Generalised Proca Theories in Teleparallel Gravity
aa r X i v : . [ g r- q c ] D ec Generalised Proca Theories in Teleparallel Gravity
Gianbattista-Piero Nicosia ∗ Institute for Theoretical Physics, University of Amsterdam, 1090 GL Amsterdam, The Netherlands andInstitute of Space Sciences and Astronomy, University of Malta, Malta
Jackson Levi Said † Institute of Space Sciences and Astronomy, University of Malta, Malta andDepartment of Physics, University of Malta, Malta
Viktor Gakis ‡ Institute of Space Sciences and Astronomy, University of Malta, Msida, Malta andDepartment of Physics, National Technical University of Athens, Zografou Campus GR 157 73, Athens, Greece
Generalised Proca theories of gravity represent an interesting class of vector-tensor theories whereonly three propagating degrees of freedom are present. In this work, we propose a new teleparallelgravity analog to Proca theories where the generalised Proca framework is extended due to thelower order nature of torsion based gravity. We develop a new action contribution and explore theexample of the Friedmann equations in this regime. We find that teleparallel Proca theories offer thepossibility of a much larger class of models in which do have an impact on background cosmology.
I. INTRODUCTION
The last decades have shown numerous successes for ΛCDM where cold dark matter is responsible for the accuratereproduction of galactic dynamics [1] while cosmological scale physics is complemented by a cosmological constant, Λ,which together with inflation produces one possible cosmology [2]. The gravitational foundations of ΛCDM is generalrelativity (GR) whereas the other contributors represent modifications to the standard model of particle physics [3].However, a number of crucial observational results have recently started to show the limits of ΛCDM cosmology, whichare in addition to well-known theoretical problems inherent in the theory [4, 5]. The most prominent of these arethe H tension, where local observations and early Universe ΛCDM dependent observations predict differing valuesof the expansion rate at current times [6], and the f σ tension which is a growing tension in the growth of large scalestructure in the Universe [7], which similarly suffers a difference in values between local observations and predictionsfrom the early Universe. Another important observation is that of the possible cosmic birefringence in the PlanckCollaboration data [8] which would constitute a clear division between GR and many of its proposed modifications.This motivates us to explore other possible ways to meet the observational challenge that the current state of theart poses. The Lovelock theorem [9] provides a clear pathway in which to probe the extra degrees of freedom that onewould have to introduce to examine other possible gravitational theories as a foundation of modifying GR in some way.One of these potential avenues is that of teleparallel gravity (TG), where the curvature associated with the Levi-Civitaconnection is exchanged with the torsion produced by the teleparallel connection [10, 11]. In this way, TG producesa novel framework in which to construct theories of gravity that does not depend on GR in its traditional form. Onesuch theory is the teleparallel equivalent of general relativity (TEGR) which is dynamically equivalent to GR in termsof its field equations [12, 13] but stems from a wholly different action in which the Ricci scalar Lagrangian is curtailedin such a way to eliminate the total divergence terms [14, 15]. Naturally, this will weaken the Lovelock theorem in ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected]
TG [16, 17] by producing a much broader plethora of paths to constructing second order theories of gravity. TGalso has a number of other attractive features such as its likeness to Yang-mills theories [11] giving it a more particlephysics perspective, its possibility of giving a definition to the gravitational energy-momentum tensor [18, 19], andthat it does not require the introduction of a Gibbons–Hawking–York boundary in order to produce a well definedHamiltonian description, among several others. TG also has a number of observationally attractive features in manyof its formulations [20–23].Gravitational wave astronomy offers the possibility of a large number of interesting novel tests of gravity that mayreveal more stringent constraints on possible fundamental physical theories of gravity. The most impactful of thesehas up to now been the measurement that gravitational waves propagation at the speed of light up to one part in 10 [24, 25] which was the result of multimessenger observations of the merger of a binary neutron star system. Up tothis point many of the proposals beyond ΛCDM had been grouped in the Horndeski gravity framework in the contextof curvature based theories [5], i.e. most modifications to gravity had been shown to be dynamically equivalent toa model in this landscape of gravitational models. Horndeski gravity is the most general curvature based theory ofgravity that contains a single scalar field but retains second order field equations [26–28]. However, the multimessengersignal constraint resulted in a severe limiting of the possible expressions of a Horndeski theory [29, 30]. Recently,a TG analog of Horndeski gravity was proposed in Ref.