients, so we expect it will serve asa useful tool in measuring the masses of GWs.In Sec. III C, we analyzed the linear perturbations of generic higher-curvature gravity (HCG) whose Lagrangianis an arbitrary polynomial of the Riemann tensor. When expanded around a flat background, the linear dynamicaldofs in this theory are identified as massless spin-2, massive spin-2 and massive spin-0 modes. The massive spin-2arises from the Weyl-squared term in the Lagrangian and the massive spin-0 from the Ricci scalar squared. Themassless spin-2 is characterized by the tensor-type (helicity-2) polarization modes. As its massive part encompassesthe identical structure to the generic MG, GWs in the generic HCG provide six massive polarizations on top of theordinary two massless modes. In parallel to MG, we found convenient representations for the scalar polarizationsdirectly connected to the coupling constants of HCG as in (54).In the analysis of HCG, we used two analysis method and showed that the two results agree. One takes fulladvantage of the partial equivalence between the generic HCG and MG at the linear level, whereas the other reliesupon a gauge-invariant formalism originally developed for cosmological perturbation theories. The present resultabout the scalar part can be compared with the case of inflationary cosmological perturbations in Einstein–Weylgravity studied in [30], where the conformal analogue of the gauge-invariant variable W = Ψ − Φ becomes dynamical.In Sec. IV, we gave a brief discussion about possible methods to determine the theory parameters by means ofGW-polarization measurements, but the full development in this direction is left to future work.3
ACKNOWLEDGMENTS
The authors are grateful to Yuki Niiyama for fruitful discussions. They thank Hideki Asada for useful comments.
Appendix A: Gauge transformations and gauge-invariant variables
In order to construct gauge-invariant variables, let us consider an active transformation of the coordinate systemunder which the coordinates of any point change according to x µ → x µ + ξ µ ( x ) , (A1)where the vector field ξ µ is as small as the perturbation. Accordingly, the spacetime metric transforms as g µν → g µν − £ ξ g µν (A2)where the arrow denotes the transformation induced by the coordinate change and £ ξ the Lie derivative along ξ µ . Itfollows that h µν → h µν − £ ξ η µν (A3)to first order in perturbations. The vector field ξ µ can be decomposed into the scalar and vector parts as( ξ µ ) = ( T, ∂ i L + L i ) (A4)with ∂ i L i = 0. It is obvious that this does not affect the tensor variable: H ij → H ij . (A5)On the other hand, the vector variables are transformed as B i → B i + ˙ L i , E i → E i − L i , (A6)so the following combination is found to be invariant:Σ i ≡ B i + ˙ E i . (A7)The transformations of the scalar variables are A → A − ˙ T , B → B − T + ˙ L , C → C , E → E − L , (A8)from which a useful set of invariant combinations is found to beΨ ≡ A − ˙ B − ¨ E , Φ ≡ C . (A9)
Appendix B: Expressions for curvature tensors and higher-curvature Lagrangian
This Appendix summarizes the necessary expressions for the linear-order curvature in the Minkowski backgroundas well as the quadratic-curvature action integrals expanded up to second order. Thanks to the topological nature ofthe Gauss–Bonnet combination in four dimensions, the Weyl-squared action can be rewritten as S C ≡ − α κ Z d x √− g C µνρσ C µνρσ = − α κ Z d x √− g (cid:18) R µν R µν − R (cid:19) (B1)up to irrelevant surface integrals. Thus, to compute its second-order expansion, we only need the first-order Riccitensor (1) R µν = − (cid:3) h µν + ∂ α ∂ ( µ h ν ) α − ∂ µ ∂ ν h , (1) R ij = − (cid:3) H ij + ∂ ( i ˙Σ j ) − ∂ i ∂ j Ψ − ∂ i ∂ j Φ − δ ij (cid:3) Φ , (1) R i = 12 △ Σ i − ∂ i ˙Φ , (1) R = △ Ψ − (1) R = ∂ µ ∂ ν h µν − (cid:3) h = − △ (Ψ − Φ) − (cid:3) Φ . (B3)The Weyl-squared action expanded up to second order is given in terms of the perturbative variables as (2) S C = − α κ Z d x (cid:18) (1) R µν (1) R µν − (1) R (cid:19) = − α κ Z d x (cid:20) (cid:3) h µν (cid:3) h µν − (cid:3) h µν ∂ α ∂ µ h αν + 13 (cid:3) h ∂ µ ∂ ν h µν + 13 ( ∂ µ ∂ ν h µν ) −
