Extra polarization states of cosmological gravitational waves in alternative theories of gravity
Marcio E. S. Alves, Oswaldo D. Miranda, Jose C. N. de Araujo
EExtra polarization states of cosmologicalgravitational waves in alternative theories of gravity
M. E. S. Alves, O. D. Miranda and J. C. N. de Araujo
INPE-Instituto Nacional de Pesquisas Espaciais - Divis˜ao de Astrof´ısica,Av.dos Astronautas 1758, S˜ao Jos´e dos Campos, 12227-010 SP, BrazilE-mail: [email protected], [email protected], [email protected]
Abstract.
Cosmological Gravitational Waves (GWs) are usually associated with thetransverse-traceless part of the metric perturbations in the context of the theory ofcosmological perturbations. These modes are just the usual polarizations ‘+’ and‘ × ’ which appear in the general relativity theory. However, in the majority of thealternative theories of gravity, GWs can present more than these two polarizationstates. In this context, the Newman-Penrose formalism is particularly suitable forevaluating the number of non-null GW modes. In the present work we intend totake into account these extra polarization states for cosmological GWs in alternativetheories of gravity. As an application, we derive the dynamical equations forcosmological GWs for two specific theories, namely, a general scalar-tensor theorywhich presents four polarization states and a massive bimetric theory which is in themost general case with six polarization states for GWs. However, the mathematical toolpresented here is quite general, so it can be used to study cosmological perturbationsin all metric theories of gravity.
1. Introduction
The future detection of gravitational waves (GWs) of cosmological origin will stronglyconstrain the possible inflationary scenarios which have been proposed in the lastdecades. Also, GWs will be useful to distinguish between the standard inflationarymodel and the alternative early Universe cosmologies, like the Pre-Big-Bang scenario,since the predicted power spectrum of each model can present very different features.In the general relativity theory (GRT) the usual procedure in order to evaluate thepower spectrum of cosmological GWs is to expand small metric perturbations aroundthe spatially homogeneous and isotropic Friedmann-Robsertson-Walker (FRW) metric.The next step is to identify GWs with the transverse-traceless (TT) part of the metricperturbations which do not couple with the perturbations of the perfect-fluid. Thus,once it was generated, this radiative gravitational field can freely travel through thespace and reach an observer today.The observational effect of GWs is to generate relative tidal accelerations betweentest particles. The Riemann tensor determines these relative accelerations and it is the a r X i v : . [ g r- q c ] A p r xtra polarization states of cosmological GWs in alternative theories of gravity P in the region. Let the observer set up an approximately Lorentz,normal coordinate system { x µ } = { t, x i } , with P as origin. For a particle with spatialcoordinates x i at rest, the relative acceleration with respect to P is: a i = − R i j x j , (1)where R i j are the so-called ‘electric’ components of the Riemann tensor due to wavesor other external gravitational influences. When the linearized theory is considered, theRiemann tensor can be split in six algebraically independent components, but for thevacuum field equations of GRT they reduce to two, which represent the two polarizationstates of free GWs. These are the ‘TT-modes’, also called the + and × polarizations.Although these are the most studied modes in the GWs physics, when theframework of an alternative theory of gravity is considered, the number of non-nullcomponents of the Riemann tensor can be greater than two and the theory presentsnot only the TT-modes, but also other polarization states can appear. This is adirect consequence of the new field equations which can generate other radiative modes.Thus, for a generic theory of gravity, GWs can present up to six polarization statescorresponding to the six independent components of R i j .Therefore, we should state that in order to work out the cosmological metricperturbations in an alternative theory of gravity, the first step is to find the numberof independent polarizations of GWs in such a theory, i.e., the number of non-nullcomponents of the Riemann tensor. This can be done by a very elegant method whichconsists in the evaluation of the non-null Newman-Penrose (NP) quantities [1, 2] of agiven theory. These quantities are the irreducible parts of the Riemann tensor writtenin a complex tetrad basis which make them very useful to evaluate the polarizations ofGWs in an unambiguous way.Thus, the present paper intends to consider the general formalism of cosmologicalperturbations in the context of alternative theories of gravity, focusing on the dynamicalequations of the GW modes. The theory of cosmological perturbations in GRT has beenlargely studied in the literature. Some classical examples are the works by Lifshiftz [3],Bardeen [4], Peebles [5], and Mukhanov, Feldman and Brandenberger [6]. For a recentreview on cosmological dynamics see, e.g.,[7, 8]. In the usual approach, the TT-modesof GWs are consistently described by the superadiabatic, or parametric, amplificationmechanism in the GRT [9]. Furthermore, it was shown by Barrow and de Garcia Maia[10, 11] that the same mechanism applies for modified theories of gravity as scalar-tensor theories and the so-called f ( R ) theories. Although they have analyzed only theevolution of the two TT-modes, it is known since the work by Eardley et al. [1] thatscalar-tensor theories present in addition at least one scalar GW polarization (moregeneral scalar-tensor theories present two scalar GW polarizations) as a consequenceof the additional degree of freedom included by the scalar field. The relic scalar GWproduction which arises from this kind of theories was recently discussed by Capozzielloet al. [12], and an upper limit was obtained from the amplitude of scalar perturbations xtra polarization states of cosmological GWs in alternative theories of gravity f ( R ) theories, using the NP formalism, it was shown that a particularclass of functions f ( R ) presents two scalar GW modes in addition to the + and × modes,thus totalizing four independent polarizations of GWs [13]. However, when the Palatiniapproach is used in the derivation of the field equations, the theory reveals only the usualtwo TT polarizations. It was also found that the scalar longitudinal mode which appearsin f ( R ) theories is a massive mode which is potentially detectable by the future spaceGW interferometer LISA [14]. Again by considering only the vacuum equations, theproduction of the relic GWs of this particular mode was also considered and constraintsusing the WMAP data was established [15].Moreover, the study of extra polarization states of cosmological GWs in the contextof alternative theories of gravity can reveal new interesting features of these theorieswhich do not appear in GRT. A remarkable example is the presence of vector longitudinalpolarization modes of GWs in some theories. These modes give rise to a non usual Sachs-Wolf effect which leaves a vector signature on the CMB polarization [16]. Otherwise,vector perturbations in GRT decay too fast and it would not leave any signature onCMB polarization.Therefore, it is clear that the future detection of GWs, and the correspondingdetermination of the number of polarization modes, are powerful tools to test theunderlying gravity theory. Thus, the goal of the present paper is to furnish a generalformalism to find the evolution equations of all the possible polarization modes whichcould appear in a generic theory of gravity. Once the number of independent polarizationmodes are found and the corresponding evolutionary equations could be obtained, one isin a position to obtain the power spectrum of each mode, finding the CMB signatures andconstraining the additional modes. In order to show the application of the formalism,and in order to find some new features of the current studied theories, we have chosento obtain the dynamical equations for GWs in the context of two particular theories,namely, a general scalar-tensor theory and a bimetric massive theory of gravity.First proposed by Brans and Dicke [17] in the aim of making the theory