# Thick branes in Born-Infeld determinantal gravity in Weitzenböck spacetime

Ke<EMAIL_ADDRESS>Hao<EMAIL_ADDRESS>Yi
<EMAIL_ADDRESS>corresponding author aSchool of Physical Science
and Technology, Southwest University, Chongqing 400715, China
bPhysics Department, Chongqing University, Chongqing 401331, China
cSchool of Physics and Electronics Science,
Hunan Provincial Key Laboratory of High-Energy Scale Physics and Applications,
Hunan University, Changsha 410082, P. R. China

###### Abstract

By adopting the idea of Born-Infeld electromagnetism, the Born-Infeld
determinantal gravity in Weitzenböck spacetime provides a way to smooth the
Big Bang singularity at the classical level. We consider a thick braneworld
scenario in the higher-dimensional extension of this gravity, and investigate
the torsion effects on the brane structure and gravitational perturbation. For
three particular parameter choices, analytic domain wall solutions are
obtained. They have a similar brane configuration that the brane thickness
becomes thinner as the spacetime torsion gets stronger. For each model, the
massless graviton is localized on the brane with the width of localization
decreasing with the enhancement of the spacetime torsion, while the massive
gravitons propagate in the bulk and contribute a correction term proportional
to ${1}/{(kr)^{3}}$ to the Newtonian potential. A sparsity constraint on the
fundamental 5-dimensional gravitational scale is estimated from the
gravitational experiment. Moreover, the parameter ranges in which the Kaluza-
Klein gravitons are tachyonic free are analyzed.

###### pacs:

04.50.+h, 04.50.Kd

##  I Introduction

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.

##  II Higher-dimensional BIDG in Weitzenböck spacetime

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

$\displaystyle S=-\frac{\lambda}{16\pi
G_{d+1}}\int{}d^{d+1}x\left[\sqrt{-|g_{MN}+{2}{\lambda}^{-1}F_{MN}|}\right.$
$\displaystyle\left.-\sqrt{-|g_{MN}|}\right]+\int{}d^{d+1}x\sqrt{-|g_{MN}|}\mathcal{L}_{\text{M}},$
(1)

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

$S\approx-\frac{1}{4}\int{}d^{d+1}x~{}e\text{Tr}(F_{MN})+\int{}d^{d+1}x~{}e\mathcal{L}_{\text{M}},$
(2)

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

$\frac{\partial\mathcal{L}_{\text{G}}}{\partial
e^{A}{}_{M}}-\partial_{S}\left(\frac{\partial\mathcal{L}_{\text{G}}}{\partial(\partial_{S}e^{A}{}_{M})}\right)=\frac{4e}{\lambda}{\Theta_{A}}^{M},$
(3)

with each term written explicitly as

$\displaystyle\frac{\partial\mathcal{L}_{\text{G}}}{\partial e^{A}{}_{M}}$
$\displaystyle=$
$\displaystyle\frac{|\mathcal{U}_{KL}|^{\frac{1}{2}}(\mathcal{U}^{-1})^{QP}}{2}\left(e_{A(P}\delta^{M}{}_{Q)}+\frac{2\partial
F_{PQ}}{\lambda\partial e^{A}{}_{M}}\right)$ (4) $\displaystyle-$
$\displaystyle{e_{A}}^{M}e,$
$\displaystyle\frac{\partial\mathcal{L}_{\text{G}}}{\partial(\partial_{S}e^{A}{}_{M})}$
$\displaystyle=$
$\displaystyle\frac{|\mathcal{U}_{KL}|^{\frac{1}{2}}(\mathcal{U}^{-1})^{QP}}{\lambda}\frac{\partial
F_{PQ}}{\partial(\partial_{S}e^{A}{}_{M})},$ (5)
$\displaystyle{\Theta_{A}}^{M}$ $\displaystyle=$
$\displaystyle\frac{1}{e}\frac{\partial(e\mathcal{L}_{\text{M}})}{\partial
e^{A}{}_{M}},$ (6)

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

$\displaystyle\frac{|\mathcal{U}_{KL}|^{\frac{1}{2}}\left({\mathcal{U}}^{-1}\right)^{QP}}{2}\left[{\delta^{M}}_{(P}g_{NQ)}+\frac{2e^{A}{}_{N}}{\lambda}\frac{\partial
F_{PQ}}{\partial{e^{A}}_{M}}\right]-{\delta^{M}}_{N}e$
$\displaystyle-\frac{e^{A}{}_{N}}{\lambda}\partial_{S}\left[|\mathcal{U}_{KL}|^{\frac{1}{2}}\left({\mathcal{U}}^{-1}\right)^{QP}\frac{\partial
F_{PQ}}{\partial(\partial_{S}{e^{A}}_{M})}\right]\\!=\\!\frac{4e}{\lambda}{\Theta_{N}}^{M},$
(7)

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

$\displaystyle\frac{\partial F_{PQ}}{\partial{e^{A}}_{M}}$ $\displaystyle=$
$\displaystyle\alpha\left({\delta^{M}}_{P}F^{(1)}_{AQ}+{\delta^{M}}_{Q}F^{(1)}_{PA}\right.$
(8) $\displaystyle+$
$\displaystyle\left.{Q^{M}}_{APKL}{T_{Q}}^{KL}-2S_{PK(A}{T_{Q}}^{KM)}\right)$
$\displaystyle+$
$\displaystyle\beta\left({Q^{M}}_{AKPL}{T^{K}}_{Q}{}^{L}-{S_{KP}}^{(M}{T^{K}}_{QA)}\right)$
$\displaystyle+$
$\displaystyle\gamma\left({\delta^{M}}_{(P}e_{AQ)}T-4g_{PQ}{F^{(2)M}}_{A}\right),$
$\displaystyle\frac{\partial F_{PQ}}{\partial(\partial_{S}{e^{A}}_{M})}$
$\displaystyle=$
$\displaystyle\alpha\left(2e_{AQ}{S_{P}}^{SM}+D_{PKLA}{}^{[SM]}{T_{Q}}^{KL}\right)$
(9) $\displaystyle+$
$\displaystyle\beta\left({S_{AP}}^{[M}{\delta^{S]}}_{Q}+D_{KPLA}{}^{[SM]}{T^{K}}_{Q}{}^{L}\right)$
$\displaystyle+$ $\displaystyle 4\gamma g_{PQ}{S_{A}}^{SM},$

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

$\displaystyle{D^{C}}_{PQB}{}^{KL}$
$\displaystyle=\frac{1}{4}\left({\delta_{P}}^{K}{\delta_{Q}}^{L}{\delta_{B}}^{C}-e^{CL}e_{B[P}\delta_{Q]}{}^{K}\right)$
$\displaystyle+\frac{1}{2}{e_{B}}^{L}{e^{C}}_{[P}\delta_{Q]}^{K},$ (10)
$\displaystyle{Q^{K}}_{A}{}^{C}{}_{PQ}$
$\displaystyle=\frac{1}{4}\left(e^{CK}T_{[PQ]A}-{\delta^{K}}_{[P}{T_{AQ]}}^{C}\right)$
$\displaystyle-\frac{1}{2}\left({\delta_{A}}^{C}{\delta^{K}}_{[P}{T^{L}}_{LQ]}-{e^{C}}_{[P}{T^{K}}_{AQ]}\right).$
(11)

##  III Thick brane scenario and first-order formalism

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

$ds^{2}=a^{2}(y)\eta_{\mu\nu}dx^{\mu}dx^{\nu}+dy^{2},$ (12)

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

$\mathcal{L}_{\text{M}}=-\frac{1}{2}\partial^{M}\phi\partial_{M}\phi-V(\phi).$
(13)

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

$\displaystyle Q^{\mu}{}_{\nu}$ $\displaystyle=$
$\displaystyle\left(-\frac{1}{2}\phi^{\prime 2}-V\right)\delta^{\mu}{}_{\nu},$
$\displaystyle Q^{y}{}_{y}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\phi^{\prime 2}-V,$ (14)

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

$\rho(y)=-Q^{0}{}_{0}-V_{0},$ (15)

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

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

$\displaystyle\frac{\left(1-BH^{2}\right)^{\frac{d-2}{2}}}{(1-AH^{2})^{\frac{1}{2}}}\left[1+(d-1)BH^{2}-dABH^{4}\right]-1$
$\displaystyle=\frac{4}{\lambda}\left(\frac{\phi^{\prime
2}}{2}-V\right),\qquad$ (16a)
$\displaystyle\frac{\left(1-BH^{2}\right)^{\frac{d-4}{2}}}{\left(1-AH^{2}\right)^{\frac{3}{2}}}\bigg{[}1+\frac{A+dB}{d}H^{\prime}-(A-(d-2)B)H^{2}$
$\displaystyle-\frac{2(2d+1)A+d(d-1)B}{d}BH^{2}H^{\prime}-(d+1)A^{2}B^{2}H^{6}H^{\prime}$
$\displaystyle-(2(d-1)A+(d-1)B)BH^{4}+(dA+(2d-1)B)ABH^{6}$
$\displaystyle+\frac{(2d^{2}+1)B+3dA}{d}ABH^{4}H^{\prime}-dA^{2}B^{2}H^{8}\bigg{]}-1$
$\displaystyle=\frac{4}{\lambda}\left(-\frac{\phi^{\prime
2}}{2}-V\right),\quad$ (16b)
$\displaystyle\phi^{\prime\prime}+4H\phi^{\prime}=V_{\phi},\qquad\qquad\qquad~{}$
(16c)

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,

$\displaystyle H\\!$ $\displaystyle\\!=\\!$ $\displaystyle\\!-\frac{W}{3},$
(17a) $\displaystyle\phi^{\prime}\\!$ $\displaystyle\\!=\\!$
$\displaystyle\\!\frac{\lambda}{4d}\frac{(9-BW^{2})^{\frac{d-4}{2}}}{3^{d}(9-AW^{2})^{\frac{3}{2}}}\Big{[}729(A+dB)-81((4d+2)A$
(17b) $\displaystyle+$ $\displaystyle(d-1)dB)BW^{2}+9(3Ad+2Bd^{2}+B)ABW^{4}$
$\displaystyle-$ $\displaystyle d(d+1)A^{2}B^{2}W^{6}\Big{]}W_{\phi},$
$\displaystyle V\\!$ $\displaystyle\\!=\\!$
$\displaystyle\\!\frac{\lambda}{4}-\frac{\lambda}{3^{d+1}}\bigg{[}\frac{\left(9-BW^{2}\right)^{\frac{d-2}{2}}}{4(9-AW^{2})^{\frac{1}{2}}}\big{[}81+9(d-1)BW^{2}$
(17c) $\displaystyle-$ $\displaystyle
dABW^{4}\big{]}+\frac{(9-BW^{2})^{\frac{d-4}{2}}}{8d(9-AW^{2})^{\frac{3}{2}}}\Big{[}2187(A+dB)$
$\displaystyle-$ $\displaystyle 243\big{[}(4d+2)A+d(d-1)B\big{]}BW^{2}$
$\displaystyle+$ $\displaystyle 27\big{[}3dA+(2d^{2}+1)B\big{]}ABW^{4}$
$\displaystyle-$ $\displaystyle
3d(d+1)A^{2}B^{2}W^{6}\Big{]}\phi^{\prime}W_{\phi}\bigg{]}.$

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.

##  IV Analytic 3-brane solutions

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.

