Thick branes in Born-Infeld determinantal gravity in Weitzenböck spacetime

Einstein’s general relativity (GR) is the cornerstone of modern cosmology, which provides the most accurate descriptions of a variety of phenomena in our universe. In GR, the gravitation is described by the curvature of spacetime, where the affine connection is symmetric and coincides uniquely with the Levi-Civita connection. A well-known alternative gravity theory dynamically equivalent to GR is the so-called teleparallel gravity Cho1976 ; Hayashi1979 , which allows us to interpret GR as a gauge theory for a translation group. In teleparallel gravity, the spacetime is characterized by the curvature-free Weitzenböck connection rather than the torsion-free Levi-Civita connection, and the dynamical field is the vielbein instead of metric in Weitzenböck spacetime Maluf2013 ; Bahamonde2021 . Following the spirit of the popular $f(R)$ gravity which generalizes the Lagrangian of GR to be some functions of the Ricci scalar $R$, the $f(T)$ gravity generalizes the Lagrangian of teleparallel gravity to be some functions of the torsion scalar $T$ Ferraro2007 ; Bengochea2009 ; Linder2010 . This impactful modification of teleparallel gravity provides possible explanations for the accelerating expansion of the universe without invoking the dark energy Bengochea2009 ; Cardone2012 and for the inflation without resorting to the inflaton field Ferraro2007 ; Ferraro2008 .

It is well-known that GR suffers from the inevitable singularities at the beginning of Big Bang and the center of black holes Hawking1970 . Some attempts to solve the singularity problem of GR at the classical level were done by replacing the Einstein-Hilbert action with the Born-Infeld-type determinantal action Deser1998 ; Vollick2004 ; Banados2010 ; Chen2016 . The Born-Infeld determinantal gravity (BIDG) in Weitzenböck spacetime was also considered in Refs. Ferraro2010 ; Fiorini2013 ; Fiorini2016 . This theory leads to second-order field equations and supports some regular cosmological solutions by replacing the possible initial singularity with a de-Sitter phase or a bounce. The tensor evolution of these cosmological solutions can hold stability in a large parameter space in the early universe, and the theoretical parameter $\lambda$ is constrained by the speed of gravitational waves Yang2019a . However, other cosmological singularities such as Big Rip, Big Bang, Big Freeze, and Sudden singularities may emerge in some regions of parameter space Bouhmadi-Lopez2014a . The Schwarzschild geometry was considered in this framework as well Fiorini2016a .

The idea that our spacetime may have hidden extra dimensions in ultraviolet regime is a long historical topic since the proposal of Kaluza-Klein (KK) theory in the 1920s. As the most plausible candidate for unifying all fundamental interactions, the superstring/M-theory requires extra dimensions for mathematical consistency. Instead of traditional compact extra dimension scenario, the braneworld scenario suggests that our universe is a 3-brane embedded into a higher-dimensional bulk, which opens a new way to solve some long-standing problems in particle physics and cosmology, such as the gauge hierarchy problem and cosmological constant problem Arkani-Hamed1998 ; Arkani-Hamed2000 ; Randall1999 ; Randall1999a ; Arvanitaki2016 . Thick brane scenario is considered as a smooth generalization of Randall-Sundrum-2 (RS2) model Randall1999a . The most interesting thick brane configuration is the domain wall, which is a global topological defect in a multidimensional bulk Dzhunushaliev2010 . The domain wall brane could be a potential carrier, onto which the Standard Model could possibly be transplanted Davies2008 .

The projected Gauss-Codazzi equations of RS2 model in the 5-dimensional teleparallel gravity contain two extra terms arising from the extra degrees of freedom in the teleparallel Lagrangian Nozari2013 . Furthermore, the brane cosmology provides an equivalent viewpoint as RS2 model in GR, however, the projected effect on the brane is determined by the projection of torsion tensor Geng2014a . The inflation and dark energy dominated stage were realized on the brane by generalizing RS2 model in 5-dimensional $f(T)$ gravity Bamba2013 . Analytic thick brane solutions were obtained in the 5-dimensional $f(T)$ gravity Yang2012b ; Menezes2014 ; Yang2018 ; Wang2018a . The gravitational perturbation against the thick branes and the corresponding resonance spectrum were investigated Guo2016 ; Tan2021 . Thick string-like brane-world models were constructed in the 6-dimensional $f(T)$ gravity, where the torsion effects on the models were analyzed as well Moreira2021 .

In this work, we will consider the thick brane scenario in the higher-dimensional BIDG in Weitzenböck spacetime and analyze the torsion effects on the brane structure and gravitational perturbation. The layout of the paper is as follows: In Sec. II, the higher-dimensional BIDG in Weitzenböck spacetime is briefly introduced. In Sec. III, the thick brane scenario is constructed in $(d+1)$-dimensional BIDG, and the first-order formalism is employed in order to solve the field equations analytically. Specifically, for $d=4$, analytical 3-brane solutions are obtained for three particular parameter choices in Sec. IV. In Sec. V, the gravitational perturbation against the domain wall backgrounds is discussed. Finally, brief conclusions and discussions are presented. Throughout the paper, the capital Latin indices $A,B,C\cdots$ and small Latin indices $a,b,c\cdots$ label the $(d+1)$-dimensional and $d$-dimensional coordinates of tangent space, respectively, while the capital Latin letters $K,L,M,N,\cdots$ and Greek letters $\mu,\nu,\rho,\sigma\cdots$ label the $(d+1)$-dimensional and $d$-dimensional spacetime indices, respectively.