of gravitycompatible with the Mach’s principle, the scalar-tensor theories are of a great interestsince, as pointed out by several authors, a coupling between a scalar field and gravityseems to be a generic outcome of the low-energy limit of string theories (see, e.g., [18]).Another interest in the scalar-tensor models is that the f ( R ) theories can be writtenas the Einstein equations plus a scalar field, and thus we could in principle extend thesame formalism applied for the scalar-tensor theories to the f ( R ) field equations. Thebimetric massive theory we consider was proposed by Visser [19] in the aim to obtaingeneral covariant field equations with massive gravitons. His method was based on theintroduction of a non-dynamical metric ( g ) µν besides the physical metric g µν . Theresulting equations appear as a small modification of the Einstein field equations for xtra polarization states of cosmological GWs in alternative theories of gravity g ) µν are present only in an additionalenergy-momentum tensor. Furthermore, our past studies have shown that the Visser’stheory is a potential explanation for the current acceleration of the expansion of theUniverse [20, 21].In deriving the equations for GWs in the two theories we will first review howto obtain the number of independent polarization modes for any theory following theEardley et al. approach [1]. In the case of the scalar-tensor models the theory presentfour polarization states in the more general case. Otherwise, the Visser’s theory is asimple example of how a weak modification of gravity can produce six polarizationmodes. The subsequent analysis show that all the polarization modes, apart fromthe usual + and × polarizations, are dynamically “coupled” to the perturbations ofthe cosmological perfect fluid. We argue that this kind of coupling and the existenceof additional polarization states could furnish distinguishable signatures of alternativetheories in the power spectrum of the relic GWs.The paper is organized as follows: in the section 2 we present an overview of theNP formalism starting from the definition of the NP quantities which define the sixpossible polarization states for GWs. Then we find the non-vanishing parameters forthe GRT, scalar-tensor theories and for the Visser’s model. In the section 3, consideringa generic theory, we find general expressions for the perturbed Einstein tensor and forthe energy-momentum tensor in the generalized harmonic coordinates. In the section 4we introduce a decomposition scheme which depends on the number of non-vanishingpolarization modes of GWs which could appear in the various alternative theories. Inthe sections 5 and 6 we apply the formalism of the preceding sections for two particulartheories, the scalar-tensor theory and the Visse’s bimetric model. Finally, we presentour conclusions and discussions in the section 7.Throughout the paper we use units such that c = 1 unless otherwise mentioned.
2. An overview of the Newman-Penrose formalism
At every point of the space one can introduce systems of four linearly independentvectors e ( a ) µ , which are known as tetrads. The index in parenthesis is the tetrad indexwhich numbers the vectors from one to four. We can define the matrix: g ( a )( b ) = e ( a ) µ e ( b ) ν g µν , (2)which is an arbitrary symmetric matrix with negative-definite determinant. Its inverse g ( c )( a ) , which is defined by: g ( c )( a ) g ( a )( b ) = δ ( b )( c ) , (3)can be used to lower the tetrad indices: e ( a ) µ = g ( a )( b ) e ( b )( µ ) , (4) xtra polarization states of cosmological GWs in alternative theories of gravity g µν : g µν = g ( a )( b ) e ( a ) µ e ( b ) ν . (5)One can write any arbitrary vector or tensor as a linear combination of the fourtetrad vectors: T µν ··· γσ ··· = T ( a )( b ) ··· ( c )( d ) ··· e µ ( a ) e ν ( b ) e ( c ) γ e ( d ) σ · ·· , (6)where the quantities T ( a )( b ) ··· ( c )( d ) ··· are the tetrad components of the tensor. They arecalculated according to: T ( a )( b ) ··· ( c )( d ) ··· = T µν ··· γσ ··· e ( a ) µ e ( b ) ν e γ ( c ) e σ ( d ) · ·· , (7)which is consistent with (2) and (4). Tetrad indices are raised and lowered with g ( a )( b ) and g ( a )( b ) respectively.The advantages offered in many cases by the use of the tetrad components becomeclear when one examines their transformation properties and when one introducestetrads which are appropriate to the particular problem being investigated. From theequation (7) one can see that the tetrad components behave like scalars under coordinatetransformations, i.e., the tetrad indices of tensors do not change under a coordinatetransformation. Therefore, we have a good way of investigating the algebraic propertiesof tensors in a coordinate-independent fashion by the choice of the tetrads.A possible choice is to identify the tetrad vectors with the base vectors of a Cartesiancoordinate system in the local Minkowski system of the point concerned: g ( a )( b ) = e µ ( a ) e ν ( b ) g µν = η ( a )( b ) = diag( − , , , . (8)The four tetrad vectors, which we shall call ( e t , e x , e y , e z ), form an orthonormalsystem of one timelike and three spacelike vectors.Another special case is the use of null vectors as tetrad vectors. A particular tetrad,known as Newman-Penrose tetrad [2] can be constructed from the orthonormal systemintroduced above, two of the four vectors k and l are real null vectors: k = 1 √ e t + e z ) , l = 1 √ e t − e z ) , (9)and the other two null vectors m and m are compex conjugates of each other: m = 1 √ e x + i e y ) , m = 1 √ e x − i e y ) . (10)It is easy to verify that the tetrad vectors obey the relations: − k · l = m · m = 1 (11) k · m = k · m = l · m = l · m = 0 , (12)and from (2) and (4) we obtain: g ( a )( b ) = −
10 0 − . (13)A null tetrad basis is especially suitable for discussing null or nearly null waves. xtra polarization states of cosmological GWs in alternative theories of gravity The Riemann tensor R λµνκ can be split in the irreducible parts: the Weyl tensor, thetraceless Ricci tensor and the curvature scalar (see, e.g., [22]), whose tetrad componentscan be named respectively as Ψ, Φ and Λ following the notation of [2]. In general, in afour dimensional space we have ten Ψ’s, nine Φ’s and one Λ which are all algebraicallyindependent. However, when we restrict ourselves to nearly plane waves, we find thatthe differential and algebraic properties of R λµνκ reduce the number of independentcomponents to six [1]. Thus, following Eardley et al. [1] we shall choose the set { Ψ , Ψ , Ψ , Φ } to describe, in a given null frame, the six independent componentsof a wave in a generic metric theory. These NP quantities are related to the followingcomponents of the Riemann tensor in the null tetrad basis described earlier:Ψ = − R lklk , (14)Ψ = − R lklm , (15)Ψ = − R lmlm , (16)Φ = − R lmlm . (17)Note that, Ψ and Ψ are complex, thus each one represents two independentpolarizations. One polarization for the real part and one for the imaginary part, thustotalizing six components. Three are transverse to the direction of propagation, with tworepresenting quadrupolar deformations and one representing a monopolar “breathing”deformation. Three modes are longitudinal, with one an axially symmetric stretchingmode in the propagation direction, and one quadrupolar mode in each of the twoorthogonal planes containing the propagation direction. The Fig. 1, which was takenfrom [23], shows the displacements induced on a ring of freely falling test particles byeach one of these modes. GRT predicts only the first two transverse quadrupolar modes(a) and (b).Other useful expressions are the following relations for the Ricci tensor: R lk = R lklk , (18) R ll = 2 R lmlm , (19) R lm = R lklm , (20) R lm = R lklm , (21)and for the curvature scalar: R = − R lk = − R lklk . (22)The overall relative accelerations in a sphere of test particles is described by therelation (1) and can be expressed in terms of the symmetric “driving-force matrix” S whose components are not but the electric components of the Riemann tensor xtra polarization states of cosmological GWs in alternative theories of gravity S ij = R i j , where the latin indices represent spatial coordinates. These componentscan be written as a combination of the NP quantities in the following way: S = − (ReΨ + Φ ) ImΨ − ImΨ (ReΨ − Φ ) 2ImΨ − − . (23) Figure 1.