###  IV.1 Case $A=B$

The first interesting case is $A=B$. Then from $A=12(\beta+2\gamma)/\lambda$,
$B=3(2\alpha+\beta+8\gamma)/\lambda$, and $\alpha+\beta+5\gamma=1$, the
parameters are fixed as $\alpha=\frac{3}{5}-3\gamma$ and
$\beta=\frac{2}{5}-2\gamma$, yet $\gamma$ left as a free parameter. It leads
to $A=B=\frac{24}{5\lambda}$. So the Eqs. (17) reduce to

$\displaystyle a^{\prime}\\!$ $\displaystyle\\!=\\!$
$\displaystyle\\!-\frac{aW}{3},$ (18a) $\displaystyle\phi^{\prime}\\!$
$\displaystyle\\!=\\!$
$\displaystyle\\!\left(\frac{1}{2}-\frac{16W^{2}}{15\lambda}\right)\left(1-\frac{8W^{2}}{15\lambda}\right)^{\frac{1}{2}}W_{\phi},$
(18b) $\displaystyle V\\!$ $\displaystyle\\!=\\!$
$\displaystyle\\!\frac{\lambda}{4}-\left(\frac{\lambda}{4}+\frac{2W^{2}}{5}-\frac{64W^{4}}{225\lambda}-\frac{\phi^{\prime}W_{\phi}}{4}+\frac{8\phi^{\prime}W_{\phi}W^{2}}{15\lambda}\right)$
(18c) $\displaystyle\\!\times\\!$
$\displaystyle\\!\left(1-\frac{8W^{2}}{15\lambda}\right)^{\frac{1}{2}}.$

(a) $~{}a(y)$

(b) $~{}\phi(y)$

(c) $~{}V(\phi)$

(d) $~{}\rho(y)$

Figure 1: The profiles of the warp factor $a(y)$, scalar field $\phi(y)$,
scalar potential $V(\phi)$, and energy density $\rho(y)$ for the case of
$A=B$.

A set of analytical solutions can be obtained with the superpotential ansatz
$W(\phi)=\frac{2k}{\sqrt{3}}\phi$, where the mass dimension one parameter $k$
is defined as $\lambda\equiv\frac{32}{45}k^{2}$. Then taking $W(\phi)$ into
Eqs. (18), the set of analytical solutions are obtained as

$\displaystyle a(y)\\!$ $\displaystyle\\!=\\!$
$\displaystyle\\!\frac{\text{sech}^{\frac{1}{3\sqrt{3}}}(ky)}{\left[\big{(}2-\sqrt{3}\big{)}\left(2+\sqrt{4-\text{sech}^{2}(ky)}\right)\right]^{\frac{1}{3\sqrt{3}}}},$
(19a) $\displaystyle\phi(y)\\!$ $\displaystyle\\!=\\!$
$\displaystyle\\!\frac{\tanh(ky)}{\sqrt{3+\tanh^{2}(ky)}},$ (19b)
$\displaystyle V(\phi)\\!$ $\displaystyle\\!=\\!$
$\displaystyle\\!\frac{k^{2}}{90}\Big{[}\phi^{2}\left(4\phi^{2}-3\right)\left(45-60\phi^{2}+16\sqrt{1-\phi^{2}}\right)$
(19c) $\displaystyle-$ $\displaystyle 16\sqrt{1-\phi^{2}}+31\Big{]},$

where the integration constants have been chosen such that $a(0)=1$ and
$\phi(0)=0$. The solutions of the warp factor $a(y)$, scalar field $\phi(y)$,
and scalar potential $V(\phi)$ are illustrated in Figs. 1(a), 1(b), and 1(c).
As shown in Fig. 1(a), the warp factor holds the $Z_{2}$ symmetry and has a
typical bell shape profile. As $y\to\pm\infty$, the scalar field
$\phi(\pm\infty)\to\pm\frac{1}{2}$, which just correspond to the two local
minima of the scalar potential, i.e.,
$V_{0}=V(\pm\frac{1}{2})=(8-6\sqrt{3})k^{2}/45$. Since the scalar field non-
trivially maps the boundaries of extra dimension into two scalar vacua, it is
a kink soliton Dzhunushaliev2010 .

With the solution (19a), the torsion scalar reads
$T=-\frac{16k^{2}\tanh^{2}(ky)}{9\left(4-\text{sech}^{2}(ky)\right)}$. It
approaches $-\frac{4k^{2}}{9}$ at the vacua $y\to\pm\infty$. Since the
absolute value of the torsion scalar is monotonically increasing with the
parameter $k$ (or $\lambda$), the torsion of spacetime is enhanced as the
parameter $k$ goes larger. As shown in Fig. 1(d), the energy density of the
brane is localized at the origin of the extra dimension, and the brane
thickness becomes thinner as the background torsion gets stronger.

By choosing the free parameter $\gamma=1/5$, one has $\alpha=\beta=0$. Then,
the action (1) reduces to a Born-Infeld-$f(T)$-type one Fiorini2013 ; Yang2018
, i.e.,

$\displaystyle S_{\text{BI}}$ $\displaystyle=$
$\displaystyle-\frac{1}{4}\int{}d^{5}x~{}ef(T)$ (20) $\displaystyle=$
$\displaystyle-\frac{\lambda}{4}\int{}d^{5}x~{}e\left[\left(1+\frac{2T}{5\lambda}\right)^{5/2}-1\right].$

In addition, the same analytic solutions (19) were obtained and the issue
about trapping fermions on the domain wall was considered in Ref. Yang2018 .

###  IV.2 Case $A=0$

(a) $~{}a(y)$

(b) $~{}\phi(y)$

(c) $~{}V(\phi)$

(d) $~{}\rho(y)$

Figure 2: The profiles of the warp factor $a(y)$, scalar field $\phi(y)$,
scalar potential $V(\phi)$, and energy density $\rho(y)$ for the case of
$A=0$.

The second interesting case is $A=0$. Then from $A=12(\beta+2\gamma)/\lambda$,
$B=3(2\alpha+\beta+8\gamma)/\lambda$, and $\alpha+\beta+5\gamma=1$, one gets
$B={6}/{\lambda}$, $\alpha=1-3\gamma$, and $\beta=-2\gamma$, with $\gamma$ a
free parameter. Now the equations (17) reduce to

$\displaystyle a^{\prime}$ $\displaystyle=$ $\displaystyle-\frac{aW}{3},$
(21a) $\displaystyle\phi^{\prime}$ $\displaystyle=$
$\displaystyle\left(\frac{1}{2}-\frac{W^{2}}{\lambda}\right)W_{\phi},$ (21b)
$\displaystyle V$ $\displaystyle=$
$\displaystyle\left(\frac{1}{4}-\frac{W^{2}}{2\lambda}\right)\phi^{\prime}W_{\phi}-\left(\frac{1}{3}-\frac{W^{2}}{3\lambda}\right)W^{2}.$
(21c)

For the superpotential ansatz $W(\phi)=2k\phi$ with $\lambda\equiv 8k^{2}$, a
set of analytical solutions are obtained

$\displaystyle a(y)$ $\displaystyle=$
$\displaystyle\text{sech}^{\frac{2}{3}}(ky),$ (22a) $\displaystyle\phi(y)$
$\displaystyle=$ $\displaystyle\tanh(ky),$ (22b) $\displaystyle V(\phi)$
$\displaystyle=$
$\displaystyle\frac{k^{2}}{6}\left(3-32\phi^{2}+64\phi^{4}\right),$ (22c)

where the integration constants have been chosen such that $a(0)=1$ and
$\phi(0)=0$. The solutions of $a(y)$, $\phi(y)$, and $V(\phi)$ are illustrated
in Figs. 2(a), 2(b), and 2(c), respectively. The warp factor is $Z_{2}$
symmetric, and the scalar field is a kink with $\phi(\pm\infty)\to\pm 1$,
which connects two local minima of the scalar potential, i.e., $V_{0}=V(\pm
1)=-{k^{2}}/{96}$.

In this case, the torsion scalar reads $T=-\frac{16k^{2}}{3}\tanh^{2}(ky)$,
and it approaches $-\frac{16k^{2}}{3}$ at the vacua $y\to\pm\infty$.
Therefore, the spacetime torsion is also enhanced as the parameter $k$ goes
larger. As shown in Fig. 2(d), the energy density of the brane is localized at
$y=0$, and its thickness becomes thinner as the spacetime torsion gets
stronger as well.

###  IV.3 Case $B=0$

(a) $~{}a(y)$

(b) $~{}\phi(y)$

(c) $~{}V(\phi)$

(d) $~{}\rho(y)$

Figure 3: The profiles of the warp factor $a(y)$, scalar field $\phi(y)$,
scalar potential $V(\phi)$, and energy density $\rho(y)$ for the case of
$B=0$.

The last interesting case is $B=0$. Then from $A=12(\beta+2\gamma)/\lambda$,
$B=3(2\alpha+\beta+8\gamma)/\lambda$, and $\alpha+\beta+5\gamma=1$, one has
$A={24}/{\lambda}$, $\alpha=-(1+3\gamma)$, and $\beta=2(1-\gamma)$, with
$\gamma$ a free parameter. In this case, the equations (17) become

$\displaystyle a^{\prime}$ $\displaystyle=$ $\displaystyle-\frac{aW}{3},$
(23a) $\displaystyle\phi^{\prime}$ $\displaystyle=$
$\displaystyle\frac{W_{\phi}}{2}\left(1-\frac{8W^{2}}{3\lambda}\right)^{-\frac{3}{2}},$
(23b) $\displaystyle V$ $\displaystyle=$
$\displaystyle\frac{1}{4}\Bigg{[}\phi^{\prime}W_{\phi}-\left(\lambda-\frac{8W^{2}}{3}\right)\left(1-\sqrt{1-\frac{8W^{2}}{3\lambda}}\right)\Bigg{]}$
(23c) $\displaystyle\times$
$\displaystyle\left(1-\frac{8W^{2}}{3\lambda}\right)^{-\frac{3}{2}}.$

By assuming the superpotential as the form
$W(\phi)=\frac{3k\sin\phi}{\sqrt{1+\sin^{2}\phi}}$, a set of analytical
solutions can be achieved as

$\displaystyle a(y)\\!$ $\displaystyle=$
$\displaystyle\\!\frac{\left(1+\sqrt{2}\right)^{\frac{\sqrt{2}}{3}}}{\left({\cosh^{\frac{1}{2}}(3ky)+\sqrt{2}\cosh\left(3ky/2\right)}\right)^{\frac{\sqrt{2}}{3}}},$
(24a) $\displaystyle\phi(y)\\!$ $\displaystyle=$
$\displaystyle\\!\arcsin\left[\tanh\left(3ky/2\right)\right],$ (24b)
$\displaystyle V(\phi)\\!$ $\displaystyle=$
$\displaystyle\\!\frac{3k^{2}}{16}\left[35+3\cos(2\phi)-32\left(1+\sin^{2}\phi\right)^{\frac{1}{2}}\right],\qquad$
(24c)

where the integration constants have been chosen such that $a(0)=1$ and
$\phi(0)=0$ as well. As shown in Fig. 3, the warp factor $a(y)$, scalar field
$\phi(y)$, scalar potential $V(\phi)$, and brane energy density $\rho(y)$ have
similar shapes with the previous two cases. The scalar field exhibits a kink
profile, whose two ends $\phi(\pm\infty)\to\pm\pi/2$ connect two vacua of the
scalar potential, i.e., $V_{0}=V(\pm\pi/2)=6\left(1-\sqrt{2}\right)k^{2}$.

Now the torsion scalar reads $T=-6k^{2}[1-\text{sech}(3ky)]$, and
$T\to-6k^{2}$ as $y\to\pm\infty$. The bulk torsion also gets stronger as the
parameter $k$ goes larger. As shown in Fig. 3(d), the energy density of the
brane is localized at the origin, and its thickness becomes thinner as the
torsion becomes stronger as the previous cases.