[31] where Bahamonde–Dialektopoulos–Levi Said (BDLS)constructed a larger class of second order theories in which only one scalar field was included but second order fieldequations were produced. This stems from the naturally lower order nature of TG. BDLS theory was then shown toproduce an organic way to revive previously ruled out models by producing a much more general gravitational wavepropagation equation in Ref.[32]. The framework was also shown to be largely compatible with solar system teststhrough its parameterized post-Newtonian formalism in Ref.[33].On the other hand, the standard model of particle physics contains several abelian and non-abelian vector fields asfundamental fields representing gauge interactions [34]. In this context, there is strong motivation to explore bosonicvector fields on the scale of cosmic evolution, for instance as the candidate for dark energy which produces the latetime accelerated expansion [35, 36]. This may have important consequences for relating beyond ΛCDM theoriesto particle physics models, and to explaining certain phenomenology such as cosmic birefringence or cosmologicalprinciple tests [37]. These Proca theories can be generalised by the equivalence principle to generalised (gravitational)Proca theories [38] where a single arbitrary vector field interacts with the gravitation sector while producing secondorder field equations that propagation three degrees of freedom. Vector-tensor theories can be shown to supportisotropic solutions [38, 39] with a temporal vector field which also feature screening mechanisms. In fact, in Ref.[40]the foundations of a possible unification of scalar and Proca fields into a single theory was explored.TG offers a novel platform on which to construct theories of gravity beyond ΛCDM. In the context of Proca fields,TG will produce the same contributions as that of the Levi-Civita connection, due to its lower order nature, in additionto further contributions through extra coupling scalars between the vector field and the gravitational sector. As instandard Proca theories [35, 38], the action for a propagating massive spin-1 field carrying three degrees of freedomthat is invariant under local Lorentz transformations and observes locality is given by S P = Z d x √− g (cid:20) − F µν F µν − m A µ A µ (cid:21) , (1)where √− g = det( g µν ) is the determinant of the metric tensor and F µν = ∂ µ A ν − ∂ ν A µ . By demanding generalcoordinate invariance, we will explore the case of a TG Proca theory framework by first introducing the foundationof TG in section II. In section III, the standard picture of Proca theories is described for curvature based gravity,while the TG scenario is explored in section IV. As an example, we explored the Friedmann equations for the theorythat we develop here in section V where the vector conservation equation is also presented. Finally, we close with asummary and conclusion of the main results in section VI. II. TELEPARALLEL GRAVITY
GR is the most prominent theory of gravity and is based on the Levi-Civita connection ˚Γ σµν (we use over-circlesto denote quantities calculated with the Levi-Civita connection throughout) which is torsion-less and satisfies themetricity condition [41, 42]. There is another formulation of GR whereby shifting the geometrical content fromcurvature to torsion at the cost of changing the Levi-Civita connection to the teleparallel connection denoted asΓ σµν . The teleparallel connection is defined as the unique connection that is torsion-ful, curvature-less and metrical[11–13, 43].GR and its modifications are based on curvature which is ultimately based on the metric g µν through the Levi-Civita connection and thus the fundamental variable. On the other hand in TG theories, torsion is defined by thetetrad field e Aµ and the spin connection ω ABµ rendering the metric a derived object [11]. We will use capital Latinletters to denote the Minkowski indices while Greek indices will denote the spacetime manifold as usual. The tetradlinks the spacetime manifold with a Minkowski manifold where the orthonormal frames (indicated by capital Latinletters) exist. In this way one can induce the spacetime metric by the Minkowski metric as g µν = e Aµ e Bν η AB , η AB = E µA E νB g µν , (2)where we denote E µA as the inverse tetrad. In addition, the tetrad and inverse tetrad are related through e Aµ E µB = δ BA , e Aµ E νA = δ νµ . (3)The TG connection can be explicitly defined using the tetrad and the spin connection as [10]Γ σνµ := E σA ∂ µ e Aν + E σA ω ABµ e Bν , (4)which is the most general linear affine connection that is flat and metrical [11, 44]. The metricity condition is alsoreflected by ω ABµ = − ω BAµ which also defines the class of Lorentz spin connections. The spin connection, in generalis introduced to attain Lorentz covariance and for TG it is totally inertial and incorporates the effects of the localLorentz transformations (LLTs) thus producing LLT invariant theories. One can always find a frame where theTG spin connection is vanishing as in the original formulation in Ref.