The six polarization modes of weak, plane, null GW permitted in anymetric theory of gravity. Also shown is the displacement that each mode induces on asphere of test particles. The wave propagates out of the plane in ( a ), ( b ) and ( c ), andit propagates in the plane in ( d ), ( e ) and ( f ). The displacement induced on the sphereof test particles corresponds to the following NewmanPenrose quantities: ReΨ ( a ),ImΨ ( b ), Φ ( c ), Ψ ( d ), ReΨ ( e ), ImΨ ( f ). See ref. [23]. Since each NP amplitude is linearly independent, we can expand the components S ij asthe sum: S ij = (cid:88) r =1 p ( r ) E ( r ) ij , (24)where we have renamed the NP quantities as follows [1]: p (1) ≡ Ψ , (25) xtra polarization states of cosmological GWs in alternative theories of gravity p (2) ≡ ReΨ , (26) p (3) ≡ ImΨ , (27) p (4) ≡ ReΨ , (28) p (5) ≡ ImΨ , (29) p (6) ≡ Φ , (30)and E ( r ) ij are the components of the “basis polarization matrices” which are given by:. E (1) = − E (2) = − E (3) = 2 E (4) = − − E (5) = E (6) = − . (31)Finally, analyzing the behavior of the set { Ψ , Ψ , Ψ , Φ } under rotations, wesee that they have the respective helicity values s = { , ± , ± , } . These are all thepossible helicity values for GWs in a general theory of gravitation. The procedure of evaluation of the number of independent polarizations involvesexamining the far-field, linearized, vacuum field equations of a theory, and then findingthe non-null NP amplitudes. For GW sources sufficiently far from the observer we hopethat the external solutions of GWs approach that of vacuum and linearized regime,in this way the method becomes an unambiguous tool for the determination of thepropagating physical modes of GWs. In what follows we are going to evaluate the NPcomponents of the Riemann tensor for three theories, namely, the GRT, a general classof scalar-tensor theories and a bimetric massive theory.In the case of the GRT, the field equations can be obtained from the Einsten-Hilbertaction: I = 116 πG (cid:90) √− gRd x + I M , (32)where I M is the action which describes the matter fields. From the Hamilton principle δI = 0 and the Einstein equations read: G µν = − πGT µν . (33) xtra polarization states of cosmological GWs in alternative theories of gravity T µν = 0 and (33) reduces to: R µν = 0 . (34)Therefore, from the relations for the Ricci tensor and for the curvature scalar inthe NP tetrad basis (18)-(22) we can show that: R lklk = R lklm = R lmlm = 0 , (35)or equivalently:Ψ = Ψ = Φ = 0 , (36)and since we have no further constraints on the components of the Riemann tensor weconclude that:Ψ (cid:54) = 0 . (37)Hence, as expected, a GW in the GRT presents two polarization states with helicity s = ± I = 116 π (cid:90) √− gd x (cid:2) − ϕR + ϕ − ω ( ϕ ) ∇ α ϕ ∇ α ϕ − U ( ϕ ) (cid:3) + I M , (38)where ϕ is a scalar field, ω ( ϕ ) is the coupling parameter and U ( ϕ ) can be interpreted asa potential associated with ϕ . The Brans-Dicke action can be obtained for the specialcase when ω is constant and U ( ϕ ) = 0. The field equations obtained by minimizing (38)with respect to the metric and with respect to the scalar field are: G µν = − πϕ T µν − ωϕ (cid:18) ϕ ; µ ϕ ; ν − g µν ϕ ; α ϕ ; α (cid:19) − ϕ ( ϕ ; µν − g µν (cid:3) ϕ ) − U g µν , (39)[3 + 2 ω ( ϕ )] (cid:3) ϕ − ϕ dUdϕ = 8 πT − dωdϕ ϕ ; α ϕ ; α , (40) T µν ; ν = 0 . (41)Note that the last condition, which express the conservation of the energy-momentum tensor, must be independently imposed since it is not a direct consequenceof the field equations as in the case of the GRT. The vacuum field equations now read: R µν = − ω ( ϕ ) ϕ ϕ ; µ ϕ ; ν − ϕ (cid:18) ϕ ; µν + 12 g µν (cid:3) ϕ (cid:19) + g µν U ( ϕ ) , (42)and R = − ωϕ ϕ ; α ϕ ; α − (cid:3) ϕϕ + 4 U. (43) xtra polarization states of cosmological GWs in alternative theories of gravity (cid:3) ϕ = 0 , (44)with the solution: ϕ = ϕ + ϕ e iq α x α , (45)where ϕ is a constant obtained from the cosmological boundary conditions, ϕ is asmall amplitude in such a way we can work only to first order in ϕ , and q α is the wavevector which is null for this particular case. It follows that the curvature scalar is null: R = 0 ⇒ R lklk = 0 , (46)and the Ricci tensor takes the form: R µν = − ϕ ϕ e iq α x α q µ q ν . (47)Hence if the wave is propagating in the + z direction, the only non-null componentsof the Ricci tensor are R zz , R zt and R tt , which means that R ll is the only non-nullcomponent in the tetrad basis, leading us to conclude that: R lklm = 0 and R lmlm (cid:54) = 0 , (48)and therefore:Ψ = Ψ = 0 , Ψ and Φ (cid:54) = 0 . (49)That is, GWs in the Brans-Dicke theory presents the two s = ± s = 0.Otherwise, if the potential U ( ϕ ) is not null we have a more general scalar-tensortheory. Now, (cid:3) ϕ is not null in general, but obeys the following relation in the linearizedregime [26]: (cid:3) ϕ − m ϕ = 0 , (50)where: m ≡ ϕ ( dU/dϕ ) ϕ = ϕ ω ( ϕ ) , (51)and ϕ is the value of ϕ for which the potential minimum is evaluated. The solution of(50) is given by (45), but now the wave vector satisfies: q α q α = − m , (52)thus the curvature scalar does not vanish but gives: R = − m ϕ ϕ e iq α x α , (53)for first order in ϕ , and the equation for the Ricci tensor is: R µν = − ϕ ϕ e iq α x α (cid:20) q µ q ν + 12 η µν m (cid:21) . (54) xtra polarization states of cosmological GWs in alternative theories of gravity = 0; Ψ , Ψ and Φ (cid:54) = 0 . (55)Now, we have a longitudinal scalar polarization mode besides the three modes whichappear in the Brans-Dicke theory. Note also that the appearance of the longitudinalmode is related to the presence of a massive scalar field in the theory.The next theory we are going to analyze is an alternative theory of gravity which takesinto account massive gravitons by the action [19]: I = (cid:90) d x (cid:20) √− gR πG + m L mass ( g, g ) (cid:21) + I M , (56)where m is the graviton mass in natural units and the Lagrangian L mass is a functionof the physical metric g and of a non-dynamical prior defined metric g . The massiveLagrangian proposed by Visser is: L mass ( g, g ) = 12 √− g (cid:110) ( g − ) µν ( g − g ) µσ ( g − ) σρ × ( g − g ) ρν − (cid:2) ( g − ) µν ( g − g ) µν (cid:3) (cid:111) . (57)By varying the action (56) with respect to the physical metric we obtain the fieldequations: G µν − m M µν = − πGT µν , (58)where the massive tensor M µν reads: M µν = ( g − ) µσ (cid:104) ( g − g ) σρ −
