On holographic braneworld cosmology

Zhong-Ying Fan1

1 Department of Astrophysics, School of Physics and Materials Science,

Guangzhou University, Guangzhou 510006, China

ABSTRACT

Recently, S. Antonini and B. Swingle builded a holographic model for
braneworld cosmology by introducing an “end-of-the-world” (ETW) brane moving
in a charged black hole spactime. In this paper, we will show that a
holographic description of braneworld cosmology is possible for a general
black hole spacetime with a pure tension brane if one implements mixed
boundary conditions on the ETW brane. As a simple example, we study AdS-
Schwarzschild black holes and show that a sensible Euclidean braneworld
solution is compatible with localization of gravity on the ETW brane.

Email<EMAIL_ADDRESS>.

###### Contents

  1. 1 Introduction
  2. 2 AdS/BCFT
  3. 3 The end-of-the-world brane in AdS
    1. 3.1 Euclidean analysis
    2. 3.2 Lorentian analysis
    3. 3.3 Boundary stress tensor
  4. 4 Gravity localization
    1. 4.1 Master equation and boundary condition
    2. 4.2 Numerical results
  5. 5 Conclusion
  6. A  More discussions on mixed boundary conditions
  7. B  Action comparison
  8. C Trapping coefficient method
  9. D Limiting behavior of the quasi bound mode
    1. D.1 Near zone solutions
    2. D.2 Far zone solutions

##  1 Introduction

The Anti-de Sitter/Coformal Field Theory (AdS/CFT) correspondence proposes
that a $(d+1)$-dimensional quantum gravity theory is dual to a $d$-dimensional
quantum filed theory without gravity residing on the asymptotic boundary of
bulk spacetime [1, 2, 3]. It provides a new approach to study both quantum
gravity and quantum field theories at strong coupling limit, becoming an
exciting research area in recent two decades. It is natural to ask whether
cosmological physics can be studied in this framework. Some earlier works on
holographic cosmology can be found in literature, for example [4, 5, 6, 7, 8,
9], where a conformal field theory (CFT) lives on the brane and is coupled to
gravity.

Recently, it was established [10] that a non-perturbative CFT description of
Friedmann-Lemaître-Robertson-Walker (FLRW) cosmology can be realized in the
Anti-de Sitter/Boundary Conformal Field Theory (AdS/BCFT) correspondence [11,
12]. The basic idea is relating the excited pure states of the CFT to a two-
sided black hole geometry with a dynamical end-of-world (ETW) brane cutting
off the left (or right) region of the black hole and replacing the asymptotic
boundary therein [13]. It is known that a two-sided wormhole geometry
describes an entangled states of two CFTs living in the left and right
asymptotic boundary respectively. Now the presence of the ETW brane was
proposed to measure the one CFT (for example the left one), destroying the
entanglement and hence leaving a pure state for the remaining CFT.

It is intriguing that the motion of the ETW brane could be interpreted as
evolution of a FLRW universe for local observers living on the brane, where
the bulk radial coordinate plays the role of the scale factor in cosmology
[14, 15, 16, 17]. Hence, this model provides a holographic description for
braneworld cosmology using the observables in the dual CFT associated to the
right asymptotic region of black holes. This is significantly different from
the earlier works [4, 5, 6, 7, 8, 9], where the CFT lives on the brane.

However, there are two things that should be considered carefully. The first
is the dual CFT should be well defined. This demands that the preparation time
of the CFT, defined in the Euclidean path integral, is positive definite [13].
The second is if the bulk is $5$-dimensional, for observers living on the
brane, gravity should be effectively $4$-dimensional and is locally localized
on the brane. In other words, ordinary gravity is observed only on non-
cosmological scales and in a limited range of times.

It was shown in [10] that for AdS Reissner-Nordström (RN) black holes with a
pure tension brane, the above two conditions are compatible by implementing
Neumann boundary condition on the brane. It provides a first proof-of-
principle result for holographic braneworld cosmology. However, the above two
conditions are not always compatible. As pointed out in [10], the result there
is not attainable in the simplest setup of an AdS-Schwarzschild black hole
with a pure tension brane. As a matter of fact, the discussions in [10] are
limited to charged black holes and cannot be extended to general black hole
backgrounds. This is unsatisfactory from the viewpoint of holographic
principle.

A possible remedy of the issue is by introducing localized matter on the ETW
brane. However, this is of great difficult in practical applications (there
are also many other issues). Instead, in this paper, we will show that the
issue can be nicely resolved by imposing a mixed boundary condition on the ETW
brane. We prove this for AdS-Schwarzschild black holes. The results are easily
generalised to different black hole backgrounds. We thus illustrate a simple
setup for braneworld cosmology in holography.

The paper is organized as follows. In section 2, we briefly review the
AdS/BCFT correspondence and the role of boundary conditions. In section 3, we
study the motion of the ETW brane in Euclidean AdS-Schwarzschild black holes
and show that sensible Euclidean solutions can be obtained by imposing a mixed
boundary condition on the brane. In section 4, we study the frequency of the
quasi bound mode for large black holes in adiabatic approximation. We find
that gravity is locally localized on the brane for parameters compatible with
sensible Euclidean solutions. We conclude in section 5.

##  2 AdS/BCFT

Boundary conditions play an essential role in the formulation of AdS/BCFT
correspondence. Consider a CFT living in a $d$-dimensional manifold $M$ with a
$(d-1)$-dimensional boundary $P$. The bulk is a $(d+1)$-dimensional AdS space
$N$, which is anchored on $M$ and has an extra boundary $Q$ in the deep
interior, with $\partial Q=\partial M=P$. In general, for a pure tension brane
$Q$, the gravitational action is given by

$I_{L}={\frac{1}{16\pi
G}}\int_{N}\sqrt{-g}\,\big{(}R-2\Lambda\big{)}+{\frac{1}{8\pi
G}}\int_{M}\sqrt{-\gamma}\,K+{\frac{1}{8\pi
G}}\int_{Q}\sqrt{-h}\,\big{(}K-(d-1)T\big{)}+{\frac{1}{8\pi
G}}\int_{P}\,\sqrt{\sigma}\,a\,,$ (1)

where $a=\mathrm{arccos}\big{(}n_{M}\cdot n_{Q}\big{)}$ is the corner term
associated to the codimension-2 boundary $P$ (from the bulk point of view).
This term is introduced for well-defined variational principle in the presence
of non-smooth boundaries [18]. Otherwise, it can be dropped in the smooth
limit, where variation of the action yields111Dirichlet boundary condition was
imposed on the asymptotic boundary $M$.

$\delta
I_{L}=\int_{Q}\sqrt{-h}\Big{(}K_{ab}-\big{(}K-(d-1)T\big{)}h_{ab}\Big{)}\delta
h^{ab}\,.$ (2)

Clearly, to have dynamics on $Q$, one should introduce Neumann boundary
condition

$K_{ab}-\big{(}K-(d-1)T\big{)}h_{ab}=0\,,$ (3)

which allows the brane moving (expanding or contracting) dynamically. However,
this case has many problems, as pointed out in [19, 20]. On one hand, since
$Q$ is a codimension-$1$ hypersurface in the bulk, it should be characterized
by a single embedding function while the Neumann boundary condition (3) is a
tensorial equation, which generally does not give a self-consistent solution
except some simple cases [11, 12]. On the other hand, the AdS/BCFT
correspondence should apply to general shape of the boundary $M$. A possible
remedy of the issue is by introducing localized matter on the brane $Q$ so
that the equation (3) (with stress tensor of matter fields introduced on the
r.h.s) can be solved consistently for a general shaped asymptotic boundary.
However, this approach is of great difficult in practical applications and
cannot be tested easily. As a consequence, whether it provides a consistent
story for the boundary theory remains open.

In [19, 20], the authors provide another solution to the issue: imposing a
mixed boundary condition on $Q$

$K=dT\,.$ (4)

This does not contradict with the variational principle since in general the
corner $P$ is non-smooth and the bulk action should pick up a joint term.
Moreover, despite that the condition looks simple, it produced correct
holographic boundary Weyl anomaly for the dual CFT. This is highly nontrivial.
It partly establishes that the above mixed boundary condition is physically
sensible in studying holographic dual of CFTs with boundaries.

Return to holographic description of braneworld cosmology. The bulk interior
boundary $Q$ corresponds to an end-of-the-world (ETW) brane, which replaces
the left/right asymptotic boundary of a two-sided black hole. In [10], the
authors first show that the AdS Reissner-Nordström (RN) black holes with a
pure tension brane are Euclidean sensible for proper parameters by imposing
Neumann boundary condition on the ETW brane222They also show that for a given
tension and a positive preparation time, large charged black holes are
dominate in the Euclidean path integral of gravity.. They further show that
for the same parameters, gravity is locally localized on the brane. This gives
a first example to holographic description of cosmology. However, they
mispointed out that the mixed boundary condition leads to the same results as
the Neumann boundary condition. As a consequence, they claimed that the above
results are not attainable in AdS-Schwarzschild black holes with a pure
tension brane because a sensible Euclidean solution mutually excluded with
localization of gravity on the brane.

In this paper, we will show that the above mixed boundary condition gives new
solutions to motion of the ETW brane, which was characterized by one
integration constant $q$, referred to as dark charge. This is a general
feature of the boundary condition, which does not rely on the black hole
background. As a simple example, we show that by tuning the dark charge $q$,
one can obtain both a sensible Euclidean solution and gravity localization on
the brane for AdS-Schwarzschild black holes with a pure tension brane.

##  3 The end-of-the-world brane in AdS

Consider a two-sided static spherically symmetric AdS black hole of the form

$ds^{2}=-f(r)dt^{2}+{\frac{dr^{2}}{f(r)}}+r^{2}d\Omega_{d-1}^{2}\,,$ (5)

and introduce an end-of-the-world (ETW) brane (spherically symmetric) on the
left side. The wormhole geometry with a ETW brane replacing the left
asymptotic boundary is dual to an excited pure state of the CFT living in the
right asymptotic boundary. The motion of the brane is characterized by
$r=r(t)$, which encodes the evolution of FLRW universe.