##  V Gravitational perturbation

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

$\displaystyle ds^{2}$ $\displaystyle=$ $\displaystyle g_{MN}dx^{M}dx^{N}$
(25) $\displaystyle=$ $\displaystyle
a^{2}(y)\big{[}\eta_{\mu\nu}+2h_{\mu\nu}+2\eta_{\mu\nu}\varphi+2\partial_{\mu}\partial_{\nu}B$
$\displaystyle+$ $\displaystyle
2\partial_{(\mu}C_{\nu)}\big{]}dx^{\mu}dx^{\nu}+a(y)(\partial_{\mu}F+G_{\mu})dx^{\mu}dy$
$\displaystyle+$ $\displaystyle(1+2\psi)dy^{2},$

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

${e^{A}}_{M}=\left({\begin{array}[]{*{20}{c}}a(y)\left(\delta^{a}{}_{\mu}+\eta^{a\nu}{h_{\nu\mu}}\right)&0\\\
0&1\end{array}}\right).$ (26)

Then, the inverse of the vielbein is given by

${e_{A}}^{M}=\left({\begin{array}[]{*{20}{c}}a^{-1}(y)\left(\delta_{a}{}^{\mu}-\eta_{a\nu}{h^{\nu\mu}}\right)&0\\\
0&1\end{array}}\right).$ (27)

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

$\displaystyle{T^{\lambda}}_{5\nu}$ $\displaystyle=$
$\displaystyle-{T^{\lambda}}_{\nu
5}=H{\delta^{\lambda}}_{\nu}+{h^{\prime}}^{\lambda}{}_{\nu},$ (28a)
$\displaystyle{T^{\lambda}}_{\mu\nu}$ $\displaystyle=$
$\displaystyle\partial_{[\mu}h^{\lambda}{}_{\nu]}.$ (28b)

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

$\displaystyle{K^{\lambda}}_{5\nu}$ $\displaystyle=$
$\displaystyle-H{\delta^{\lambda}}_{\nu}-{h^{\prime}}^{\lambda}{}_{\nu},$
(29a) $\displaystyle{K^{5}}_{\mu\nu}$ $\displaystyle=$ $\displaystyle
a^{2}\left(H\eta_{\mu\nu}+2Hh_{\mu\nu}+{h^{\prime}}_{\mu\nu}\right),$ (29b)
$\displaystyle{K^{\lambda}}_{\mu\nu}$ $\displaystyle=$
$\displaystyle\partial^{\lambda}h_{\mu\nu}-\partial_{\mu}{h^{\lambda}}_{\nu}.$
(29c)

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

$\displaystyle{S_{\lambda}}^{\mu 0}$ $\displaystyle=$
$\displaystyle-{S_{\lambda}}^{0\mu}=\frac{1}{2}\left[(d-1)H\delta^{\mu}{}_{\lambda}-{h^{\prime}}^{\mu}{}_{\lambda}\right],$
(30a) $\displaystyle{S_{\lambda}}^{\mu\nu}$ $\displaystyle=$
$\displaystyle\frac{1}{2a^{2}}\partial^{[\mu}h^{\nu]}{}_{\lambda}.$ (30b)

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

$\displaystyle F^{(1)}_{\mu\nu}$ $\displaystyle=$
$\displaystyle(1-d)a^{2}H^{2}\left[\eta_{\mu\nu}+2h_{\mu\nu}+\frac{d-2}{d-1}\frac{h^{\prime}_{\mu\nu}}{H}\right],\qquad$
(31a) $\displaystyle F^{(2)}_{\mu\nu}$ $\displaystyle=$
$\displaystyle\frac{1-d}{2}a^{2}H^{2}\left[\eta_{\mu\nu}+2h_{\mu\nu}+\frac{d-2}{d-1}\frac{h^{\prime}_{\mu\nu}}{H}\right],$
(31b) $\displaystyle F^{(2)}_{55}$ $\displaystyle=$
$\displaystyle\frac{d(1-d)}{2}H^{2},$ (31c) $\displaystyle F^{(3)}_{\mu\nu}$
$\displaystyle=$ $\displaystyle
d(1-d)a^{2}H^{2}\left(\eta_{\mu\nu}+2h_{\mu\nu}\right),$ (31d) $\displaystyle
F^{(3)}_{55}$ $\displaystyle=$ $\displaystyle T=d(1-d)H^{2}.$ (31e)

Further, the perturbations of $\frac{\partial F_{PQ}}{\partial{e^{A}}_{M}}$
and $\frac{\partial F_{PQ}}{\partial(\partial_{S}{e^{A}}_{M})}$ can be
assembled by $\frac{\partial F_{PQ}}{\partial{e^{A}}_{M}}=\alpha\frac{\partial
F^{(1)}_{PQ}}{\partial{e^{A}}_{M}}+\beta\frac{\partial
F^{(2)}_{PQ}}{\partial{e^{A}}_{M}}+\gamma\frac{\partial
F^{(3)}_{PQ}}{\partial{e^{A}}_{M}}$ and $\frac{\partial
F_{PQ}}{\partial(\partial_{S}{e^{A}}_{M})}=\alpha\frac{\partial
F^{(1)}_{PQ}}{\partial(\partial_{S}{e^{A}}_{M})}+\beta\frac{\partial
F^{(2)}_{PQ}}{\partial(\partial_{S}{e^{A}}_{M})}+\gamma\frac{\partial
F^{(3)}_{PQ}}{\partial(\partial_{S}{e^{A}}_{M})}$. Because there are too many
items, the non-vanishing components of $\frac{\partial
F^{(i)}_{PQ}}{\partial{e^{A}}_{M}}$ and $\frac{\partial
F^{(i)}_{PQ}}{\partial(\partial_{S}{e^{A}}_{M})}$ are listed in A.

With all of these perturbed terms, the linear perturbation of the field
equations (7) is obtained finally

$F_{2}{h^{\prime\prime}}_{\mu\nu}+F_{1}{h^{\prime}}_{\mu\nu}+F_{0}\Box^{(4)}h_{\mu\nu}=0,$
(32)

where $\Box^{(4)}=\eta^{\rho\sigma}\partial_{\rho}\partial_{\sigma}$ is the
4-dimensional d’Alembert operator, and the functions $F_{2}$, $F_{1}$ and
$F_{0}$ are defined as

$\displaystyle F_{2}\\!$ $\displaystyle\\!=\\!$
$\displaystyle\\!\frac{A}{d\left(1-AH^{2}\right)}+\frac{B}{1-BH^{2}}-\frac{(d-1)C^{2}H^{2}}{2\left(1-BH^{2}\right)^{2}},$
(33a) $\displaystyle F_{1}\\!$ $\displaystyle\\!=\\!$
$\displaystyle\\!\frac{AH}{1-AH^{2}}-\frac{(d-1)\left[2-(2B-dC)H^{2}\right]CH}{2\left(1-BH^{2}\right)^{2}}$
(33b) $\displaystyle\\!+\\!$
$\displaystyle\\!\frac{\left[dB+(d-1)C\right]H}{1-BH^{2}}-\frac{2ABHH^{\prime}}{\left(1-AH^{2}\right)\left(1-BH^{2}\right)}$
$\displaystyle\\!+\\!$
$\displaystyle\\!\frac{A^{2}HH^{\prime}}{d\left(1-AH^{2}\right)^{2}}-\frac{\left[(d-2)B^{2}+(d-1)C^{2}\right]HH^{\prime}}{\left(1-BH^{2}\right)^{2}}$
$\displaystyle\\!+\\!$
$\displaystyle\\!\left[\frac{AC^{2}}{2\left(1-AH^{2}\right)}+\frac{(d-4)BC^{2}}{2\left(1-BH^{2}\right)}\right]\frac{(d-1)H^{3}H^{\prime}}{\left(1-BH^{2}\right)^{2}},\qquad$
$\displaystyle F_{0}\\!$ $\displaystyle\\!=\\!$
$\displaystyle\\!\frac{2(d-1)}{\lambda
a^{2}}\left(\frac{\gamma}{1-AH^{2}}+\frac{1-\gamma}{1-BH^{2}}\right),$ (33c)

with $C=(d-2)(2\alpha+\beta)/\lambda$.

In the low energy regime $\lambda\to\infty$, the leading orders of the
coefficient functions $F_{2}$, $F_{1}$ and $F_{0}$ read
$F_{2}\to{2(d-1)}{\lambda}^{-1}$, $F_{1}\to{2(d-1)dH}{\lambda}^{-1}$, and
$F_{0}\to{2(d-1)}{\lambda}^{-1}a^{-2}$, respectively. Then, the evolution
equation (32) reduces to the standard form in GR, i.e.,

${h^{\prime\prime}}_{\mu\nu}-4H{h^{\prime}}_{\mu\nu}+a^{-2}\Box^{(4)}h_{\mu\nu}=0.$
(34)

If $F_{2}/F_{0}>0$, we can make a coordinate transformation
$dz=dy/\sqrt{F_{2}/F_{0}}$ to eliminate the prefactors $F_{2}$ and $F_{0}$ in
Eq. (32). Then, the evolution equation (32) is recast as

$\ddot{h}_{\mu\nu}+F_{3}\dot{h}_{\mu\nu}+\Box^{(4)}h_{\mu\nu}=0,$ (35)

where the dot denotes the derivative with respect to the coordinate $z$, and
$F_{3}=\frac{F_{1}}{\sqrt{F_{0}F_{2}}}+\frac{1}{2}\left(\frac{\dot{F}_{0}}{F_{0}}-\frac{\dot{F}_{2}}{F_{2}}\right)$.

In order to get a Schrödinger-like equation, we employ a KK decomposition as
the from
$h_{\mu\nu}=\epsilon_{\mu\nu}(x^{\mu})\Phi(z)e^{-\int{\frac{F_{3}}{2}dz}}$.
Then, the evolution equation (35) can be decompose into two equations. The
first one is a Klein-Gordon equation $(\Box^{(4)}-m^{2})\epsilon_{\mu\nu}=0$
owing to the preserved 4-dimensional Poincaré invariance, where $m$ is the
observed 4-dimensional effective mass of KK gravitons. The other is the aimed
Schrödinger-like equation

$-\ddot{\Phi}(z)+U(z)\Phi(z)=m^{2}\Phi(z),$ (36)

where the effective potential $U(z)$ is given by

$U(z)=\frac{\dot{F}_{3}}{2}+\frac{F_{3}^{2}}{4}.$ (37)

The Hamiltonian can be further factorized as a supersymmetric quantum
mechanics form, i.e., $\mathcal{H}=\mathcal{A}^{\dagger}\mathcal{A}=$
$\left(\partial_{z}+\frac{F_{3}}{2}\right)\left(-\partial_{z}+\frac{F_{3}}{2}\right)$.
With the Neumann boundary condition $\partial_{z}h_{\mu\nu}=0$, the self-
adjoint Hamiltonian leads to non-negative eigenvalues $m^{2}\geq 0$ Yang2017 .
Therefore, the KK gravitons are tachyonic free. The wave function of the
massless graviton can be directly derived from the equation
$\left(-\partial_{z}+\frac{F_{3}}{2}\right)\Phi_{0}(z)=0$, yielding

$\Phi_{0}(z)=N_{0}e^{\int{\frac{F_{3}}{2}dz}},$ (38)

where $N_{0}$ is the normalization constant.

###  V.1 Case $A=B$

(a) $~{}U(z)$

(b) $~{}\Phi_{0}(z)$

Figure 4: The profiles of the effective potential $U(z)$ and gravitational
zero mode $\Phi_{0}(z)$ for the case of $A=B$.