12 ( g ) σρ ( g − ) αβ ( g − g ) αβ (cid:105) ( g − ) ρν . (59)Among the existing bimetric theories of gravity (Rosen’s bimetric theory [27], forexample), the most common criterion for the choice of the non-dynamical backgroundmetric is to impose the Riemann-flat condition R λµνκ ( g ) = 0 [23, 28]. Thus, thesimpler choice is a flat metric which recovers the Minkowski metric by going to cartesiancoordinates [19, 20]. By introducing this background metric, Visser has constructed theLagrangian (57) aiming a general covariant description of massive gravitons in such away that the model circumvents the van Dam-Veltman-Zakharov discontinuity [29, 30],a inconsistency which plagues other massive terms.Furthermore, the Visser’s model could be an alternative explanation to the currentacceleration of the expansion of the Universe as indicated by the recession of distantSupernova [21]. In this sense, the massive tensor M µν mimics the dark energy effects inlarge scales while the theory passes all the local tests of gravity. Hence, it is a matter ofinterest to distinguish between the Visser’s theory and the Einstein gravity by dynamicaltests such as GW observations. xtra polarization states of cosmological GWs in alternative theories of gravity R µν = m (cid:18) M µν − η µν M (cid:19) , (60)and for the weak field approximation the right-hand-side of (60) is simply m h µν where h µν is a metric perturbation around the flat background metric g µν = ( g ) µν + h µν with | h µν | (cid:28)
1. Therefore, since there are no further restrictions on R µν , we conclude thatall its components are non-null and we find that: R lklk , R lmlm , R lklm and R lklm (cid:54) = 0 , (61)and then: Ψ , Ψ , Ψ and Φ (cid:54) = 0 . (62)Thus, GWs in the Visser’s theory present all the six possible polarization statesshowed in the Fig. 1. This result was first obtained by de Paula et al. [31] in a differentbut equivalent approach. Note that in the limit m →
0, the polarization modes Ψ , Ψ and Φ vanish and we recover the GRT with the only non-null mode Ψ .
3. Cosmological Perturbations without decomposition
Let us consider a general theory of gravity for which the field equations can be writtenin the form: G µν + F µν = − πGT µν , (63)where besides the Einstein tensor G µν and the energy-momentum tensor for the matterfields T µν , we have a general tensor F µν = F µν ( g αβ , γ αβ , (cid:36) α , ϕ, . . . ) which could be afunction of the physical metric g µν , of some prior defined metric γ µν , of vector fields (cid:36) µ , of scalar fields ϕ and of derivatives of these quantities. A prior defined metric is ingeneral considered in bimetric theories of gravity, for which besides the dynamical metric g µν , a kind of “absolute” geometry is specified through γ µν . One of the most knowntheory of this kind is the Rosen’s theory [27] for which the second metric takes intoaccount the effects of inertial forces. Another example is the bimetric massive theoryconsidered by Visser which was introduced in the last section and we will analyze inmore detail in the section 6.Supposing that there is a cosmological solution of such a theory, we can work out theperturbations on a cosmological background metric g µν . We adopt the metric g µν as theRobertson-Walker metric written in Cartesian coordinates with zero spatial curvature k = 0. With these considerations the line element reads: ds = a ( η ) η µν dx µ dx ν , (64)where the scale factor a ( η ) is a function of the conformal time x = η . The cosmic time t is related to the conformal time by the relation a ( η ) dη = dt and the Minkowski metricis η µν = diag( − , +1 , +1 , +1). xtra polarization states of cosmological GWs in alternative theories of gravity δg µν = a ( η ) h µν ( x α ) with | h µν | (cid:28) ds = a ( η )( η µν + h µν ) dx µ dx ν . (65)In this form, the indices of h µν are raised and lowered by the metric η µν . With theline element (65) in the equations (63) we can obtain the perturbed field equations: δG µν + δF µν = − π [ GδT µν + δGT µν ] , (66)where, in order to take into account the theories with varying Newtonian “constant”,we have included the perturbation δG .Hereafter we will use the generalized harmonic coordinates discussed by Bicak andKatz [32]. For this coordinate system, we have the condition: g µν δ Γ λµν = 0 , (67)where δ Γ λµν is the metric connection perturbation. It is easy to show that, the condition(67) is equivalent to: ∇ ν δ ¯ g µν = 0 , (68)where δ ¯ g µν = δg µν − g µν δg/ δg = g αβ δg αβ . One can show that δ ¯ g µν = a ¯ h µν wherethe ¯ h µν is the trace-reverse perturbation:¯ h µν = h µν − η µν h, (69)with h = η µν h µν and ¯ h = − h .With the metric perturbations given by (65) and with the condition (68) we canevaluate the components of the perturbed Einstein tensor δG νµ for first order in h µν . Astraightforward calculation leads to: δG = 12 a (cid:104) − ¯ h (cid:48)(cid:48) + ∇ ¯ h − H ¯ h (cid:48) + 3(3 H − H (cid:48) )¯ h − ( H − H (cid:48) )¯ h ii + 4 a H ∂ i ¯ h i (cid:105) , (70) δG i = 12 a (cid:104) − ¯ h i (cid:48)(cid:48) + ∇ ¯ h i − H ¯ h i (cid:48) + ( H − H (cid:48) )¯ h i + 2 a − H η ij ( ∂ j ¯ h − ∂ k ¯ h kj ) (cid:105) , (71) δG ji = 12 a (cid:104) − ¯ h j (cid:48)(cid:48) i + ∇ ¯ h ji − H ¯ h j (cid:48) i + ( H + H (cid:48) )¯ h kk δ ji − ( H − H (cid:48) )¯ h δ ji − a H η jk ∂ ( k ¯ h i )0 (cid:105) . (72)The prime in the above expressions denotes derivatives with respect to theconformal time η , and we have defined the Hubble parameter for the conformal time as H ≡ a (cid:48) /a .Now, it is necessary to evaluate the perturbations of the energy-momentum tensorfor a perfect fluid: T µν = ( ρ + P ) U µ U ν + P g µν , (73) xtra polarization states of cosmological GWs in alternative theories of gravity ρ and P are the energy density and the pressure, and U ν = − U ν = (1 , , , T µν → ˜ T µν = T µν + δT µν , (74)where: δT µν = ( ρ + P ) g λν ( U µ δU λ + δU µ U λ ) + ( δρ + δP ) U µ U ν + δP δ µν − ( ρ + P ) δg λν U µ U λ . (75)And considering the perturbed metric as defined earlier we find the components ofthe perturbed quantity δT µν : δT = − δρ, δT i = ( ρ + P )( V i + a − ¯ h i ) , δT ji = δP δ ji , (76)where, for completeness, we have considered V i = δU i (cid:28) U ν = U ν + δU ν = (1 − δg , V i ) . (77)It is easy to see that the four-velocity is approximately the unit timelike vectorsince we assume V i (cid:28)
1, and terms proportional to V and V h can thus be neglected.Then, from the calculation of the first order perturbation of the tensor δF µν andusing the perturbed Einstein tensor and the perturbed energy-momentum tensor givenabove, one can write the perturbed field equations for a generic theory (see eq. 66).As we will see in the following sections, the above equations are written in asuitable way to consider extra-polarization states of the GWs in any alternative theoryof gravity with the field equations (63). They are also useful to find solutions withoutany decomposition as it was presented in the interesting work by Bicak et al. [33]. Notethat, despite of the difference in notation, a similar set of equations can be found inthat reference.
