Generating primordial fluctuations from modified teleparallel gravity with local Lorentz-symmetry breaking

In the context of modified teleparallel gravity, we study the generation of primordial density fluctuations in a general scalar-torsion theory whose Lagrangian density is an arbitrary function $f(T,\phi)$ of the torsion scalar $T$ and a scalar field $\phi$, plus the kinetic term of this latter. It is well known that generic modifications of teleparallel gravity are not invariant under six-parameter local Lorentz transformations. In order to restore the local Lorentz symmetry, we have incorporated the six additional degrees of freedom in the form of Goldstone modes of the symmetry breaking through a Lorentz rotation of the tetrad field. After integrating out all the auxiliary modes, we obtain a second order action for the scalar and tensor propagating modes and their power spectrum generated during inflation. It is found that an explicit mass term emerges in the second order action for curvature perturbation, describing the effects of local Lorentz violation at first-order of slow-roll. In a possible scenario of slow-roll inflation, we show that for nonminimal coupling functions $f(T,\phi)$ which are non-linear in $T$, including the particular case of $f(T)$ gravity with minimally coupled scalar field, there is a successful generation of primordial fluctuations with sub-horizon perturbation modes freely propagating and matching the Bunch-Davies vacuum which can survive by time of horizon crossing. At super-horizon scales the curvature perturbation modes freeze up to a slight logarithmic time-dependence proportional to slow-roll parameters, and tachyonic instability is avoided.


Introduction
Lorentz invariance is considered as one of the most fundamental symmetries in physics, which provides a fundamental support for the principles of general relativity (GR) and the standard model of particle physics [1]. Nevertheless, at a sufficiently high-energy scale (Planck scale), it is expected that these two field theories merge into a single unified and quantum-consistent theory, under a possible breaking of local Lorentz symmetry [2][3][4][5][6][7]. Furthermore, if primordial density fluctuations were generated during inflation [8][9][10][11], they give us an unique opportunity of learn about physics at energy scales that would not otherwise be accessible, since it is believed that inflation occurs near the scale of grand unification, and therefore not too far from scales where quantum gravity is relevant [12][13][14].
An alternative description of gravity based on torsion is the so-called teleparallel equivalent of GR or simply teleparallel gravity (TG) [15][16][17][18][19][20][21][22]. The dynamical variable is the tetrad field rather than the Einstein's metric tensor, and the usual torsionless Levi-Civita connection is replaced by the Weitzenböck connection, which has torsion but no curvature [23][24][25]. So, the Lagrangian density is proportional to the torsion scalar T which is different from the curvature scalar R by a total derivative term and, 1 manuel.gonzalez@pucv.cl 2 giovanni.otalora@pucv.cl thus, the two theories are fully equivalent in the level of field equations [26]. Although being equivalents, notable conceptual differences arise, and it has been shown that the purely inertial spin connection of TG leads to a classical gauge theory for gravitation based on the translation group [27]. In modifying gravity from the viewpoint of TG there are several different ways that can be followed. In the same line of scalar-tensor theories, a natural extension for TG is a non-minimally coupled scalar-torsion gravity theory where the scalar field is non-minimally coupled to gravity through an interaction term F (φ)T , where F (φ) is a coupling function, and hence, also leading us to a teleparallel "Horndeski-type" theory [28][29][30][31][32][33][34]. Another way it is to extend the torsion scalar T to a function f (T ), in the same analogy with f (R) gravity [35][36][37][38]. These modifications of gravity based on torsion stablish a new class of modified gravity theories with a rich phenomenology, explaining inflation and dark energy [39,40].
Related to the richness of its structure, it has been shown that theories of generalized teleparallel gravity break the local Lorentz symmetry [41,42]. This has caused great interest in the study of cosmic inflation and the effects of local Lorentz violation on the inflationary observables from the framework of modified teleparallel gravity theories [43][44][45][46][47]. Particularly, in [44] the authors have investigated power-law and intermediate inflation in f (T ) gravity, whereas that in [46] it has been studied the slow-roll inflation in a generalized non-minimally coupled scalar-torsion gravity theory with a Galileon-type selfinteraction. On the other hand, in [45] the authors have concentrated their efforts in the investigation of the consequences of local Lorenz violation to the generation of primordial density fluctuations in a class of teleparallel scalar-torsion gravity theories, whose Lagrangian density is linear in the torsion scalar T . The most outstanding result from this latter work is that due to local Lorentz violation no subhorizon scalar-perturbation mode can survive by the time of horizon crossing, and thus these theories are incapable of generate enough primordial density inhomogeneity, even if it brings some de Sitter background solution. Moreover, in [47], where it has also been included a nonminimal coupling to the vector torsion, the authors have corroborated the above result of [45], and they have also concluded that for some particular coupling functions to torsion scalar and vector torsion, the scalar field can source the linear perturbations.
In the present paper, we study the generation of primordial density fluctuations in a larger class of generalized teleparallel scalar-torsion f (T, φ) gravity theories with local Lorentz symmetry breaking. We confirm that obtained in [45] in the case of a particular limit of a nonminimal coupling function f (T, φ) lineal in T , but our most important result is that in the more general case of a nonminimal coupling function f (T, φ), which is non-linear in T , there is a successful generation of primordial fluctuations, in a possible slow-roll inflation scenario. The paper is organized as follows. In section 2, we give a concise introduction to TG. In section 3, we develop the framework of generalized f (T, φ) gravity theory, calculating the background equations for a Friedmann-Robertson-Walker (FRW) metric and then analysing the de-Sitter limit. In the sections 4 and 5 , we investigate the cosmological perturbations using the ADM formalism for the tetrad fields and using the Maldacena's method of expanding the action until second order for the perturbations [48]. Finally, in Section 6, we summarize our findings and present our main conclusions and final remarks.