In this case, the parameters are $A=B=\frac{24}{5\lambda}$ and
$C=\frac{16-80\gamma}{5\lambda}$, with $\gamma$ a free parameter. It is easily
verified that

$\frac{F_{2}}{F_{0}}=\frac{a^{2}}{1-\phi^{2}}\left[1-\left(\frac{40\gamma^{2}}{3}-\frac{16\gamma}{3}+\frac{23}{15}\right)\phi^{2}\right]\geq
a^{2}.$ (39)

So the coordinate transformation $dz=dy/\sqrt{F_{2}/F_{0}}$ is robust.
Following the above procedure, the KK gravitons are tachyonic free. This
conclusion can also be seen directly by recasting Eq. (32) into the form,
${h^{\prime\prime}}_{\mu\nu}+\frac{F_{1}}{F_{2}}{h^{\prime}}_{\mu\nu}+\mathcal{M}^{2}h_{\mu\nu}=0$,
with $\mathcal{M}^{2}\equiv m^{2}{F_{0}}/{F_{2}}$ acting as an effective
squared mass Fukushima2019 .

Due to the free parameter $\gamma$, the expressions can be simplified a lot by
fixing $\gamma=1/5$, which corresponds to $C=0$. In this case,
$F_{2}/F_{0}=a^{2}$. With the domain wall solutions (19), the functions
$F_{2}$, $F_{1}$ and $F_{0}$ in Eqs. (33) are given explicitly as

$\displaystyle F_{2}\\!\\!$ $\displaystyle\\!=\\!$
$\displaystyle\\!\\!\frac{45}{16k^{2}}\left(3+\tanh^{2}(ky)\right),$ (40)
$\displaystyle F_{1}\\!\\!$ $\displaystyle\\!=\\!$
$\displaystyle\\!\\!-\frac{5}{16k}\left[27\text{sech}^{2}(ky)+8\sqrt{9+3\tanh^{2}(ky)}\right]$
(41) $\displaystyle\times$ $\displaystyle\tanh(ky),$ $\displaystyle
F_{0}\\!\\!$ $\displaystyle\\!=\\!$
$\displaystyle\\!\\!\frac{45\left(3+\tanh^{2}(ky)\right)}{16k^{2}}\left[\left(2-\sqrt{3}\right)\cosh(ky)\right.$
(42)
$\displaystyle\times\left.\left(2+\sqrt{4-\text{sech}^{2}(ky)}\right)\right]^{\frac{2}{3\sqrt{3}}}.$

Since the coordinate transformation $dz=dy/a(y)$ cannot be integrated out
analytically in this case, we cannot obtain the analytical expression of the
effective potential in $z$ coordinate. However, the effective potential in $y$
coordinate can be expressed explicitly as

$\displaystyle U(y)\\!$ $\displaystyle\\!=\\!$
$\displaystyle\\!\frac{3a^{2}}{4\left(4k^{2}-27H^{2}\right)^{2}}\Big{[}5H^{2}\left(16k^{4}+729H^{4}\right)$
(43) $\displaystyle-$ $\displaystyle
270H^{4}\left(4k^{2}-27H^{\prime}\right)+8k^{2}\left(4k^{2}-27H^{\prime}\right)H^{\prime}$
$\displaystyle\\!-\\!$
$\displaystyle\\!81H^{2}H^{\prime}\left(16k^{2}-9H^{\prime}\right)-54\left(4k^{2}-27H^{2}\right)HH^{\prime\prime}\Big{]}$
$\displaystyle=$
$\displaystyle-\frac{k^{2}\text{sech}^{8}(ky)}{72\sqrt{4-\text{sech}^{2}(ky)}\left(3+\tanh^{2}(ky)\right)^{2}}$
$\displaystyle\times$
$\displaystyle\frac{\cosh(ky)^{-\frac{2}{3\sqrt{3}}}\left(2+\sqrt{4-\text{sech}^{2}(ky)}\right)^{-\frac{2}{3\sqrt{3}}}}{\left(2-\sqrt{3}\right)^{\frac{2}{3\sqrt{3}}}\left(2+\sqrt{4-\text{sech}^{2}(ky)}\right)^{2}}$
$\displaystyle\times$
$\displaystyle\Bigg{[}4\left(1+2\cosh(2ky)\right)\Big{[}5\left(28+15\sqrt{3}\right)\cosh(2ky)$
$\displaystyle-$
$\displaystyle\left(113+6\sqrt{3}\right)\cosh(4ky)-5\cosh(6ky)+302$
$\displaystyle+$ $\displaystyle
57\sqrt{3}\Big{]}+\sqrt{4-\text{sech}^{2}(ky)}\Big{[}2\left(593+150\sqrt{3}\right)$
$\displaystyle+$ $\displaystyle
2\left(701+222\sqrt{3}\right)\cosh(2ky)-10\cosh(8ky)$ $\displaystyle-$
$\displaystyle
3\left(23-44\sqrt{3}\right)\cosh(4ky)-\left(12\sqrt{3}+241\right)$
$\displaystyle\times$ $\displaystyle\cosh(6ky)\Big{]}\Bigg{]}.$

Correspondingly, the wave function of the massless graviton in $y$ coordinate
is given by

$\displaystyle\\!\\!\\!\\!\Phi_{0}(y)$ $\displaystyle=$ $\displaystyle
N_{0}a^{\frac{3}{2}}\left(1-\frac{27H^{2}}{4k^{2}}\right)^{3/4}$ (44)
$\displaystyle=$
$\displaystyle\frac{N_{0}3^{3/4}\cosh^{\frac{3}{2}}(ky)}{\left(2-\sqrt{3}\right)^{\frac{1}{2\sqrt{3}}}\left(1+2\cosh(2ky)\right)^{3/4}}$
$\displaystyle\times$
$\displaystyle\left(2\cosh(ky)+\sqrt{1+2\cosh(2ky)}\right)^{-\frac{1}{2\sqrt{3}}}.$

By numerically integrating out the coordinate transformation, the effective
potential $U(z)$ and the wave function $\Phi_{0}(z)$ of the massless graviton
in $z$ coordinate are illustrated in Fig. 4. As shown in Fig. 4(b), the
massless graviton is normalizable and its width decreases with the enhancement
of the spacetime torsion. With the normalization condition
$\int^{\infty}_{-\infty}{\Phi_{0}^{2}dz}=\int^{\infty}_{-\infty}{\frac{\Phi_{0}^{2}dy}{\sqrt{F_{2}/F_{0}}}}=1$,
the normalization constant can be approximately calculated, yielding
$N_{0}\approx\sqrt{\frac{k}{4.681}}$. The normalizable massless mode ensures
that the effective 4-dimensional gravity can be recovered on the brane.
Further, by counting the contribution of the massless mode sector in the
action (1), the 4-dimensional gravitational constant can be derived from the
reduction
$\frac{1}{G_{4}}=\frac{1}{G_{5}N_{0}^{2}}\int^{\infty}_{-\infty}{\Phi_{0}^{2}dz}$.
So the fundamental 5-dimensional gravitational constant $G_{5}$ reads
$G_{5}={G_{4}}/N_{0}^{2}\approx 4.681{G_{4}}/{k}$.

As shown in Fig. 4(a), the effective potential is volcano-like and the width
of potential well decreases with the enhancement of the spacetime torsion.
Therefore, the massless graviton is the only bound state, and the massive
excited states possess a continuous mass spectrum. The massive states will
contribute a correction to the Newtonian potential at short distance.
Quantitatively, for two point-like sources separated by a distance $r$ on the
brane, a volcano-like potential with $U(z)\sim\alpha(\alpha+1)/z^{2}$ for
$z\gg 1$ will yield a correction proportional to ${1}/{(kr)^{2\alpha}}$ to the
Newtonian potential Csaki2000 ; Bazeia2009 . For the current case, the
effective potential behaves like $U(z)\sim\frac{15}{4z^{2}}$ for $z\gg 1$, so
it leads to a correction proportional to ${1}/{(kr)^{3}}$ to the Newtonian
potential.

###  V.2 Case $A=0$

For case $A=0$, the parameters are $B=\frac{6}{\lambda}$ and
$C=\frac{4-16\gamma}{\lambda}$, with $\gamma$ a free parameter. It is easily
shown that

$\frac{F_{2}}{F_{0}}=\frac{9-\left(32\gamma^{2}-16\gamma+5\right)\phi^{2}}{\left(3-\phi^{2}\right)\left(3-\gamma\phi^{2}\right)}a^{2}.$
(45)

Due to $-1\leq\phi(y)\leq 1$, the ratio $F_{2}/F_{0}$ is positive everywhere
in the intervals $\frac{1-\sqrt{3}}{4}<\gamma<\frac{1+\sqrt{3}}{4}$ or
$\gamma>3$. So the coordinate transformation $dz=dy/\sqrt{F_{2}/F_{0}}$ is
robust and the KK gravitons are tachyonic free in these parameter intervals.

The corresponding expressions can be simplified a lot by setting $C=0$, which
is realized by fixing the free parameter $\gamma=1/4$. In this case,
$F_{2}/F_{0}=\frac{12a^{2}}{12-\phi^{2}}$. With the solutions (22), the
functions $F_{2}$, $F_{1}$ and $F_{0}$ in Eqs. (33) are written explicitly as

$\displaystyle F_{2}$ $\displaystyle=$
$\displaystyle\frac{9}{4k^{2}\left(3-\tanh^{2}(ky)\right)},$ (46)
$\displaystyle F_{1}$ $\displaystyle=$
$\displaystyle-\frac{3(11+4\cosh(2ky))\sinh(2ky)}{4k(2+\cosh(2ky))^{2}},$ (47)
$\displaystyle F_{0}$ $\displaystyle=$
$\displaystyle\frac{3(13+11\cosh(2ky))}{32k^{2}(2+\cosh(2ky))\text{sech}^{\frac{4}{3}}(ky)}.$
(48)

Then, the effective potential in $y$ coordinate is given by

$\displaystyle U(y)\\!$ $\displaystyle\\!=\\!$
$\displaystyle\\!12a^{2}k^{2}\bigg{[}\frac{5H^{2}}{16k^{2}-3H^{2}}$ (49)
$\displaystyle\\!-\\!$
$\displaystyle\\!\frac{9\left(512k^{6}-240k^{4}H^{2}+24k^{2}H^{4}+9H^{6}\right)H^{\prime
2}}{\left(4k^{2}-3H^{2}\right)^{2}\left(16k^{2}-3H^{2}\right)^{3}}$
$\displaystyle\\!+\\!$
$\displaystyle\\!\frac{8\left(16k^{4}-48k^{2}H^{2}+9H^{4}\right)H^{\prime}}{\left(4k^{2}-3H^{2}\right)\left(16k^{2}-3H^{2}\right)^{2}}$
$\displaystyle\\!-\\!$
$\displaystyle\\!\frac{18\left(4k^{2}-H^{2}\right)HH^{\prime\prime}}{\left(4k^{2}-3H^{2}\right)\left(16k^{2}-3H^{2}\right)^{2}}\bigg{]}$
$\displaystyle\\!=\\!$
$\displaystyle\\!-\frac{k^{2}\text{sech}^{\frac{4}{3}}(ky)}{4(2+\cosh(2ky))^{2}(13+11\cosh(2ky))^{3}}$
$\displaystyle\times\\!$
$\displaystyle\\!\Big{[}295290+365898\cosh(2ky)+56976\cosh(4ky)$
$\displaystyle\\!-\\!$ $\displaystyle\\!24669\cosh(6ky)-8602\cosh(8ky)$
$\displaystyle\\!-\\!$ $\displaystyle\\!605\cosh(10ky)\Big{]}.$

(a) $~{}U(z)$

(b) $~{}\Phi_{0}(z)$

Figure 5: The profiles of the effective potential $U(z)$ and gravitational
zero mode $\Phi_{0}(z)$ for the case of $A=0$.