[10], and this choice of frame is dubbed theWeitzenb¨ock gauge (WG). The TG spin connection can be fully represented as ω ABµ = Λ AC ∂ µ Λ CB [11], where thefull breadth of the LLTs (Lorentz boosts and rotations) are represented by Λ AB . Hence there is no unique tetrad thatinduces a specific spacetime metric Eq. (2).Using the TG connection we can define the torsion tensor as T σµν := − σ [ µν ] , (5)where square brackets denote the usual antisymmetrization. This tensor field also acts as the field strength of TG[11], which transforms covariantly under both diffeomorphisms and LLTs. We can consequently build the contorsiontensor as K σµν := Γ σµν − ˚Γ σµν = 12 (cid:0) T σµ ν + T σν µ − T σµν (cid:1) , (6)that links the Levi-Civita connection and the TG connection.This has an important role to play in relating TG with GR and its modifications, as will become apparent later on.Another core component of TG is the superpotential defined as [12] S µνλ := 12 (cid:16) K µνλ + δ νλ T µ − δ µλ T ν (cid:17) , (7)where T ν := T α να = − T ανα . This tensor is related with the energy-momentum tensor for gravitation [45] but theissue remains open [46]. Contracting the torsion tensor with the superpotential we define the torsion scalar as [13] T := S µνλ T λµν = 14 T + 12 T − T , (8)where T := T µνρ T µνρ , T := T µνρ T ρνµ , T := T ρ T ρ . (9)The torsion scalar plays the role of the Lagrangian density just like the Ricci scalar. The standard Ricci scalar ◦ R (computed with the Levi-Civita connection) can be expressed, by using contorsion tensor Eq.(6), as the torsion scalarplus some boundary term [47, 48] ◦ R = − T + B , (10)where the boundary term is defined as B := 2˚ ∇ µ ( T µ ) . (11)It is a well known fact that Eq.(10) is the evidence of dynamical equivalence between GR and TEGR and serves asa justification of using torsion as our primary geometrical object [11]. Following this paradigm shift, we define theTEGR action as S TEGR := − κ Z d x eT + Z d x e L m , (12)where e = det (cid:0) e Aµ (cid:1) = √− g is the tetrad determinant, κ = 8 πG and L m is the regular matter Lagrangian.The (linear) boundary term difference at the level of the Lagrangians plays no role regarding the field equationsbut this is not the case when it is introduced in a non-linear manner like in f ( T, B ) gravity [49–54, 54, 55]. In fact,Eq.(10) could be seen as a split of the Ricci scalar into a contribution of first order derivatives of the tetrad(Torsionscalar) and a contribution of only second order derivatives of the tetrad(Boundary term).A common misconception in the realm of TG gravity is the use of the WG, where one chooses a tetrad field in sucha way that the spin connection is trivialised. Upon trivializing, the spin connection the manifest LLT covariance islost. This does not mean that the theory is Lorentz violating but rather that it is presented in a non-manifest LLTcovariant form. We remind the reader that one can do exactly the same for the case of the Levi-Civita connectionbut only locally, i.e, choosing a particular coordinate system around a point where ˚Γ σµν ≡ e Aµ e ′ Aµ = Λ AB e Bµ , (13) ω ′ ABµ = Λ AC ∂ µ (Λ − ) C B . (14)Hence one can straight-forwardly restore Local Lorentz covariance. One last, quite potent, argument in favour ofthe use of the WG gauge in TG is that the spin connection is non-dynamical and thus it is just a gauge degrees offreedom, hence if one formulates the action by LLT invariant scalars then the spin connection can be safely trivialised.In the light of these arguments, we will work in the WG throughout this paper having implicitly trivialised the spinconnection without loss of generality.We close off this section by presenting the irreducible decomposition of the torsion tensor, under the action of theLLT, into a vector v µ , a pseudo-vector a µ and a pure tensor part t αµν as v µ := T µ , (15a) a µ := 16 ǫ µνσρ T νσρ , (15b) t αµν := 12 ( T αµν + T µαν ) + 16 ( g να v µ + g νµ v α ) − g αµ v ν , (15c)where ǫ µνσρ denotes the Levi-Civita tensor and the tensorial part enjoys the properties t αµν = t µαν , (16a) t αµν + t ναµ + t µνα = 0 , (16b) t αµα = t α µα = t µαα = 0 . (16c)In this irreducible representation we can express the torsion scalar as T = 14 T + 12 T − T = 32 T axi + 23 T ten − T vec , (17)where T axi := a µ a µ = 118 ( T − T ) , (18a) T vec := v µ v µ = T , (18b) T ten := t σµν t σµν = 12 ( T + T ) − T . (18c)Both of these representations are equivalent in general but one is more suitable than the other in specific analyses. III. GENERALISED PROCA FIELDSA. Generalised Proca in flat space
In this section, we review generalised Proca theories in standard gravity [35, 38, 39]. The goal is to generalise theProca action in Eq.