Gravitational waves in modified teleparallel theories of gravity
aa r X i v : . [ g r- q c ] J un Gravitational waves in modified teleparallel theories of gravity
Habib Abedi ∗ and Salvatore Capozziello
2, 3, 4, 5, † Department of Physics, University of Tehran, North Kargar Ave, Tehran, Iran. Dipartimento di Fisica, Universit`a di Napoli ”Federico II”, Via Cinthia, I-80126, Napoli, Italy, Istituto Nazionale di Fisica Nucleare (INFN), Sez. di Napoli, Via Cinthia, Napoli, Italy, Gran Sasso Science Institute, Via F. Crispi 7, I-67100, L’Aquila, Italy, Tomsk State Pedagogical University, ul. Kievskaya, 60, 634061 Tomsk, Russia. (Dated: June 5, 2018)Teleparallel theory of gravity and its modifications have been studied extensively in literature.However, gravitational waves has not been studied enough in the framework of teleparallelism. In thepresent study, we discuss gravitational waves in general theories of teleparallel gravity containingthe torsion scalar T , the boundary term B and a scalar field φ . The goal is to classify possiblenew polarizations generalizing results presented in Ref.[16]. We show that, if the boundary termis minimally coupled to the torsion scalar and the scalar field, gravitational waves have the samepolarization modes of General Relativity. PACS numbers: 04.50.Kd, 04.30.-w, 98.80.-kKeywords: Teleparallel gravity; modified gravity; gravitational waves.
I. INTRODUCTION
The observed late expansion of the universe can be de-scribed by either introducing an exotic form of energy(dark energy) or modifying gravity. In this framework,several modifications have been proposed [1–4] and,among them, the possibility to consider teleparallel grav-ity [5]. Einstein introduced the idea of teleparallelismsoon after General Relativity (GR) [6]. Teleparallel La-grangian coincides with Einstein-Hilbert Lagrangian upto a boundary term, i.e. T = − R + B , where T is thescalar torsion, R is the Ricci scalar and B is a boundaryterm. Therefore, GR and Teleparallel Equivalent Gen-eral Relativity (TEGR) result in the same equations ofmotion.However, difference between them arise in modifiedLagrangians, where scalar fields coupled nonminimallyto gravity or arbitrary functions of T or R are takeninto account [5]. Such modifications of TEGR violatethe local Lorentz symmetry invariance and result in sixextra degrees of freedom [7]. In a more general case,the Lagrangian can be a function of both T and R , i.e. f ( T, R ) [8, 9]. This theory can be studied as f ( T, B ),where B is the boundary term [10, 11].In both GR and TEGR, gravitational waves (GW)have two independent polarizations, usually denoted asplus and cross modes. However, extra polarizations ap-pear in modified theories. The perturbation theory inthe post-Minkowski limit is a way to study the numberof GW polarizations.The other way is the Newman-Penrose (NP) formal-ism [12, 13]. Adopting the NP formalism in a genericmetric theory, plane GWs have six independent modes ∗ [email protected] † [email protected] of polarization: considering the z -direction as the prop-agation direction of GWs, they are plus (+), cross ( × ),breathing ( b ), longitudinal ( l ), vector- x ( x ) and vector-y( y ) modes. These modes can be described by the inde-pendent NP quantities { Ψ , Ψ , Ψ , Φ } , where Ψ andΨ are complex and each one describes two polarizationmodes. The extra polarization modes can be used to dis-criminate among modified theories of gravity beyond GR(see, e.g. Ref. [14, 15]). As shown in [16], GWs in f ( T ),and in its scalar-tensor representation, are equivalent tothat in GR and TEGR [17]. In f ( R ) gravity, where theLagrangian is an arbitrary function of Ricci scalar, threemodes exist [18–20]. Models f ( R, Θ) and f ( R, Θ φ ) werealso studied in Ref. [21], where Θ and Θ φ are the tracesof the energy-momentum tensors of standard matter andof a scalar field, respectively. The Authors studied differ-ent form of function f , and have shown that the numberof