Moreover, the wave function of the massless graviton in $y$ coordinate is
obtained as

$\displaystyle\Phi_{0}(y)$ $\displaystyle=$ $\displaystyle
N_{0}a^{\frac{3}{2}}\left(1-\frac{3H^{2}}{4k^{2}}\right)^{\frac{1}{2}}\left(1-\frac{3H^{2}}{16k^{2}}\right)^{\frac{1}{4}}$
(50) $\displaystyle=$ $\displaystyle
N_{0}\text{sech}(ky)\left(1-\frac{\tanh^{2}(ky)}{3}\right)^{\frac{1}{2}}$
$\displaystyle\times$
$\displaystyle\left(1-\frac{\tanh^{2}(ky)}{12}\right)^{\frac{1}{4}},$

where the normalization constant reads
$N_{0}=\sqrt{\frac{91k}{453\sqrt{\pi}}\frac{\Gamma\left({1}/{6}\right)}{\Gamma\left({2}/{3}\right)}}\approx\sqrt{\frac{k}{2.146}}$.
Therefore, the fundamental 5-dimensional gravitational constant is
${G_{5}}=G_{4}/N_{0}^{2}\approx 2.146{G_{4}}/{k}$ in this model. The effective
potential $U(z)$ and the wave function of the massless graviton $\Phi_{0}(z)$
in $z$ coordinate are illustrated in Fig. 5. It is shown that the widths of
the potential well and wave function are both decreasing with the enhancement
of the spacetime torsion. The volcano-like effective potential also behaves
like $U(z)\sim\frac{15}{4z^{2}}$ for $z\gg 1$, so it leads to a correction
proportional to ${1}/{(kr)^{3}}$ to the Newtonian potential as the previous
case.

###  V.3 Case $B=0$

In this case, the parameters are $A=\frac{24}{\lambda}$ and
$C=-\frac{16\gamma}{\lambda}$, with $\gamma$ a free parameter. Then, the ratio
$F_{2}/F_{0}$ reads

$\frac{F_{2}}{F_{0}}=\frac{3+\left(6-8\gamma^{2}\right)\sin^{2}\phi+3\sin^{4}\phi}{3\left(1+\sin^{2}\phi\right)\left(1+\gamma\sin^{2}\phi\right)}a^{2}.$
(51)

Due to $-\frac{\pi}{2}\leq\phi(y)\leq\frac{\pi}{2}$, one has
$0\leq\sin^{2}\phi(y)\leq 1$. Thus, the ratio $F_{2}/F_{0}$ is positive
everywhere only if $\gamma<-\sqrt{\frac{3}{2}}$ or
$-1<\gamma<\sqrt{\frac{3}{2}}$. The coordinate transformation
$dz=dy/\sqrt{F_{2}/F_{0}}$ is robust and the KK gravitons are tachyonic free
in these two parameter intervals.

In order to simplify the expressions, it is convenient to set $\gamma=0$,
which vanishes the parameter $C$. Now the ratio (51) reduces to
$F_{2}/F_{0}=a^{2}\left(1+\sin^{2}\phi\right)$. With the solutions (24), the
functions $F_{2}$, $F_{1}$ and $F_{0}$ are written explicitly as

$\displaystyle F_{2}$ $\displaystyle=$
$\displaystyle\frac{1+\tanh^{2}\left(3ky/2\right)}{4k^{2}},$ (52)
$\displaystyle F_{1}$ $\displaystyle=$
$\displaystyle-\frac{\tanh\left(3ky/2\right)}{8k}\Big{[}8\left(1+\tanh^{2}\left(3ky/2\right)\right)^{\frac{1}{2}}$
(53) $\displaystyle-$ $\displaystyle
3\text{sech}^{2}\left(3ky/2\right)\Big{]},$ $\displaystyle F_{0}$
$\displaystyle=$
$\displaystyle\frac{1}{4k^{2}}\left(\cosh^{\frac{1}{2}}(3ky)+\sqrt{2}\cosh\left(3ky/2\right)\right)^{\frac{2\sqrt{2}}{3}}$
(54) $\displaystyle\times$
$\displaystyle\left(\sqrt{2}-1\right)^{\frac{2\sqrt{2}}{3}}.$

(a) $~{}U(z)$

(b) $~{}\Phi_{0}(z)$

Figure 6: The profiles of the effective potential $U(z)$ and gravitational
zero mode $\Phi_{0}(z)$ for the case of $B=0$.

Then, the effective potential in $y$ coordinate can be expressed explicitly as

$\displaystyle\\!\\!U(y)\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\frac{3k^{2}a^{2}\left(5k^{2}H^{2}-5H^{4}+2k^{2}H^{\prime}\right)}{4\left(k^{2}-H^{2}\right)^{2}}$
(55) $\displaystyle=$
$\displaystyle-\frac{5-5\cosh(3ky)+6\left(1+\tanh^{2}\left(3ky/2\right)\right)}{\left(\cosh^{\frac{1}{2}}(3ky)+\sqrt{2}\cosh\left(3ky/2\right)\right)^{\frac{2\sqrt{2}}{3}}}$
$\displaystyle\times$
$\displaystyle\frac{3k^{2}\left(1+\sqrt{2}\right)^{\frac{2\sqrt{2}}{3}}}{4(1+\cosh(3ky))}.$

Correspondingly, the wave function of the massless graviton in $y$ coordinate
is obtained as

$\displaystyle\Phi_{0}(y)\\!$ $\displaystyle=$
$\displaystyle\\!N_{0}a^{\frac{3}{2}}$ (56) $\displaystyle=$
$\displaystyle\\!\frac{N_{0}\left(1+\sqrt{2}\right)^{\frac{1}{\sqrt{2}}}}{\left(\cosh^{\frac{1}{2}}(3ky)+\sqrt{2}\cosh\left(3ky/2\right)\right)^{\frac{1}{\sqrt{2}}}},$

where the normalization constant is estimated as
$N_{0}\approx\sqrt{\frac{k}{1.684}}$. Hence, the fundamental 5-dimensional
gravitational constant reads ${G_{5}}=G_{4}/N_{0}^{2}\approx
1.684{G_{4}}/{k}$. The effective potential $U(z)$ and the wave function of the
massless graviton $\Phi_{0}(z)$ in $z$ coordinate are illustrated in Fig. 6.
It is shown that the widths of the potential well and wave function are both
decreasing with the enhancement of the spacetime torsion as the previous
cases. The volcano-like effective potential behaves like
$U(z)\sim\frac{15}{4z^{2}}$ for $z\gg 1$ as well, which contributes a
correction proportional to ${1}/{(kr)^{3}}$ to the Newtonian potential.

##  VI Conclusions

We studied thick brane scenario in the context of the higher-dimensional BIDG
in Weitzenböck spacetime. In order to solve the equations of motion (7)
analytically, we resorted to the first-order formalism, which transforms the
equations of motion into first-order equations by introducing a
superpotential. Three analytic 3-brane solutions were obtained corresponding
to some particular cases of parameter choices. It was found that all the three
models describe the domain wall braneworld, where the kink scalar maps the
boundaries of the extra dimension into two scalar vacua non-trivially. In
every model, the thickness of the domain wall brane becomes thinner as the
spacetime torsion gets stronger. Furthermore, by introducing a Yukawa coupling
between the Dirac field and scalar field in the bulk, the massless chiral
fermion can be localized on the domain wall brane Yang2018 ; Liu2017 .

Further, we analyzed the tensor perturbation against the domain wall
backgrounds and derived the Schrödinger-like equation (36) of KK modes. The
effective potential of the Schrödinger-like equation is volcano-like, which is
universal in flat brane scenario. The wave functions of the massless graviton
are localized on the brane, so the effective 4-dimensional gravity can be
recovered in each model. The widths of the potential well and wave function
decrease with the enhancement of the spacetime torsion. The KK gravitons are
tachyonic free for any $\gamma$ in the first model ($A=B$), for
$\gamma\in\left(\frac{1-\sqrt{3}}{4},\frac{1+\sqrt{3}}{4}\right)\cup(3,\infty)$
in the second model ($A=0$), and for
$\gamma\in\left(-\infty,-\sqrt{\frac{3}{2}}\right)\cup\left(-1,\sqrt{\frac{3}{2}}\right)$
in the third model ($B=0$).

Since the effective potentials of the Schrödinger-like equation have the same
behavior like $U(z)\sim\frac{15}{4z^{2}}$ for $z\gg 1$, the correction to the
Newtonian potential is proportional to ${1}/{(kr)^{3}}$ in all modes. This is
the same as the correction generated by RS2 model, but is different from the
correction generated by domain wall branes in Eddington-inspired Born-Infeld
gravity, where the correction is proportional to ${1}/{(kr)^{3+4n}}$ with $n$
a positive integer Liu2012 ; Fu2014 . The experimental test suggests that the
length scale deviating from the gravitational inverse-square law is at least
less than 48 $\mu$m Tan2020 , so the parameter $k$ is roughly estimated to be
$k>10^{-3}$eV. From the relations ${G_{5}}\sim{G_{4}}/{k}$ and
$G_{d+1}=\left(1/M_{d+1}\right)^{d-1}$, a sparsity constraint on the
fundamental 5-dimensional gravitational scale is obtained as
${M_{5}}>10^{5}$TeV.

Although the current scope of this works is to build the braneworld models and
investigate the gravitational perturbation, the (pseudo-)scalar and
(pseudo-)vector perturbations are also interesting and important. Due to the
complexity of the field equations (7), the calculation and analysis on these
modes are left for our further investigation.

## ACKNOWLEDGMENTS

This work was supported by the National Natural Science Foundation of China
under Grant No. 12005174. K. Yang acknowledges the support of Natural Science
Foundation of Chongqing, China under Grant No. cstc2020jcyj-msxmX0370. H. Yu
acknowledges the support of the Postdoctoral Science Foundation of Chongqing,
China under Grant No. tc2021jcyj-bsh0124. Y. Zhong acknowledges the supported
of the Natural Science Foundation of Hunan Province, China (Grant No.
2022JJ40033) and the Fundamental Research Funds for the Central Universities
(Grants No.531118010195).