Teleparallel Gravity
The dynamical variable of TG is the tetrad field e A (x µ ) [26], and it connects the spacetime metric g µν and the Minkowski tangent space metric η AB = diag (−1, 1, 1, 1) thorough the local relation where e A µ are the tetrad components in a coordinate base and then satisfying the orthogonality conditions e A µ e ν A = δ ν µ and e A µ e µ B = δ A B , with e µ B the inverse components. The action functional of TG is given by being T the torsion scalar, e = det e A µ = √ −g, and M 2 pl = (8πG) −1 the reduced Planck mass. The torsion scalar is defined as where are the components of torsion tensor, and is the so-called super-potential, with the contorsion tensor. The purely inertial spin connection of TG is with Λ A D (x) a local (point-dependent) Lorentz transformation. For this connection the curvature tensor vanishes identically, whereas that the torsion tensor is nonvanishing [26].
The corresponding spacetime-indexed linear connection is which is the so-called Weitzenböck connection. It is related to the Levi-Civita connection of GR through Using this latter equation, it can be shown that where R is the curvature scalar of Levi-Civita connection [26]. So, the above equation show that TG and GR are equivalent theories in the level of field equations. However, when one modifies gravity from the viewpoint of TG, by introducing a non-minimally coupled matter field, as for example a scalar field, or by adding into the action, nonlinear terms in the torsion scalar T , as for example in f (T ) gravity, it is obtained a new class of modified gravity theories with a rich phenomenology and not equivalents to their corresponding counterpart in the curvature approach [40]. Below, we are going to study the generation of primordial density fluctuations in generalized teleparallel scalartorsion gravity theories.