4. Metric perturbations with extra polarization states of GWs
Instead of considering the minimal decomposition which appears, for example, in thereference [6], let us write the components of δg µν in a more general form which was firstpresented by Bessada and Miranda [16]: δ ¯ g µν = a ( η ) (cid:32) φ S i + ∂ i BS i + ∂ i B ¯ h ij (cid:33) , (78)where φ and B are scalar perturbations and S i are the components of a divergencelessvector perturbation ∂ i S i = 0. In our approach, the quantities φ , B and S i are dynamicalquantities which appear in any theory. On the other hand, the way the perturbation ¯ h ij must be decomposed depends on the number of independent GW polarization modes,and hence, it is theory dependent. Thus, the first step is to evaluate the number ofnon-null NP quantities and then we can decompose ¯ h ij identifying the GW amplitudes xtra polarization states of cosmological GWs in alternative theories of gravity h ij can beexpanded in terms of six components which represent the six metric amplitudes of GWs:¯ h ij ( x , η ) = (cid:88) r =1 (cid:15) ( r ) ij ¯ h ( r ) ( x , η ) , (79)where (cid:15) ( r ) µν are the polarization tensors.In what follows, without loss of generality, we can consider the wave vector of theGW oriented in the + z direction. Thus, we can construct the six (cid:15) ( r ) µν by combinationsof the three ortonormal vectors: (cid:96) i = (1 , , m i = (0 , , n i = (0 , , , (80)in the following way: (cid:15) ij (1) = n i n i , (81) (cid:15) ij (2) = (cid:96) i n j + (cid:96) j n i , (82) (cid:15) ij (3) = m i n j + m j n i , (83) (cid:15) ij (4) = (cid:96) i (cid:96) j − m i m j , (84) (cid:15) ij (5) = (cid:96) i m j + (cid:96) j m i , (85) (cid:15) ij (6) = (cid:96) i (cid:96) j + m i m j . (86)One can verify that the tensors (81) - (86) are linearly independent and form anortogonal basis. Writing in a matricial form we have: (cid:104) (cid:15) ij (1) (cid:105) = (cid:104) (cid:15) ij (2) (cid:105) = (cid:104) (cid:15) ij (3) (cid:105) = (cid:104) (cid:15) ij (4) (cid:105) = − (cid:104) (cid:15) ij (5) (cid:105) = (cid:104) (cid:15) ij (6) (cid:105) = . (87)We can see that, apart from a constant, these are just the basis polarization matrices(31) which we have described earlier. Now, the expansion (79) can be written in a moreintuitive form:¯ h ij = π ij + τ ij + χ ij + ψ ij , (88) xtra polarization states of cosmological GWs in alternative theories of gravity s : π ij ≡ ¯ h (1) ij → s = 0 τ ij ≡ ¯ h (2) ij + ¯ h (3) ij → s = ± χ ij ≡ ¯ h (4) ij + ¯ h (5) ij → s = ± ψ ij ≡ ¯ h (6) ij → s = 0 (89)And from the very definition of the quantities (89), we have the following properties: χ ii = τ ii = 0 , ∂ j χ ji = ∂ j ψ ji = 0 . (90)Their associated helicity values can be evaluated from the behaviour of theperturbations under rotations. Notice also that we have chosen to define separatelythe two tensors for the s = 0 modes of GWs since one of them is a longitudinal mode( π ij ) and the other is transversal to the direction of propagation ( ψ ij ). Furthermore,we have seen in the section 2 that these two modes can appear for some theories (ina general scalar-tensor theory, for example), but other theories have a structure suchthat only one scalar mode appears (this is the case of the Brans-Dicke theory). This isa particular feature of the scalar modes, since it is not possible that a certain theorypresents only one of the two tensor modes and not the other. In fact, there is notheory for which the + and × polarizations appear separated for vacuum GWs as itwas evidenced by the construction of the NP amplitude Ψ (see section 2). The samehappens with the two vector modes which generate the same NP quantity Ψ .It is instructive to write the NP quantities in terms of these metric perturbationsconsidering an observer today located at the local Minkowskian space-time. Thus,considering the Cartesian tetrad basis we have:Ψ = −
112 ( π zz, − π zz,zz + φ ,zz ) + 124 (¯ h , − ¯ h ,zz ) , (91)Ψ = 14 √ τ xz,zz − τ xz, ) + i ( τ yz, − τ yz,zz )] , (92)Ψ = 14 [ χ yy, − χ xx, + 2 iχ xy, ] , (93)Φ = −
14 ( ψ xx, + ψ yy, − ¯ h , ) , (94)where: ¯ h = − φ + ψ ii + π ii . (95)Now, since the polarization modes of GWs are linearly independent, we cansummarize the procedure to calculate the perturbed field equations for a given generaltheory of the form (63) in the following way: • Scalar metric perturbations xtra polarization states of cosmological GWs in alternative theories of gravity δ ¯ g ( s ) µν = a ( η ) (cid:32) φ ∂ i B∂ i B h ( s ) ij (cid:33) . (96)If the theory has Ψ (cid:54) = 0 and Φ (cid:54) = 0, it means that an observer today can measurethe two scalar GW modes ( s = 0). Hence, in order to describe the evolution of thesemodes we write the term h ( s ) ij in the form: h ( s ) ij = π ij + ψ ij , (97)with the constraint ∂ j ψ ji = 0 which guarantees the transversal propagation of the tensor ψ ji . If the theory has Ψ = 0 and Φ (cid:54) = 0, no longitudinal scalar GW amplitude canbe measured. Then π ij is suppressed, and in order to do not change the number ofdegrees of freedom of the metric perturbations, a scalar longitudinal component, say D , must be added. This new scalar component represents a dynamical perturbation ofthe cosmic gravitational potential rather than a radiative GW field. Consequently, anobserver today measures only one scalar GW mode whose evolution can be describedby ψ ij , and h ( s ) ij takes the form: h ( s ) ij = ∂ i ∂ j D + ψ ij , (98)where we have introduced D using the spatial partial derivatives in order to representthe longitudinal behaviour of this quantity. This is the same procedure of introducinglongitudinal scalars in the minimal decomposition [6]. For those theories which canbe found in this case (e.g., the Brans-Dicke theory), the only non-null scalar degree offreedom (in the local Minkowskian frame) is ψ ij . In this sense, it is always possibleto make D = 0 by gauge transformations in the reference frame of the local farfield observer, although in general D does not cancel out when considering the wholecosmological evolution of the perturbations.On the other hand, if the theory has Ψ = Φ = 0, the perturbations must bedecomposed in the usual way, since there are no scalar GWs which could be observedtoday, that is: h ( s ) ij = Eδ ij + ∂ i ∂ j D, (99)where we have added the new scalar quantity E to describe the evolution of theperturbations of the gravitational potential. The most important theory which can befound in this case is the Einstein theory for which we have only the two “pure tensor”degrees of freedom for vacuum GWs. In Minkowski coordinates, it is always possible tocancel out all the scalar components by gauge transformations, but in the presence ofthe cosmic fluid they are important to describe the evolution of the perturbations. • Vector metric perturbations xtra polarization states of cosmological GWs in alternative theories of gravity δ ¯ g ( v ) µν = a ( η ) (cid:32) S i S i h ( v ) ij (cid:33) , (100)where the vector S i and h ( v ) ij satisfies: ∂ i S i = h i ( v ) i = 0 (101)If the NP quantity Ψ is non-null (the Visser’s theory is an example), an observertoday can measure the two vector modes ( s = ±