##  Appendix A Perturbations of $\frac{\partial F_{PQ}}{\partial{e^{A}}_{M}}$
and $\frac{\partial F_{PQ}}{\partial(\partial_{S}{e^{A}}_{M})}$

The perturbation of $\frac{\partial F_{PQ}}{\partial{e^{A}}_{M}}$ can be
assembled by $\frac{\partial F_{PQ}}{\partial{e^{A}}_{M}}=\alpha\frac{\partial
F^{(1)}_{PQ}}{\partial{e^{A}}_{M}}+\beta\frac{\partial
F^{(2)}_{PQ}}{\partial{e^{A}}_{M}}+\gamma\frac{\partial
F^{(3)}_{PQ}}{\partial{e^{A}}_{M}}$, with the non-vanishing components of
$\frac{\partial F^{(1)}_{PQ}}{\partial{e^{A}}_{M}}$, $\frac{\partial
F^{(2)}_{PQ}}{\partial{e^{A}}_{M}}$, and $\frac{\partial
F^{(3)}_{PQ}}{\partial{e^{A}}_{M}}$ given by

$\displaystyle\frac{\partial{F}^{(1)}_{\mu\nu}}{\partial e^{5}{}_{5}}\\!$
$\displaystyle\\!=\\!$
$\displaystyle\\!2(d-1)a^{2}H^{2}\Big{[}\eta_{\mu\nu}+2h_{\mu\nu}+\frac{d-2}{d-1}\frac{{h^{\prime}}_{\mu\nu}}{H}\Big{]},$
(57)
$\displaystyle\frac{\partial{F}^{(1)}_{5\nu}}{\partial{e}^{5}{}_{\lambda}}\\!$
$\displaystyle\\!=\\!$
$\displaystyle\\!(1-d)H^{2}\delta^{\lambda}{}_{\nu}+(2-d)H{h^{\prime}}^{\lambda}{}_{\nu},$
(58) $\displaystyle\frac{\partial{F}^{(1)}_{\mu 5}}{\partial{e}^{a}{}_{5}}\\!$
$\displaystyle\\!=\\!$
$\displaystyle\\!\frac{\partial{F}^{(1)}_{5\mu}}{\partial{e}^{a}{}_{5}}=(1-d)aH^{2}\Big{[}\eta_{a\mu}+h_{a\mu}+\frac{d-2}{d-1}\frac{{h^{\prime}}_{a\mu}}{H}\Big{]},$
$\displaystyle\frac{\partial{F}^{(1)}_{\mu\nu}}{\partial{e}^{0}{}_{\lambda}}\\!$
$\displaystyle\\!=\\!$
$\displaystyle\\!H\left[{\partial_{\nu}h^{\lambda}{}_{\mu}}-(d-2){\partial_{\mu}h^{\lambda}{}_{\nu}}+(d-3)\partial^{\lambda}h_{\mu\nu}\right],$
(60)
$\displaystyle\frac{\partial{F}^{(1)}_{\mu\nu}}{\partial{e}^{a}{}_{5}}\\!$
$\displaystyle\\!=\\!$
$\displaystyle\\!aH\left[\Big{(}\frac{3}{2}-d\Big{)}{\partial_{\mu}h_{a\nu}}+\frac{1}{2}{\partial_{\nu}h_{a\mu}}+(d-2){\partial_{a}h_{\mu\nu}}\right],$
$\displaystyle\frac{\partial{F}^{(1)}_{5\nu}}{\partial{e}^{a}{}_{\lambda}}\\!$
$\displaystyle\\!=\\!$
$\displaystyle\\!\frac{H}{a}\left(\frac{1}{2}\partial^{\lambda}h_{a\nu}+\frac{1}{2}{\partial_{a}h^{\lambda}{}_{\nu}}-{\partial_{\nu}h_{a}{}^{\lambda}}\right),$
(62)
$\displaystyle\frac{\partial{F}^{(1)}_{\mu\nu}}{\partial{e}^{a}{}_{\lambda}}\\!$
$\displaystyle\\!=\\!$
$\displaystyle\\!aH^{2}\left[\Big{(}\frac{1}{2}-d\Big{)}\delta^{\lambda}{}_{\nu}\eta_{a\mu}-\frac{1}{2}\delta^{\lambda}{}_{\mu}\eta_{a\nu}+\delta_{a}{}^{\lambda}\eta_{\mu\nu}\right]$
(63) $\displaystyle\\!+\\!$
$\displaystyle\\!aH^{2}\Bigg{[}\Big{(}\frac{1}{2}-d\Big{)}\delta^{\lambda}{}_{\nu}h_{a\mu}-\frac{1}{2}\delta^{\lambda}{}_{\mu}h_{a\nu}-\delta_{\mu\nu}h_{a}{}^{\lambda}$
$\displaystyle\\!+\\!$
$\displaystyle\\!2\delta_{a}{}^{\lambda}h_{\mu\nu}\Bigg{]}-aH\Bigg{[}\frac{1}{2}\eta_{a\mu}{h^{\prime}}^{\lambda}{}_{\nu}+\eta_{a\nu}{h^{\prime}}^{\lambda}{}_{\mu}$
$\displaystyle\\!-\\!$
$\displaystyle\\!\eta_{\mu\nu}{h^{\prime}}_{a}{}^{\lambda}-\delta_{a}{}^{\lambda}{h^{\prime}}_{\mu\nu}+\Big{(}d-\frac{3}{2}\Big{)}\delta^{\lambda}{}_{\nu}{h^{\prime}}_{a\mu}\Bigg{]},$
$\displaystyle\frac{\partial F^{(2)}_{\mu\nu}}{\partial{e}^{5}{}_{5}}\\!$
$\displaystyle\\!=\\!$
$\displaystyle\\!(d-1)a^{2}H^{2}\left[\eta_{\mu\nu}+2h_{\mu\nu}+\frac{d-2}{d-1}\frac{{h^{\prime}}_{\mu\nu}}{H}\right],$
(64) $\displaystyle\frac{\partial
F^{(2)}_{5\nu}}{\partial{e}^{5}{}_{\lambda}}\\!$ $\displaystyle\\!=\\!$
$\displaystyle\\!\frac{1}{2}\left[(1-d)H^{2}\delta^{\lambda}{}_{\nu}+(2-d)H{h^{\prime}}^{\lambda}{}_{\nu}\right],$
(65) $\displaystyle\frac{\partial F^{(2)}_{\mu
5}}{\partial{e}^{5}{}_{\lambda}}\\!$ $\displaystyle\\!=\\!$
$\displaystyle\\!(1-d)H^{2}\delta^{\lambda}{}_{\mu}+(2-d)H{h^{\prime}}^{\lambda}{}_{\mu},$
(66) $\displaystyle\frac{\partial F^{(2)}_{\mu 5}}{\partial e^{a}{}_{5}}\\!$
$\displaystyle\\!=\\!$ $\displaystyle\\!\frac{\partial
F^{(2)}_{5\mu}}{\partial
e^{a}{}_{5}}=\frac{1-d}{2}aH^{2}\Big{[}\eta_{a\mu}+h_{a\mu}+\frac{d-2}{d-1}\frac{{h^{\prime}}_{a\mu}}{H}\Big{]},$
$\displaystyle\frac{\partial F^{(2)}_{55}}{\partial e^{a}{}_{\lambda}}\\!$
$\displaystyle\\!=\\!$
$\displaystyle\\!(d-1)\frac{H^{2}}{a}\left[{\delta_{a}}^{\lambda}-{h_{a}}^{\lambda}+\frac{d-2}{d-1}\frac{{h^{\prime}}_{a}{}^{\lambda}}{H}\right],$
(68) $\displaystyle\frac{\partial F^{(2)}_{\mu\nu}}{\partial
e^{5}{}_{\lambda}}\\!$ $\displaystyle\\!=\\!$
$\displaystyle\\!\frac{H}{2}\bigg{[}{\partial_{\mu}h^{\lambda}{}_{\nu}}-(d-2){\partial_{\nu}h^{\lambda}{}_{\mu}}+(d-3)\partial^{\lambda}h_{\mu\nu}\bigg{]},$
$\displaystyle\frac{\partial F^{(2)}_{\mu\nu}}{\partial e^{a}{}_{5}}\\!$
$\displaystyle\\!=\\!$
$\displaystyle\\!\frac{aH}{2}\bigg{[}\Big{(}\frac{3}{2}-d\Big{)}{\partial_{\nu}h_{a\mu}}+\frac{1}{2}{\partial_{\mu}h_{a\nu}}+(d-2){\partial_{a}h_{\mu\nu}}\bigg{]},$
$\displaystyle\frac{\partial F^{(2)}_{\mu 5}}{\partial{e^{a}}_{\lambda}}\\!$
$\displaystyle\\!=\\!$
$\displaystyle\\!\frac{H}{2a}\left[(d-3){\partial_{\mu}{h_{a}}^{\lambda}}-(d-2){\partial_{a}h^{\lambda}{}_{\mu}}+\partial^{\lambda}h_{a\mu}\right],$
$\displaystyle\frac{\partial F^{(2)}_{5\nu}}{\partial e^{a}{}_{\lambda}}\\!$
$\displaystyle\\!=\\!$
$\displaystyle\\!\frac{H}{2a}\bigg{[}(d-2){\partial_{\nu}h_{a}{}^{\lambda}}-\left(d-\frac{3}{2}\right){\partial_{a}h^{\lambda}{}_{\nu}}+\frac{1}{2}\partial^{\lambda}h_{a\nu}\bigg{]},$
$\displaystyle\frac{\partial
F^{(2)}_{\mu\nu}}{\partial{e}^{a}{}_{\lambda}}\\!$ $\displaystyle\\!=\\!$
$\displaystyle\\!\frac{aH^{2}}{4}\Big{[}2{\delta_{a}}^{\lambda}\eta_{\mu\nu}-(2d-1)\delta_{\mu}{}^{\lambda}\eta_{a\nu}-\delta_{\nu}{}^{\lambda}\eta_{a\mu}\Big{]}$
(73) $\displaystyle-$
$\displaystyle\frac{aH^{2}}{4}\Big{[}(2d-1){\delta_{\mu}}^{\lambda}h_{a\nu}+{\delta_{\nu}}^{\lambda}h_{a\mu}+2\eta_{\mu\nu}{h_{a}}^{\lambda}$
$\displaystyle-$ $\displaystyle
4{\delta_{a}}^{\lambda}h_{\mu\nu}\Big{]}-\frac{aH}{4}\Big{[}\eta_{a\mu}{h^{\prime}}^{\lambda}{}_{\nu}+2(d-1){\delta_{\mu}}^{\lambda}{h^{\prime}}_{a\nu}$
$\displaystyle+$
$\displaystyle{\delta_{\nu}}^{\lambda}{h^{\prime}}_{a\mu}-2\eta_{\mu\nu}{h^{\prime}}_{a}{}^{\lambda}-2{\delta_{a}}^{\lambda}{h^{\prime}}_{\mu\nu}\Big{]},$
$\displaystyle\frac{\partial F^{(3)}_{\mu\nu}}{\partial{e^{5}}_{5}}\\!$
$\displaystyle\\!=\\!$
$\displaystyle\\!2d(d-1)a^{2}H^{2}\left(\eta_{\mu\nu}+2h_{\mu\nu}\right),$
(74) $\displaystyle\frac{\partial F^{(3)}_{\mu
5}}{\partial{e^{5}}_{\lambda}}\\!$ $\displaystyle\\!=\\!$
$\displaystyle\\!\frac{\partial
F^{(3)}_{5\mu}}{\partial{e^{5}}_{\lambda}}=d(1-d)H^{2}\delta^{\lambda}{}_{\mu},$
(75) $\displaystyle\frac{\partial F^{(3)}_{\mu 5}}{{\partial e}^{a}{}_{5}}\\!$
$\displaystyle\\!=\\!$ $\displaystyle\\!\frac{\partial
F^{(3)}_{5\mu}}{{\partial
e}^{a}{}_{5}}=d(1-d)aH^{2}\left(\eta_{a\mu}+h_{a\mu}\right),$ (76)
$\displaystyle\frac{\partial F^{(3)}_{55}}{{\partial e}^{a}{}_{\lambda}}\\!$
$\displaystyle\\!=\\!$
$\displaystyle\\!2(d-1)\frac{H^{2}}{a}\left[\delta_{a}{}^{\lambda}-h_{a}{}^{\lambda}+\frac{d-2}{d-1}\frac{{h^{\prime}}_{a}{}^{\lambda}}{H}\right],$
(77) $\displaystyle\frac{\partial
F^{(3)}_{\mu\nu}}{\partial{e^{a}}_{\lambda}}\\!$ $\displaystyle\\!=\\!$
$\displaystyle\\!(1-d)aH^{2}\left(d\delta^{\lambda}{}_{(\mu}\eta_{\nu)a}-2\delta^{\lambda}{}_{a}\eta_{\mu\nu}\right)$
(78) $\displaystyle+$
$\displaystyle(1-d)aH^{2}\left(d\delta^{\lambda}{}_{(\mu}h_{\nu)a}+2\eta_{\mu\nu}h_{a}{}^{\lambda}-4\delta_{a}{}^{\lambda}h_{\mu\nu}\right)$
$\displaystyle+$ $\displaystyle
2(d-2)aH\eta_{\mu\nu}{h^{\prime}}_{a}{}^{\lambda}.$