GW-modes depends on the form of it [22].An important remark is necessary at this point. Modi-fied theories of gravity are taken into account to achieve acomprehensive picture of cosmic dynamics ranging fromearly inflation, up to large scale structure formation andcurrent acceleration of the universe [1–4]. The approachis aimed to give, in principle, a full geometric descriptionof cosmic history consisting, for example, in extensionsof GR, like f ( R ), or of TEGR, like f ( T ). The maintask is explaining dynamics by further degrees of free-dom of gravitational field (with respect to GR or TEGR)instead of invoking dark components [23]. However, toachieve a self-consistent description, further scalar fieldscould be necessary. For example, as discussed in [24], theflat rotation curve of galaxies is better fitted consideringa theory like f ( R, φ ), instead of a pure f ( R ), because,in such a case, it is possible to reproduce the so-calledSanders potential with better precision. In this case, bya conformal transformation, it is shown that a modellike f ( R, φ ) is analogue to f ( R, (cid:3) R ) so that the scalarfield has a straightforward geometric interpretation too.In general, terms like (cid:3) R , (cid:3) R and so on appear as UVcorrections that have effects also at IR scales (see [2] for adetailed discussion of this topic). In this perspective, fur-ther scalar fields, having a geometric or a matter origin,could be useful to describe coherently cosmic dynamicsat any scale. Here, we consider TEGR extensions assum-ing not only general functions of the torsion scalar T , butalso boundary terms B and a scalar field φ that, accord-ing to the discussion in [24], could be geometrically in-terpreted. In particular, considering further scalar fieldsis important for a full classification of GW modes andpolarizations.The present paper is organized as follows. The fieldequations of our modified teleparallel theory are derivedin Sec. II. Sec. III is devoted to study GWs in two mod-ifications of teleparallel gravity; first, we study the caseof scalar field nonminimally coupled to both the scalartorsion and the boundary term. Then, we assume a La-grangian as a nonlinear function of the scalar torsion andthe boundary term. In Sec. IV, we obtain the number ofGW-polarizations when the scalar field kinetic term iscoupled to the scalar torsion. We show that, due to thelocal Lorentz Invariance violation, such a coupling is notviable because of the extra degrees of freedom. In Sec.V, we discuss the results and draw conclusions. II. TELEPARALLEL GRAVITY AND ITSEXTENSIONS
In teleparallel theories, vierbein fields describe gravity.Considering a set of orthonormal basis in each point of ageneric manifold, the metric is given by g µν = η AB e Aµ e Bν , (1)where e Aµ are vierbein fields and η AB is the Minkowskimetric. Then, one can write e Aµ e νA = δ νµ . With therule of absolute transport ˜ ∇ µ e νA = 0, the Weitzenb¨ockconnection with vanishing Riemann tensor is defined byΓ ανµ := e αA ∂ µ e Aν . (2)˜ ∇ µ is the covariant derivative is defined by theWeitzenb¨ck connection. This connection results in non-vanishing torsion tensor as follows T αµν = e αA (cid:0) ∂ µ e Aν − ∂ ν e Aµ (cid:1) . (3)Defining contorsion and superpotential, respectively, K µν := − (cid:0) T µνρ − T νµρ − T µνρ (cid:1) , (4) S µνρ := 12 (cid:0) K µνρ + δ µρ T ανα − δ νρ T αµα (cid:1) , (5)scalar torsion is T := S µνρ T ρµν . (6) The scalar torsion (6) is related to the Ricci scalar con-structed by the Levi-Civita connection as follows T = − R + B, (7)where B = 2 ∇ µ T νµν is a boundary term in the teleparal-lel Lagrangian. If a scalar field is nonminimally coupledto the torsion scalar, the Einstein frame can be recov-ered by considering the boundary term B coupled to thescalar field [25]. Let us now take into account the follow-ing action S = 12 Z d x e [ f ( T, B, φ ) − ∂ µ φ ∂ µ φ − V ( φ ) + 2 L m ] , (8)where e = det (cid:0) e Aν (cid:1) = √− g , V ( φ ) is a generic potentialand L m is the matter Lagrangian. The variation of action(8) with respect to the vierbein fields yields the followingfield equations e µA (cid:3) f B − e νA ∇ µ ∇ ν f B + 12 Bf B e µA + 2 ∂ ν ( f B + f T ) S νµA + 2 e − ∂ ν (cid:16) eS νµA (cid:17) f T − f T T ανA S µνα − e µA [ f − ∂ α φ ∂ α φ − V ( φ )] = Θ µA , (9)where Θ µA = − δ L m /δh Aµ is the stress-energy tensor ofmatter. Eqs. (9) in spacetime indices become − f T G µν + ( g µν (cid:3) − ∇ µ ∇ ν ) f B + 12 ( f B B + f T T − f ) g µν + 2 S αν µ ∂ α ( f T + f B ) − g µν (cid:20) ∂ α φ ∂ α φ + V ( φ ) (cid:21) + ∂ µ φ ∂ ν φ = Θ µν , (10)where we have used G νσ = − (cid:18) e − ∂ µ ( eS µνA ) − T ρµA S νµρ − e νA T (cid:19) e Aσ . (11)The variation of the action (8) with respect to the scalarfield results in (cid:3) φ + 12 f ′ − V ′ = 0 , (12)where prime denotes the derivative with respect to thescalar field φ . In the weak field approximation, the metriccan be written as g µν = η µν + h µν , (13)where h µν is small and first order, O (cid:0) h (cid:1) ≪ e Aµ = δ Aµ + h Aµ . (14)and R (1) µν = 12 (cid:0) ∂ ρ ∂ ν h ρµ + ∂ ρ ∂ µ h νρ − (cid:3) h µν − ∂ µ ∂ ν h (cid:1) , (15) R (1) = ∂ ρ ∂ µ h ρµ − (cid:3) h. (16)where h = η µν h µν and (cid:3) = η µν ∂ µ ∂ ν . The indices arelowered and raised by the Minkowski background metric η µν . The boundary term B is second order in perturba-tions; therefore, up to first order we have R (1) = − T (1) . III. NONMINIMAL COUPLINGA. The role of scalar field
In order to develop our considerations, we can specifythe function in (8) as f ( T, B, φ i ) = [ − ξ F ( φ )] T + χ E ( φ ) B, (17)where F and E are two arbitrary functions of scalar field.For ξ = 0 = χ it reduced to TEGR. Field equations getthe following form( − ξF ) G µν + χ ( g µν (cid:3) − ∇ µ ∇ ν ) E + 2 S αν µ ∂ α ( ξF + χE ) − g µν (cid:18) ∂ α φ ∂ α φ + V (cid:19) + ∂ µ φ ∂ ν φ = Θ µν . (18)At first order we have( − ξF ) (cid:18) R (1) µν − η µν R (1) (cid:19) + χE ′ (cid:0) η µν ∂ − ∂ µ ∂ ν (cid:1) δφ − h µν V − η µν V ′ δφ = Θ (1) µν . (19)Taking the trace of Eq. (19), we get − ( − ξF ) R (1) + 3 ξE ′ (cid:3) δφ − hV − V ′ δφ = Θ (1) . (20)According to these considerations, we can define¯ h µν = h µν − η µν h + χE ′ − ξF η µν δφ, (21)¯ h = − h + 4 χE ′ − ξF δφ, (22) h µν =¯ h µν − η µν ¯ h + χE ′ − ξF η µν δφ, (23)and, in vacuum, we have (cid:3) ¯ h µν = 0 . (24)With the plane wave ans¨atz, its solution in Fourier spaceis ¯ h µν ( k ) = A µν ( k ) exp ( ik α x α ) + c . c . (25) One can assume ¯ φ as the minimum of the potential, i.e. V ≃ V + 12 γ ( δφ ) . (26)The above scalar field equation, with the choice (17), getsthe following form (cid:3) φ + 12 ( ξT F ′ + χBE ′ ) − V ′ = 0 . (27)At first order, it becomes (cid:3) δφ − ξF ′ R (1) − V ′′ δφ = 0 , (28)where we have used B = O ( h ) and T (1) = − R (1) . Then,we get (cid:0) (cid:3) − m (cid:1) δφ = 0 , m = 2 V ′′ ( − ξF )2( − ξF ) − ξχF ′ E ′ , , (29)where m defines an effective mass. We assumed V ′ = 0.The solution at first order is then δφ ( q ) = a ( q ) exp ( iq α x α ) + c . c . (30)Let us now consider z as the direction of wave traveling.Taking Ω as the angular frequency, we have q = (cid:16) Ω , , , p Ω − m (cid:17) , (31)and the group velocity is v G = √ Ω − m Ω . (32)Assuming the speed v G constant, we get m = q (1 − v )Ω . (33)The effect of gravitational polarization can be studied bythe geodesic deviation,¨ x i = − R itjt x j . (34)Only the ”electric part” of the Riemann tensor, i.e. R itjt ,affects the geodesic deviation. In absence of modes thatare described by Eq. (25), i.e. ¯ h ij = 0, we have h µν = χE ′ − ξF η µν δφ. (35)Then, geodesic deviation becomes¨ x i = χE ′ − ξF ) (cid:16) η ij ¨ δφ + ( δφ ) ,ij (cid:17) x j . (36)Expressing (36) in components, one gets¨ x = − χE ′ Ω − ξF ) δφ x, ¨ y = − χE ′ Ω − ξF ) δφ y, ¨ z = − χE ′ m − ξF ) δφ z. (37)If Ω ≫ m , the displacement in longitudinal direction issmaller than the transverse one, ¨ z/z = ( m/ Ω) ¨ x/x . Invery low frequency band, l and b modes can be of thesame order. Considering the weak field limit, we canadopt the NP formalism. To obtain the independent NPquantities, one can use the solution (30), that is R (1) µν = (cid:20)(cid:18) − − ξF + 32 (cid:19) η µν (cid:3) + 1 − ξF ∂ µ ∂ ν (cid:21) χE ′ δφ. (38)Defining a set of tetrads ( e t , e x , e y , e z ), the null tetradsare k = 1 √ e t + e z ) , l = 1 √ e t − e z ) , m = 1 √ e x + i e y ) , ¯m = 1 √ e x − i e y ) , (39)where m and ¯m are complex but l and k are real. Thenull tetrads satisfy following relations − k · l = ¯m · m = 1 , k · l = k · ¯m = l · m = l · ¯m = 0 . (40)Then the non-vanishing NP quantities becomeΨ = − R l ¯ml ¯m ∼ + and × modes , (41)Ψ = − R l ¯m ∼ x and y modes , (42)Ψ = 16 R lk ∼ l mode , (43)Φ = − R ll ∼ b mode . (44)Then, we haveΨ =0 , (45)Ψ = χE ′ m a exp ( iq α x α )12 (cid:20) − ξF − (cid:21) (46)Φ = − χE ′ − ξF ) exp ( iq α x α ) ( q t − q z ) (47)therefore, in general, we have four independent polariza-tions: × , +, b and l modes. However, the NP formalismcan be used for massless waves. Considering V ′′ = 0 wehave Ψ =0 = Ψ , Ψ = 0 = Φ (48)therefore there exists just three modes: × , + and b . Thecase in which χ = 0 results in Φ = 0, consequently,the two polarization modes of GR remain. Consider thatthese two polarizations are obtained also in TEGR. Itis worth noticing that the massless scalar field, coupledwith the boundary term, leads to the breathing mode. B. The f ( T, B ) theory Let us consider now the following action S = 12 Z d x e f ( T, B ) . (49)The field equations are − f T G µν + ( g µν (cid:3) − ∇ µ ∇ ν ) f B + 12 g µν ( f B B + f T T − f )+ 2 S αν µ ∂ α ( f T + f B ) = 0 . (50)Supposing f ( T, B ) being an analytic function of T and B , one can expand it as follows f ( T, B ) = f ( T , B ) + f T ( T , B ) T + f B ( T , B ) B + f T B ( T , B ) T B + · · · . (51)Then the field equations at first order become − f T G (1) µν + f T B ( η µν (cid:3) − ∂ µ ∂ ν ) T (1) = 0 . (52)Up to first order we have again R (1) = − T (1) . Therefore,we get f T (cid:18) R (1) µν − η µν R (1) (cid:19) + f T B ( η µν (cid:3) − ∂ µ ∂ ν ) R (1) = 0 . (53)Using the transformation h µν = ¯ h µν −
12 ¯ hη µν − f T B f T R (1) η µν , (54)we get (cid:3) ¯ h µν = 0 . (55)The trace of Eq.(52) is f T R (1) − f T B (cid:3) R (1) = 0 . (56)Then we have (cid:3) R (1) + m R (1) = 0 , (57)where m = − f T f T B , (58)is the effective mass. The solution of this equation is R (1) = ˆ R ( q ρ ) exp ( iq ρ x ρ ) . (59)One can study different cases: • If f T B = 0 (for example F ( T )+ G ( B )), then, fromEq. (56), we get R (1) = 0 . (60) • In order to respect the local Lorentz symmetry in-variance, we have to consider f ( T, B ) = F ( R ). Inthis case, the field equations reduce to F R G µν + ( g µν (cid:3) − ∇ µ ∇ ν ) F R + 12 g µν ( F R R − F ) = 0 . (61)By considering a situation similar to the paper [26], F ( R ) = R + αR + βR , (62)the mass (58) reduces to m = − α and then re-sults for F ( R ) gravity can be easily recovered. Furthermore, the action (49) can be written as S = 12 Z d x e [ f ,φ T + f ,ψ − U ( φ, ψ )] , (63)where the new potential is 2 U ( φ, ψ ) = f ,φ + f ,ψ ψ − f ( φ, ψ ). Varying the action with respect to φ and ψ byassuming f ,φφ = 0 and f ,ψψ = 0, we get the identifica-tions φ = T and ψ = B that can be used as Lagrangemultipliers, that is S = 12 Z d x e [ f + f ,φ ( T − φ ) + f ,ψ ( B − ψ )]= 12 Z d x e h f − f ,φ (cid:16) (3) R + φ (cid:17) − f ,ψ ψ − f ,φ (cid:0) ¯Σ ij ¯Σ ij − ¯Σ (cid:1) + ( f ,φ + f ,ψ ) D T + f ,ψ D R (cid:3) . (64)Finally, we get S = Z d x N √ h ( f − f ,φ (cid:16) (3) R + φ (cid:17) − f ,ψ ψ − f ,φ (cid:0) ¯Σ ij ¯Σ ij − ¯Σ (cid:1) + ¯Σ N (cid:16) N j ¯ D j f ψ − f ,ψψ ˙ ψ − f ,ψφ ˙ φ (cid:17) ¯ D j f ψ ¯ D j ln N + h ij T αjα ¯ D i ( f ,φ + f ,ψ ) − ¯ D j ( f ,φ + f ,ψ ) ¯ D j ln N + A µ ∇ µ ( f ,φ + f ,ψ ) ) , (65)where A µ = n µ ¯ D i ω i + n µ N ¯ D i (cid:0) N b B ib (cid:1) + n µ (cid:0) B ij ¯ D