However, a faithful holographic description for braneworld cosmology demands
that the dual CFT state is well defined, at least in the Hilbert space of
states. Let us turn to analyze Euclidean CFT, which is dual to Euclidean black
holes, using imaginary time $\tau=it$. The boundary state will evolve for an
finite amount of time $\tau_{0}$, referred to as preparation time

$|\Psi\rangle=e^{-\tau_{0}H}|B\rangle\,,$ (6)

where the ETW brane can be viewed as bulk extension of the past boundary of
the CFT, corresponding to the past boundary condition in the Euclidean path
integral. Clearly, the preparation time, which is a property of the CFT state,
should be positive definite. Otherwise, the state will be non-normalizable and
hence is not in the Hilbert space.

In the bulk, the total range of the Euclidean time $\tau$ is given by the the
inverse temperature $\beta$ of the black hole. However, for a given
preparation time $\tau_{0}>0$, the brane trajectory excises part of this total
range of time: it begins to contract from the asymptotic boundary at
$\tau=-\tau_{0}$, reaches a minimal radius $r=r_{0}$ and then returns to the
asymptotic boundary again at $\tau=\tau_{0}$, see Fig. 1. Hence, the CFT state
is defined only in the interval $\tau\in[-\tau_{0}\,,\tau_{0}]$. The
preparation time can be evaluated as

$\tau_{0}=\beta/2-\Delta\tau\,,$ (7)

where $\Delta\tau$ is the Euclidean time associated to half trajectory of the
brane.

###  3.1 Euclidean analysis

Figure 1: Euclidean brane trajectory (red). The brane contracts from the
asymptotic boundary at $\tau=-\tau_{0}$, reaches a minimal radius $r=r_{0}$
and then returns to the asymptotic boundary again at $\tau=\tau_{0}$. The
shaded region is excised by the brane.

Let us study the motion of the ETW brane and calculate the preparation time
explicitly. The unit dual normal vector of the brane is given by

$n_{\mu}=N(-r^{\prime}\,,1\,,\mathbf{0})\,,\qquad
N=\sqrt{{\frac{f(r)}{f^{2}(r)+(r^{\prime})^{2}}}}\,,$ (8)

where a prime denotes the derivative with respect to $\tau$. The induced
metric on the brane turns out to be

$ds^{2}_{\mathrm{ETW}}\equiv
h_{ab}dx^{a}dx^{b}=N^{-2}d\tau^{2}+r^{2}d\Omega_{d-1}^{2}\,.$ (9)

The extrinsic curvature can be calculated from the definition
$K_{ab}=e^{\mu}_{a}e^{\nu}_{b}\nabla_{\mu}n_{\nu}$, where
$e^{\mu}_{a}=dx^{\mu}/dy^{a}$ and the embedding function is
$x^{\mu}=x^{\mu}(y^{a})$. With $e^{\mu}_{\tau}=(1\,,r^{\prime}\,,\mathbf{0})$,
one finds

$K_{\tau\tau}=N\Big{(}-r^{\prime\prime}+{\frac{3f_{r}(r)}{2f(r)}}(r^{\prime})^{2}+{\frac{f(r)f_{r}(r)}{2}}\Big{)}\,,\qquad
K_{ij}={\frac{Nf(r)}{r}}h_{ij}\,,$ (10)

where $f_{r}=df/dr$ and $i\,,j$ labels the angular coordinates. According to
the mixed boundary condition (4), one finds333There are a variety of choices
for the mixed boundary condition imposed on the brane, for example
$C^{ab}\Big{[}K_{ab}-\big{(}K-(d-1)T\big{)}h_{ab}\Big{]}=0\,,$ (11) where
$C^{ab}$ is a projector. In the main text, we study the simplest case
$C_{ab}=h_{ab}$. Different choices of the boundary condition is discussed in
the Appendix A.

$r^{\prime\prime}-{\frac{3f_{r}(r)}{2f(r)}}(r^{\prime})^{2}-{\frac{f(r)f_{r}(r)}{2}}-{\frac{(d-1)f(r)}{N^{2}r}}+{\frac{dT}{N^{3}}}=0\,.$
(12)

This is a second order nonlinear ODE, describing the motion of the brane.
However, using the following relations

$r^{\prime}=\pm\sqrt{{\frac{f(r)}{N^{2}}}-f^{2}(r)}\,,\qquad
r^{\prime\prime}={\frac{1}{2}}\partial_{r}\Big{(}{\frac{f(r)}{N^{2}}}-f^{2}(r)\Big{)}\,,$
(13)

the above equation simplifies to

$0=\sqrt{g}\,\big{(}K-dT\big{)}=\partial_{r}\big{(}f(r)r^{d-1}N-Tr^{d}\big{)}\,.$
(14)

It leads to a first integral

$f(r)r^{d-1}N-Tr^{d}=q^{d-1}\,,$ (15)

where $q$ is an integration constant, referred to as dark charge. We deduce

${\frac{dr}{d\tau}}=\pm{\frac{f(r)}{\varphi(r)}}\sqrt{f(r)-\varphi(r)^{2}}\,,\qquad\varphi(r)=Tr+\big{(}{\frac{q}{r}}\big{)}^{d-1}\,,$
(16)

where the $\pm$ sign corresponds to expansion or contraction of the brane
($\tau$ is an increasing or decreasing function of $r$). We are aware of that
the above solution includes the result of Neumann boundary condition [10] as a
special case $q=0$. However, to have a sensible Euclidean solution, one needs
a very large $q$: $q/r_{H}\gg 1$, as will be shown later. This partly explains
why the results in [10] are not attainable for AdS-Schwarzschild black holes.

The inversion point of the ETW brane $r_{0}$ is determined by

$f(r_{0})=\varphi^{2}(r_{0})\,,$ (17)

where $r_{0}$ stands for the minimal real root of the equation. Gravity
localization on the brane demands that the brane should be far away from the
event horizon: $r_{0}/r_{H}\gg 1$. Otherwise, gravitons will leak into the
bulk, fall into the horizon and hence delocalize the gravity. We may take
$r_{0}$ close to the asymptotic AdS boundary such that
$\varphi(r_{0})\rightarrow Tr_{0}$. This implies $T^{2}<1/L^{2}$, where $L$ is
the AdS radius. We shall focus on a positive tension brane $0<T<1/L$ since
physical meaning of the solutions with a negative tension brane are unclear
[10, 13].

In the following, we consider the simplest case: AdS-Schwarzschild black holes

$f(r)={\frac{r^{2}}{L^{2}}}+1-{\frac{2\mu}{r^{d-2}}}\,,$ (18)

where $\mu$ is the mass parameter of the black hole (the black hole mass is
$M={\frac{(d-1)\Omega_{d-1}}{8\pi G}}\,\mu$). In Fig. 2, we plot the ratio
$r_{0}/r_{H}$ as a function of the brane tension. The brane with a near
critical tension $T\rightarrow T_{crit}=1/L$ can have a very large ratio
$r_{0}/r_{H}$ by increasing the dark charge $q$. For example, for
$r_{H}=100\,,T=0.99999\,,q=200000$, $r_{0}/r_{H}\sim 200$. For larger black
holes with the same tension brane, the dark charge will be larger to have the
same ratio. This is consistent with naive expectations since gravity
localization becomes more difficult for larger black holes and hence needs a
larger $q$.

Explicitly, the preparation time is given by

$\tau_{0}={\frac{\beta}{2}}-\int_{r_{0}}^{\infty}dr\,{\frac{\varphi(r)}{f(r)\sqrt{f(r)-\varphi^{2}(r)}}}\,.$
(19)

The numerical results are shown in Fig. 3. For a given $q$, the preparation
time is a decreasing function of the brane tension, and is positive definite
for a non-critical brane, see the left panel. On the other hand, for a near
critical brane, a positive $\tau_{0}$ demands a sufficiently large $q$,
depending on the black hole sizes, as shown in the right panel.

It is worth emphasizing that one can always find parameters $(T\,,r_{H}\,,q)$
allowing both a large ratio $r_{0}/r_{H}\gg 1$ and a positive preparation
time. We may expect that for AdS-Schwarzschild black holes, the boundary CFT
state is well defined and is consistent with locally localization of gravity
on the ETW brane. We will show this explicitly in section 4.

Figure 2: The minimal radius $r_{0}$ is an increasing function of the brane
tension $T$. We have set $L=1$, $r_{H}=100\,,q=200000$.

Last but not least, we show in the Appendix B that for a positive preparation
time $\tau_{0}$, the large black hole solution is the dominant phase, in the
thermodynamic ensemble with a fixed dark charge $q$.

###  3.2 Lorentian analysis

Figure 3: The preparation time $\tau_{0}$ as a function of $T$ (left panel) or
$q$ (right panel) for a large black hole with $r_{H}=100$.

In real time, the CFT state can be obtained by analytically continuing
$\tau\rightarrow it$ in Eq.(6). Correspondingly, the bulk geometry is the
maximally extended AdS-Schwarzschild black holes with the ETW brane cutting
off part of the left asymptotic region. However, changing to Lorentzian
signature will flip the sign of the square root in Eq.(16). It implies that
the minimum radius in Euclidean signature now becomes the maximal one. Hence,
the brane contracts/expands in the bulk interior $r\leq r_{0}$, crossing the
horizon and reaching/leaving the singularity. This is the physical picture
relevant to braneworld cosmology.