(1) to include self-interacting terms without changing the propagating degrees of freedom. We willstart by defining the Lagrangian for a generalised Proca (GP) vector field as L GP = − F + X n =2 α n L n , (19)where F ≡ F µν F µν , α n are arbitrary constants and L n are different self-interacting Lagrangian terms (which willdiffer in the number of derivatives of the vector field). In order to define the various L n contributions, one startswith the most general possible combination of vector field and its derivatives, and then reduce their form by imposingphysical constraints. In this case, the constraint is given by the fact that we require the generalised theory to propagate3 degrees of freedom of the the vector field, i.e. there should be no propagation of the zeroth component. To ensurethat this is true, the Hessian matrix, a symmetric n × n matrix of second-order partial derivatives of the scalar field H µν L n = ∂ L n ∂ ˙ A µ ∂ ˙ A ν , with ˙ A µ = ∂ A µ , (20)is calculated for each L n and then we impose that the determinant of the matrix vanishes. These conditions ensurethat there are no eigenvalues corresponding to the A component in the kinetic matrix, thus making the A componentnon-dynamical. The latter statement can also be viewed as requiring H = H i = 0 [39]. Thus, one finds that thevarious L n are given by L = f ( X, F, Y ) , (21) L = f ( X ) ∂ · A , (22) L = f ( X )[( ∂ · A ) − ∂ µ A ν ∂ ν A µ ] + c ˜ f ( X ) F , (23) L = f ( X )[( ∂ · A ) − ∂ · A ) ∂ µ A ν ∂ ν A µ + 2 ∂ ρ A σ ∂ α A ρ ∂ σ A α ] + d ˜ f ( X ) ˜ F µρ ˜ F νρ ∂ µ A ν , (24) L = e f ( X ) ˜ F µν ˜ F γρ ∂ µ A γ ∂ ν A ρ , (25)where ∂ · A ≡ ∂ µ A µ will be used hereafter, X = − A µ A µ / F = F µν F µν / Y = A µ A ν F αµ F να , f , , , , are arbitraryfunctions and c , d , e are arbitrary constants. The argument of the arbitrary functions f , , , is fixed by the factthat these functions are multiplied by derivatives of the vector field, implying that they themselves cannot containany derivatives of the vector field to avoid any cancellations through partial integration. On the other hand, f is notmultiplied by derivatives of the vector field and thus differs from the other functions. In fact, this function containsall the possible terms which have U (1) symmetry. Also, X, F, Y are the independent contractions from which all theother terms can be obtained. One should also notice that all of these give rise to second order equations of motionfor the vector field, as needed.The first Lagrangian term which appears in Eq.(21), L , represents the simplest modification that can be imple-mented to the action, i.e. promoting the mass term m to a function of the vector field. This term, in fact, includes,among others, the mass term and the potential interactions of the field, V ( A ) ⊂ f .The other Lagrangian terms are a broader generalisation of the theory and, in order to understand how thesewere derived, we will take L as an example. We start by considering the most general Lagrangian containing twoderivatives of the vector field L = f ( X )[ c ( ∂ · A ) + c ∂ µ A ν ∂ µ A ν + c ∂ µ A ν ∂ ν A µ ] , (26)where, c , c , and c are arbitrary constants. Thus, we find the Hessian matrix H µν L = f c + c + c ) 00 − c δ ij ! . (27)There exist two possibilities in order to obtain a vanishing determinant. The first is c = 0, but this would imply thepropagation of only the temporal component of the vector, which is exactly the opposite of what is required. Thus,we opt for the second option which is the parameter constraint c + c + c = 0. Without loss of generality, we canset c = 1, implying c = − (1 + c ). Thus, our Lagrangian can then be re-written as L = f [( ∂ · A ) + c ∂ µ A ν ∂ µ A ν − (1 + c ) ∂ µ A ν ∂ ν A µ ]= f [( ∂ · A ) − ∂ µ A ν ∂ ν A µ + c F ] , (28)where, the second equality was obtained by simplifying the derivatives. Now, one notices that the term proportionalto c which together with an independent separate function ˜ f ( X ) could be included in f . In this case we shall leavethe term explicitly in this form, as shown in (23). Repeating this process for all the other Langrangian terms, weobtain the Lagrangian setup shown in Eqs.(21–25). B. Generalised Proca in curved background
The Proca Lagrangian contributions that result in the previous section were obtained on a flat background. Wenow want to generalise this to a general non-flat background. When going to such a space and making the metric, g µν , dynamical one should also impose that the field equations of the metric have to be, at most, second-order.Furthermore, the derivative self-interactions could lead to possible excitations of the temporal polarisation of thevector field via the formation of non-minimal couplings [56]. Thus, when converting the partial derivatives intocovariant ones, one should be careful and add non-minimal couplings to the graviton as counter terms to prevent this,and in order to maintain the second order nature of the equations of motion.For the construction of these non-minimal couplings, the divergenceless tensors of the gravity sector are of vitalimportance. Thus, the total Lagrangian density is given by [56] L curvedGP = − F + X n =2 β n L n , (29)with [38, 56] L = G ( X, F, Y ) , (30a) L = G ( X ) ◦ ∇ · A , (30b) L = G ( X ) ◦ R + G ,X ( X )[( ◦ ∇ · A ) − ◦ ∇ µ A ν ◦ ∇ ν A µ ] , (30c) L = G ( X ) ◦ G µν ◦ ∇ µ A ν − G , X ( X )[( ◦ ∇ · A ) − ◦ ∇ · A ) ◦ ∇ µ A ν ◦ ∇ ν A µ +2 ◦ ∇ µ A ν ◦ ∇ α A µ ◦ ∇ ν A α ] − ˜ G ( X ) ˜ F αµ ˜ F βµ ◦ ∇ α A β , (30d) L = G ( X ) ◦ L µναβ ◦ ∇ µ A ν ◦ ∇ α A β + 12 G , X ( X ) ˜ F αµ ˜ F βγ ◦ ∇ α A β ◦ ∇ µ A γ , (30e)where f ,X := ∂f /∂X , ◦ G µν is the Einstein tensor and ◦ L µναβ is the double dual Riemann tensor ◦ L µναβ := 14 ǫ µνρσ ǫ αβγδ ◦ R ρσγδ , (31)which inherits the same symmetry properties as the Riemann tensor, i.e. ◦ L µναβ = ◦ L αβµν , ◦ L µναβ = − ◦ L νµαβ and ◦ L µναβ = − ◦ L µνβα [39]. Note that the couplings in the last term in L , ˜ G ( X ) ˜ F αµ ˜ F βµ ◦ ∇ α A β , and L do not require anynon-minimal counter terms, as in these cases the coupling to the connection is linear [39, 56]. Furthermore, note thatterms such as ◦ G µν A µ A ν , which do not propagate the temporal component of the vector field, are already included inthe above interactions after integration by parts [35].In summary, in this section we have seen how a general theory for Proca fields can be constructed for a flat spacetimeand then generalised to a curved one. In the next section, we provide a way to translate this to a TG setting. IV. PROCA THEORIES IN TELEPARALLEL GRAVITY