The perturbation of $\frac{\partial
F_{PQ}}{\partial(\partial_{S}{e^{A}}_{M})}$ can be assembled by
$\frac{\partial
F_{PQ}}{\partial(\partial_{S}{e^{A}}_{M})}=\alpha\frac{\partial
F^{(1)}_{PQ}}{\partial(\partial_{S}{e^{A}}_{M})}+\beta\frac{\partial
F^{(2)}_{PQ}}{\partial(\partial_{S}{e^{A}}_{M})}+\gamma\frac{\partial
F^{(3)}_{PQ}}{\partial(\partial_{S}{e^{A}}_{M})}$,where the non-vanishing
components of $\frac{\partial
F^{(1)}_{PQ}}{\partial(\partial_{S}{e^{A}}_{M})}$, $\frac{\partial
F^{(2)}_{PQ}}{\partial(\partial_{S}{e^{A}}_{M})}$, and $\frac{\partial
F^{(3)}_{PQ}}{\partial(\partial_{S}{e^{A}}_{M})}$ are given by

$\displaystyle\frac{\partial{F}^{(1)}_{\mu
5}}{\partial(\partial_{5}e^{5}{}_{\lambda})}\\!$ $\displaystyle\\!=\\!$
$\displaystyle\\!(1-d)H\delta^{\lambda}{}_{\mu}+{h^{\prime}}^{\lambda}{}_{\mu},$
(79) $\displaystyle\frac{\partial F^{(1)}_{\mu
5}}{\partial({\partial_{\gamma}e^{5}{}_{5}})}\\!$ $\displaystyle\\!=\\!$
$\displaystyle\\!(d-1)H\delta^{\gamma}{}_{\mu}-{h^{\prime}}^{\gamma}{}_{\mu},$
(80)
$\displaystyle\frac{\partial{F}^{(1)}_{\mu\nu}}{\partial({\partial_{5}e^{5}{}_{\lambda})}}\\!$
$\displaystyle\\!=\\!$
$\displaystyle\\!\partial^{\lambda}h_{\mu\nu}-{\partial_{\mu}h^{\lambda}{}_{\nu}},$
(81) $\displaystyle\frac{\partial
F^{(1)}_{\mu\nu}}{\partial({\partial_{\gamma}e^{5}{}_{5}})}\\!$
$\displaystyle\\!=\\!$
$\displaystyle\\!{\partial_{\mu}h^{\gamma}{}_{\nu}}-\partial^{\gamma}h_{\mu\nu},$
(82) $\displaystyle\frac{\partial
F^{(1)}_{5\nu}}{\partial({\partial_{5}e^{a}{}_{\lambda}})}\\!$
$\displaystyle\\!=\\!$
$\displaystyle\\!\frac{1}{2a}\left({\partial_{a}h^{\lambda}{}_{\nu}}-\partial^{\lambda}h_{a\nu}\right),$
(83) $\displaystyle\frac{\partial
F^{(1)}_{5\nu}}{\partial(\partial_{\gamma}e^{a}{}_{5})}\\!$
$\displaystyle\\!=\\!$
$\displaystyle\\!\frac{1}{2a}\left(\partial^{\gamma}h_{a\nu}-{\partial_{a}h^{\gamma}{}_{\nu}}\right),$
(84) $\displaystyle\frac{\partial F^{(1)}_{\mu
5}}{\partial(\partial_{\gamma}e^{5}{}_{\lambda})}\\!$ $\displaystyle\\!=\\!$
$\displaystyle\\!\frac{1}{a^{2}}\left(\partial^{\gamma}h^{\lambda}{}_{\mu}-\partial^{\lambda}h^{\gamma}{}_{\mu}\right),$
(85) $\displaystyle\frac{\partial
F^{(1)}_{5\nu}}{\partial(\partial_{\gamma}e^{5}{}_{\lambda})}\\!$
$\displaystyle\\!=\\!$
$\displaystyle\\!\frac{1}{2a^{2}}\left(\partial^{\gamma}h^{\lambda}{}_{\nu}-\partial^{\lambda}h^{\gamma}{}_{\nu}\right),$
(86) $\displaystyle\frac{\partial
F^{(1)}_{\mu\nu}}{\partial({\partial_{5}e^{a}{}_{\lambda}})}\\!$
$\displaystyle\\!=\\!$
$\displaystyle\\!-a\bigg{[}H\delta_{a}{}^{\lambda}\eta_{\mu\nu}+\left(d-\frac{3}{2}\right)H\delta^{\lambda}{}_{\mu}\eta_{a\nu}$
(87) $\displaystyle-$
$\displaystyle\frac{1}{2}H\delta^{\lambda}{}_{\nu}\eta_{a\mu}\bigg{]}+a\bigg{[}H\eta_{\mu\nu}h_{a}{}^{\lambda}-2H\delta_{a}{}^{\lambda}h_{\mu\nu}$
$\displaystyle+$
$\displaystyle\frac{1}{2}H\delta_{\nu}{}^{\lambda}h_{a\mu}-\left(d-\frac{3}{2}\right)H\delta^{\lambda}{}_{\mu}h_{a\nu}+\eta_{a\nu}{h^{\prime}}^{\lambda}{}_{\mu}$
$\displaystyle-$
$\displaystyle\delta_{a}{}^{\lambda}{h^{\prime}}_{\mu\nu}+\frac{1}{2}\delta^{\lambda}{}_{\mu}{h^{\prime}}_{a\nu}+\frac{1}{2}\eta_{a\mu}{h^{\prime}}^{\lambda}{}_{\nu}\bigg{]},$
$\displaystyle\frac{\partial
F^{(1)}_{\mu\nu}}{\partial(\partial_{\gamma}e^{a}{}_{5})}\\!$
$\displaystyle\\!=\\!$
$\displaystyle\\!-aH\bigg{[}\delta_{a}{}^{\gamma}\eta_{\mu\nu}-\frac{1}{2}\delta^{\gamma}{}_{\nu}\eta_{a\mu}+\left(d-\frac{3}{2}\right)\delta^{\gamma}{}_{\mu}\eta_{a\nu}\bigg{]}$
(88) $\displaystyle+$ $\displaystyle
a\bigg{[}H\eta_{\mu\nu}h_{a}{}^{\gamma}-2H\delta_{a}{}^{\gamma}h_{\mu\nu}+\frac{1}{2}H\delta^{\gamma}{}_{\nu}h_{a\mu}$
$\displaystyle-$
$\displaystyle\Big{(}d-\frac{3}{2}\Big{)}H\delta^{\gamma}{}_{\mu}h_{a\nu}+\eta_{a\nu}{h^{\prime}}^{\gamma}{}_{\mu}-\delta_{a}{}^{\gamma}{h^{\prime}}_{\mu\nu}$
$\displaystyle+$
$\displaystyle\frac{1}{2}\delta_{\mu}{}^{\gamma}{h^{\prime}}_{a\nu}+\frac{1}{2}\eta_{a\mu}{h^{\prime}}^{\gamma}{}_{\nu}\bigg{]},$
$\displaystyle\frac{\partial
F^{(1)}_{\mu\nu}}{\partial(\partial_{\gamma}e^{5}{}_{\lambda})}\\!$
$\displaystyle\\!=\\!$
$\displaystyle\\!\frac{1}{2}\Big{(}H\delta^{\gamma}{}_{\mu}\delta^{\lambda}{}_{\nu}-H\delta^{\gamma}{}_{\nu}\delta^{\lambda}{}_{\mu}+\delta^{\gamma}{}_{\mu}{h^{\prime}}^{\lambda}{}_{\nu}$
(89) $\displaystyle-$
$\displaystyle\delta^{\lambda}{}_{\mu}{h^{\prime}}^{\gamma}{}_{\nu}\Big{)},$
$\displaystyle\frac{\partial
F^{(1)}_{5\nu}}{\partial(\partial_{\gamma}e^{a}{}_{\lambda})}\\!$
$\displaystyle\\!=\\!$
$\displaystyle\\!\frac{1}{a}\Big{(}H\delta_{a}{}^{\lambda}\delta^{\gamma}{}_{\nu}-H\delta_{a}{}^{\gamma}\delta^{\lambda}{}_{\nu}+H\delta_{\nu}{}^{\lambda}h^{\gamma}{}_{a}$
(90) $\displaystyle-$ $\displaystyle
H\delta_{\nu}{}^{\gamma}h^{\lambda}{}_{a}+\delta_{a}{}^{\lambda}{h^{\prime}}^{\gamma}{}_{\nu}-\delta_{a}{}^{\gamma}{h^{\prime}}^{\lambda}{}_{\nu}\Big{)},$
$\displaystyle\frac{\partial
F^{(1)}_{\mu\nu}}{\partial(\partial_{\gamma}e^{a}{}_{\lambda})}\\!$
$\displaystyle\\!=\\!$
$\displaystyle\\!\frac{1}{a}\bigg{[}\frac{1}{2}\delta^{\lambda}{}_{\mu}\left(\partial^{\gamma}h_{a\nu}-{\partial_{a}h^{\gamma}{}_{\nu}}\right)$
(91) $\displaystyle-$
$\displaystyle\frac{1}{2}\delta^{\gamma}{}_{\mu}\big{(}\partial^{\lambda}h_{a\nu}-{\partial_{a}h^{\lambda}{}_{\nu}}\big{)}$
$\displaystyle+$
$\displaystyle\frac{1}{2}\eta_{a\mu}\left(\partial^{\gamma}h^{\lambda}{}_{\nu}-\partial^{\lambda}h^{\gamma}{}_{\nu}\right)$
$\displaystyle+\eta_{a\nu}\big{(}\partial^{\gamma}h^{\lambda}{}_{\mu}-\partial^{\lambda}h^{\gamma}{}_{\mu}\big{)}$
$\displaystyle+$
$\displaystyle\delta_{a}{}^{\lambda}\left({\partial_{\mu}h^{\gamma}{}_{\nu}}-\partial^{\gamma}h_{\mu\nu}\right)$
$\displaystyle-$
$\displaystyle\delta_{a}{}^{\gamma}\left({\partial_{\mu}h^{\lambda}{}_{\nu}}-\partial^{\lambda}h_{\mu\nu}\right)\bigg{]},$
$\displaystyle\frac{\partial F^{(2)}_{\mu
5}}{\partial(\partial_{5}e^{5}{}_{\lambda})}\\!$ $\displaystyle\\!=\\!$
$\displaystyle\\!\frac{1}{2}\left[(d-1)H\delta^{\lambda}{}_{\mu}-{h^{\prime}}^{\lambda}{}_{\mu}\right],$
(92) $\displaystyle\frac{\partial F^{(2)}_{\mu
5}}{\partial(\partial_{\gamma}e^{5}{}_{5})}\\!$ $\displaystyle\\!=\\!$
$\displaystyle\\!\frac{1}{2}\left[(1-d)H\delta^{\gamma}{}_{\mu}+{h^{\prime}}^{\gamma}{}_{\mu}\right],$
(93) $\displaystyle\frac{\partial
F^{(2)}_{55}}{\partial(\partial_{5}e^{a}{}_{\lambda})}\\!$
$\displaystyle\\!=\\!$
$\displaystyle\\!\frac{1-d}{a}\left(H\delta_{a}{}^{\lambda}-Hh_{a}{}^{\lambda}-\frac{{h^{\prime}}_{a}{}^{\lambda}}{d-1}\right),$
(94) $\displaystyle\frac{\partial
F^{(2)}_{55}}{\partial(\partial_{\gamma}e^{a}{}_{5})}\\!$
$\displaystyle\\!=\\!$