Defining the proper time $d\lambda^{2}=\Big{(}f(r)-{\frac{r^{\prime
2}(t)}{f(r)}}\Big{)}dt^{2}$, the reduced metric on the brane can be written as

$ds^{2}_{\mathrm{ETW}}=-d\lambda^{2}+r^{2}(\lambda)d\Omega^{2}_{d-1}\,.$ (20)

It corresponds to a FLRW universe with curvature $K=0$. The bulk radial
coordinate plays the role of the scale factor in cosmology. From the motion of
the brane, it is straightforward to derive the Friedmann equation. For
Schwarzschild black holes, one finds

$\Big{(}{\frac{\dot{r}}{r}}\Big{)}^{2}=\Big{(}T^{2}-1/L^{2}\Big{)}-{\frac{1}{r^{2}}}+{\frac{2\big{(}\mu+Tq^{d-1}\big{)}}{r^{d}}}+{\frac{q^{2(d-1)}}{r^{2d}}}\,,$
(21)

where a dot denotes the derivative with respect to the proper time. It implies
that comoving observer interprets the motion of the brane as expansion or
contraction of a Big Bang-Big Crunch FLRW cosmology. In $d=4$ dimension, this
describes a radiation dominated universe with a (very small) negative
cosmological constant $3(T^{2}-1/L^{2})$ and energy density proportional to
the black hole mass plus the contribution from the dark charge. It is
remarkable that the dark charge $q$ gives an extra contribution to the energy
density which is naively not visible to bulk observers. However, observers on
the brane can definitely find this part of energy via cosmological
measurements (This can be said even more precisely by calculating the energy
measured by observers on the brane directly [14, 15, 16, 17]. However, we
prefer explaining the energy terms in the Friedmann equation using the stress
tensor in the boundary CFT, see the next subsection).

In the AdS/CFT correspondence, black hole mass is usually interpreted as total
energy in the boundary theory. However, the above Friedmann equation implies
that this may not be true in braneworld cosmology. The black hole mass is only
a part of the total energy driving the cosmological evolution. This inspires
us that if one interprets the black hole mass as visible matter in our
universe, the dark charge may be identified with dark matter in FLRW
cosmology. However, this holographic argument neither gives any hint on
microstructures of dark matter nor explains its gravitational effects in
galactic scales.

Notice that the last term in the Friedmann equation is unconventional. It was
not found in earlier literature on braneworld cosmology. However, this term
disappears for a different mixed boundary condition. In this case, the dark
charge plays completely the same role of black hole mass, see the Appendix A.

###  3.3 Boundary stress tensor

To examine the physical meaning of the dark charge in the boundary CFT, we
would like to evaluate the stress tensor in the boundary in Euclidean
signature. As usual, we needs renormalizing the gravitational action by
introducing proper counter terms. In the $d=4$ dimensional case, the
renormalized action is given by

$\displaystyle I_{ren}$ $\displaystyle=$ $\displaystyle-{\frac{1}{16\pi
g}}\int_{N}\sqrt{g}\,\big{(}R-2\Lambda\big{)}-{\frac{1}{8\pi
G}}\int_{ETW}\sqrt{h}\,\big{(}K-(d-1)T\big{)}$ (22)
$\displaystyle-{\frac{1}{8\pi
G}}\int_{M}\sqrt{\gamma}\,\big{(}K_{M}-3-{\frac{1}{4}}R_{M}\big{)}-{\frac{1}{8\pi
G}}\int_{P}\sqrt{\sigma}\,\big{(}\theta-\theta_{0}-{\frac{1}{2}}K_{P}-\alpha
R_{P}\big{)}\,,$

where $K_{M}\,,R_{M}$ are extrinsic and intrinsic curvatures of the asymptotic
boundary $M$ respectively. $K_{P}$ is the extrinsic curvature of $P$, which is
introduced as the Gibbons-Hawking-York (GHY) surface term for $R_{M}$.
$\theta_{0}=\theta|_{r\rightarrow\infty}$ and
$\alpha={\frac{T}{4\sqrt{1-T^{2}}}}$. Variation of the action yields

$\delta
I_{ren}=\int_{N}E.O.M-{\frac{1}{2}}\int_{M}T^{M}_{\mu\nu}\delta\gamma^{\mu\nu}-{\frac{1}{2}}\int_{Q}T^{Q}_{\mu\nu}\delta
h^{\mu\nu}-{\frac{1}{2}}\int_{P}T^{P}_{ab}\delta\sigma^{ab}\,,$ (23)

where

$\displaystyle T^{M}_{\mu\nu}={\frac{1}{8\pi
G}}\Big{[}K^{M}_{\mu\nu}-\big{(}K_{M}-3\big{)}g^{M}_{\mu\nu}-{\frac{1}{2}}\mathcal{G}^{M}_{\mu\nu}\Big{]}\,,$
$\displaystyle T^{Q}_{\mu\nu}={\frac{1}{8\pi
G}}\Big{[}K_{\mu\nu}-\big{(}K-3T\big{)}g_{\mu\nu}\Big{]}\,,$ $\displaystyle
T^{P}_{ab}=-{\frac{1}{8\pi
G}}\Big{[}{\frac{1}{2}}\big{(}K^{P}_{ab}-K_{P}\sigma_{ab}\big{)}+\big{(}\theta-\theta_{0}\big{)}\sigma_{ab}+2\alpha\,\mathcal{G}_{ab}^{P}\Big{]}\,,$
(24)

where $\mathcal{G}^{M}_{\mu\nu}\,,\mathcal{G}^{P}_{ab}$ are the Einstein
tensor of $M$ and $P$ respectively. The result for $T^{M}_{\mu\nu}$ is the
usual holographic stress tensor of the dual CFT living on the manifold $M$
without extra boundaries. However, now the manifold $M$ has a past boundary
and generally there exists a nonzero stress tensor on $P$, depending on the
boundary conditions.

Evaluation of the stress tensors on $Q$ yields ( we set $G=1/8\pi$)

$T_{\tau\tau}^{Q}=-{\frac{3q^{3}}{r^{4}}}{\frac{f^{2}(r)}{\phi^{2}(r)}}\,,\qquad
T_{ab}^{Q}={\frac{q^{3}}{r^{2}}}\,\zeta_{ab}\,,$ (25)

where $\zeta_{ab}$ is the metric of a unit $\mathbb{S}^{3}$. It is clear that
for $q=0$, corresponding to the Neumann boundary condition, there exists no
energy flux on the ETW brane, as expected. On the other hand, on the past
boundary $P$, the vev of the stress tensor can be read off as

$\langle
T_{ab}^{P}\rangle=\lim_{r\rightarrow\infty}r^{2}T_{ab}^{P}=\Big{(}{\textstyle{\frac{\scriptstyle
T(3-2T^{2})}{\scriptstyle 8(1-T^{2})^{3/2}}}}+{\textstyle{\frac{\scriptstyle
q^{3}+\mu T}{\scriptstyle\sqrt{1-T^{2}}}}}\Big{)}\zeta_{ab}\,.$ (26)

It implies that the mixed boundary condition imposed on the ETW brane will
increase the stress tensor $\langle T^{P}_{ab}\rangle$ on the past boundary
$P$. In particular, the dark charge $q$ plays the same role of the black hole
mass, similar to the situation in the Friedmann equation, despite that it
never contributes to the bulk stress tensor $\langle T_{\mu\nu}^{M}\rangle$.
This supports our previous argument in part.

##  4 Gravity localization

We are left to show gravity localization: whether observers on the brane see
approximately four dimensional gravity. In the Randall-Sundrum II model [21],
four dimensional gravity is reproduced on a Minkowski brane embedded in a
warped AdS5 spacetime due to the graviton zero-mode that is bound on the
brane. The bound mode is lost in the presence of a black hole in the bulk
[22]. Instead, a resonant quasi bound mode persists if the brane is static and
far from the black hole horizon [23]. This quasi bound mode is meta-stable
because of a finite life time. Hence, localization of gravity on the brane is
a local effect, valid to spatial and time scales smaller than the cosmological
ones.

In [23], the authors focused on small AdS Schwarzschild black holes with
$r_{H}<L$. In this case, there exists a potential well for graviton wave
function outside the event horizon (see Fig. 4), leading to a discrete set of
quasi bound modes, referred to as overtones. On the other hand, for large
black holes $r_{H}>L$, the potential well disappears and there is only a
fundamental quasi bound mode. It was argued in [23] that this fundamental mode
approaches to the RS II bound mode when the brane is close to asymptotic
infinity, irrespective of the black hole sizes. That is in this limit the
imaginary part of the frequency of the quasi bound mode vanishes whereas its
real part tends to the frequency
$\omega_{GR}=\sqrt{\ell(\ell+2)f(y_{b})}/y_{b}$, expected for a four-
dimensional gravitational perturbation of the Einstein static universe.

However, this topic deserves further investigations in current paper. There
are two reasons. The first is the brane is static in [23] while in our case it
moves dynamically. It is highly nontrivial to implement a proper boundary
condition in this case. The second is the above limiting behavior of the
fundamental quasi bound mode was criticized in [10], in which the authors
found that when the brane is sufficiently far from the black hole horizon, the
quasi bound mode becomes very short-lived and hence gravity localization is
lost. We will re-examine this issue using both numerical and half-analytical
methods.

###  4.1 Master equation and boundary condition

Consider a linear perturbation around a AdS-Schwarzschild black hole $\delta
g_{\mu\nu}=g_{\mu\nu}-g^{(0)}_{\mu\nu}$. The perturbation can be decomposed
into scalar, vector and tensor components [24]. The latter corresponds to
graviton mode of interest which has $\delta g_{r\mu}=0=\delta g_{t\mu}$. As
usual, we work in transverse-traceless gauge $\delta
g^{\mu}_{\,\,\,\,\mu}=0=\nabla^{\mu}\delta g_{\mu\nu}$. Then the graviton mode
can be expanded as ($i\,,j=1\,,2\,,\cdots\,,d-1$)

$\delta
g_{ij}(t\,,r\,,x^{i})=\sum_{\omega\,,k}r^{{\frac{5-d}{2}}}\psi(r)e^{i\omega
t}\,Y_{ij}^{(k)}(x^{i})\,,$ (27)

where $Y_{ij}^{(k)}$ stands for hyperspherical harmonics, satisfying
$\Delta_{d-1}Y_{ij}^{(k)}=-k^{2}Y_{ij}^{(k)}$, where $\Delta_{d-1}$ is the
Laplacian operator on the unit $(d-1)$-sphere and $k^{2}=\ell(\ell+d-2)-2$,
with $\ell=1\,,2\,,\cdots$ is the generalized angular momentum.