j ω i + ω j ¯ D i B ji (cid:1) . (66)We have used the integration by parts. One can simplywrite the momentum conjugates of degrees of freedom as π φ = ∂S∂ ˙ φ = √ h (cid:2) − ¯Σ f ,ψφ + A N ( f ,φφ + f ,ψφ ) (cid:3) , (67) π ψ = ∂S∂ ˙ ψ = √ h (cid:2) − ¯Σ f ,ψψ + A N ( f ,φψ + f ,ψψ ) (cid:3) , (68) π N = ∂S∂ ˙ N = 0 , π N i = ∂S∂ ˙ N i = 0 . (69)The only term that contains time derivative of telepar-allel extra degrees of freedom is the second one in thesecond line of the action; according to our definition oftorsion, we have T αjα = T j + T iji = − N ∂ j ( N + N a ω a ) + ω a N ∂ j (cid:0) N a + ω a N + N b B ab (cid:1) + 1 N ∂ ω j − ω a N ∂ (cid:0) h aj + B aj (cid:1) + (cid:18) ω i + N i N (cid:19) ( ∂ j ω i − ∂ i ω j ) + (cid:18) B ia + N i N ω a + h ia (cid:19) (cid:2) ∂ i (cid:0) h aj + B aj (cid:1) − ∂ j ( h ai + B ai ) (cid:3) . (70)Then, using ∂T αjα ∂ ˙ ω k = 1 N δ kj , ∂T αjα ∂ ˙ B bk = − ω b N δ kj . (71) we have π ω k = ∂S∂ ˙ ω k = √ hh ik ¯ D i ( f ,φ + f ,χ ) , (72) π B ak = ∂S∂ ˙ B bk = −√ hh ik ω a ¯ D i ( f ,φ + f ,χ ) . (73)The momentum conjugate of h ij becomes π kl = ∂S∂ ˙ h kl = √ h (cid:20) − f ,φ (cid:0) ¯Σ kl − h kl ¯Σ (cid:1) + h kl N (cid:16) N j ¯ D j f ,ψ − f ,ψψ ˙ ψ − f ,ψφ ˙ φ (cid:17) − h ik h al ω a ¯ D i ( f ,φ + f ,ψ ) (cid:21) . (74)It is worth noticing that quantities constructed from h ij do not contain any extra degrees of freedom. Its tracebecomes π = √ h (cid:20) f ,φ ¯Σ + 3 N (cid:16) N j ¯ D j f ,ψ − f ,ψψ ˙ ψ − f ,ψφ ˙ φ (cid:17) − h al ω a ¯ D i ( f ,φ + f ,ψ ) (cid:21) . (75)In summary, we have classified all possible momenta re-lated to the degrees of freedom. IV. KINETIC COUPLING
In action (8) we have considered that gravity couplesminimally to kinetic term. In this section, we study suchcoupling in view of GW polarizations. Let us considerthe ADM line element,d s = − N d t + h ij (cid:0) d x i + N i d t (cid:1) (cid:0) d x j + N j d t (cid:1) , (76)where N , N i and h ij are the lapse function, the shiftfunction and the metric of three-dimensional space, re-spectively. One can write extrinsic curvature as follows¯Σ ij = 1 N (cid:16) ˙ h ij − ¯ D i N j − ¯ D j N i (cid:17) . (77)where ¯ D i the 3-Levi-Civita covariant derivative. Thenthe Ricci scalar is given by R = (3) R + ¯Σ ij ¯Σ ij − ¯Σ + D R , (78)where ¯Σ = ¯Σ ij h ij is the trace of the extrinsic curvatureand D R = 2 N √ γ ∂ t (cid:0) √ γ ¯Σ (cid:1) − N ¯ D i (cid:0) ¯Σ N i + γ ij ∂ j N (cid:1) . (79)In GR, R coupled to ∂ µ φ ∂ µ φ changes the number ofdynamical degrees of freedom (see [27] for a discussion).In view of this, let us onsider the action with the followingterm S ⊃ Z d x √− gRX, (80) where X = ∂ α φ ∂ α φ is the kinetic term. By using theADM decomposition, we have X = − N ˙ φ + N i N ˙ φ ∂ i φ + 12 (cid:18) h ij − N i N j N (cid:19) ∂ i φ ∂ j φ, (81)and then the action contains the following term S ⊃ Z d x √− g (cid:18) − N ˙ φ (cid:19) (cid:2) − ∇ µ ( ¯Σ n µ ) (cid:3) = Z d x ˙ φ ˙¯Σ N . (82)According to this development, the lapse function is adynamical variable. Therefore, it is unstable and hencenot viable for GWs. However, some fine tuned combina-tion of geometry and scalar field derivatives exists whichincludes G µν ∂ µ φ∂ ν φ where G µν is the Einstein tensor(see [28]). These extra degrees of freedom cancel outand allow the models to be stable and avoiding the Os-trogradskij instability. In the teleparallel approach, thevierbein fields, related to the ADM line element (76) canbe written as [29] e µ =( N, ) , e aµ =( N a , h ai ) ,e µ =(1 /N, − N i /N ) , e µa =(0 , h ia ) . (83)The torsion becomes T = − (3) R − ¯Σ ij ¯Σ ij + ¯Σ + D T , (84)where [29] D T = − N ¯ D k ( N T i ki ) . (85)is the boundary terms in T . Therefore we can split B ina curvature and torsion component, that is B = D R + D T . (86)Clearly D T has no time derivative while D R contains timederivative of ¯Σ. This means, in general, that the bound-ary term B contains time derivative. One can concludethat the coupling