Tensor stability in Born-Infeld determinantal gravity

We consider the transverse-traceless tensor perturbation of a spatial flat homogeneous and isotropic spacetime in Born-Infeld determinantal gravity, and investigate the evolution of tensor mode for two regular solutions in the early universe. For the first solution where the initial singularity is replaced by a geometric de Sitter inflation of infinite duration, the evolution of tensor mode is stable for most regions of the parameter space. For the second solution where the initial singularity is replaced by a primordial brusque bounce, the evolution of tensor mode is stable for all regions of the parameter space. Our calculation suggests that the stability of tensor mode in a very early radiation filled universe is a remarkable property of Born-Infeld determinantal gravity.


I. INTRODUCTION
The teleparallel equivalent of general relativity, also called teleparallel gravity or teleparallelism for short, can be traced back to an attempt by Einstein to unify the electromagnetism and gravity on the mathematical structure of distant parallelism [1]. In this theory, there is a set of dynamical vierbein (or tetrad) fields which form the orthogonal bases for the tangent space of each spacetime point. Instead of the Levi-Civita connection { ρ µν }, a spacetime is characterized by a curvature-free Weitzenböck connection Γ ρ µν = e A ρ ∂ ν e A µ , with e A µ the vierbein, which refers to the metric through the relation g µν = e A µ e B ν η AB , where η AB = diag(1, −1, −1, −1) is the Minkowski metric for the tangent space. Although the Weitzenböck spacetime is flat, it possesses torsion, which is defined as T ρ µν = Γ ρ νµ − Γ ρ µν . The well-known action of teleparallel gravity reads where e = |e A M | = −|g µν |, and T ≡ S ρ µν T ρ µν is torsion scalar contracted by the torsion tensor and a new tensor S ρ µν defined by S ρ µν = 1 2 (K µν ρ + δ µ ρ T σν σ − δ ν ρ T σµ σ ), with K µν ρ the contorsion tensor related to the difference between the Levi-Civita connection and Weitzenböck connection, i.e., K ρ µν = Γ ρ µν − { ρ µν } = 1 2 (T µ ρ ν + T ν ρ µ − T ρ µ ν ). Gravitational interaction is described by the curved spacetime geometry in general relativity, however, the contorsion tensor can be * keyang@swu.edu.cn † zhangyupeng14@lzu.edu.cn ‡ liuyx@lzu.edu.cn, corresponding author regarded as a gravitational force acting on particles in teleparallel gravity. Nevertheless, no matter which description of gravity we use, with the identity T = −R+2e −1 ∂ ν (eT µ µν ) between the torsion scalar and Ricci scalar, the equivalence between teleparallel gravity and general relativity is manifest from the action (1).
As the cornerstone of modern cosmology, general relativity provides precise descriptions to a variety of phenomena in our universe. However, it is well-known that general relativity suffers from various troublesome theoretical problems, such as the dark matter problem [2], dark energy problem [3] and the unavoidable singularity problem [4]. One of the attempts to solve the singularity problem in classical level, first suggested by Deser and Gibbons [5], follows the spirit of Born-Infeld electromagnetic theory, which regularises the divergent self-energy of the electron in classical dynamics [6]. One of the Born-Infeld type generalized gravity can be written as the form [7][8][9][10][11][12][13] where the rank-2 tensor F µν is a function of certain fields ψ i and their derivatives, λ is the Born-Infeld constant with mass dimension 2, and ∆ is a constant.
In order to recover a proper low-energy theory, the simplest case is to choose F µν to be the Ricci tensor R µν (ψ i ), then general relativity is recovered. However, If ψ i is the metric field, namely working in a pure metric formalism, it will lead to fourth order field equations with ghost in-stabilities [5]. If ψ i is the connection field, namely working in the Palatini formalism, the theory is free from the ghost problem [7,8], and is now dubbed Eddingtoninspired Born-Infeld (EiBI) gravity. An intriguing property of EiBI theory is that it may avoid the initial Big Bang singularity of the universe [8,14,15]. However, although the background evolution maybe free from an initial singularity, by including the linear perturbations, the overall evolution may still be singular [16,17].
Another interesting choice is to simply require Tr(F µν ) to be the torsion scalar [9][10][11]. Thus, the low-energy theory recovers the teleparallel gravity or general relativity equivalently. This can be fulfilled with F µν = αF µν = g µν T , and α + β + (d + 1)γ = 1. This theory leads to a second-order field equations, and is dubbed Born-Infeld determinantal gravity. It supports some regular cosmological solutions by replacing the possible initial singularity with a de-Sitter phase or a bounce [11,18]. The authors in Ref. [19] pointed out that although the theory is singularity-free in some regions of the parameter space, nevertheless, the Big Rip, Big Freeze, or Sudden singularities may still emerge in some other regions of the parameter space. The equations of motion were analyzed and Schwarzschild geometry was studied in Ref. [20]. If α = β = 0, the theory reduces to an f (T ) type theory, a spatially flat cosmology and a 5-dimensional domain wall have been considered in this reduced Born-Infeldf (T ) theory [21,22].
In this work, we investigate the evolution of transversetraceless (TT) tensor mode in early high energy regime of the spatially flat Friedmann-Robertson-Walker (FRW) cosmology in Born-Infeld determinantal gravity. The evolution of TT tensor mode, which is pure gravitational and is irrelevant to matter density perturbations, reveals the overall stability of the singularity-free background solutions against the tensor perturbation. Through the paper, the capital Latin indices A, B, · · · and small Latin indices a, b, · · · label the four-dimensional and three-dimensional coordinates of tangent space, respectively, and Greek indices µ, ν, · · · and small Latin indices i, j, · · · label the four-dimensional spacetime and threedimensional space coordinate, respectively. For simplicity, we set a vanishing cosmological constant by fixing ∆ = 1 in the following analysis.
The paper is organized as follows. In Sec. II, we introduce the equations of motion of Born-Infeld determinantal gravity. In Sec. III, we consider the tensor perturbation in a spatial flat FRW cosmological background and get the evolution of tensor mode. In Sec. IV, we investigate the evolution of tensor mode in very early cosmology for two regular cosmic solutions. Finally, brief conclusions are presented.