It was established in [24] that using the above expansion, the linearized
Einstein equation for each tensor mode of the perturbations can be recast into
a one-dimensional Schrödinger equation for $\psi(r)$. For later convenience,
we work directly with adimensional coordinates and parameters. We introduce
$y=r/r_{h}\,,\tilde{t}=t/r_{h}\,,\gamma=L/r_{h}$ and define the adimensional
tortoise coordinate $y_{*}(y)=-\int_{y}^{\infty}d\tilde{y}/f(\tilde{y})$,
where we have chosen $y_{*}(\infty)=0$. The Schrödinger equation for the wave
function reads

$-{\frac{d^{2}}{dy_{*}^{2}}}\psi(y_{*})+V_{k}(y_{*})\psi(y_{*})=\omega^{2}\psi(y_{*})\,,$
(28)

where both the potential $V_{k}$ and the frequency $\omega$ are dimensionless,
normalized by the horizon radius. One has

$V_{k}(y_{*})=f(y)\Big{[}{\frac{d^{2}-1}{4\gamma^{2}}}+{\frac{4k^{2}+d^{2}-4d+11}{4y^{2}}}+{\frac{(d-1)^{2}\big{(}1+{\textstyle{\frac{\scriptstyle
1}{\scriptstyle\gamma^{2}}}}\big{)}}{4y^{d}}}\Big{]}\,,$ (29)

where for AdS-Schwarzschild black holes

$f(y)=1-{\frac{1+{\textstyle{\frac{\scriptstyle
1}{\scriptstyle\gamma^{2}}}}}{y^{d-2}}}+{\frac{y^{2}}{\gamma^{2}}}\,.$ (30)

Figure 4: The potential $V_{1}$ for $d=4$ dimensional black holes with angular
momentum $\ell=1$. The left panel is for large black holes with $\gamma=0.01$
whereas the right panel is for small black holes with $\gamma=10$.

In $d=4$ dimension, the tortoise coordinate can be solved as

$y_{*}={\textstyle{\frac{\scriptstyle\gamma^{2}}{\scriptstyle
2\big{(}\gamma^{2}+2\big{)}}}}\Big{[}\log{\big{(}{\textstyle{\frac{\scriptstyle
y-1}{\scriptstyle
y+1}}}\big{)}}+2\sqrt{\gamma^{2}+1}\,\,\mathrm{arctan}\Big{(}{\textstyle{\frac{\scriptstyle
y}{\scriptstyle\sqrt{\gamma^{2}+1}}}}\Big{)}-\pi\sqrt{\gamma^{2}+1}\,\Big{]}\,.$
(31)

Close to the event horizon $y\rightarrow 1$, $y_{*}\rightarrow-\infty$, the
potential vanishes exponentially as $V_{k}(y_{*})\sim e^{2\kappa y_{*}}$,
where $\kappa=1+2/\gamma^{2}$ is the surface gravity of black holes, whereas
at asymptotic infinity $y\rightarrow\infty\,,y_{*}\rightarrow 0$, the
potential diverges in a power-law $V_{k}(y)\sim 15/4y_{*}^{2}$. This is
universal to both small and large black holes. However, for large black holes
$\gamma<1$, there is only one meta-stable mode while for small black holes
$\gamma>1$, there exists an overtones for the quasi bound mode due to
existence of the potential barrier outside the event horizon, see Fig. 4.

The quasinormal modes of small black holes were carefully studied in [23] for
AdS-Schwarzschild black holes with a Einstein static brane. However, in our
case, the ETW brane moves dynamically. It is unclear how to impose a proper
boundary condition on the brane $y_{*}=y_{b*}$ in this case. Inspired by [10],
we adopt adiabatic approximation: taking the ETW brane as quasi-static with
respect to the time scale of the gravitational perturbation. Precisely
speaking, if the quasinormal mode frequency is given by
$\omega=\bar{\omega}+i\Gamma/2$, then the oscillation time
$t_{o}=1/\bar{\omega}$ of the perturbation should be much smaller than the
“Hubble time” of the brane $T_{H}=y(t)/y^{\prime}(t)$, namely

$t_{o}\ll T_{H}={\frac{y\,\phi(y)}{f(y)\sqrt{\phi^{2}(y)-f(y)}}}\,,$ (32)

where we have used the Lorential version of (16). We will first compute the
quasinormal mode frequency and then test the above condition in the final.

For a static brane with $y=y_{b}$, the dual normal vector is given by
$n_{\mu}=(dy)_{\mu}/\sqrt{f(y)}$ and $e^{\mu}_{a}=\delta^{\mu}_{a}$.
Evaluation of the linearized extrinsic curvature, one finds from the mixed
boundary condition (4)

$\partial_{y_{*}}\psi(y_{*})\Big{|}_{y_{*}=y_{*}^{b}}={\frac{(d-1)f(y)}{2y}}\,\psi(y_{*})\,.$
(33)

This is exactly the same as the boundary condition imposed in [10]. However,
derivation of the condition in [10] further required that $y^{\prime
2}/f^{2}(y)\ll 1$, which guarantees that interpretation of gravitons for a
bulk observer is equivalent to that of a comoving observer on the brane.
However, in our case, we do not need it any longer.

As usual, we impose ingoing boundary condition on the event horizon

$\psi(y_{*})|_{y\rightarrow 1}\sim e^{i\omega y_{*}}\,.$ (34)

The quasinormal mode frequencies are generally discrete since we impose two
boundary conditions.

###  4.2 Numerical results

Figure 5: The squared trapping coefficient $\xi(\omega)$ for $d=4$ dimensional black holes with $\gamma=0.01$ and angular momentum $\ell=1$. One has $\omega_{GR}=173.205$. Left panel: $y_{b}=34.12$. $\bar{\omega}=173.143\,,\Gamma=17.181$. Middle panel: $y_{b}=67.69$. $\bar{\omega}=173.194\,,\Gamma=4.366$. Right panel: $y_{b}=149.41$. $\bar{\omega}=173.202\,,\Gamma=0.896$. As the brane stays farther from the black hole, the peak becomes higher and the width becomes narrower. In the limit $y_{b}\rightarrow\infty$, $\bar{\omega}\rightarrow\omega_{GR}$ and $\Gamma\rightarrow 0$, the mode reduces to the RS II bound mode. $y_{b}$ | $t_{o}$ | $t_{d}$ | $T_{H}$ | $\lambda(y_{b})/\lambda_{tot}$
---|---|---|---|---
60 | $4.997601\cdot 10^{-6}$ | 0.1406596 | $4.549750\cdot 10^{-5}$ | 0.7689702
70 | $4.997576\cdot 10^{-6}$ | 0.1914732 | $5.679376\cdot 10^{-5}$ | 0.6770077
80 | $4.997564\cdot 10^{-6}$ | 0.2482983 | $7.362785\cdot 10^{-5}$ | 0.5603646
90 | $4.997551\cdot 10^{-6}$ | 0.3154222 | $1.085232\cdot 10^{-4}$ | 0.4007155
99 | $4.997541\cdot 10^{-6}$ | 0.3901243 | $3.526517\cdot 10^{-4}$ | 0.1277716
99.9 | $4.997538\cdot 10^{-6}$ | 0.3900052 | $1.117740\cdot 10^{-3}$ | 0.0404218
100 | $4.997537\cdot 10^{-6}$ | 0.3908119 | $\infty$ | 0
Table 1: Time scales comparison for $d=4$ dimensional black holes with
$\gamma=0.01$ and angular momentum $\ell=2000$. We have set
$L=1\,,T=0.9999\,,q=10000$ so that the maximum radius of the brane is
$y_{b}^{max}=100$. The second and third columns report the oscillation time
$t_{o}=1/\bar{\omega}$ and decay time $t_{d}=2/\Gamma$ (or the lift time) of
the quasi bound mode. The last column gives the ratio of the proper time
$\lambda(y_{b})$ needed for the brane to expand from $y_{b}$ to $y_{b}^{max}$
to the total proper time needed to complete the trajectory of expansion.

We are interested in the large black hole case $\gamma\ll 1$, which has only
one meta-stable mode. We adopt trapping coefficient method [23] to extract the
frequency. For self-consistency, we review the method in the Appendix C. It
turns out that the real and imaginary parts of the quasi bound mode can be
read off from the squared trapping coefficient $\xi(\omega)$, which takes the
form of

$\xi(\omega)=R^{2}(\omega){\frac{\,\Gamma_{n}^{2}}{4(\omega-\bar{\omega}_{n})^{2}+\Gamma_{n}^{2}}}\,,$
(35)

where $R(\omega)$ is a slowly varying function of $\omega$. The real part of
the frequency of the quasi bound mode corresponds to the Lorentzian (Breit-
Wigner) peak while half of its imaginary part $\Gamma/2$ gives rise to half-
width of the peak.

In Fig. 5, we show the squared trapping coefficient for four dimensional black
holes with $\gamma=0.01$ and angular momentum $\ell=1$. As the location of the
brane $y_{b}$ increases, the Breit-Wiger peak becomes narrower and higher,
implying that localization of gravity becomes more efficient for a larger
radius brane. On the other hand, fixing $y_{b}$ and $\gamma$, we find that for
larger angular momentum $\ell$, the gravity localization becomes more
efficient as well. This is in accordance with a effective description of
gravity on the brane locally since gravitons with larger angular momentum
$\ell$ probe shorter distance scales. The shorter the spatial scale of
gravitational perturbation is, the better the graviton mode is bound on the
brane. Our numerical results are consistent with [23]: in the limit
$y_{b}\rightarrow\infty$, $\bar{\omega}\rightarrow\omega_{GR}$,
$\Gamma\rightarrow 0$, the quasi bound mode reduces to the RS II bound mode
when the brane approaches to asymptotic infinity. In the Appendix C, we study
the limiting behavior of the quasi bound mode half-analytically using the
large dimensional limit. The result again supports our numerics.

In Table 1, we compare time scales for a large black hole $\gamma=0.01$ in
$d=4$ dimensions. For a given brane tension and dark charge, the adiabatic
approximation is valid very well near the inversion point of the brane
trajectory irrespective of the angular momentum. For a larger angular
momentum, for example $\ell=2000$, it holds also for a significant portion in
the trajectory of the brane. For example, if one accepts $t_{o}\sim T_{H}/10$
as the threshold for the validity of the approximation, then it holds for more
than $70\%$ in the brane trajectory. The larger the angular momentum is, the
more portion of the brane trajectory the approximation is valid to. We also
observe that the decay time (or lift time) $t_{d}=2/\Gamma$ of the quasi bound
mode is much larger than the Hubble time. This implies that our analysis will
break down before the quasi bound mode leaks into the bulk.