Field equations and local Lorentz invariance
The relevant action is given by where f is an arbitrary function of φ and T , and also X = −∂ µ φ∂ µ φ/2. This general action includes nonminimally coupled scalar-torsion gravity models, and f (T ) gravity, plus scalar field. For f (T, φ) = −M 2 pl T /2 − V (φ), we recover TG, with V (φ) the scalar potential [12].
Varying the action with respect to the tetrad field e A µ we find the corresponding field equations which have been expressed in a general coordinate basis, and G µ ν = e µ A G A ν is the Einstein tensor, and the tensor [26]. The equation (12) has an antisymmetric part associated with the tensor S µν ρ in the second term. It is an expected result as the action (11) is not local Lorentz invariant [41,42]. To see this explicitly, let us to consider the infinitesimal point-dependent Lorentz

Under this transformation the variation of the action is
where ξ µν = e µ A e ν B ξ AB . The condition δS = 0 for arbitrary ξ AB leads us to the following constraint For TG, f ∼ T , one has ∂ ρ f ,T = 0, and thus the left hand side of this latter equation becomes identically equal to zero, and local Lorentz invariances is restored. For modified teleparallel gravity, ∂ ρ f ,T = 0, it corresponds to a set of six equations for six additional degrees of freedom, due to violation of local Lorentz symmetry.

Cosmological background
We impose the standard homogeneous and isotropic background geometry by choosing which corresponds to a flat Friedmann-Robertson-Walker (FRW) universe with metric where a is the scale factor which is a function of the cosmic time t.
Replacing this tetrad field in the field equations (12), we obtain the background equations (19) where H ≡ȧ/a is the Hubble rate, and a dot represents derivative with respect to t. Also, a comma denotes derivative with respect to φ or T .
To analyse the de-Sitter limit [49], we use the values φ = φ * and H = H * (t) in the background equations, so we obtainḢ * = 0 and f ,φ (T * , φ * ) = 0. After that, we study the perturbations of this de-Sitter limit, using φ = φ * +δφ(t) and H = H * +δH(t). Then, we expand equation (17) to first order and we find δH = 0. Next, expanding equation (19) to first order, we find the equation for δφ(t), whose solution is given by, where C 1 and C 2 are integration constants, and Therefore, the perturbation δφ is stable only for For non-phantom scalar fields, it is required P (φ * ) > 0, and then the above constraint becomes f ,φφ (T * , φ * ) < 0. For f (T ) gravity, without dynamical scalar field, one has f ,φφ (T * , φ * ) = 0, and then the eigenvalues are µ − = 0 and µ + = −3H * . Therefore, in this latter particular case, the de-Sitter background is always (marginally) stable [40]. In order to realize the slow-roll approximation into the present scenario, from Eqs. (17) and (18) we obtain where we have introduced the slow-roll parameters Also, it is useful to split the parameter δ f,T as where we define The time dependence of these parameters is calculated asδ P X where it has also been defined the slow-roll parameters During slow-roll inflationδ P X ∼δ f,T ∼δ f X ∼ O(2), and similarly for the other parameters.