As stated in section II, the TEGR and GR approach differ in the underlying geometry since the former is a torsionbased theory, while the latter is a curvature based one. As seen in section III B, in order to covariantise a theory, oneneeds to find a connection between the neighbouring tangent spaces to a manifold. Thus, since the geometries aredifferent (implying different connections), the covariantisation procedure differs between the two theories. Nonetheless,since the teleparallel connection is linked to the usual Lorentzian geometry on which GR is based upon via Eq.(6),the covariantisation procedure is simplified.Keeping as a reference Eq.(6), it can be noted that the coupling prescriptions of GR and TG can be taken asequivalent [11]. In a similar manner the BDLS theory was formulated [31, 32] which we will follow. Since the couplingprescriptions between the teleparallel connection and the Levi-Civita connection are equivalent meaning e Aµ (cid:12)(cid:12) flat → e Aµ , (32) ∂ µ → ◦ ∇ µ , (33)we use the usual Levi-Civita connection prescription for simplicity. An implication of this fact is that the form of theEqs.(30a-30e) remains the same under the covariantisation scheme although we will re-write them again with someslight adjustments L = G ( X, F, Y ) , (34a) L = G ( X ) ◦ ∇ · A , (34b) L = G ( X )( − T + B ) + G ,X ( X )[( ◦ ∇ · A ) − ◦ ∇ µ A ν ◦ ∇ ν A µ ] , (34c) L = G ( X ) ◦ G µν ◦ ∇ µ A ν − G , X ( X )[( ◦ ∇ · A ) − ◦ ∇ · A ) ◦ ∇ µ A ν ◦ ∇ ν A µ +2 ◦ ∇ µ A ν ◦ ∇ α A µ ◦ ∇ ν A α ] − ˜ G ( X ) ˜ F αµ ˜ F βµ ◦ ∇ α A β , (34d) L = G ( X ) ◦ L µναβ ◦ ∇ µ A ν ◦ ∇ α A β + 12 G , X ( X ) ˜ F αµ ˜ F βγ ◦ ∇ α A β ◦ ∇ µ A γ , (34e)where we have substituted the Ricci scalar in L from its TG equivalent form as presented in Eq.(10). One could alsouse the TG form of the Einstein tensor in the WG as ◦ G µν = e − e aµ g νρ ∂ σ ( eS ρσa ) − S σb ν T bσµ + 14 T g µν , (35)and further express the Riemann tensor in terms of quantities related to TG using the fundamental Eq.(10) as ◦ R ρσµν = − ◦ ∇ µ K ρσν + ◦ ∇ ν K ρσµ − K βσν K ρβµ + K βσµ K ρβν . (36)Note that while these Lagrangians were enough to describe the general Proca theory when using the Levi-Civitaconnection prescription, the same cannot be said for the teleparallel connection prescription. In fact, when theteleparallel connection is used, the Lovelock’s theorem is weakened [16, 17], hence dramatically increasing the poolof available scalar invariants constructed via the torsion tensor. These new scalars lead to a new Teleparallel ProcaLagrangian contribution, L T P , hence the new total lagrangian density of the theory L T eleGP will be given by L TeleGP = − F + X n =2 β n L n + L T P . (37)The form of L T P can be specified from the scalars we are going to construct out of the irreducible components ofthe torsion tensor Eq.(15a-15c) and the vector field A µ . Note that our available set of possible scalars is quite large,and thus for the needs of this work we will only focus on scalars built obeying the following rules:1. The resulting field equations must, at most, be of second order in both e Aµ and A µ ;2. A µ must have maximum 3 degrees of freedom, A being not dynamical;3. Cannot be parity violating;4. Must be linear in the torsion tensor;5. Up to fourth order derivatives on A µ , i.e, ∂A∂A∂A∂A ∼ ( ∂A ) .The first condition guarantees that we avoid Ostrogradsky ghosts by restricting the Lagrangian to include at mostfirst order covariant derivatives wrt the tetrad and the vector field [57, 58]. The first condition is not enough sincesecond order equation of motion for the temporal component would mean that A is a dynamical ghost degree offreedom. Also, it is known that the massive spin-1 representation of the Lorentz group should only carry threedynamical fields and we demand the inclusion of derivative self-interactions does not alter this property. The previouscriteria can be combined into the second condition. The fourth and fifth conditions just restrict the number ofavailable scalars that one can include since just including linear in torsion scalars immediately increases the numberof available scalars to the order of hundreds. n Vectorial ( v ) Axial ( a ) Purely tensorial ( t )0 vA - tAAA vA ◦ ∇ A ǫaA ◦ ∇ A tA ◦ ∇ A vAF F , vA e F e F ǫaAF F, ǫaA ˜ F ˜ F tAF F , tA e F e F vAF F F , vA e F e F F ǫaAF F F , ǫaA ˜ F ˜ F F tAF F F , tA e F e F F vAF F F F , vA e F e F e F e F , vA e F e F F F ǫaAF F F F , ǫaA ˜ F ˜ F ˜ F ˜ F , ǫaA ˜ F ˜ F F F tAF F F F , tA e F e F e F e F , tA e F e F F F
TABLE I: Generators of scalars – These are the independent components from which all the other terms can be obtained bypermuting the indices.