$\displaystyle\\!\frac{d-1}{a}\left(H\delta_{a}{}^{\gamma}-Hh_{a}{}^{\gamma}-\frac{{h^{\prime}}_{a}{}^{\gamma}}{d-1}\right),$
(95) $\displaystyle\frac{\partial
F^{(2)}_{\mu\nu}}{\partial(\partial_{5}e^{5}{}_{\lambda})}\\!$
$\displaystyle\\!=\\!$
$\displaystyle\\!\frac{1}{2}\left(\partial^{\lambda}h_{\mu\nu}-{\partial_{\nu}h^{\lambda}{}_{\mu}}\right),$
(96) $\displaystyle\frac{\partial
F^{(2)}_{\mu\nu}}{\partial(\partial_{\gamma}e^{5}{}_{5})}\\!$
$\displaystyle\\!=\\!$
$\displaystyle\\!\frac{1}{2}\left(\partial^{\gamma}h_{\mu\nu}-{\partial_{\nu}h^{\gamma}{}_{\mu}}\right),$
(97) $\displaystyle\frac{\partial F^{(2)}_{\mu
5}}{\partial({\partial_{5}e^{a}{}_{\lambda}})}\\!$ $\displaystyle\\!=\\!$
$\displaystyle\\!\frac{1}{2a}\left({\partial_{\mu}h_{a}{}^{\lambda}}-\partial^{\lambda}h_{a\mu}\right),$
(98)
$\displaystyle\frac{\partial{F}^{(2)}_{5\nu}}{\partial(\partial_{5}e^{a}{}_{\lambda})}\\!$
$\displaystyle\\!=\\!$
$\displaystyle\\!\frac{1}{4a}\left(2{\partial_{\nu}h_{a}{}^{\lambda}}-\partial^{\lambda}h_{a\nu}-{\partial_{a}h^{\lambda}{}_{\nu}}\right),$
(99) $\displaystyle\frac{\partial F^{(2)}_{\mu
5}}{\partial(\partial_{\gamma}e^{a}{}_{5})}\\!$ $\displaystyle\\!=\\!$
$\displaystyle\\!\frac{1}{2a}\left(\partial^{\gamma}h_{a\mu}-{\partial_{\mu}h_{a}{}^{\gamma}}\right),$
(100) $\displaystyle\frac{\partial
F^{(2)}_{5\nu}}{\partial(\partial_{\gamma}e^{a}{}_{5})}\\!$
$\displaystyle\\!=\\!$
$\displaystyle\\!\frac{1}{4a}\left(\partial^{\gamma}h_{a\nu}+{\partial_{a}h^{\gamma}{}_{\nu}}-2{\partial_{\nu}h_{a}{}^{\gamma}}\right),$
(101) $\displaystyle\frac{\partial
F^{(2)}_{5\nu}}{\partial(\partial_{\gamma}e^{5}{}_{\lambda})}\\!$
$\displaystyle\\!=\\!$
$\displaystyle\\!\frac{1}{4a^{2}}\left(\partial^{\gamma}h^{\lambda}{}_{\nu}-\partial^{\lambda}h^{\gamma}{}_{\nu}\right),$
(102) $\displaystyle\frac{\partial
F^{(2)}_{\mu\nu}}{\partial(\partial_{\gamma}e^{5}{}_{\lambda})}\\!$
$\displaystyle\\!=\\!$
$\displaystyle\\!\frac{1}{4}\Big{(}H\delta_{\mu}{}^{\lambda}\delta_{\nu}{}^{\gamma}-H\delta_{\mu}{}^{\gamma}\delta_{\nu}{}^{\lambda}+\delta_{\mu}{}^{\lambda}{h^{\prime}}^{\gamma}{}_{\nu},$
(103) $\displaystyle-$
$\displaystyle\delta_{\mu}{}^{\gamma}{h^{\prime}}^{\lambda}{}_{\nu}\Big{)}$
$\displaystyle\frac{\partial{F}^{(2)}_{\mu\nu}}{\partial(\partial_{5}e^{a}{}_{\lambda})}\\!$
$\displaystyle\\!=\\!$
$\displaystyle\\!-\frac{a}{2}\bigg{[}H\delta_{a}{}^{\lambda}\eta_{\mu\nu}+\Big{(}d-\frac{3}{2}\Big{)}H\delta^{\lambda}{}_{\nu}\eta_{a\mu}$
(104) $\displaystyle-$
$\displaystyle\frac{1}{2}H\delta^{\lambda}{}_{\mu}\eta_{a\nu}\bigg{]}-\frac{a}{2}\bigg{[}2H\delta_{a}{}^{\lambda}h_{\mu\nu}-H\eta_{\mu\nu}h_{a}{}^{\lambda}$
$\displaystyle-$
$\displaystyle\frac{1}{2}H\delta^{\lambda}{}_{\mu}h_{a\nu}+\Big{(}d-\frac{3}{2}\Big{)}H\delta^{\lambda}{}_{\nu}h_{a\mu}+\delta_{a}{}^{\lambda}{h^{\prime}}_{\mu\nu}$
$\displaystyle-\delta^{\lambda}{}_{\nu}{h^{\prime}}_{a\mu}-\frac{1}{2}\delta^{\lambda}{}_{\mu}{h^{\prime}}_{a\nu}-\frac{1}{2}\eta_{a\mu}{h^{\prime}}^{\lambda}{}_{\nu}\bigg{]},$
$\displaystyle\frac{\partial
F^{(2)}_{\mu\nu}}{\partial(\partial_{\gamma}e^{a}{}_{5})}\\!$
$\displaystyle\\!=\\!$
$\displaystyle\\!\frac{a}{2}\bigg{[}H\delta_{a}{}^{\gamma}\eta_{\mu\nu}+\left(d-\frac{3}{2}\right)H\delta^{\gamma}{}_{\nu}\eta_{a\mu}$
(105) $\displaystyle-$
$\displaystyle\frac{1}{2}H\delta^{\gamma}{}_{\mu}\eta_{a\nu}\bigg{]}+\frac{a}{2}\bigg{[}2H\delta_{a}{}^{\gamma}h_{\mu\nu}-H\eta_{\mu\nu}h_{a}{}^{\gamma}$
$\displaystyle-$
$\displaystyle\frac{1}{2}H\delta_{\mu}{}^{\gamma}h_{a\nu}+\left(d-\frac{3}{2}\right)H\delta_{\nu}{}^{\gamma}h_{a\mu}+\delta_{a}{}^{\gamma}{h^{\prime}}_{\mu\nu}$
$\displaystyle-$
$\displaystyle\delta_{\nu}{}^{\gamma}{h^{\prime}}_{a\mu}-\frac{1}{2}\delta_{\mu}{}^{\gamma}{h^{\prime}}_{a\nu}-\frac{1}{2}\eta_{a\mu}{h^{\prime}}^{\gamma}{}_{\nu}\bigg{]},$
$\displaystyle\frac{\partial
F^{(2)}_{\mu\nu}}{\partial(\partial_{\gamma}e^{a}{}_{\lambda})}\\!$
$\displaystyle\\!=\\!$
$\displaystyle\\!\frac{1}{4a}\Big{[}2\delta_{a}{}^{\lambda}\big{(}{\partial_{\nu}h^{\gamma}{}_{\mu}}-\partial^{\gamma}h_{\mu\nu}\big{)}$
(106) $\displaystyle-$ $\displaystyle
2\delta_{a}{}^{\gamma}\big{(}{\partial_{\nu}h^{\lambda}{}_{\mu}}-\partial^{\lambda}h_{\mu\nu}\big{)}$
$\displaystyle+$
$\displaystyle\eta_{a\mu}\big{(}\partial^{\gamma}h^{\lambda}{}_{\nu}-\partial^{\lambda}h^{\gamma}{}_{\nu}\big{)}$
$\displaystyle+\delta_{\mu}{}^{\gamma}\big{(}2{\partial_{\nu}h_{a}{}^{\lambda}}-\partial^{\lambda}h_{a\nu}-{\partial_{a}h^{\lambda}{}_{\nu}}\big{)}$
$\displaystyle-$
$\displaystyle\delta_{\mu}{}^{\lambda}\big{(}2{\partial_{\nu}h_{a}{}^{\gamma}}-\partial^{\gamma}h_{a\nu}-{\partial_{a}h^{\gamma}{}_{\nu}}\big{)}$
$\displaystyle+$ $\displaystyle
2\delta^{\gamma}{}_{\nu}\big{(}{\partial_{\mu}h_{a}{}^{\lambda}}-\partial^{\lambda}h_{a\mu}\big{)}$
$\displaystyle-$ $\displaystyle
2\delta^{\lambda}{}_{\nu}\big{(}{\partial_{\mu}h_{a}{}^{\gamma}}-\partial^{\gamma}h_{a\mu}\big{)}\Big{]},$
$\displaystyle\frac{\partial
F^{(3)}_{55}}{\partial({\partial_{5}e^{a}{}_{\lambda}})}\\!$
$\displaystyle\\!=\\!$
$\displaystyle\\!\frac{2(1-d)}{a}\left[H\delta_{a}{}^{\lambda}-Hh_{a}{}^{\lambda}-\frac{{h^{\prime}}_{a}{}^{\lambda}}{d-1}\right],$
(107) $\displaystyle\frac{\partial
F^{(3)}_{55}}{\partial(\partial_{\gamma}e^{a}{}_{5})}\\!$
$\displaystyle\\!=\\!$
$\displaystyle\\!\frac{2(d-1)}{a}\left[H\delta_{a}{}^{\gamma}-Hh_{a}{}^{\gamma}-\frac{{h^{\prime}}_{a}{}^{\gamma}}{d-1}\right],$
(108) $\displaystyle\frac{\partial
F^{(3)}_{55}}{\partial({\partial_{\gamma}e^{a}{}_{\lambda}})}\\!$
$\displaystyle\\!=\\!$
$\displaystyle\\!\frac{2}{a^{3}}\left(\partial^{\gamma}h_{a}{}^{\lambda}-\partial^{\lambda}h_{a}{}^{\gamma}\right),$
(109) $\displaystyle\frac{\partial
F^{(3)}_{\mu\nu}}{\partial({\partial_{5}e^{a}{}_{\lambda}})}\\!$
$\displaystyle\\!=\\!$
$\displaystyle\\!2a\eta_{\mu\nu}\left[(1-d)H\delta_{a}{}^{\lambda}-(1-d)Hh_{a}{}^{\lambda}+{h^{\prime}}_{a}{}^{\lambda}\right]$
(110) $\displaystyle+$ $\displaystyle
4(1-d)aH\delta_{a}{}^{\lambda}h_{\mu\nu},$ $\displaystyle\frac{\partial
F^{(3)}_{\mu\nu}}{\partial({\partial_{\gamma}e^{a}{}_{5}})}\\!$
$\displaystyle\\!=\\!$
$\displaystyle\\!2a\eta_{\mu\nu}\left[(d-1)H\delta_{a}{}^{\gamma}-(d-1)Hh_{a}{}^{\gamma}-{h^{\prime}}_{a}{}^{\gamma}\right]$
(111) $\displaystyle+$ $\displaystyle
4(d-1)aH\delta_{a}{}^{\gamma}h_{\mu\nu},$ $\displaystyle\frac{\partial
F^{(3)}_{\mu\nu}}{\partial({\partial_{\gamma}e^{a}{}_{\lambda}})}\\!$
$\displaystyle\\!=\\!$
$\displaystyle\\!\frac{2}{a}\eta_{\mu\nu}\left(\partial^{\gamma}h_{a}{}^{\lambda}-\partial^{\lambda}h_{a}{}^{\gamma}\right).$
(112)

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