1) of GWs, and the metric perturbations h ( v ) ij can be identified to the corresponding GW amplitude: h ( v ) ij = τ ij , (102)with τ ii = 0.On the other hand, if Ψ = 0, there are no vector GWs today and we have theusual representation in terms of the vector quantity Q i : h ( v ) ij = ∂ i Q j + ∂ j Q i , (103)where, from (101) Q i is divergenceless ∂ i Q i = 0 and again, it is easy to verify thatthe number of the degrees of freedom of the metric perturbations does not change.The quantity Q i is a vector perturbation of the gravitational potential for which thedynamical equations gives, in general, a decaying mode in the context of the GRT(see, e.g. [4, 6]). As in the case of scalar perturbations we have discussed earlier, itis always possible to cancel Q i by gauge transformations in the local Minkowski framewhen Ψ = 0. • Tensor metric perturbations
Finally, the tensor metric perturbations are constructed using a symmetric tensor χ ij which satisfies the constraints: χ ii = ∂ j χ ji = 0 . (104)Thus, the tensor component which corresponds to GWs with s = ± δ ¯ g ( t ) µν = a ( η ) (cid:32) χ ij (cid:33) . (105)Counting the number of independent components we have used to construct δg µν ,and the number of constraints, we can see that we have four functions for scalarperturbations, four functions for vector perturbations, and two functions for tensorperturbations. Thus, as expected, we have ten independent components of δg µν . xtra polarization states of cosmological GWs in alternative theories of gravity is the onlynon-null NP quantity, thus it is direct to verify that the metric perturbations have theusual minimal decomposition: δ ¯ g µν = a ( η ) (cid:32) φ S i + ∂ i BS i + ∂ i B Eδ ij + ∂ i ∂ j D + ∂ i Q j + ∂ j Q i + χ ij (cid:33) , (106)and GWs are described only by the quantity χ ij whose evolution equation can be derivedfrom δG ( t ) µν = − πGT ( t ) µν to obtain the very known result: χ j (cid:48)(cid:48) i + 2 H χ j (cid:48) i − ∇ χ ji = 0 . (107)In the following sections we will exemplify the above decomposition scheme for thetwo other theories which we have treated in the section 2, namely, a general scalar-tensortheory and the Visser’s bimetric theory with massive gravitons. For each theory, afteridentifying the form of the metric perturbations we will find the dynamical equationsfor GWs in the coordinate system defined by (67).
5. GWs in scalar-tensor theories
In this section we will consider perturbations of the general class of scalar-tensor theoriesintroduced in the section 2. With a glance at the field equations (39) we identify it withthe general form (63) where G ( ϕ ) = φ − and the generic function F µν takes the form: F µν = ω ( ϕ ) ϕ (cid:18) ϕ ; µ ϕ ; ν − g µν ϕ ; α ϕ ; α (cid:19) + 1 ϕ ( ϕ ; µν − g µν (cid:3) ϕ ) + g µν U ( ϕ ) . (108)Thus, disturbing the Newtonian “constant”: δG = dGdϕ δϕ = − δϕϕ , (109)and evaluating the components of the perturbation δF µν , we can find the perturbed fieldequations (66) which for this case reads: δG µν = − πϕ (cid:18) δT µν − δϕϕ T µν (cid:19) + δF µν . (110)As we have already shown in the section 2, the evaluation of the NP parameters forthe most general scalar-tensor theory lead us to conclude that an observer today wouldmeasure: Ψ (cid:54) = 0 , Ψ = 0 , Ψ (cid:54) = 0 and Φ (cid:54) = 0 , (111)remembering that Ψ = 0 if the potential associated to the scalar field is null U ( ϕ ) = 0.Thus, following the procedure of the last section we can write the perturbations for thegeneral case which reads: δ ¯ g µν = a ( η ) (cid:32) φ S i + ∂ i BS i + ∂ i B π ij + χ ij + ψ ij + ∂ i Q j + ∂ j Q i (cid:33) . (112)Now, we can introduce the perturbations given above in the perturbed Einsteintensor and in the perturbed energy-momentum tensor calculated without decomposition xtra polarization states of cosmological GWs in alternative theories of gravity δF µν . Thus, after astraightforward calculation we find the dynamical equations which describe scalar andtensorial GWs in the context of the scalar-tensor theories: Scalar φ (cid:48)(cid:48) + 2 H φ (cid:48) − (9 H − H (cid:48) ) φ − ∇ φ − (cid:104) ω (cid:16) ϕ (cid:48) ϕ (cid:17) + ϕ (cid:48)(cid:48) ϕ (cid:105) ( φ + ξ ii ) −
12 [( H (cid:48) − H )] ξ ii − a H∇ B = 16 πa ϕ − (cid:16) − δρ + δϕϕ ρ (cid:17) − , (113)( ∂ i B ) (cid:48)(cid:48) + 4 H ( ∂ i B ) (cid:48) − ∇ ( ∂ i B ) + ( H (cid:48) − H ) ∂ i B + (cid:104) ϕ (cid:48)(cid:48) ϕ + ω (cid:16) ϕ (cid:48) ϕ (cid:17) (cid:105) ∂ i B − H a − (2 ∂ i φ + ∂ k π ik ) = 16 πGa ( ρ + P )( V i (cid:107) + ∂ i B ) + 2∆ i , (114) ξ j (cid:48)(cid:48) i + 2 H ξ j (cid:48) i − ∇ ξ ji + ( H (cid:48) + H ) ξ kk δ ji − H a ( ∂ i ∂ j B )+( H (cid:48) − H ) φδ ji = 16 πa ϕ (cid:104) δP δ ji − δϕϕ P δ ji (cid:105) + 2∆ ij + ϕ (cid:48) ϕ (cid:104) H (cid:16) ξ ji + 12 δ ji φ (cid:17) + 14 ω ϕ (cid:48) ϕ ( φ + ξ kk ) δ ji (cid:105) (115) Tensor χ j (cid:48)(cid:48) i + (cid:18) H + ϕ (cid:48) ϕ (cid:19) χ j (cid:48) i + 2 H ϕ (cid:48) ϕ χ ji − ∇ χ ji = 0 . (116)In the above equations we defined ξ ji = π ji + ψ ji , and V i (cid:107) is the component of V i which is parallel to the direction of propagation. The perturbed quantities ∆ , ∆ i and∆ ij take into account the perturbation of the scalar field ϕ and its derivatives. Theyare defined as follows:∆ = ω ϕ (cid:48) ϕ (cid:18) δϕϕ (cid:19) (cid:48) + 12 (cid:18) ϕ (cid:48) ϕ (cid:19) δω + 3 H ϕ (cid:48) ϕ δϕϕ + a (cid:20) δ ( ϕ ;00 ) ϕ + δ ( (cid:3) ϕ ) ϕ (cid:21) , (117)∆ i = ω ϕ (cid:48) ϕ ∂ i (cid:18) δϕϕ (cid:19) + δ ij δ ( ϕ ;0 j ) ϕ , (118)∆ ij = ω ϕ (cid:48) ϕ (cid:16) δϕϕ (cid:17) (cid:48) δ ji + 12 (cid:16) ϕ (cid:48) ϕ (cid:17) δωδ ji + (cid:16) a (cid:3) ϕϕ + H ϕ (cid:48) ϕ (cid:17) δϕϕ δ ji + 1 ϕ (cid:104) δ ( ϕ ; il ) δ lj − a δ ( (cid:3) ϕ ) δ ji (cid:105) . (119)The equation for tensor perturbations in scalar-tensor theories (116) was studiedin the reference [10]. It represents the evolution of free GWs with helicity s = ± xtra polarization states of cosmological GWs in alternative theories of gravity ϕ contributes for thecosmological potential which generates