Keeping these properties in mind, we gather all the possible contractions schematically in table I by denoting as n the parameter that indicates the expansion of the product Q n ◦ ∇ µ n A ν n = ◦ ∇ µ A ν ◦ ∇ µ A ν , .., ◦ ∇ µ n A ν n where all possibleindex configurations are implied for each non-indexed expression. For example the generator n vA ◦ ∇ A o spans thefollowing index configurations I = v β A α ◦ ∇ α A β , (38) I = v β A α ◦ ∇ β A α , (39) I = v α A α ◦ ∇ β A β , (40)one can refer to the appendix A, for the full expansion of every generator. Note that for n > ∇ µ A ν are substituted by F µν in order to keep the A component non-dynamical. We stress that these scalars will onlyeffect non-trivial geometries, away from Minkowski spacetime, where the torsion tensor is not trivialised. Thus, theresulting scalars are trivialised in flat spacetime reducing the theory to the standard generalised Proca one [35, 38].Table I was partially generated with the help of the xAct packages [59–65].Having produced the scalars we give the explicit form of L T P , which we will strictly keep linear in any torsionargument, by defining L T P := G T P ( X, F, Y, I , I , .., I ) (41)where the I ’s are found in the appendix A for brevity’s sake. Let us point out that this function in a Minkowski settingwill be absorbed into L from Eq.(34a) since all the I ’s will be trivialized, thus reducing the theory back to usualgeneralised Proca. At first glance one could argue that there are a lot of I arguments in G T P ( X, F, Y, I , I , .., I )but not all of them survive in symmetric backgrounds since most of them are exhaustive permutations of specificforms. This is evident in section V where for a flat cosmological background only 4 of them survive. V. COSMOLOGICAL BACKGROUND IN TELEPARALLEL PROCA THEORIES
We now explore the scenario of a flat Friedmann–Lemaˆıtre–Robertson–Walker (FLRW) cosmology which will exposepossible dynamical effects of the vector field at background level. We consider the FLRW metric as ds = − dt + a ( t ) ( dx + dy + dz ) where a ( t ) is the scale factor in terms of cosmic time. This can be described by the tetrad e Aµ = diag(1 , a ( t ) , a ( t ) , a ( t )) , (42)which is compatible with the so-called WG in which the spin connection components vanish [12], i.e. ω ABµ = 0. Thiscan be shown to produce the FLRW metric by considering the relations in Eq.(2).By taking the definition of the torsion scalar in Eq.(8), this turns out to be T = 6 H , (43)while the boundary term is given by B = 6 (cid:16) H + ˙ H (cid:17) . This is important because by Eq.(10), we recover the regularRicci scalar term as ◦ R = − T + B = 6 (cid:16) H + ˙ H (cid:17) .In the present study, we consider the case of a homogeneous and time dependent vector field given by A µ = ( A ( t ) , , , , (44)where A ( t ) is a scalar function that depends only on cosmic time. By considering the full breadth of scalars that areproduced from table I (or more specifically by the scalar definitions in appendix (A)), we find that the nonvanishingscalars are I = 3 AH , (45) I = − A ˙ AH = I = I , (46)0whereas the purely Proca scalars are then given by X = 12 A , (47) Y = 0 = F (48)This means that the additional Lagrangian contribution will only be effected at background level by these scalars,meaning that G T P = G T P ( X, I , I , I , I ).By considering a Universe filled by a perfect fluid with energy density ρ and pressure p . The ensuing Friedmannequations turn out to be A TP + Σ i =2 A i = ρ , (49) B TP + Σ i =2 B i = p , (50)where A = G − A G ,X , (51) A = − HA G ,X , (52) A = 6 H G − (cid:0) G ,X + G ,XX A (cid:1) H A , (53) A = G ,XX H A + 5 G ,X H A , (54) A TP = G TP − A ˙ AHG TP ,I + 3 A H G TP ,I + 12 AH (cid:16) ˙ A + 3 AH (cid:17) G TP ,I − A (cid:16) ˙ H + H (cid:17) G TP ,I + 9 A H h − A H + AH (cid:16) AH + 6 A (cid:16) ˙ H + H (cid:17)(cid:17) + ˙ A H + A ¨ AH + A ˙ A (cid:16) ˙ H + H (cid:17)i G TP ,I I + 6 A ˙ AHG TP ,I + 3 A H G TP ,I − A (cid:16) ˙ H + H (cid:17) G TP ,I − A H (cid:16) A ˙ AH − ˙ A H − A ¨ AH − A ˙ A (cid:16) ˙ H + H (cid:17)(cid:17) G TP ,I I + 9 A H h − A + AH (cid:16) AH + 6 A (cid:16) ˙ H + H (cid:17)(cid:17) + ˙ A H + A ¨ AH + A ˙ A (cid:16) ˙ H + H (cid:17)i G TP ,I I − A H h A ˙ AH − ˙ A H − A ¨ AH − A ˙ A (cid:16) ˙ H + H (cid:17)i G TP ,I I + 6 A ˙ AHG TP ,I + 3 A H G TP ,I − A (cid:16) ˙ H + H (cid:17) G TP ,I − A H [ G TP ,I + G TP ,I + G TP ,I ] − A H h A ˙ AH − ˙ A H − A ¨ AH − A ˙ A (cid:16) ˙ H + H (cid:17)i G TP ,I I + 9 A H h − A H + AH (cid:16) AH + 6 A (cid:16) ˙ H + H (cid:17)(cid:17) + ˙ A H + A ¨ AH + A ˙ A (cid:16) ˙ H + H (cid:17)i G TP ,I I − A H h A ˙ AH − ˙ A H − A ¨ AH − A ˙ A (cid:16) ˙ H + H (cid:17)i G TP ,I I − A H h A ˙ AH − ˙ A H − A ¨ AH − A ˙ A (cid:16) ˙ H + H (cid:17)i G TP ,I I − AHG TP ,I I − A H h − AH + ˙ AH + A (cid:16) ˙ H + H (cid:17)i G TP ,I I − A H h − AH + ˙ AH + A (cid:16) ˙ H + H (cid:17)i G TP ,I I − A H h − AH + ˙ AH + A (cid:16) ˙ H + H (cid:17)i G TP ,I I − A G TP ,X − A ˙ AHG TP ,XI − A ˙ AHG TP ,XI − A ˙ AHG TP ,XI , (55)for the first Friedmann equation, where commas denote derivatives with respect to the appearing scalars, and B = G , (56) B = − A ˙ AG ,X , (57) B = 2 G (cid:16) H + 2 ˙ H (cid:17) − G ,XA (cid:16) H A + 2 H ˙ A + 2 ˙ HA (cid:17) − G ,XX HA ˙ A , (58) B = G ,XX H A ˙ A + G ,X HA (cid:16) HA + 2 H A + 3 H ˙ A (cid:17) , (59)1 B TP = G TP + h − A H + ˙ A + A ¨ A + A (cid:16)