Second order action
In order to study primordial density fluctuations, we start from the Arnowitt-Deser-Misner (ADM) decomposition of the tetrad field [43] where Using the uniform field gauge, δφ = 0, a convenient ansatz for the fields is which gives the corresponding perturbed metric [50] The additional degrees of freedom due to local Lorentz violation can be incorporated in the form of Goldstone modes of the symmetry breaking, by performing a Lorentz rotation of the tetrad field [5]. So, under the transformation and keeping fixed the zero spin connection for the cosmological background, the full tetrad field is written as The matrix χ AB = −χ BA is parametrized as where χ a = η ab χ b and B ab = −B ba . It is defined the spatial vector χ i = h i a χ a = ∂ i β + χ Following [48], the next step is to expand the action (11) up to second order to obtain where we have defined the functions From action (39), it can be seen that the scalar modes α, ψ and β are auxiliary fields and do not propagate. Varying this action with respect to ∂ 2 ψ leads us to whereas that variation with respect to ∂ 2 β gives −4w 4Ṙ + 4w 4 Hα + w 5 R − 2 w 6 a 2 ∂ 2 ψ +2w 6 ∂ 2 β = 0, and for α we have Solving the above three equations for α, ∂ 2 ψ and ∂ 2 β, and after substituting these results in Eq. (39), the second order action for curvature fluctuation can be written as where, The first and second term in action (44) are the usual terms appearing in the quadratic action of perturbations, while the third term is a new explicit mass term, that represents the effects of local Lorentz-symmetry breaking. The origin of this mass term is the Lorentz violating coupling term f (T, φ) in action (11), which is non-linear in torsion scalar T . The emergence of this propagating massive scalar mode could be related to an alternative gravitational Higgs mechanism [3,5].
For any theory to be physically viable, it must be free of ghosts and Laplacian instabilities by requiring Q s > 0 and c 2 s > 0. These two conditions are satisfied by equations (45) (for P > 0) and (46). Moreover, in the presence of an explicit mass term, there is an additional condition that is the non-occurrence of tachyonic instability [52]. There are two situations in which the tachyonic instability can be avoided. The first possibility is that the mass squared must be positive, m 2 > 0, and, the second one, if we have m 2 < 0, then it is required that |m 2 | H 2 [52,53].
In terms of slow roll parameters we can write and it is also useful to define Similarly, the mass term can be written as For f (T, φ) non-linear in T , and either δ f X = 0 or δ f X = 0, one has that η R ∼ O(1) is non-zero (and finite). Furthermore, tachyonic instability is avoided as long as |η R | 1.
In the absence of coupling between T and φ, one has δ f X = 0, and thus δ f,T = δ fḢ . So, from Eq. (50), we find η R = 2δ fḢ ∼ O(1). This is the explicit mass term arising in f (T ) gravity, plus scalar field. For TG, f ∼ T , one has δ fḢ = 0, and then η R = 0, which is an expected result as in this case there is not local Lorentz violation. Now, let us consider the case δ f X = 0, and f (T, φ) a linear function in T . This is precisely the non-minimally coupled scalar-torsion theory of Ref. [45]. For this model one has δ fḢ = 0, δ f,T = δ f X , and then |η R | = ∞. The physical meaning of this is that there are no nonzero-momentum solutions for the scalaron, as in this case one would have ∂ 2 R = 0, from action (39), and then, spoiling the generation of primordial density fluctuations. This latter result is consistent with that obtained in [45].

Mukhanov-Sasaki equation
It is introduced the canonically-normalized Mukhanov where we also have defined Making the change to conformal time dτ = dt/a, and using the above variables, action (44) can be written as where it has been defined the effective mass term as where m 2 = 3H 2 η R , with η R given by (50), and z ′′ /z is the usual effective mass term coming from the interaction between R and the cosmological background.
The scalar power spectrum of curvature perturbation is calculated as Given that η R ∼ O(1), the consequence of local Lorentz violation is a slight logarithmic time-dependence of the curvature perturbation and its power spectrum at superhorizon scales. Thus, as a satisfactory approximation, we can evaluate this latter at the horizon crossing [54]. Finally, the scale-dependence of the scalar power spectrum is This carries out the effects of local Lorentz violation on the scalar power spectrum through the new mass term 2η R , at first-order in slow-roll approximation.

Tensor perturbations
From the ADM decomposition for the tetrad field presented in Eqs. (32) and (33), and using the uniform field gauge, δφ = 0, we take [43] Then the induced 3−metric is where we have defined and γ a j = γ i j δ a i . Given that the γ 2 term has contribution only in cubic calculations of the Lagrangian [48], we keep only until the second term h ij in the induced metric. Also, the tensor γ ij can be splitted in the form γ ij = γ (i,j) +γ [i,j] . The symmetric part h ij = γ (i,j) fulfills the transverse and traceless conditions, ∂ i h ij = h i i = 0, to be gauge invariant [12]. On the other hand, the antisymmetric part matches the gauge degrees of freedom in the local Lorentz invariant theory, and then we identify B ij with γ [i,j] .
Then, using the tetrad formalism we find the secondorder action for the tensor modes, h ij = h + e + ij + h × e × ij , in the way where two polarization states are given by λ = +, ×. We also have defined and the squared tensor propagation speed is The non-ghost condition is satisfied only for f ,T < 0. Besides the usual transverse massless graviton modes, propagating at speed of light, there are no additional propagating modes in the quadratic action (70), which is consistent with local Lorentz invariance of tensor perturbations [46]. The power spectrum for tensor perturbations becomes with H k and Q T k the values of H and Q T at k = aH. Thus, the spectral index is Tensor-to-scalar ratio, evaluated at the horizon crossing, is given by Using the Eqs. (74) and (75), we obtain the consistency relation This is agreement with the standard inflation limit where r = −8n T . The quantity δ f,T appears as a small correction to the value of standard inflation.