II. EQUATIONS OF MOTION
We start from the action in which gravity is minimally coupled to a matter field [11] where L M represents the Lagrangian of a matter field coupling only to the vierbein field or to metric equivalently. By varying with respect to the vierbein, one gets the Euler-Lagrange equation where L G represents the gravitational Lagrangian, and If the action of the matter field is local Lorentz invariant, then the energymomentum tensor is symmetric and conserved [23].
With some algebra, the left two terms in Euler-Lagrange equation can be written explicitly as where U µν = g µν + 2λ −1 F µν . After contracting the index A of tangent space via multiplying a vierbein e A ν , the equations of motion read [20] |U µν | 1 2 where the energy-momentum tensor for a perfect fluid reads Θ µ ν = (ρ + P )u µ u ν − P δ µ ν , and the two partial derivative terms are written explicitly as with the tensors S C αβ and Q λ ∂e A λ , and given by

III. LINEAR TENSOR PERTURBATION
We consider a 4-dimensional perturbed spatial flat FRW spacetime (d = 3) with the metric to be of the form where a(t) is the scale factor and h ij (t, x) a TT tensor perturbation, i.e., ∂ i h ij = δ ij h ij = 0. The corresponding perturbed vierbein reads where h a i = δ a j h j i . With the perturbed vierbein, the nonvanishing components of perturbed torsion tensor are where H =ȧ/a is the Hubble parameter. Then, the nonvanishing components of the perturbed contorsion tensor read and the nonvanishing components of the perturbed tensor S P MN read The perturbation of F µν can be assembled by F µν = αF (1) µν +βF (2) µν +γF µν , and F µν given as The expressions for the perturbations of ∂F αβ /∂e A µ and ∂F αβ /∂(∂ γ e A µ ) are listed in appendix A. By only focusing on the TT tensor mode, we can shut down all the scalar and vector modes in the perturbed perfect fluid, since the scalar, vector and tensor modes are decoupled from each other and evolve separately. So the nonvanishing components of the perturbed energymomentum tensor are simply given by Substituting the above perturbed variables into the field equation (8), we can get the background equations and linear perturbed equation via counting the orders of perturbations. With some cumbersome algebra, the background equations read where the constants A = 6(β + 2γ)λ −1 and B = 2(2α + β + 6γ)λ −1 . Further, by counting the first-order perturbations of field equation (8), the only non-vanishing equation reads where the Laplace operator ∇ 2 = δ ij ∂ i ∂ j , and the coefficients are given by with the constants C = (2α + β)λ −1 and D = 3γλ −1 . In low energy regime (λ → ∞), the evolution equation (31) of the TT tensor mode reduces to the standard one in general relativity as expected, i.e., where we have replaced the Laplace operator with −k 2 with k the wave vector.