##  5 Conclusion

In this paper, we study a holographic model of braneworld cosmology proposed
in [10]. The main new contributions we made in this work is proving that the
model can be extended to general black hole backgrounds while the results in
[10] are limited to charged black holes.

We focus on the simplest setup of AdS-Schwarzschild black holes with a pure
tension end-of-the-world (ETW) brane. By imposing a mixed boundary condition
on the brane, we find that for a given tension, the motion of the brane is
characterized by an integration constant $q$, referred to as dark charge.
Tuning the dark charge, we show that in Euclidean signature the parameters
allowing a large radius brane far from the black hole horizon are compatible
with those admitting a positive preparation time. The latter is a least
condition for sensible Euclidean solutions, which are dual to well-defined CFT
states. Furthermore, by numerically evaluating the frequency of the quasi
bound mode, we show that gravity is locally localized on the brane for spatial
and times scales smaller than the cosmological ones. Therefore, sensible
Euclidean solutions and localization of gravity on the brane can accommodate
with each other in our setup.

Our analysis can be easily generalised to different black hole backgrounds.
Since the dark charge plays an essential role in a consistent holographic
description of braneworld cosmology, it is of great importance to explore its
physical meaning. There are two hints:
$\bullet$ In Euclidean signature, the dark charge $q$ contributes to a part of
the stress tensor on the past boundary of the CFT. This is similar to black
hole mass. However, it never contributes to the bulk stress tensor.
$\bullet$ In real time, physical effects of $q$ was encoded in the motion of
the brane, which corresponds to evolution of FLRW cosmology. By deriving the
Friedmann equation, we find that the dark charge gives an exra contribution to
the energy density in FLRW cosmology. Again, this is similar to the black hole
mass.

Based on these observations, we argue that if the black hole mass is
interpreted as visible matter in our universe, the dark charge may be
identified with dark matter in cosmology. However, this neither gives any hint
on the microstructures of dark matter nor explains its gravitational effects
in galactic scales. Despite that the argument is far from a consistent story,
it still provides a new possibility to study dark matter in holography.

## Acknowledgments

Z.Y. Fan was supported in part by the National Natural Science Foundations of
China with Grant No. 11805041 and No. 11873025.

##  Appendix A  More discussions on mixed boundary conditions

In general, one can impose a mixed boundary condition on the ETW brane as

$C^{ab}\Big{[}K_{ab}-\big{(}K-(d-1)T\big{)}h_{ab}\Big{]}=0\,,$ (36)

where $C_{ab}=\lambda_{1}h_{ab}+\lambda_{2}K_{ab}+\lambda_{3}R_{ab}+\cdots$.
It was argued in [19, 20] that this general choice of boundary condition does
not produce correct boundary Weyl anomaly for general shapes of the boundary
$P$. However, this does not effect our discussions since in our case $P$ is a
past boundary so that its spatial curvature and the boundary Weyl anomaly are
trivial.

We first work in Euclidean signature. The unit dual normal vector of the brane
is still given by (8), namely

$n_{\mu}=N(-r^{\prime}\,,1\,,\mathbf{0})\,,\qquad
N=\sqrt{{\frac{f(r)}{f^{2}(r)+(r^{\prime})^{2}}}}\,.$ (37)

By letting $N\equiv\varphi(r)/f(r)$, one finds

${\frac{dr}{d\tau}}=\pm{\frac{f(r)}{\varphi(r)}}\sqrt{f(r)-\varphi(r)^{2}}\,,$
(38)

where the function $\varphi(r)$ should be solved from the boundary condition.
We are interested in the case of

$C_{ab}=\alpha T\,h_{ab}+K_{ab}\,,$ (39)

where $\alpha$ is an adimensional constant. Without loss of generality, we
take $0\leq\alpha<+\infty$, where the limit $\alpha\rightarrow+\infty$
corresponds to the simplest case $C_{ab}=h_{ab}$ studied in the main text.

Figure 6: Left panel: the minimum radius for a $d=4$ dimensional black hole
with $r_{H}=100\,,T=0.9\,,q=8\times 10^{7}$. Right panel: the preparation time
for the same black hole with $T=0.999\,,q=2\times 10^{6}$. Both $r_{0}$ and
$\tau_{0}$ are decreasing functions of $\alpha$.

It follows that

$\displaystyle 0$ $\displaystyle=$ $\displaystyle\sqrt{g}\,\big{(}\alpha
T\,h^{ab}+K^{ab}\big{)}\Big{[}K_{ab}-\big{(}K-(d-1)T\big{)}h_{ab}\Big{]}$ (40)
$\displaystyle=$
$\displaystyle-(d-1)\partial_{r}\Big{[}r^{d-2}\big{(}\varphi(r)-Tr\big{)}\big{(}\varphi(r)+\alpha
T\,r\big{)}\Big{]}\,.$

Again, we obtain a first integral

$r^{d-2}\big{(}\varphi(r)-Tr\big{)}\big{(}\varphi(r)+\alpha T\,r\big{)}\equiv
q^{d-2}\,,$ (41)

where $q$ is an integration constant, still referred to as dark charge. We
deduce

$\varphi(r)={\frac{1}{2}}\Big{(}\sqrt{T_{+}^{2}r^{2}+{\textstyle{\frac{\scriptstyle
q^{d-2}}{\scriptstyle r^{d-2}}}}}+T_{-}r\Big{)}\,,$ (42)

where $T_{\pm}=T(1\pm\alpha)$. The brane tension is bounded above as $T<1/L$
since $\varphi(r)\rightarrow Tr$ in the asymptotic limit. Notice that the
above solution cannot reduce to (16) in the limit $\alpha\rightarrow+\infty$
(but one can rediscover the solution directly from (41) despite that the dark
charge cannot be identified with one another). However, it turns out that for
a given $\alpha$, basic features of the above solution, such as the minimum
radius $r_{0}$ of the ETW brane and the preparation time $\tau_{0}$ of the
boundary theory as functions of the brane tension $T$ or the dark charge $q$
are qualitatively similar to those established in sec.3.1. Moreover,
linearizing the mixed boundary condition (36) around a Schwarzschild black
hole leads to the same boundary condition (33) on the brane for the quasi
bound modes of gravitons (in adiabatic approximation). Therefore, under the
boundary condition (36), a positive preparation time in the boundary CFT is
again compatible with locally localization of gravity on the ETW brane. In
Fig. 6, we show the minimum radius $r_{0}$ and the preparation time $\tau_{0}$
as a function of $\alpha$. Interestingly, they both decrease as $\alpha$
increases. It implies that a smaller $\alpha$ is better for realizing
braneworld cosmology.

In real time, the Friedmann equation is given by

$\Big{(}{\frac{\dot{r}}{r}}\Big{)}^{2}=\Big{(}{\textstyle{\frac{\scriptstyle
T_{+}^{2}+T_{-}^{2}}{\scriptstyle
4}}}-1/L^{2}\Big{)}-{\frac{1}{r^{2}}}+{\frac{2\mu+{\textstyle{\frac{\scriptstyle
1}{\scriptstyle
4}}}q^{d-2}}{r^{d}}}+{\frac{T_{-}}{2}}\,\sqrt{T_{+}^{2}+{\textstyle{\frac{\scriptstyle
q^{d-2}}{\scriptstyle r^{d}}}}}\,.$ (43)

Again it describes a radiation dominated universe with a negative cosmological
constant $3\big{(}T^{2}-1/L^{2}\big{)}$ and energy density proportional to the
black hole mass plus the contribution from the dark charge. In particular, for
$\alpha=1$, the Friedmann equation greatly simplifies to

$\Big{(}{\frac{\dot{r}}{r}}\Big{)}^{2}=\Big{(}T^{2}-1/L^{2}\Big{)}-{\frac{1}{r^{2}}}+{\frac{2\mu+{\textstyle{\frac{\scriptstyle
1}{\scriptstyle 4}}}q^{d-2}}{r^{d}}}\,,$ (44)

where the last unconventional term in (43) disappears. In this case, the dark
charge plays completely the same role of black hole mass in cosmology.

##  Appendix B  Action comparison

For a given ETW brane with a fixed tension $T$, a dark charge $q$ and a
preparation time $\tau_{0}$, there will be different black hole solutions
$r_{H}$, including the pure AdS. We shall compute the Euclidean action to
establish which phase has least action and hence is dominant in the path
integral. It is worth emphasizing that for a given $q\neq 0$, there exists
nonzero energy flux passing through the ETW brane. While this energy is
invisible in the bulk, it plays an indispensable role in our discussions. The
action that we would like to compare is the one for different phases with this
same “dark energy”. This defines a certain thermodynamic ensemble.