In Weitzenböck spacetime, the fundamental dynamic field is the vielbein ${e^{A}}_{M}$, which refers to the metric through the relation $g_{MN}={e^{A}}_{M}{e^{B}}_{N}\eta_{AB}$, with $\eta_{AB}=\text{diag}(-1,1,\cdots,1)$ the Minkowski metric for the tangent space. The torsion tensor ${T^{P}}_{MN}={\Gamma^{P}}_{NM}-{\Gamma^{P}}_{MN}$ is constructed in terms of the Weitzenböck connection ${\Gamma^{P}}_{MN}={e_{A}}^{P}\partial_{N}{e^{A}}_{M}$. With the torsion tensor, the contorsion tensor is defined as ${K^{P}}_{MN}=\frac{1}{2}({{T_{M}}^{P}}_{N}+{{T_{N}}^{P}}_{M}-{{T^{P}}_{M}}_{N})$. By defining the superpotential torsion tensor ${S_{P}}^{MN}=\frac{1}{2}(K^{MN}{}_{P}+\delta^{N}_{P}T_{Q}{}^{QM}-\delta^{M}_{P}T_{Q}{}^{QN})$, the torsion scalar is given by $T={S_{P}}^{MN}{T^{P}}_{MN}$.

The action of $(d+1)$-dimensional BIDG in Weitzenböck spacetime takes the form Fiorini2013

where the rank-2 tensor $F_{MN}$ is a function of vielbein field ${e^{A}}_{M}$ and its derivatives, $\lambda$ the Born-Infeld constant with mass dimension 2, and $\mathcal{L}_{\text{M}}$ the Lagrangian of matter fields. The $(d+1)$-dimensional gravitational constant $G_{d+1}$ will be set to $4\pi G_{d+1}=1$ for later convenience.

In the low-energy limit $\lambda\rightarrow\infty$, the above action approximates to

where $e=|{e^{A}}_{M}|=\sqrt{-|g_{MN}|}$. Thus, the teleparallel gravity can be recovered in the case of $\text{Tr}(F_{MN})=T$. Generally, $F_{MN}$ is given by $F_{MN}=\alpha F^{(1)}_{MN}+\beta F^{(2)}_{MN}+\gamma F^{(3)}_{MN}$ with $F^{(1)}_{MN}={S_{M}}^{PQ}T_{NPQ}$, $F^{(2)}_{MN}={S_{PM}}^{Q}{T^{P}}_{NQ}$, and $F^{(3)}_{MN}=g_{MN}T$ Fiorini2013 . The three dimensionless parameters $\alpha$, $\beta$ and $\gamma$ satisfy the condition $\alpha+\beta+(d+1)\gamma=1$ in order to yield $\text{Tr}(F_{MN})=T$.

By varying the action with respect to the vielbein, one gets the Euler-Lagrange equation

with each term written explicitly as

where $\mathcal{L}_{\text{G}}$ represents the gravitational Lagrangian and $\mathcal{U}_{KL}=g_{KL}+2\lambda^{-1}F_{KL}$. After contracting the index $A$ of tangent space via multiplying a vielbein ${e^{A}}_{N}$, the equations of motion read Fiorini2016a

where the symbol $(~{})$ denotes the symmetric tensor components. The energy-momentum tensor ${\Theta}_{N}{}^{M}={e^{A}}_{N}{\Theta_{A}}^{M}$ is symmetric and conserved if the action of the matter fields is local Lorentz invariant Li2011d . With some algebra, these two partial derivative terms are written explicitly as

where $[~{}]$ denotes the skew-symmetric tensor components, and the tensors ${D^{C}}_{PQB}{}^{KL}$ and ${Q^{K}}_{A}{}^{C}{}_{PQ}$ are expressed as

To investigate the braneworld scenario, the $(d+1)$-dimensional line element is described by

where $y$ denotes the extra dimension coordinate perpendicular to the brane, and $a(y)$ is the so-called warp factor. Correspondingly, the proper vielbein reads ${e^{A}}_{M}=\text{diag}\left(a(y),\cdots,\right.$ $\left.a(y),1\right)$.

In order to construct a braneworld configuration, a single background scalar field is included, of which the Lagrangian reads

The scalar field should depend on the coordinate $y$ only, to be consistent with the $d$-dimensional Poincaré invariance of the metric ansatz. Then, the energy-momentum tensor is written explicitly as

where the prime denotes the derivative with respect to the coordinate $y$. Since the brane is constructed by the scalar field, the brane configuration is more easily seen from its effective energy density, which is defined as Liu2012 ; Yang2012a

where $V_{0}$ is the vacuum energy of the scalar potential.