of D R or B to the kinetic term willresult in instability.Let us consider now the following action S = Z d x e (cid:20) R + 12 ∂ µ φ ∂ µ φ − V ( φ )+ 12 ( ξT + χB ) ∂ µ φ ∂ µ φ (cid:21) , (87)where ξ and χ represent coupling constant to the torsionscalar and the boundary term. ξ + χ = 0 is the casethat has been studied in Ref [28], then it was assumed χ = ξ = 0. However, in action (87), it is enough toconsider χ = 0, in order to avoid ghost instabilities. Theaction we are going to study contains a torsion scalarnonminimaly coupled to the kinetic term as follows S = Z d x e (cid:20) − T ξ∂ µ φ ∂ µ φ ) + 12 ∂ µ φ ∂ µ φ − V ( φ ) + L m (cid:21) . (88)For ξ = 0, the action (88) is equivalent to GR minimallycoupled to a scalar field. Varying with respect to thevierbein fields yields − ξ∂ µ φ ∂ µ φ ) h e − ∂ α (cid:0) eS ανA (cid:1) − T ρβA S νβρ i + 12 e νA T − ξS ανA ∂ α (cid:0) ∂ γ φ ∂ γ φ (cid:1) − e νA (cid:20) ∂ γ φ ∂ γ φ − V ( φ ) (cid:21) + ∂ ν φ ∂ A φ = Θ νA . (89)contracting with e Aσ , we get G νσ (1 + ξ∂ µ φ ∂ µ φ ) − ξS ανσ ∂ α (cid:0) ∂ γ φ ∂ γ φ (cid:1) − δ νσ (cid:20) ξT ∂ γ φ ∂ γ φ + 12 ∂ γ φ ∂ γ φ − V ( φ ) (cid:21) + ∂ ν φ ∂ σ φ = Θ νσ . (90)The trace of Eq. (89) is − ξ∂ µ φ ∂ µ φ ) (cid:2) e − e Aµ ∂ α ( eS ανA ) + T (cid:3) + 2 T − ξS ανν ∂ α ( ∂ γ φ ∂ γ φ ) − ∂ γ φ ∂ γ φ + 4 V = Θ . (91)This modification is not local Lorentz invariant. Vari-ation of the action with respect to the scalar field alsoresults in (cid:3) φ + V ,φ = ξ ∂ µ φ ∂ µ T. (92)Action (88) has been studied in Ref [30]. In a Friedman-Robertson-Walker background, we have T = G = 6 H .This implies that the derivative coupling T ∂ µ φ ∂ µ φ , onsuch a background, gives the same cosmological evolutionas the derivative coupling of the scalar field to the Ein-stein tensor G µν ∂ ν φ ∂ µ φ . However, beyond backgroundlevel, they will differ (see also [31]). Eqs. (90) and (92),at first order, results in (cid:20) G νσ − δ νσ (cid:18) ∂ γ φ ∂ γ φ − V ( φ ) (cid:19) + ∂ ν φ ∂ σ φ (cid:21) (1) = (Θ νσ ) (1) , (93)and ( (cid:3) φ + V ,φ ) (1) = 0 . (94)These equations are exactly the same as equations ofmotion for a scalar field minimally coupled to the Ricciscalar. Therefore, the number of GW polarizations arethe same as in the Einstein gravity. Under local Lorentz transformation e Aµ =Λ AB ( x ν ) e B , some quantities of teleparallel grav-ity are not invariant, e.g. torsion tensor be-comes T αµν + Λ AB e αA (cid:0) e Cν ∂ µ − e Cµ ∂ ν (cid:1) Λ C . Theinfinitesimal local Lorentz transformation isΛ AB ( x ) = ( e ω ) AB ≃ δ AB + ω AB . By breakingthis symmetry, six extra degrees of freedom appear [32],i.e. ω B =(0 , ω B ) , ω aB =( ω a , B ab ) , (95)where B ab is antisymmetric. Considering these new de-grees of freedom, the vierbein fields (83), up to first order,get the following form e µ =( N + N a ω a , ω i ) ,e aµ =( N a + N ω a + N b B ab , h ai + B ai ) ,e µ =(1 /N, − N i /N − ω i ) ,e µa =( − ω a /N, h ia + B ia + N i N ω a ) . (96)Up to second order in extra degrees of freedom, aftersome simple calculations, one gets [32] T = − (3) R + ¯Σ − ¯Σ ij ¯Σ ij + 2 N ¯ D i ¯ D i N − N ¯ D i (cid:0) h ij N T αjα (cid:1) − ∇ µ (cid:20) n µ ¯ D i ω i + n µ N ¯ D i ( N b B ib ) (cid:21) − ¯ ∇ µ (cid:2) n µ ( B ij ¯ D j ω i + ω j ¯ D i B ij ) (cid:3) . (97)Now, let us consider an action with the following couplingterm Z d x eT X. (98)The action contains S ⊃ Z d x ˙ φ N ∂ (cid:20) ¯ D i ω i N + ¯ D ( N b B ib ) N + B ij ¯ D j ω i N + ω j ¯ D i B ij N (cid:21) . (99)Therefore, on considering extra degrees of freedom, thetorsion scalar coupled into kinetic term results in insta-bility. A. G µν coupled to field derivatives Finally, let us consider the action containing the fol-lowing term G µν ∇ µ φ ∇ ν φ . (100)At first order, we get (cid:0) (cid:3) − m (cid:1) δφ =0 , (101) R (1) µν − η µν R (1) =0 . (102)where the massive term is m = V ,φ φ . Therefore, thenumber of GW polarization is the same as in GR plus ascalar mode related to the presence of the scalar field. V. CONCLUSIONS