Conclusions
We have studied the generation of primordial fluctuations in generalized teleparallel scalar-torsion gravity theories whose Lagrangian density is an arbitrary function f (T, φ) of the torsion scalar T and a scalar field φ, plus the kinetic term of this latter. To develop primordial density perturbations, we started from the Arnowitt-Deser-Misner (ADM) formalism of the tetrad field, and we choose the uniform gauge [14]. The tetrad field has sixteen degrees of freedom and local Lorentz invariance of TG allows us to eliminate six degrees of freedom, yielding the same number of independent components of the metric tensor [26]. However, it is well known that the action for modified teleparallel gravity is no longer a local Lorentz invariant, and thus, the field equations are not completely symmetric [41,42]. In order to restore the local Lorentz invariance, we have introduced six additional degrees of freedom in the form of Goldstone modes of the symmetry breaking, through a Lorentz rotation of the tetrad field [5]. So, the antisymmetric part of field equations constitutes a set of six equations for six extra modes, that is, a scalar, a transverse 3-vector, and a spatial antisymmetric tensor modes.
Putting all these pieces together, and after integrating out the auxiliary fields, we have calculated the second order action for the propagating modes [48]. As usual, we have treated the scalar and tensor modes separately since they are not coupled. Vector modes decay rapidly with the cosmic expansion, and thus they can be ignored [12]. Furthermore, we have verified that the corresponding additional tensor modes are completely cancelled out from the second order action for tensor perturbations, remaining only the usual transverse massless graviton modes, propagating at speed of light, and therefore indicating the local Lorentz invariance in the tensor perturbations sector [46].
In the second order action for curvature perturbation, it is observed the emergence of an explicit mass term, which represent the effects of local Lorentz violation. This explicit mass term is of first-order in slow-roll, and it is always nonzero (and finite), for nonminimal coupling functions f (T, φ), which are non-linear in T , including the f (T ) gravity, plus scalar field, as a particular example. The arising of this propagating massive scalar mode could be related to an alternative Higgs mechanism that has no direct analogue in nonabelian gauge theory [3]. As expected, in the case of TG this explicit mass term is equal to zero, because the local Lorentz invariance [26]. On the other hand, when the nonminimal coupling function f (T, φ) is linear in T , like the action considered in [45], it becomes divergent, which necessarily leads us to ∂ 2 R = 0, or equivalently, R k = 0 for all Fourier mode k, and hence, it is immediate to conclude that no subhorizon scalar mode could propagate and survive by the time of horizon crossing. This latter result is consistent with that obtained in [45].
Thus, our results indicate that only for modified teleparallel gravity theories with non-linear coupling functions f (T, φ), including the f (T ) gravity, plus scalar field, there is a successful generation of primordial density fluctuations in a possible slow-roll inflation scenario. At sub-horizon scales the solution for the primordial quantum fluctuations matches the Bunch-Davies vacuum boundary condition, and then at scales deep inside the horizon, the effects of the explicit mass term can be neglected. At super-horizon scales the curvature perturbation modes freeze up to a slight logarithmic time-dependence proportional to slowroll parameters, and thus the spectral index of scalar power spectrum, evaluated at the horizon crossing, carries out the effects of local Lorentz violation. For large field models, and using the non-ghost condition f ,T < 0, the explicit mass term could contribute with the red tilt of the scalar spectrum, but tachyonic instability is avoided as long as the instability rate is less than the Hubble rate [52,53].