We work in Euclidean signature and focus on the metric of the form

$ds^{2}=f(r)d\tau^{2}+{\frac{dr^{2}}{f(r)}}+r^{2}d\Omega^{2}_{d-1}\,.$ (45)

The induced metric on the ETW brane is given by

$ds^{2}_{\mathrm{ETW}}=\Big{(}f(r)\big{(}{\frac{d\tau}{dr}}\big{)}^{2}+{\frac{1}{f(r)}}\Big{)}dr^{2}+r^{2}d\Omega^{2}_{d-1}\,.$
(46)

The total Euclidean action reads

$\displaystyle I$ $\displaystyle=$ $\displaystyle-{\frac{1}{16\pi
G}}\int_{N}d^{d+1}x\,\sqrt{g}\,\big{(}R-2\Lambda\big{)}-{\frac{1}{8\pi
G}}\int_{\mathrm{ETW}}d^{d}y\,\sqrt{h}\,\Big{(}K-(d-1)T\Big{)}$ (47)
$\displaystyle-{\frac{1}{8\pi
G}}\int_{M}d^{d}y\,\sqrt{\gamma}\,K-{\frac{1}{8\pi
G}}\int_{P}d^{d-1}x\,\sqrt{\sigma}\,\theta\,,$

where $M$ is the asymptotic boundary at infinity and $P=M\bigcap\mathrm{ETW}$
stands for the corners at asymptotic infinity. For the Schwarzschild like
metric (45), one always has $R-2\Lambda=-2dL^{-2}$. On the ETW brane, $K=dT$
owing to the boundary condition. Moreover, $\sqrt{g}=r^{d-1}$ and

$\sqrt{h}=r^{d-1}\sqrt{f(r)\big{(}{\frac{d\tau}{dr}}\big{)}^{2}+{\frac{1}{f(r)}}}={\frac{r^{d-1}}{\sqrt{f(r)-\varphi^{2}(r)}}}=\pm
r^{d-1}{\frac{f(r)}{\varphi(r)}}{\frac{d\tau}{dr}}\,,$ (48)

where we have used the equation of motion for the brane

${\frac{d\tau}{dr}}=\pm{\frac{\varphi(r)}{f(r)\sqrt{f(r)-\varphi^{2}(r)}}}\,.$
(49)

In addition, for $r=\mathrm{cons}$ hypersurface, the dual normal vector is
given by $\tilde{n}_{\mu}={\frac{(dr)_{\mu}}{\sqrt{f(r)}}}$ and
$\sqrt{\gamma}=\sqrt{f(r)}\,r^{d-1}$. The extrinsic curvature is evaluated as

$K\Big{|}_{r=\mathrm{cons}}={\frac{f^{\prime}(r)}{2\sqrt{f(r)}}}+{\frac{(d-1)\sqrt{f(r)}}{r}}\,.$
(50)

For the corners at asymptotic infinity, a simple calculation gives
$\theta=\arccos{T}$, which does not depend on bulk interior geometries.

In the black hole phase, one has on the brane trajectory

$\displaystyle\tau(r)=\int_{r_{0}}^{r}d\hat{r}\,{\frac{\varphi(\hat{r})}{f(\hat{r})\sqrt{f(\hat{r})-\varphi^{2}(\hat{r})}}}\,.$
(51)

Hence, the preparation time of the dual CFT is given by

$\tau_{0}={\frac{\beta}{2}}-\int_{r_{0}}^{\infty}dr\,{\frac{\varphi(r)}{f(r)\sqrt{f(r)-\varphi^{2}(r)}}}\,,$
(52)

where for Schwarzschild black holes,

$f(r)={\frac{r^{2}}{L^{2}}}+1-{\frac{r_{H}^{d-2}}{r^{d-2}}}\Big{(}{\frac{r_{H}^{2}}{L^{2}}}+1\Big{)}\,,\quad\beta={\frac{4\pi
r_{H}L^{2}}{dr_{H}^{2}+(d-2)L^{2}}}\,.$ (53)

For pure AdS, the result depends on whether $q$ is vanishing or nonvanishing.
When $q=0$, the configuration has two disconnected branches bounded by the ETW
brane. This case has been studied carefully in [13]. The conclusion is for
given $(T\,,\tau_{0})$, the largest black hole phase is dominant.

We shall focus on a nonvanishing $q\neq 0$. In this case, the brane has a
minimum radius even in pure AdS. The situation is similar to the black hole
phase except that the Euclidean time periodicity $\beta$ is chosen by hand
(precisely speaking, by matching asymptotic geometries of pure AdS with that
of black holes)

$\tau_{0}={\frac{\hat{\beta}}{2}}-\int_{\hat{r}_{0}}^{\infty}dr\,{\frac{\varphi(r)}{f_{AdS}(r)\sqrt{f_{AdS}(r)-\varphi^{2}(r)}}}\,,$
(54)

where $f_{AdS}=r^{2}/L^{2}+1$. In the following, we will set $L=1$ for
simplicity.

Black hole phase: For the black hole phase, the bulk action is given by the
total action for the Euclidean black hole (up to some UV cut-off $r_{max}$)
minus the action for the excised part. This gives

${\frac{\Omega_{d-1}}{8\pi
G}}\Big{[}\beta\int_{r_{H}}^{r_{max}}dr\,dr^{d-1}-\int_{r_{0}}^{r_{max}}dr\,dr^{d-1}\,2\tau(r)\Big{]}\,.$
(55)

The brane action is given by

$-{\frac{\Omega_{d-1}}{8\pi
G}}\int_{r_{0}}^{r_{max}}dr\,2Tr^{d-1}{\frac{f(r)}{\varphi(r)}}{\frac{d\tau}{dr}}\,.$
(56)

These two parts lead to

$\displaystyle{\frac{\Omega_{d-1}}{4\pi
G}}\Big{[}{\frac{\beta}{2}}r^{d}\Big{|}_{r_{H}}^{r_{max}}-\int_{r_{0}}^{r_{max}}dr\,\Big{(}dr^{d-1}\tau(r)+Tr^{d-1}{\frac{f(r)}{\varphi(r)}}{\frac{d\tau}{dr}}\Big{)}\Big{]}$
$\displaystyle={\frac{\Omega_{d-1}}{4\pi
G}}\Big{[}{\frac{\beta}{2}}r^{d}\Big{|}_{r_{H}}^{r_{max}}-r^{d}\tau(r)\Big{|}^{r_{max}}_{r_{0}}-\int_{r_{0}}^{r_{max}}dr\,\Big{(}{\textstyle{\frac{\scriptstyle
T}{\scriptstyle r}}}-{\textstyle{\frac{\scriptstyle\varphi(r)}{\scriptstyle
f(r)}}}\Big{)}r^{d}{\frac{f(r)}{\varphi(r)}}{\frac{d\tau}{dr}}\Big{]}\,,$ (57)

where in the second line we have used
$\big{(}r^{d}\tau(r)\big{)}^{\prime}=dr^{d-1}\tau(r)+r^{d}d\tau/dr$.

Combined with the GHY surface term evaluated at asymptotic infinity and the
corner terms, we deduce

$\displaystyle I_{BH}$ $\displaystyle=$
$\displaystyle{\frac{\Omega_{d-1}}{4\pi
GL^{2}}}\Big{[}\tau_{0}\,r^{d-1}\big{(}r-{\textstyle{\frac{\scriptstyle
f^{\prime}(r)}{\scriptstyle
2}}}-{\textstyle{\frac{\scriptstyle(d-1)f(r)}{\scriptstyle
r}}}\big{)}_{r=r_{max}}-\theta\,r_{max}^{d-1}$ (58)
$\displaystyle\qquad\qquad-{\frac{1}{2}}\beta
r_{H}^{d}-\int_{r_{0}}^{r_{max}}dr\,\Big{(}{\textstyle{\frac{\scriptstyle
T}{\scriptstyle r}}}-{\textstyle{\frac{\scriptstyle\varphi(r)}{\scriptstyle
f(r)}}}\Big{)}{\frac{r^{d}}{\sqrt{f(r)-\varphi^{2}(r)}}}\Big{]}\,.$

Pure AdS: Since $q\neq 0$, the ETW brane in pure AdS will also have a
connected configuration similar to the black hole case. The action for
Euclidean black holes (58) is valid to pure AdS as well, with all the
quantities replaced by those of thermal AdS $r_{H}\rightarrow
0\,,r_{\max}\rightarrow\hat{r}_{max}\,,r_{0}\rightarrow\hat{r}_{0}\,,f(r)\rightarrow
f_{AdS}(r)\,,\tau_{0}\rightarrow\hat{\tau}_{0}\,,\beta\rightarrow\hat{\beta}\,,\theta\rightarrow\hat{\theta}$.
One has

$\displaystyle I_{AdS}$ $\displaystyle=$
$\displaystyle{\frac{\Omega_{d-1}}{4\pi
G}}\Big{[}\hat{\tau}_{0}\,r^{d-1}\big{(}r-{\textstyle{\frac{\scriptstyle
f_{AdS}^{\prime}(r)}{\scriptstyle
2}}}-{\textstyle{\frac{\scriptstyle(d-1)f_{AdS}(r)}{\scriptstyle
r}}}\big{)}_{r=\hat{r}_{max}}$ (59)
$\displaystyle\qquad\qquad-\hat{\theta}\,\hat{r}_{max}^{d-1}-\int_{\hat{r}_{0}}^{\hat{r}_{max}}dr\,\Big{(}{\textstyle{\frac{\scriptstyle
T}{\scriptstyle r}}}-{\textstyle{\frac{\scriptstyle\varphi(r)}{\scriptstyle
f_{AdS}(r)}}}\Big{)}{\frac{r^{d}}{\sqrt{f_{AdS}(r)-\varphi^{2}(r)}}}\Big{]}\,.$

Notice that the Euclidean periodicity $\hat{\beta}$ can be chosen arbitrarily
so that the preparation time $\hat{\tau}_{0}$ matches with the black hole case
at asymptotic infinity
$\tau_{0}=\hat{\tau}_{0}|_{\hat{r}_{max}\rightarrow\infty}$.

Before explicitly evaluating the action difference, we shall point out that
for Schwarzschild black holes, the Euclidean periodicity is up bounded as
$\beta\leq\beta_{c}={\frac{2\pi L}{\sqrt{d(d-2)}}}$, where the equality
corresponds to a critical horizon $r_{H,c}=\sqrt{{\frac{d-2}{d}}}L$ and a
temperature at which the Hawking-Page transition occurs. For a given Euclidean
periodicity $\beta$, in general there exists two phases: a smaller black hole
with $r_{H}<r_{H,c}$ and a larger black hole with $r_{H}>r_{H,c}$. It follows
that the preparation time $\tau_{0}$ shares a similar behavior to $\beta$ for
given $T\,,q$: it is up bounded as $\tau_{0}\leq\tau_{0}(r_{H\,,c})$ and for a
given $\tau_{0}$ below the maximum value, there exists a smaller black hole
phase with $r_{H}<r_{H,c}$ and a larger black hole phase with $r_{H}>r_{H,c}$,
respectively. For our purpose, we are interested in the larger black hole
phase. The preparation time $\tau_{0}$ and $\beta$ are in a one-to-one map
with the horizon radius. In other words, fixing $T\,,q\,,\beta$ is equivalent
to fixing $T\,,q\,,\tau_{0}$ for the regions of parameters of interests.