The number of GW polarizations depends on the con-sidered theory of gravity. In present work we have studiedGWs in extended teleparallel gravity where a boundaryterm B and a further scalar field φ are taken into ac-count beside the torsion scalar T . The conclusions wereached are the following. There is no extra polariza-tion in TEGR and in f ( T ) theory with respect to GRas already shown in [16]. Here we demonstrated that ascalar field, non-minimally coupled to torsion, has onlythe two polarization of GR plus the scalar mode relatedto the scalar field itself. However, new polarizations ap-pear when the scalar field is coupled to the boundaryterm B , beside the standard two modes of GR. One canalso write the Lagrangian as a function of scalar torsion T and Ricci scalar R , however in order to study GWpolarizations, it is better to decouple the Ricci scalar R = − T + B and then using f ( T, B ). In f ( T, B ), extramassless and massive modes arise when the scalar tor-sion and the boundary term are non-minimally coupledas in the theory of f ( R ) = f ( − T + B ). The detectionof these extra modes could be a fundamental feature todiscriminate between metric and teleparallel approaches(see [5] for a discussion).In this perspective, the GW170817 event [33] has setimportant constraints and upper bounds on viable the-ories of gravity. In fact, besides the multi-messenger issues, the event provides constraints on the differencebetween the speed of electromagnetic and gravitationalwaves. This fact gives a formidable way to fix themass of further gravitational modes which results verylight (see [34] for details). Furthermore the GW170817event allows the investigation of equivalence principle(through Shapiro delay measurement) and Lorentz in-variance. The limits of possible violations of Lorentzinvariance are reduced by the new observations, by upto ten orders of magnitude [34]. This fact is extremelyrelevant to discriminate between metric and teleparallelformulation of gravitational theories. Finally, GW170817seems to exclude some alternatives to GR, including somescalar-tensor theories like Brans-Dicke gravity, Horava-Lifshitz gravity, and bimetric gravity [35]. Consideringthe present study, the reported data seem in favor of thetensor modes excluding the scalar ones. This means that f ( T ) gravity, showing the same gravitational modes asGR [16], should be favored with respect to other telepar-allel theories involving further degrees of freedom. Start-ing from these preliminary results, it seems possible acomplete classification of modified theories by gravita-tional waves. However, more events like GW170817 arenecessary in order to fix precisely gravitational parame-ters and not giving just upper bounds. In this context,the present study could constitute a sort of paradigm inorder to classify gravitational modes and polarizations(see also [14, 15]). In a forthcoming paper, the compar-ison with gravitational wave data will be developed indetail. ACKNOWLEDGMENTS
SC acknowledges the support of INFN ( iniziative speci-fiche
TEONGRAV and QGSKY). This paper is basedupon work from COST action CA15117 (CANTATA),supported by COST (European Cooperation in Scienceand Technology). [1] S. Nojiri and S. D. Odintsov, “Unified cosmic his-tory in modified gravity: from F(R) theory to Lorentznon-invariant models, Phys. Rept. (2011) 59doi:10.1016/j.physrep.2011.04.001 [arXiv: 1011.0544][2] S. Capozziello and M. De Laurentis, “ExtendedTheories of Gravity,” Phys. Rept. (2011) 167doi:10.1016/j.physrep.2011.09.003 [arXiv: 1108.6266][3] S. Nojiri, S. D. Odintsov and V. K. Oikonomou, “Modi-fied Gravity Theories on a Nutshell: Inflation, Bounceand Late-time Evolution, Phys. Rept. (2017) 1doi:10.1016/j.physrep.2017.06.001 [arXiv: 1705.11098][4] T. Clifton, P. G. Ferreira, A. Padilla and C. Sko-rdis, Modified Gravity and Cosmology, Phys. Rept. (2012) 1 doi:10.1016/j.physrep.2012.01.001[arXiv: 1106.2476][5] Y. F. Cai, S. Capozziello, M. De Laurentis andE. N. Saridakis, “f(T) teleparallel gravity and cosmology, Rept. Prog. Phys. (2016) 106901. doi:10.1088/0034-4885/79/10/106901 [arXiv:1511.07586 ][6] A. Einstein, Math. Ann. 102 (1930) 685.