11 ˙ AH + 6 A (cid:16) ˙ H + H (cid:17)(cid:17)i G TP ,I + ˙ A G TP ,I − A ˙ AHG TP ,I + A ¨ AG TP ,I ˙ A G TP ,I − A ˙ AHG TP ,I + A ¨ AG TP ,I − ˙ AG TP ,I + AHG TP ,I + 4 AH h(cid:16) ˙ A + 6 AH (cid:17) G TP ,I + ˙ AG TP ,I + ˙ AG TP ,I − G TP ,I i − A (cid:16) ˙ A + 6 AH (cid:17) " (cid:16) − A H + AH (cid:16) AH + 6 A (cid:16) ˙ H + H (cid:17)(cid:17) + ˙ A H + AH ¨ A + A ˙ A (cid:16) ˙ H + H (cid:17) (cid:17) G TP ,I I + 3 (cid:16) − A ˙ AH + ˙ A H + AH ¨ A + A ˙ A (cid:16) ˙ H + H (cid:17)(cid:17) G TP ,I I + 3 (cid:16) − A ˙ AH + ˙ A H + A ¨ AH + A ˙ A (cid:16) ˙ H + H (cid:17)(cid:17) G TP ,I I + 3 (cid:16) − AH + ˙ AH + A (cid:16) ˙ H + H (cid:17)(cid:17) G TP ,I I − A ˙ AG TP ,XI − A ˙ A " (cid:16) − A H + AH (cid:16) AH + 6 A (cid:16) ˙ H + H (cid:17)(cid:17) + ˙ A H + A ¨ AH + A ˙ A (cid:16) ˙ H + H (cid:17) (cid:17) G TP ,I I + 3 (cid:16) − A ˙ AH + ˙ A H + A ¨ AH + A ˙ A (cid:16) ˙ H + H (cid:17)(cid:17) G TP ,I I + 3 (cid:16) − A ˙ AH + ˙ A H + A ¨ AH + A ˙ A (cid:16) ˙ H + H (cid:17)(cid:17) G TP ,I I − (cid:16) − AH + ˙ AH + A (cid:16) ˙ H + H (cid:17)(cid:17) G TP ,I I − A ˙ AG TP ,XI − A ˙ A " (cid:16) − A H + AH (cid:16) AH + 6 A (cid:16) ˙ H + H (cid:17)(cid:17) + ˙ A H + A ¨ AH + A ˙ A (cid:16) ˙ H + H (cid:17)(cid:17) G TP ,I I + 3 (cid:16) − A ˙ AH + ˙ A H + A ¨ AH + A ˙ A (cid:16) ˙ H + H (cid:17)(cid:17) G TP ,I I + 3 (cid:16) − A ˙ AH + ˙ A H + A ¨ AH + A ˙ A (cid:16) ˙ H + H (cid:17)(cid:17) G TP ,I I − (cid:16) AH + ˙ AH + A (cid:16) ˙ H + H (cid:17)(cid:17) G TP ,I I − A ˙ AG TP ,XI + A " (cid:16) − A H + AH (cid:16) AH + 6 A (cid:16) ˙ H + H (cid:17)(cid:17) + ˙ A H + AH ¨ A + A ˙ A (cid:16) ˙ H + H (cid:17) (cid:17) G TP ,I I + 3 (cid:16) − A ˙ AH + ˙ A H + A ¨ AH + A ˙ A (cid:16) ˙ H + H (cid:17)(cid:17) G TP ,I I + 3 (cid:16) − A ˙ AH + ˙ A H + A ¨ AH + A ˙ A (cid:16) ˙ H + H (cid:17)(cid:17) G TP ,I I − (cid:16) − AH + ˙ AH + A (cid:16) ˙ H + H (cid:17)(cid:17) G TP ,I I − A ˙ AG TP ,XI , (60)for the second Friedmann equation. Similarly, the vector conservation equation can be obtained, giving P TP + Σ i =2 P i = 0 , (61)where P = AG ,X (62) P = 3 A HG ,X (63) P = 6 AH G ,X + 6 H A G ,XX (64) P = − H A G ,X − H A G ,XX (65)2 P TP = 54 A H ˙ H ( G TP ,I + G TP ,I I + G TP ,I I ) + 3 HG TP ,I + A " − H (cid:16) G TP ,I − G TP ,I + G TP ,I )+ 3 ˙ A ( G TP ,I I + 2 G TP ,I + G TP ,I I + 2 ( G TP ,I I + G TP ,I I ) + G TP ,I I ) − A ( G TP ,I I + G TP ,I I + G TP ,I I ) (cid:17) + 3 (cid:16) ˙ H + H (cid:17) ( G TP ,I + G TP ,I + G TP ,I ) + G TP ,X + 3 A H " − H ¨ A ( G TP ,I I + 2 G TP ,I I + G TP ,I I + 2 ( G TP ,I I + G TP ,I I ) + G TP ,I I ) − HG TP ,I I − HG TP ,I I − HG TP ,I I + ˙ A (cid:16) H ( − G TP ,I I − G TP ,I I + G TP ,I I − G TP ,I I + 2 G TP ,I I + G TP ,I I ) − (cid:16) ˙ H + H (cid:17) (cid:16) G TP ,I I + 2 G TP ,I I + G TP ,I I + 2 ( G TP ,I I + G TP ,I I + G TP ,I I )+ G TP ,XI + G TP ,XI + G TP .XI (cid:17)(cid:17) (66)Altogether, Eqs.(49,50,61) represent the background evolution equations that describe the teleparallel Proca theorybeing proposed in this work. In the limit where the new terms vanish, this tends to the standard gravity version ofProca gravity [38], as expected. Despite the low number of nonvanishing scalars, the resulting cosmology immediatelybecomes very rich even at background level indicating even more possible impacts of these new scalars at perturbativeorder. VI. CONCLUSION
Proca theories of gravity offer an interesting framework in which to produce gravitational theories that includes amassive spin-1 vector field. In terms of second order equations of motion, this lays out the Lagrangian contributionsin Eqs.(21–25), which together describe the generalised Proca theory on Minkowski spacetime. By using the minimalcoupling prescription, this can be directly extended to an arbitrary curvature spacetime that results in Eqs.(30a–30e). These equations describe the covariantised version of the generalised Proca theory which is made up of scalarsconstructed from the vector field and coupling functions.The TG analog of the standard gravity version of Proca theories is obtained by using the TG coupling prescriptionto raise the Minkowski spacetime contributions in Eqs.(21–25) which would produce an infinite number of terms due tothe lower order nature of torsional theories of gravity. In this context, we introduce the five guiding principles on whichwe build our theory, namely that (i) second order equations of motion are produced, in order to avoid Ostrogradskyinstabilities; (ii) only three degrees of freedom are expressed via the vector field, which is a core requirement ofProca theories; (iii) does not violate parity, which is a good base for healthy theories; (iv) Lagrangian contributionsmust be, at most, linear in the torsion tensor, since higher order contractions may not contribute to leading orderterms; (v) vector field derivatives must appear less than fourth order so that it is related to the structure of thegeneralised Proca theory thus providing a reasonable cutoff. Altogether these conditions produce a well defined TGgeneralised Proca theory where the standard gravity terms are complemented by an additional contribution in theform of the Lagrangian term in Eq.