To evaluate the action difference, we need match the geometries in the
asymptotic region for both cases. First, we choose the asymptotic cutoffs
$r_{max}\,,\hat{r}_{max}$ such that they match with the cutoff $z=\epsilon$ in
Fefferman-Graham coordinates

${\frac{Ldz}{z}}={\frac{dr}{\sqrt{f(r)}}}\,,$ (60)

where the integration constant in the above equation is chosen such that
$r\sim 1/\epsilon$ to leading order. It is easily found that for pure AdS in
general dimension

$\hat{r}_{max}={\frac{1}{\epsilon}}-{\frac{\epsilon}{4}}\,,$ (61)

and for $d=4$ dimensional black holes

$r_{max}={\frac{1}{\epsilon}}-{\frac{\epsilon}{4}}+{\frac{1}{8}}r_{H}^{2}(r_{H}^{2}+1)\epsilon^{3}+{\frac{1}{32}}r_{H}^{2}(r_{H}^{2}+1)\epsilon^{5}+O(\epsilon^{7})\,.$
(62)

Moreover, we need match the proper Euclidean preparation times in the
asymptotic region444For the matching conditions we imposed, the GHY surface
term evaluated at asymptotic infinity has nontrivial contributions in the
action difference. However, if one extends the relation
$\tau_{0}=\hat{\tau}_{0}$ at asymptotic infinity to any finite cutoff, the GHY
surface term will no longer contribute to the action difference. The paper
[10] has a minor mistake in this sense.

$2\tau_{0}\sqrt{f(r_{max})}=2\hat{\tau}_{0}\sqrt{f_{AdS}(\hat{r}_{max})}\,.$
(63)

The lowest order in $\epsilon$ gives (for Schwarzschild black holes)

$\hat{\tau}_{0}(\epsilon)=\tau_{0}\big{(}1-{\textstyle{\frac{\scriptstyle(d-1)\mu}{\scriptstyle
d}}}\,\epsilon^{d}\big{)}\,.$ (64)

Combining all the results above, we deduce the action difference

$\displaystyle\delta I$ $\displaystyle\equiv$ $\displaystyle{\frac{4\pi
G}{\Omega_{d-1}}}\big{(}I_{AdS}-I_{BH}\big{)}$ (65) $\displaystyle=$
$\displaystyle\tau_{0}\Big{(}(d-1)\big{(}r_{max}^{d}+r_{max}^{d-2}-\hat{r}_{max}^{d}-\hat{r}_{max}^{d-2}+{\textstyle{\frac{\scriptstyle(d-1)\mu}{\scriptstyle
d}}}\big{)}-d\mu\Big{)}$ $\displaystyle+\big{(}\theta
r_{max}^{d-1}-\hat{\theta}\hat{r}_{max}^{d-1}\big{)}+{\frac{1}{2}}\beta
r_{H}^{d}$
$\displaystyle-\int_{\hat{r}_{0}}^{\hat{r}_{max}}dr\,\Big{(}{\textstyle{\frac{\scriptstyle
T}{\scriptstyle r}}}-{\textstyle{\frac{\scriptstyle\varphi(r)}{\scriptstyle
f_{AdS}(r)}}}\Big{)}{\frac{r^{d}}{\sqrt{f_{AdS}(r)-\varphi^{2}(r)}}}$
$\displaystyle+\int_{r_{0}}^{r_{max}}dr\,\Big{(}{\textstyle{\frac{\scriptstyle
T}{\scriptstyle r}}}-{\textstyle{\frac{\scriptstyle\varphi(r)}{\scriptstyle
f_{BH}(r)}}}\Big{)}{\frac{r^{d}}{\sqrt{f_{BH}(r)-\varphi^{2}(r)}}}\,.$

Figure 7: The action difference $\delta I$ and the preparation time $\tau_{0}$
for a large black hole with $r_{H}=100$. Left panel: For a fixed charge $q$
(we choose $q=500$), the preparation time is positive for a brane tension not
too close to the critical value $T_{crit}=1/L$ and the action difference is
always positive for any $\tau_{0}\geq 0$. It implies that for a sensible
Euclidean solution, the black hole phase is dominant in the path integral.
Right panel: For a near critical tension (we choose $T=0.99999$), the
preparation time becomes positive for a sufficiently large charge $q$ and the
action difference is already positive before this happens, again implying that
the black hole phase is dominant for a sensible Euclidean solution.

It should be emphasized that the GHY surface term evaluated at asymptotic
infinity gives nonvanishing contributions.

In the $d=4$ dimension, one has

$\delta I={\frac{\pi
r_{H}^{5}}{2r_{H}^{2}+1}}+{\frac{5}{8}}r_{H}^{2}(r_{H}^{2}+1)\tau_{0}-{\frac{T\big{(}r_{0}-\hat{r}_{0}\big{)}}{\sqrt{1-T^{2}}}}+I_{4,AdS}+I_{4\,BH}\,,$
(66)

where

$\displaystyle
I_{4,AdS}=-\int_{\hat{r}_{0}}^{\infty}dr\,\Big{[}\big{(}{\textstyle{\frac{\scriptstyle
T}{\scriptstyle r}}}-{\textstyle{\frac{\scriptstyle\varphi(r)}{\scriptstyle
f_{AdS}(r)}}}\big{)}{\frac{r^{4}}{\sqrt{f_{AdS}(r)-\varphi^{2}(r)}}}-{\frac{T}{\sqrt{1-T^{2}}}}\Big{]}\,,$
$\displaystyle
I_{4,BH}=\int_{r_{0}}^{\infty}dr\,\Big{[}\big{(}{\textstyle{\frac{\scriptstyle
T}{\scriptstyle r}}}-{\textstyle{\frac{\scriptstyle\varphi(r)}{\scriptstyle
f_{BH}(r)}}}\big{)}{\frac{r^{4}}{\sqrt{f_{BH}(r)-\varphi^{2}(r)}}}-{\frac{T}{\sqrt{1-T^{2}}}}\Big{]}\,.$
(67)

Numerical results for $d=4$ dimensions are shown in Fig. 7. For a given $q\neq
0$, the preparation time is positive for a brane tension not too close to the
critical value $T_{crit}=1/L$. In this case, the action difference is always
positive for any $\tau_{0}\geq 0$, see the left panel. On the other hand, for
a near critical tension (for example $T=0.99999$), the preparation time can
still be positive for a sufficiently large $q$. In the right panel, one finds
that the action difference becomes already positive before this happens. These
results strongly imply that the large black hole phase is dominant for a
sensible Euclidean solution even in the presence of nonzero energy flux
passing through the ETW brane.

##  Appendix C Trapping coefficient method

In the near horizon region, the potential $V_{k\,,\omega}$ vanishes and the
wave function $\psi_{k\,,\omega}$ will take the form of a plane wave

$\psi_{k\,,\omega}={\frac{A_{h}}{2}}e^{-i\delta(\omega)}\Big{(}e^{-i\omega
y_{*}}+S(\omega)e^{i\omega y_{*}}\Big{)}\,,$ (68)

where $S(\omega)=e^{2i\delta(\omega)}$ is the scattering matrix and
$\delta(\omega)$ is the scattering phase shift. Since the solution is
infalling on the horizon, the scattering matrix should have some poles (at
complex frequencies) so that the second term in the bracket is dominant.
Recall that we impose an extra boundary condition on the brane, these poles
will be discrete, giving rise to a discrete set of quasi-bound modes (labelled
by $\omega_{n}$). The leading Laurent expansion for the scattering matrix is
given by [25]

$S(\omega)=e^{2i\delta_{0}(\omega)}{\frac{\omega-\omega_{n}^{*}}{\omega-\omega_{n}}}\,,$
(69)

where $\delta_{0}(\omega)$ is a slowly varying function of $\omega$. Using the
definition $\omega_{n}=\bar{\omega}_{n}+i\Gamma_{n}/2$, one finds

$\delta(\omega)=\delta_{0}(\omega)+\mathrm{arcsin}\Big{[}{\frac{\Gamma_{n}}{\sqrt{4(\omega-\bar{\omega}_{n})^{2}+\Gamma_{n}^{2}}}}\Big{]}\,.$
(70)

On the other hand, in the asymptotic region, the wave function is just a
slowly-varying function of $\omega$. If its magnitude at the brane is denoted
by $A_{b}$, one can write [23]

$\displaystyle\psi_{k\,,\omega}=\left\\{\begin{array}[]{ll}A_{h}\,\mathrm{Re}\big{[}e^{i\delta(\omega)}e^{i\omega
y_{*}}\big{]}\,,\quad y_{*}\rightarrow-\infty\\\ \\\ A_{b}\equiv
A_{h}R(\omega)\,\mathrm{Re}\big{[}e^{i\delta(\omega)}e^{i\theta(\omega)}\big{]}=-A_{h}R(\omega)\sin\big{(}\delta(\omega)+\theta(\omega)-{\textstyle{\frac{\scriptstyle\pi}{\scriptstyle
2}}}\big{)}\,,\quad y_{*}=y_{b*}\,,\end{array}\right.$ (74)

where $R(\omega)\,,\theta(\omega)$ are slowly-varying functions of $\omega$.
The trapping coefficient is defined as

$\eta(\omega)={\frac{A_{b}}{A_{h}}}=-R(\omega)\sin\big{(}\delta(\omega)+\theta(\omega)-{\textstyle{\frac{\scriptstyle\pi}{\scriptstyle
2}}}\big{)}\,.$ (75)

In practical calculations, one can choose a proper normalization for the wave
function such that $\psi_{k\,,\omega}(y_{b*})=A_{b}=1$. This does not change
the trapping coefficient but is convenient for numerical calculations.
Naively, if the quasi bound mode is locally localized on the brane due to the
presence of an attractive delta potential, we may expect the squared trapping
coefficient has a peak at the frequency of the quasi bound mode.