With the vielbein and matter energy-momentum tensor, the field equations (7) become

where $H\equiv a^{\prime}/a$, $V_{\phi}\equiv\frac{dV}{d\phi}$, $A=d(d-1)(\beta+2\gamma)/\lambda$, and $B=(d-1)(2\alpha+\beta+2d\gamma)/\lambda$.

It is noted that the system is underdetermined since there are only two of the three equations of motion are independent, but we have three unknown variables $a$, $\phi$, and $V(\phi)$. In order to solve the system analytically, we employ the first-order formalism DeWolfe2000 ; Gremm2000a ; Afonso2006 , which can transform the equations of motion (16) into first-order equations by introducing a superpotential $W(\phi)$, namely,

where $W_{\phi}\equiv\frac{dW}{d\phi}$.

In order to recover the massless chiral fermions on the brane, the bulk should be $Z_{2}$-symmetric along the coordinate $y$. Thus, the warp factor $a(y)$ must be an even function of extra dimension. From Eq. (17a), the superpotential $W(\phi)$ must be chosen as an odd function of the scalar $\phi$. With an appropriate ansatz of superpotential, the variables can be solved directly from the above equations.

Specifically, we focus on solving the 3-brane solutions in a 5-dimensional bulk ($d=4$) in the rest of the work. The higher-dimensional brane solutions can be solved similarly. However, Eqs. (17a) and (17b) can not be integrated out generally due to their complicated form. Therefore, it is convenient to consider some particular cases by fixing the values of free parameters, which can simplify the equations greatly.

The full perturbed metric against the domain wall backgrounds takes the form

where $h_{\mu\nu}$ is the transverse-traceless (TT) tensor mode, $G_{\mu}$ and $C_{\nu}$ are the transverse vector modes, and $\psi$, $\varphi$, $F$, and $B$ are the scalar modes. Due to the broken 5-dimensional local Lorentz invariance of current theory, the broken gauge freedom in tangent frame will release 10 extra degrees of freedom in the vielbein Li2011d . Thus, one can generally write the perturbed vielbein as the form $e^{A}{}_{M}=\bar{e}^{A}{}_{M}+\ae^{A}{}_{M}$ Wu2012a ; Izumi2013 . The degrees of freedom of the perturbed metric $g_{MN}$ are encoded in ${\bar{e}}^{A}{}_{M}$, satisfying the condition $g_{MN}=\eta_{AB}{\bar{e}}^{A}{}_{M}{\bar{e}}^{A}{}_{N}$. Then, all the 10 extra degrees of freedom are included in ${\ae^{A}}_{M}$, which can be decomposed explicitly as ${\ae^{5}}_{5}=0$, ${\ae^{a}}_{5}=\delta^{a\mu}(\partial_{\mu}\beta+D_{\mu})$, ${\ae^{5}}_{\mu}=0$, ${\ae^{a}}_{\mu}=\delta^{a\nu}\epsilon_{\mu\nu\lambda}(\partial^{\lambda}\sigma+V^{\lambda})$, with $\beta$ a scalar, $D_{\mu}$ a transverse vector, $\sigma$ a pseudo-scalar, and $V^{\lambda}$ a transverse pseudo-vector. After taking the scalar-vector-tensor decomposition in the linear perturbed equations of motion, the TT tensor, transverse (pseu-)vector, and (pseu-)scalar modes can be decoupled from each other, and they can be dealt with separately.

Here, we focus on studying the property of 4-dimensional KK gravitons in current models. This is done by considering the TT tensor perturbation against the domain wall backgrounds. So we close all the transverse (pseu-)vector and (pseu-)scalar modes in the perturbed metric (25). Now, the non-vanishing components of the perturbed inverse metric are $g^{\mu\nu}=a^{-2}\left(\eta^{\mu\nu}-2h^{\mu\nu}\right)$ and $g^{55}=1$.

Correspondingly, the perturbed vielbein with respect to the perturbed metric reads

Then, the inverse of the vielbein is given by

With the perturbed vielbein and metric, the expressions for the non-vanishing components of the perturbed torsion tensor $T^{P}{}_{MN}$ read

Then, the non-vanishing components of the perturbed contorsion tensor $K^{P}{}_{MN}$ are given by

With the expressions of the perturbed torsion tensor $T^{P}{}_{MN}$ and contorsion tensor $K^{P}{}_{MN}$, the non-vanishing components of the perturbed superpotential torsion tensor ${S_{P}}^{MN}$ read

By taking the torsion tensor $T^{P}{}_{MN}$ and superpotential torsion tensor ${S_{P}}^{MN}$ into consideration, the perturbed tensor $F_{PQ}=\alpha F^{(1)}_{PQ}+\beta F^{(2)}_{PQ}+\gamma F^{(3)}_{PQ}$ can be calculated by the non-vanishing components of $F^{(1)}_{PQ}$, $F^{(2)}_{PQ}$, and $F^{(3)}_{PQ}$, given by