(41). TG is very sensitive to the selection rules that forms the scalars since eventhese seemingly selective rules produce a large number of potential scalars.One may loosen the selection criteria and produce arbitrarily large additional contributions to the standard Procaaction which may take the form of further contributions of the vector coupling terms. Another possibility is theinclusion of quadractic(or even arbitrary higher) TG scalars as arguments, like the irreducible torsion scalars Eqs.(18a–18c) [47, 66]. In this context, the present new Lagrangian term in Eq.(41) is simply one of the myriad of potential3expressions of generalised Proca theories in TG which offers a large class of potential avenues for producing viablemodels that may produce a fundamental field theory explanation for a number of important phenomena such as darkenergy and dark matter.Finally, in section V we present a simple example of a homogeneous and isotropic Universe in which we derivethe Friedmann equations. These background level equations are important for determining the expansion rate of theUniverse, but also to assess the sensitivity of the equations of motion to the various scalars being proposed in thiswork. We find that only four scalars contribute in a nonvanishing way to the equations of motion, which are presentedin Eqs.(45–46) where a vector field that changes only in time is assumed. This means that the vector field scalars onlyproduce a nonvanishing X contribution. Despite this overly simplified scenario, the Friedmann equations still turnout to be quite involved. The first and second Friedmann equations are presented in Eq.(49–50) whereas the vectorequation in given in Eq.(61). The combination of these three equations fully determines the cosmology at backgroundlevel.It would be interesting probe this new class of generalised Proca theories and investigate how the models fromstandard gravity are extended in this framework. This will have ramifications beyond background level and maydrastically change the cosmological perturbations for such theories. Acknowledgements
JLS would also like to acknowledge funding support from Cosmology@MALTA which is supported by the Universityof Malta. The authors would like to acknowledge networking support by the COST Action CA18108. V.G would liketo thank J. Beltran for useful and fruitful discussions.4
Appendix A: Teleparallel Proca scalars
In this appendix we will expand all the generators from the table I in all their possible index configurations. We willdenote the generator or groups of generators with brackets like in the example after the table I, n vA ◦ ∇ A o . Note thatthe following sets of scalars are the full list of possible independent scalars. Torsion vector component v µ { vA } I := v µ A µ , (A1) n vA ◦ ∇ A o I := A α v β ◦ ∇ α A β , (A2) I := A α v β ◦ ∇ β A α , (A3) I := A α v α ◦ ∇ β A β , (A4) n vAF F, vA e F e F o I := A α F v α , (A5) I := A α F αγ F βγ v β , (A6) n vAF F F, vA e F e F F o I := A α F αµ F βγ F µγ v β , (A7) I := A α F αβ F v β , (A8) n vAF F F F, vA e F e F e F e F , vA e F e F F F o I := A α F βν F γβ F γµ F µν v α , (A9) I := A α F v α , (A10) I := A α F αµ F βγ F γν F µν v β , (A11) I := A α F αγ F βγ F v β , (A12)5 Torsion axial component a µ n ǫaA ◦ ∇ A o I := A α a β ǫ αβγµ ◦ ∇ µ A γ , (A13) n ǫaAF F, ǫaA ˜ F ˜ F o I := A α a β ǫ βµνγ F αµ F νγ , (A14) I := A α a β ǫ αµνγ F βµ F νγ , (A15) I := A α a α ǫ γµνβ F γµ F νβ , (A16) n ǫaAF F F, ǫaA ˜ F ˜ F F o I := A α a β ǫ αβµν F γρ F γµ F ρν , (A17) I := A α a β ǫ µργν F αµ F βρ F γν , (A18) I := A α a β ǫ βργν F αµ F γν F µρ , (A19) I := A α a β ǫ αργν F βµ F γν F µρ , (A20) I := A α a β ǫ νργµ F αβ F γµ F νρ , (A21) I := A α a β ǫ αβνρ F νρ F , (A22) n ǫaAF F F F, ǫaA ˜ F ˜ F ˜ F ˜ F, ǫaA ˜ F ˜ F F F o I := A α a β ǫ βµρσ F αµ F γσ F νγ F ν ρ , (A23) I := A α a β ǫ αµρσ F βµ F γσ F νγ F ν ρ , (A24) I := A α a β ǫ ργνσ F αµ F βρ F µγ F νσ , (A25) I := A α a α ǫ µβνσ F γρ F γµ F νσ F ρβ , (A26) I := A α a β ǫ µγνσ F αµ F βρ F νσ F ργ , (A27) I := A α a β ǫ βγνσ F αµ F µρ F νσ F ργ , (A28) I := A α a β ǫ αγνσ F βµ F µρ F νσ F ργ , (A29) I := A α a β ǫ σγνρ F αµ F βµ F νρ F σγ , (A30) I := A α a α ǫ νρσβ F νρ F σβ F , (A31) I := A α a β ǫ βρσγ F αρ F σγ F , (A32) I := A α a β ǫ αρσγ F βρ F σγ F , (A33)6 Purely tensorial component t αβγ { tAAA } I := A α A β A γ t αβγ , (A34) n tA ◦ ∇ A o I := A α t αβγ ◦ ∇ γ A β , (A35) I := A α t αγβ ◦ ∇ γ A β , (A36) I := A α t βγα ◦ ∇ γ A β , (A37) n tAF F, tA e F e F o I := A α F βµ F βγ t αµγ , (A38) I := A α F αβ F γµ t βγµ , (A39) I := A α F βµ F βγ t γµα , (A40) n tAF F F, tA e F e F F o I := A α F βµ F βγ F µν t αγν , (A41) I := A α F βγ F t αβγ , (A42) I := A α F αβ F γν F γµ t βνµ , (A43) I := A α F αβ F βγ F µν t γµν , (A44) I := A α F αβ F γν F γµ t µνβ , (A45) n tAF F F F, tA e F e F e F e F , tA e F e F F F, tAF F F F o I := A α F βµ F βγ F µν F ν ρ t αργ , (A46) I := A α F βµ F βγ F t αµγ , (A47) I := A α F αβ F γν F γµ F νρ t βµρ , (A48) I := A α F αβ F γµ F t βγµ , (A49) I := A α F αβ F βγ F µρ F µν t γρν , (A50) I := A α F αβ F βγ F γµ F νρ t µνρ , (A51) I := A α F βµ F βγ F µν F ν ρ t γρα , (A52) I := A α F βµ F βγ F t γµα , (A53) I := A α F αβ F βγ F µρ F µν t νργ . (A54) [1] L. Baudis, “Dark matter detection,” J. Phys.
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