If the relation

$\delta_{0}(\omega)+\theta(\omega)={\frac{\pi}{2}}\,,{\frac{3\pi}{2}}\,,$ (76)

holds, the squared trapping coefficient will be given by

$\xi(\omega)=\eta^{2}(\omega)=R^{2}(\omega)\,{\frac{\Gamma_{n}^{2}}{4(\omega-\bar{\omega}_{n})^{2}+\Gamma_{n}^{2}}}\,.$
(77)

This is the well-known Lorentzian (Breit-Wigner) distribution. The peak is
centered at the real part of the frequency of the quasi-bound mode whereas its
half-width is equal to the half of the imaginary part of the frequency
$\Gamma_{n}/2$. Of course, if the condition (76) is not satisfied, the
relation (77) is invalid and the squared trapping coefficient will take a more
complicated form. However, it was shown in [23] that (76) is a good
approximation and our numerical results confirm this again. As was pointed out
in [23], to obtain the Lorentzian distribution, one generally needs subtract
from the plot of $\xi(\omega)$ a baseline function, which accounts for the
slow variation of $\delta_{0}(\omega)\,,\theta(\omega)$ with frequency.
However, this is not necessary in our situation since we are interested in the
large black hole case, in which only the zero mode is present.

It turns out that the trapping coefficient method is a straightforward
algorithm to extract the real and imaginary parts of the quasi bound mode: (1)
Numerically solving the Schrödinger equation for a range of real frequencies,
given the boundary condition on the brane, together with the normalization
$\psi_{k\,,\omega}(y_{b*})=1$; (2) Find numerically the amplitude of the wave
function $A_{h}$ in the near horizon region. The square of its inverse
$1/A_{h}^{2}$ gives the squared trapping coefficient $\xi(\omega)$; (3) Plot
$\xi(\omega)$ and subtract a baseline function; (4) Fit the numerical data
with Breit-Wigner distrubtion and read off the real and imaginary parts of the
frequency of the quasi bound mode.

##  Appendix D Limiting behavior of the quasi bound mode

In general, quasinormal modes of black holes can only be numerically studied.
However, in the large dimensional limit ($D=d+1$), the gravitational field of
a black hole is well localized in a region near the horizon so that the
frequency of quasinormal modes can be half-analytically extracted [26, 27,
28]. The quasinormal modes can be classified into two types according to their
frequency: decoupling modes and non-decoupling modes [26, 27]. The first
refers to the modes with smaller frequency, which are trapped in the near
horizon region whereas the latter stands for the larger frequency modes which
can penetrate the gravitational potential barrier and escape to asymptotic
infinity with certain probability. For example, for asymptotically flat black
holes, decoupling modes has $\omega\sim O(1)$ whereas nondecoupling modes has
$\omega\sim O(D)$.

In our case, we are interested in the limiting behavior of the quasi bound
mode for a far brane close to the asymptotic infinity. The quasi bound mode of
course corresponds to the non-decoupling modes. Without confusion, we send the
brane to asymptotic infinity and will not distinguish the two in the
following. In addition, we focus on large black holes which has $\gamma\ll 1$
and the frequency of the quasi bound mode should obey
$\mathrm{Re}\,\omega<\omega_{GR}$, where the two should be in the same order
of $D$.

According to these considerations, we choose $\omega\sim O(n^{3})\,,\gamma\sim
O(1/n)\,,\ell\sim O(n^{2})$, where $n=D-3$. It is convenient to introduce
$\hat{\omega}=\omega/n^{3}\,,\hat{\gamma}=n\gamma\,,\hat{\ell}=\ell/n^{2}$ and
a new radial coordinate $R=y^{n}$. It turns out that the gravitational
potential approaches to a plateau for $R\gg 1$ before arriving at asymptotic
infinity, see Fig. 8

$V_{k}(R)\Big{|}_{R\gg 1}\longrightarrow
n^{6}\omega_{c}^{2}\,,\quad\omega_{c}=\sqrt{\hat{\omega}_{GR}^{2}+{\frac{1}{4\hat{\gamma}^{4}}}}\,,$
(78)

where $\hat{\omega}_{GR}\equiv\omega_{GR}/n^{3}=\hat{\ell}/\hat{\gamma}$.
Clearly, smaller frequency modes with $\omega\ll n^{3}\omega_{c}$ will be
trapped inside the potential barrier, staying in the near horizon region since
their wave functions will vanish exponentially in $n$ outside the barrier.
These are the decoupling modes of AdS-Schwarzschild black holes, which are
studied carefully in [28]. The modes that we are interested in are the
nondecoupling ones which have higher frequency $\omega\sim n^{3}\omega_{c}$.
For our purpose, we would like to take $\omega_{GR}\sim n^{3}\omega_{c}$ so
that $\mathrm{Re}\,\omega\sim\omega_{GR}$. This demands $\hat{\ell}\gg
1/\hat{\gamma}$ and hence $\hat{\omega}_{GR}\simeq\omega_{c}$.

Here it should be emphasized that the large black hole considered above
features a resonant cavity in the bulk. In this case, there will exist an
overtones for each $\ell$ for the quasi bound mode. This is significantly
different from the $d=4$ dimensional case. The frequency of the quasi bound
mode can be extracted by matching the near horizon solutions and the far zone
solutions in the overlap region $1/n\ll y-1\ll 1$ (or $1\ll R\ll e^{n}$).

Figure 8: The gravitational potential for a large black hole in $n=100$
dimensions with $\gamma=1/100$ and angular momentum $\ell=100^{2}$.

###  D.1 Near zone solutions

In the near horizon region with $y-1\ll 1$ (or $R\ll e^{n}$), the metric
function $f(R)$ and the potential $V_{k}(R)$ simplify to

$\displaystyle
f(R)={\frac{n^{2}}{\hat{\gamma}^{2}}}\Big{(}1-{\frac{1}{R}}\Big{)}\,,$
$\displaystyle
V_{k}(R)=f(R)n^{4}\Big{(}\hat{\gamma}^{2}\omega_{c}^{2}+{\frac{1}{4\hat{\gamma}^{2}R}}\Big{)}\,,$
(79)

where we keep only the leading order in the large $D$ limit. The ingoing
boundary condition implemented on the horizon is
$\psi(R)=\big{(}R-1\big{)}^{i\hat{\omega}\hat{\gamma}^{2}}\phi(R)$, where
$\phi(R)$ is a regular function at $R=1$. Solving the master equation yields

$\psi(R)=\big{(}R-1\big{)}^{i\hat{\omega}\hat{\gamma}^{2}}\sqrt{R}\,\,{}_{2}F_{1}\big{(}\alpha_{+}\,,\alpha_{-}\,,1\,;R\big{)}\,,$
(80)

where

$\alpha_{\pm}={\frac{1}{2}}+i\hat{\omega}\hat{\gamma}^{2}\pm\hat{\gamma}^{2}\sqrt{\omega_{c}^{2}-\hat{\omega}^{2}}\,.$
(81)

In the overlap region $R\gg 1$, the solution asymptotes to

$\psi(R)\longrightarrow
A_{+}(\omega)\,R^{\hat{\gamma}^{2}\sqrt{\omega_{c}^{2}-\hat{\omega}^{2}}}+A_{-}(\omega)\,R^{-\hat{\gamma}^{2}\sqrt{\omega_{c}^{2}-\hat{\omega}^{2}}}\,,$
(82)

where explicit expressions for the coefficients $A_{\pm}(\omega)$ are
unimportant except that their ratio is of order unity

$\Big{|}{\frac{A_{+}(\omega)}{A_{-}(\omega)}}\Big{|}\sim O(1)\,.$ (83)

Figure 9: The solution to (90) gives a continuous spectrum for the quasi bound
mode. The spectrum should be discretized into separate overtones when
subleading order corrections in $1/n$ expansion are included.

###  D.2 Far zone solutions

In the far zone where $y-1\gg 1/n$, the term $1/y^{n}$ is exponentially small
in $n$ and hence can be dropped in the metric function. In addition, the
spatial curvature is also considerably small compared to the leading term
$y^{2}/\gamma^{2}$ in asymptotic infinity so that
$f(y)=n^{2}y^{2}/\hat{\gamma}^{2}$ to leading order. The gravitational
potential simplifies to

$V_{k}(y)=n^{6}\Big{(}\omega_{c}^{2}+{\frac{y^{2}-1}{4\hat{\gamma}^{4}}}\Big{)}\,.$
(84)

In the large $D$ limit, the mixed boundary condition (33) imposed on the ETW
brane becomes

$\psi(y)\Big{|}_{y\gg 1}\longrightarrow y^{n/2}\,.$ (85)

The solution to the master equation, which obeys the above condition is given
by the Hankel function

$\psi(y)=y^{-1/2}\,H_{n/2}^{(1)}\Big{(}{\textstyle{\frac{\scriptstyle
n\hat{\gamma}^{2}\sqrt{\hat{\omega}^{2}-\hat{\omega}_{GR}^{2}}}{\scriptstyle
y}}}\Big{)}\,.$ (86)

In the overlap region $1\ll R\ll e^{n}$, the solution behaves asymptotically
as555The asymptotic behavior of the Hankel function can be found in the
appendix in [27].

$\psi(R)\longrightarrow
A_{+}(\omega)\,R^{\hat{\gamma}^{2}\sqrt{\omega_{c}^{2}-\hat{\omega}^{2}}}+A_{-}(\omega)\,R^{-\hat{\gamma}^{2}\sqrt{\omega_{c}^{2}-\hat{\omega}^{2}}}\,,$
(87)

where

$\Big{|}{\frac{A_{+}(\omega)}{A_{-}(\omega)}}\Big{|}=\mathrm{exp}\Big{[}n\,\mathrm{Re}\,g(x)\Big{]}\,,\quad
x=2\hat{\gamma}^{2}\sqrt{\hat{\omega}^{2}-\hat{\omega}_{GR}^{2}}\,,$ (88)

where

$g(x)=\log\Big{(}{\frac{1+\sqrt{1-x^{2}}}{x}}\Big{)}-\sqrt{1-x^{2}}\,.$ (89)

Matching the above far zone solutions to the near zone ones leads to

$\mathrm{Re}\,g\Big{(}2\hat{\gamma}^{2}\sqrt{\hat{\omega}^{2}-\hat{\omega}_{GR}^{2}}\,\Big{)}=0\,.$
(90)

This gives rise to a continuous spectrum for the quasi bound mode, see Fig. 9.
However, this is a artifact in the large $D$ limit. Inclusion of subleading
order corrections should discretize the spectrum into separate overtones.
Nonetheless, this half-analytical result supports our numerical results for
the limiting behavior of the quasi bound mode.

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