16 ( (cid:3) h ) (cid:21) = − α κ Z d x (cid:20) (cid:3) H ij ) + ( ∂ i ˙Σ j ) − ( △ Σ i ) + 43 [ △ (Ψ − Φ)] (cid:21) , (B4)where surface terms have been discarded. The computation of the second-order expansion of the Ricci-squared action S R ≡ β κ Z d x √− g R (B5)is straightforward: (2) S R = β κ Z d x (1) R = β κ Z d x (cid:2) ( ∂ µ ∂ ν h µν ) − (cid:3) h ∂ µ ∂ ν h µν + ( (cid:3) h ) (cid:3) = β κ Z d x △ (Ψ − Φ) + 3 (cid:3) Φ] . (B6) Appendix C: Decoupling dofs in quadratic curvature gravity on arbitrary Einstein manifolds
In this Appendix, we describe the equivalence of the quadratic curvature gravity (QCG) and GR “minus” MG atthe linear level on arbitrary Einstein manifolds [24]. Let us begin with the generic quadratic curvature action with acosmological constant S QCG [ g µν ] = 12 κ Z d x √− g (cid:0) R − − α C µνρσ C µνρσ + β R (cid:1) , (C1)where we have dropped the topological Gauss–Bonnet term. This theory admits any metric ¯ g µν satisfying R µν [¯ g µν ] = Λ ¯ g µν (C2)as a solution to the eom. It is useful to introduce a Lovelock tensor G µν ≡ G µν + Λ g µν , (C3)which vanishes when evaluated with ¯ g µν . The action can be rewritten as S QCG [ g µν ] = χ κ Z d x √− g − G − ˜ α G µν G µν + ˜ β G ! , (C4)where G = g µν G µν , χ ≡ β + 4 α/
3) Λ, ˜ α ≡ α/χ and ˜ β ≡ (2 β + 4 α/ /χ and where we have again discarded theGauss–Bonnet term. Taking ¯ g µν as the background and expanding the action up to quadratic order in h µν ≡ g µν − ¯ g µν ,we obtain the second-order action for h µν (2) S QCG [ h µν ] = χ κ Z d x √− ¯ g (cid:16) − h µν (1) G µν [ h µν ] − ˜ α (1) G µν [ h µν ] (1) G µν [ h µν ] + ˜ β (1) G [ h µν ] (cid:17) , (C5)where (1) G µν [ h µν ] ≡ (1) G µν [ h µν ] + Λ h µν (C6)5and (1) G [ h µν ] ≡ ¯ g µν (1) G µν [ h µν ]. As usual, the tensor indices are raised and lowered with the background metric ¯ g µν .Replacing (1) G µν [ h µν ] in (C5) with an auxiliary variable A µν and adding a constraint leads to (2) S QCG [ h µν , A µν , λ µν ] = χ κ Z d x √− ¯ g (cid:16) − h µν A µν − ˜ α A µν A µν + ˜ β A + λ µν ( A µν − (1) G µν [ h µν ]) (cid:17) , (C7)where A ≡ ¯ g µν A µν and λ µν is a Lagrange multiplier. The variation of the above action with respect to A µν gives analgebraic constraint λ µν = h µν + 2 ˜ α A µν − β A ¯ g µν , (C8)which can be substituted back to the action to eliminate A µν to give (2) S QCG [ h µν , λ µν ] = χ κ Z d x √− ¯ g (cid:20) − λ µν (1) G µν [ h µν ] + m (cid:0) ( h µν − λ µν ) ( h µν − λ µν ) − (1 − ǫ ) ( h − λ ) (cid:1)(cid:21) , (C9)where ǫ ≡ β/ (˜ α − β ) = 9 β/ (2 α + 12 β ), m ≡ / ˜ α = χ/ (2 α ) and λ ≡ ¯ g µν λ µν . Finally, by transforming h µν → φ µν + ˜ φ µν , λ µν → φ µν − ˜ φ µν , (C10)we arrive at (2) S QCG [ φ µν , ˜ φ µν ] = χ κ Z d x √− ¯ g (cid:20) − φ µν (1) G µν [ φ µν ] + ˜ φ µν (1) G µν [ ˜ φ µν ] + m (cid:16) ˜ φ µν ˜ φ µν − (1 − ǫ ) ˜ φ (cid:17)(cid:21) , (C11)where ˜ φ ≡ ¯ g µν ˜ φ µν . Equation (46) is obtained as the Minkowski version of this. Appendix D: χ = 0 This class contains conformal gravity and R gravity. At first glance one might expect this is the massless limit,but it is not. The action is S (T)HCG [ H ij ] = 12 κ Z d x (cid:2) − α (cid:3) H ij (cid:3) H ij (cid:3) , (D1) S (V)HCG [Σ i ] = 12 κ Z d x h − α (cid:16) ∂ j ˙Σ i ∂ j ˙Σ i − △ Σ i △ Σ i (cid:17)i , (D2) S (S)HCG [Φ , Θ , Ξ] = 12 κ Z d x (cid:2) − α Θ − β Ξ − β Ξ (cid:3) Φ + 6 β Ξ Θ (cid:3) , (D3)where we have already introduced Ξ to replace (1) R .The eoms for the tensor and vector variables can be easily found: α (cid:3) H ij = 0 , α (cid:3) △ Σ i = 0 . (D4)When α = 0, these eoms admit plane-wave solutions H ij = A ij e i ω A ( z − t ) + t B ij e i ω B ( z − t ) , Σ i = C i e i ω C ( z − t ) , (D5)with A ij , B ij and C i arbitrary constants. The tensor wave indicates the emergence of an instability.The scalar part is more involved. The variations of the action with respect to each scalar variable give a set ofequations β (cid:3) Ξ = 0 , − β (cid:3) Φ + 3 β Θ − β Ξ = 0 , α Θ − β Ξ = 0 . (D6)If β = 0, then we have Θ = 0 hence Φ = Ψ, but these cannot be determined. If β = 0 but α = 0, then Ξ = 0 and onecannot determine Φ and Θ. If both α and β are non-zero but α = 3 β , we obtain equations for Φ and Ψ as (cid:3) Φ = (cid:3) △ Ψ = 0 . (D7)Finally, in the most generic case when both α and β are non-zero and α = 3 β , we can eliminate Ξ and Θ to have theeom for Φ as (cid:3) Φ = 0 , (D8)which suggests that Φ is unstable.6 Appendix E: Detector responses
In this Appendix, we just summarize the angular pattern functions for the six polarizations defined by F α ( Ω ) ≡ D : e α ( Ω ) , (E1)where D is the so-called detector tensor, e α the polarization tensor, Ω the unit vector pointing the impinging directionof a GW and the symbol : denotes contraction between tensors.In the main text, we have kept using an inertial coordinate system such that a gravitational wave propagates inthe z direction. We call it the gravitational-wave frame and denote its orthonormal basis as ( m , n , Ω ), where Ω isthe unit vector along the z direction. Note that there is a rotation degree of freedom along the Ω axis which will bedenoted as the polarization angle ψ . The polarization tensors for α ∈ { + , × , x, y, B , L , T , T } can be written using theunit vectors as e + = m ⊗ m − n ⊗ n , e × = m ⊗ n + n ⊗ m , e x = m ⊗ Ω + Ω ⊗ m , e y = n ⊗ Ω + Ω ⊗ n , e B = m ⊗ m + n ⊗ n , e L = √ Ω ⊗ Ω , e T = r
23 ( m ⊗ m + n ⊗ n + Ω ⊗ Ω ) , e T = 1 √ m ⊗ m + n ⊗ n − Ω ⊗ Ω ) . (E2)To characterize ground-based interferometers or pulsar timing arrays, we introduce an inertial coordinate systemspecified by an orthonormal basis ( u , v , w ) such that w is the upward normal to the Earth’s surface. We call it thedetector frame and introduce the usual polar angles ( θ, φ ) in this frame to point the GW propagation direction. Wethen rotate the detector frame ( u , v , w ) to ( u ′ , v ′ , w ′ ) so that w ′ points towards the GW propagation as shown inFig. 2 of Ref. [31]. Their relationship is u ′ = cos θ cos φ u + cos θ sin φ v − sin θ w , v ′ = − sin φ u + cos φ v , w ′ = sin θ cos φ u + sin θ sin φ v + cos θ w . (E3)Finally, introducing ψ as the angle from u ′ to m in the plane perpendicular to w ′ , we arrive at the relationshipbetween the mediation coordinates and the GW coordinates m = cos ψ u ′ + sin ψ v ′ , n = − sin ψ u ′ + cos ψ v ′ , Ω = w ′ . (E4)Let us consider L-shaped interferometers such as LIGO, Virgo and KAGRA. The detector tensor in this case is [31] D = 12 ( u ⊗ u − v ⊗ v ) . (E5)7The antenna pattern functions F α are calculated as F + ( θ, φ, ψ ) = 12 (cid:0) θ (cid:1) cos 2 φ cos 2 ψ − cos θ sin 2 φ sin 2 ψ ,F × ( θ, φ, ψ ) = − (cid:0) θ (cid:1) cos 2 φ sin 2 ψ − cos θ sin 2 φ cos 2 ψ ,F x ( θ, φ, ψ ) = sin θ (cos θ cos 2 φ cos ψ − sin 2 φ sin ψ ) ,F y ( θ, φ, ψ ) = − sin θ (cos θ cos 2 φ sin ψ + sin 2 φ cos ψ ) ,F B ( θ, φ ) = −
12 sin θ cos 2 φ ,F L ( θ, φ ) = 1 √ θ cos 2 φ ,F T = 0 ,F T ( θ, φ ) = − √
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