A. Solution I
An interesting solution discussed in Ref. [11] is achieved by choosing B = 0. Combining the normalization condition α + β + 4γ = 1, one has β = α + 3 and γ = −(1 + α)/2 with α a free parameter. It leads to A = 12λ −1 . In this case, the background equations (29) and (30) reduce to where q = − a ′′ aH 2 is the deceleration parameter. The square roots in the equations restrict the parameter λ to be positive in this case.
By considering a perfect fluid with the state equation P = ωρ, one can find that for every barotropic index ω > −1, the solution describes a geodesically complete spacetime without the big bang singularity and possesses a geometrical de Sitter inflationary stage naturally [9,11,24]. From the conservation equationρ + 3(ρ + P )H = 0, one has with the constants a 0 and ρ 0 relating to the present day values.
This implies that the tensor mode is highly suppressed in this regime, and hence, the solution is stable against the tensor perturbation in the asymptotic past. b) If α = −1 and −1 < ω < −2/3, we know that So the evolution equation (31) reduces to h i j ≈ 0 just as the case a. Thus, the tensor mode is stable in this case.
c) If α = −1 and ω = −2/3, we have F 0 /F 2 ≈ 4ε. Then, the evolution equation (31) is rewritten as The solution is given by where p 1 = 1 − 48ε λ and c i are the integration constants. So the tensor mode blows up as t → −∞, and hence, the spacetime renders an instability within the tensor perturbation in this case. d ) If α = −1, ω > −2/3 and ω = 0, F 0 /F 2 ≈ 12ε(1 + ω)e √ λ 3 (2+3ω)t is negligible comparing with F 1 /F 2 . So the asymptotic behavior of the evolution equation (31) is which can be solved as As t → −∞, the tensor mode is convergent and stable for ω > 0, but divergent and unstable for −2/3 < ω < 0. e) If α = −1 and ω = 0, we have and F 0 /F 2 ∝ e √ Thus the solution reads and the tensor mode is linearly divergent in the asymptotic past. In brief, for most regions of the parameter space of α and ω, the evolution of the tensor mode is regular in the very early cosmic stage. However, the tensor mode is unstable in the asymptotic past for α = −1 and −2/3 ≤ ω ≤ 0.

B. Solution II
Another interesting solution studied in Ref. [18] is achieved by setting A = B, which implies α = β and leads to A = B = 3λ −1 with γ a free parameter. In this case, the background equations (29) and (30) reduce to For a perfect fluid with the Born-Infeld parameter λ < 0, the solution depicts an irregular spacetime with the Hubble rate diverges as the scale factor goes to zero, whereas for λ > 0 and the barotropic index ω > −1, a brusque bounce solution can be obtained with the approximate behavior around the bounce point given by where the minus sign corresponds to t > 0 and the positive sign to t < 0. Then the Hubble rate is given by As |t| → 0, the Hubble rate reaches a maximum H(0) ≈ √ λ/3, where the maximum energy density is ρ m ≈ (48πG) −1 λ and the scale factor reaches a mini- With the asymptotic solution, the coefficients of the evolution equation (31) behave like where K 1 = 48γ 2 −24γ −29, K 2 = K 1 (3+2ω)+96(1+ω), and the minus and positive signs in F 1 correspond to t > 0 and t < 0, respectively, whereas the lowest orders of F 0 and F 2 are the same for both positive and negative time.
for the barotropic index −1 < ω ≤ 0, then the asymptotic behavior of evolution equation (31) reduces tö Note that K 1 is negative in this case. Thus, the solution is given by where . Therefore, the spacetime is stable against the tensor perturbation around the bounce point.
In brief, for all regions of the parameter space of γ and ω, the evolution of tensor mode is regular. Thus, this solution is stable against the TT tensor perturbation.

V. CONCLUSIONS
We linearized the field equations of Born-Infeld determinantal gravity and investigated the tensor stability of two regular solutions in the early universe. For solution I, which supports a regular λ-driven de Sitter evolution of infinite duration, we found that the solution is stable against the tensor perturbation in general parameter space, yet the solution renders an instability for α = −1 and −2/3 ≤ ω ≤ 0. For α = −1, the parameter γ = 0, then the term F (3) µν = g µν T in F µν vanishes. So this implies F (3) µν may be crucial for maintaining the tensor stability in this case. For solution II, which supports a regular brusque bounce universe, we found the solution is stable against the tensor perturbation around the bounce point. Anyway, for a very early radiation filled universe (ω = 1/3), about which we are most concerned, the cosmic evolution is stable against the tensor perturbation.
It is well-known that general relativity suffers the singularity problem in early universe, and the Born-Infeld type gravity may avoid the Big Bang singularity, such as the EiBI theory and the Born-Infeld determinantal gravity. However, although the background solution is singularity-free in EiBI theory, the overall evolution is singular because of the unstable tensor perturbation in Eddington regime. Therefore, our calculation suggests that the stable cosmic evolutions against tensor perturbation in the majority of cases is a remarkable property of Born-Infeld determinantal gravity. The full linear perturbations, including scalar modes and vector modes, are quite more complicated and left for our future work.
The perturbation of