the amplification.Regarding scalar perturbations, the driven equations for the NP modes Ψ and Φ are given by the equations (113), (114) and (115). These set of equations, considerablymore complicated than the tensorial case, show some new physical features whencompared with the usual metric decomposition. The most important difference is thatthese equations represent the evolution of radiative fields coupled to the evolution ofthe generalized Newtonian gravitational potential. The radiative fields are representedby the quantities π ij and ψ ij while the generalized Newtonian potential is described bythe dynamical perturbations φ and B or some particular combination of them. Thus, insome sense, we can say that if the scalar-tensor theory is the “correct” theory we wouldhave a new kind of cosmological GW background. On the contrary to the tensorial case,this new background is coupled to the matter perturbations of the perfect fluid. Thisis expected since the usual scalar perturbations are also coupled to the fluid dynamics.But, in the usual sense, GWs would be coupled to the matter only if a component ofanisotropic stress would be present.A direct consequence of such a coupling is that the whole evolution of the densityperturbations of the cosmic fluid would depend not only on the evolution of φ and B but also on the evolution of the amplitudes of the scalar GWs π ij and ψ ij . Thus, from(113), for example, we are lead to conclude that: δρρ = δρρ ( φ, B, ξ ii , δϕ ) , (120)where we have also included the dependence on the perturbations of the scalar field δϕ . Notice that, in fact, the dependence of δρ/ρ with the GWs amplitudes appears asthe dependence on the trace of the overall contribution of the scalar GW amplitudes ξ ii = π ii + ψ ii . Therefore, a complete understanding of the evolution of the GWs withhelicity s = 0 requires the knowledge of the evolution of the scalar perturbations and,similarly, these GW modes affect the evolution of the density perturbations.Another issue of particular interest is how the presence of the scalar GWs wouldaffect the angular pattern of the CMB. Since the GW amplitudes π ij and ψ ij now enterthe geodesic equation for photons, it is expected that they leave a signature on theCMB due the so-called Sachs-Wolfe effect, which can be understood as the shift ofphoton frequency along the line of sight. The small fluctuations in the CMB may beconveniently described by perturbations of the temperature parameter T in the Planckdistribution f . The computation of the contribution of the scalar GWs to the angulartemperature inhomogeneities δT /T of the CMB is out of the scope of the present paperand a rigorous treatment will appear elsewhere. xtra polarization states of cosmological GWs in alternative theories of gravity
6. GW modes in a bimetric theory of gravity
Now, let us turn our attention to the massive bimetric theory first considered by Visser[19] which we have introduced in the section 2. Comparing the field equations (58) withthe generic form (63) we identify F µν = − m M µν .Our explicit calculations of the section 2, and the previous result by de Paula etal. [31], have led us to the conclusion that all the NP quantities are non-null for theVisser’s model:Ψ (cid:54) = 0 , Ψ (cid:54) = 0 , Ψ (cid:54) = 0 and Φ (cid:54) = 0 . (121)Thus, following the procedure of the section 4, the perturbations now should bewritten in the form: δ ¯ g µν = a ( η ) (cid:32) φ S i + ∂ i BS i + ∂ i B π ij + τ ij + χ ij + ψ ij (cid:33) . (122)With (122) in the perturbed field equations (66), calculating the components of δF µν and with the help of the perturbed components of the Einstein tensor and of theenergy-momentum tensor calculated in the section (3), we obtain the perturbed fieldequations for Visser’s theory for each group of perturbations: Scalar φ (cid:48)(cid:48) + 2 H φ (cid:48) − ∇ φ − (9 H − H (cid:48) − m a ) φ − [( H (cid:48) − H ) − m a ( a − ξ ii − a H∇ B = − πGa δρ, (123)( ∂ i B ) (cid:48)(cid:48) + 4 H ( ∂ i B ) (cid:48) − ∇ ( ∂ i B ) + 12 [2( H (cid:48) − H ) + m a (3 − a )] ∂ i B − H a − (2 ∂ i φ + ∂ k π ik ) = 16 πGa ( ρ + P )( V i (cid:107) + ∂ i B ) , (124) ξ j (cid:48)(cid:48) i + 2 H ξ j (cid:48) i − ∇ ξ ji − ( H (cid:48) + H ) ξδ ji + 12 m a ( a + 1) ξ ji +4 H a ( ∂ i ∂ j B ) −
12 [2( H (cid:48) − H ) − m a ( a − φδ ji = 16 πGa δP δ ji . (125) Vector S i (cid:48)(cid:48) + 4 H S i (cid:48) − ∇ S i + 12 [2( H (cid:48) − H ) − m a ( a − S i +2 H a − ∂ k τ ik = 16 πGa ( ρ + P )( V i ⊥ + S i ) , (126) τ j (cid:48)(cid:48) i + 2 H τ j (cid:48) i − ∇ τ ji + 12 m a ( a + 1) τ ji + 4 H aη kj ∂ ( i S k ) = 0 . (127) xtra polarization states of cosmological GWs in alternative theories of gravity Tensor χ j (cid:48)(cid:48) i + 2 H χ j (cid:48) i − ∇ χ ji + 12 m a ( a + 1) χ ji = 0 . (128)Similarly to the tensorial equations for the GRT and for the scalar-tensor theory,this last equation describe the evolution of free GWs related to the NP mode Ψ .But now it is the new term which contains m that contributes to the parametricamplification of GWs.The equations for the scalar modes ( s = 0) are now given by the set (123), (124)and (125). The argumentation for the scalar modes π ij and ψ ij is similar to the caseof the scalar-tensor theories, except for the absence of the scalar field perturbation δϕ .Again, the three equations must be solved simultaneously in order to find the evolutionof the GWs with helicity s = 0 and to find the evolution of the density perturbationwhich is related to the metric perturbations through the equation (123): δρρ = δρρ ( φ, B, ξ ii ) . (129)Conversely, we have now GWs with helicity s = ± . The equation which describes the evolution of the GW amplitudes for this mode isthe equation (127). Note the similarity of this equation to the equation (128), exceptfor the presence of the term containing S i in the equation (127). The presence of thisterm makes the vector GW modes coupled with the vector perturbations since S i iscoupled with the fluid vector perturbations through equation (126). Furthermore, theperpendicular part of the vector perturbation (which is a pure vector) is a function ofthe quantities S i and τ ij : V i ⊥ = V i ⊥ ( S i , τ ij ) . (130)Regarding the CMB anisotropy, the presence of the longitudinal vector modesof GWs do not yield the well know version of the Sachs-Wolfe effect which appearsin the GRT or in the scalar-tensor theory [34]. Theories which present vector GWs(Ψ (cid:54) = 0) give rise to a nontrivial Sachs-Wolfe effect which leaves a vector signature ofthe quadrupolar form Y , ± on the CMB polarization (see detailed discussion in [16]).