[7] M. Li, R. -X. Miao and Y. -G. Miao, ”Degrees offreedom of f(T) gravity”, JHEP 1107, 108 (2011)[arXiv:1105.5934].[8] R. Myrzakulov, FRW Cosmology in F(R,T) gravity,Eur. Phys. J. C72 (2012) 2203, [arXiv:1207.1039].[9] S. Capozziello, M. De Laurentis andR. Myrzakulov, “Noether Symmetry Ap-proach for teleparallel-curvature cosmology,”Int. J. Geom. Meth. Mod. Phys. 12 (2015) 1550095,[arXiv:1412.1471].[10] S. Bahamonde, C. G. B¨ohmer, and M.Wright, Modified teleparallel theories of gravity,Phys. Rev. D , 104042 (2015), [arXiv:1508.05120]. [11] S. Bahamonde and S. Capozziello, Noether SymmetryApproach in f ( T, B ) teleparallel cosmology,’ Eur. Phys.J. C (2017) no.2, 107 doi:10.1140/epjc/s10052-017-4677-0 [arXiv: 1612.01299].[12] D. M. Eardley, D. L. Lee, A. P. Lightman, R. V.Wagonerand C. M. Will, Phys. Rev. Lett. 30, 884 (1973).[13] D. M. Eardley, D. L. Lee and A. P. Lightman,Phys. Rev. D 8, 3308 (1973).[14] C. Bogdanos, S. Capozziello, M. De Laurentis, andS. Nesseris, Massive, massless and ghost modesof gravitational waves from higher-order gravity,Astropart.Phys. 34 (2010) 236-244, [arXiv:0911.3094].[15] X. Calmet, S. Capozziello and D. Pryer, GravitationalEffective Action at Second Order in Curvature and Grav-itational Waves, Eur. Phys. J. C (2017) no.9, 589doi:10.1140/epjc/s10052-017-5172-3 [arXiv:1708.08253].[16] K. Bamba, S. Capozziello, M. De Laurentis, S. No-jiri, and D. S. Gomez, No further gravitational wavemodes in F(T) gravity, Phys.Lett. B727 (2013) 194-198,[arXiv:1309.2698].[17] L.C. Garcia de Andrade, Gravitational and torsion wavesin linearized teleparallel gravity, [gr-qc/0206007 ].[18] M. E. S. Alves, O. D. Miranda and J. C. N. de Araujo,Phys. Lett. B 679, 401 (2009), [arXiv:0908.0861].[19] S. Capozziello, C. Corda and M. F. De Laurentis,Phys. Lett. B 669, 255 (2008), [arXiv:0812.2272].[20] H. Rizwana Kausar, L. Philippoz and P. Jet-zer, Phys. Rev. D 93, no. 12, 124071 (2016),[arXiv:1606.07000].[21] M. E. S. Alves, P. H. R. S. Moraes, J. C.N. de Araujo, and M. Malheiro, Gravitationalwaves in f ( R, T ) and f ( R, T φ ) theories of gravity,Phys.Rev. D 94 (2016) 024032, [arXiv:1604.03874].[22] O. Bertolami, C. Gomes, and F. S. N. Lobo, Gravita-tional waves in theories with a non-minimal curvature-matter coupling, [arXiv:1706.06826].[23] S. Capozziello and M. Francaviglia, Extended The-ories of Gravity and their Cosmological and Astro-physical Applications, Gen. Rel. Grav. (2008) 357,[arXiv: 0706.1146].[24] A. Stabile and S. Capozziello, Galaxy rotation curvesin f ( R, φ ) gravity, Phys. Rev. D (2013) 064002,[arXiv:1302.1760].[25] S. Bahamonde and M. Wright, Teleparallel quintessence with a nonminimal coupling to a boundary term,Phys. Rev. D , 084034 (2015), [arXiv:1508.06580].[26] H. R. Kausar, L. Philippoz, and P. Jetzer,Gravitational Wave Polarization Modes in f(R)Theories, Phys.Rev. D93 (2016) no.12, 124071,[arXiv:1606.07000].[27] S. Capozziello, G. Lambiase and H. J. Schmidt, Non-minimal derivative couplings and inflation in general-ized theories of gravity, Annalen Phys. (2000) 39,[arXiv:gr-qc/9906051].[28] C. Germani, and A. Kehagias, New Model of In-flation with Non-minimal Derivative Couplingof Standard Model Higgs Boson to Gravity,Phys.Rev.Lett. 105 (2010) 011302, [arXiv:1003.2635].[29] Y.-P. Wu, and C. -Q. Geng, Primordial Fluctuationswithin Teleparallelism, Phys.Rev. D86 (2012) 104058,[arXiv:1110.3099].[30] G. Kofinas, E. Papantonopoulos, and E. N.Saridakis, Self-Gravitating Spherically Sym-metric Solutions in Scalar-Torsion Theories,Phys. Rev. D 91, 104034 (2015), [arXiv:1501.00365].[31] S. Capozziello, M. De Laurentis, S. Nojiri andS. D. Odintsov, Evolution of gravitons in acceleratingcosmologies: The case of extended gravity, Phys. Rev. D95