7. Final Remarks
In the present work we have studied the evolution equations of cosmological GWs inalternative theories of gravity. Since the most part of the alternative theories presentmore than the two usual + and × polarizations of GWs, we have addressed the problemof how one could take into account the new polarization states in the cosmological metricperturbations.First of all, we have presented an overview of the NP formalism since it isparticularly suitable for evaluating the number of non-null GW modes of any theory. xtra polarization states of cosmological GWs in alternative theories of gravity s = 0) and two tensor modes ( s = ±
2) of GWs.In the case of the bimetric theory, GWs have in addition the two vector modes ( s ± π ij and ψ ij is the possible signaturesthat this quantities would leave in the angular pattern of the CMB. Such a signaturemight impose strong limits in the amplitudes of the scalar GWs by the analysis of theCMB data. Moreover, a remarkable effect on the CMB which have already been studiedin the literature [16] is a non-usual Sachs-Wolfe effect which appears due the presenceof the longitudinal vector GW modes on the geodesic equation for photons. Such vectorGW modes are present, for example, in the massive bimetric theory analyzed here.The detection of GWs is a particularly challenging issue and it may be the finalanswer to the “correct” theory of gravity. If GWs present non-tensorial polarizationmodes as discussed in the present paper, we will have a stochastic cosmologicalbackground of GWs which is a mixture of all the polarization modes. If, in analyzingsuch a background, scalar and/or vector GWs could be found, the result would bedisastrous for the Einstein theory.The evaluation of the response function of the non-tensor polarization modes forinterferometric GW detectors was carried out in the references [35] and [36]. Particularly,Nishizawa et al. [35] have found that more than three detectors can separate the mixtureof polarization modes in the detector outputs. But they have considered only separationbetween the three groups: scalar, vector and tensor modes of GWs. Furthermore, theyhave found that, statistically, the GW detectors have almost the same sensitivity toeach polarization mode of the stochastic background of GWs. In the work by Corda[37], the dectability of a particular polarization was discussed, namely, the longitudinalscalar component. It was also shown that the angular dependence of such a mode could,in principle, allows discriminating this polarization with respect to that of GRT.A positive detection for certain modes and a negative detection for others mayexclude a particular theory or, at least, establish strong constraints to the alternativetheories of gravity. But it is important to emphasize that, the confirmation by theobservation of the number of non-null GW modes is not enough to determine the“correct” theory, since a number of theories can have the same number of non-nullmodes. It is also necessary to evaluate the spectrum of GWs for each mode and for each xtra polarization states of cosmological GWs in alternative theories of gravity Acknowledgments
The authors would like to thank D. Bessada for helpful discussions and the refereeswhose comments and criticisms help to improve significantly the first version of thepaper. MESA would like to thank also the Brazilian Agency FAPESP for support(grant 06/03158-0). ODM and JCNA would like to thank the Brazilian agency CNPqfor partial support (grants 305456/2006-7 and 307424/2007-3 respectively).
References [1] Eardley, D.M., Lee, D.L., and Lightman, A.P., 1973,
Phys. Rev. D
8, 3308.[2] Newman, E., and Penrose, R., 1962,
J. Math Phys.
3, 566; see errata, ibid.
Zh. Eksp. Teor. Phys.
16, 587[4] Bardeen, J.M., 1980,
Phys. Rev. D
22, 1882[5] Peebles, P.J.E., 1993, Principles of Physical Cosmology (Princeton Univ. Press, Princeton)[6] Mukhanov, V.F., Feldman, H.A., and Brandenberger, R.H., 1992,
Phys. Reports
Phys. Reports
Zh. Eksp. Teor. Fiz.
67, 825[10] Barrow, J.D., Mimoso, J.P., and de Garcia Maia, M.R., 1993,
Phys. Rev. D
48, 3630[11] de Garcia Maia, M.R., and Barrow, J.D., 1994,
Phys. Rev. D
50, 6262[12] Capozziello, S., Corda, C., and de Laurentis, 2007,
Mod. Phys. Lett. A
22, 2647[13] Alves, M.E.S., Miranda, O.D., and de Araujo, J.C.N., 2009,
Phys. Lett. B
Phys. Lett. B
Astropart. Phys.
30, 209[16] Bessada, D., and Miranda, O.D., 2008,
Class. Quantum Grav.
26, 045005[17] Brans, C., and Dicke, R.H., 1961,
Phys. Rev.
Nucl. Phys. B
Gen. Relativ. Gravit.
30, 1717[20] Alves, M.E.S., Miranda, O. D., and de Araujo, J.C.N, 2007,
Gen. Relativ. Gravit.
39, 777[21] Alves, M.E.S., Miranda, O. D., and de Araujo, J.C.N, 2009, arXiv:0907.5190v1[22] Weinberg, S., 1972, Gravitation and cosmology: principles and applications of the general theoryof relativity, New York: John Wiley & Sons[23] C. M. Will, Living Reviews in Relativity (2006) http://relativity.livingreviews.org/Articles/Irr-2006-3[24] Wagoner, R.V., 1970,
Phys. Rev. D
1, 3209[25] Nordtvedt, K., 1970,
Astrophys. J.
Phys. Rev. D
62, 024004[27] Rosen, N., 1973,
Ann. of Phys.
84, 455[28] Will, C.M., 1993, Theory and experiment in gravitational physics, Cambridge: CambridgeUniversity Press xtra polarization states of cosmological GWs in alternative theories of gravity [29] van Dam, H., Veltman, M., 1970, Nucl. Phys. B
22, 397[30] Zakharov, V.I., 1970,
JETP Lett.
12, 312[31] de Paula, W.L.S., Miranda, O.D. and Marinho, R.M., 2004,
Class. Quantum Grav.
21, 4595[32] Bicak, J., and Katz, J., 2005,
Czech. J. Phys.
55, 105[33] Bicak, J., Katz, J., and Lynden-Bell, D., 2007,
Phys. Rev. D
76, 063501[34] Giovannini, M., 2005,
IJMP D
14, 363[35] Nishizawa, A., Taruya, A., Hayama, K., Kawamura, S., and Sakagami, M., 2009,
Phys. Rev. D