# The dark bubbleography

Souvik Banerjee  Ulf Danielsson  and Maximilian Zemsch

###### Abstract

We present the holographic construction of the dark bubble model of dark
energy and highlight the pivotal role played by the non-normalizable modes.
Following the route of holographic renormalization, we show that the non-
normalizable modes are essential for having a vanishing mass for the induced
graviton in any braneworld model. We then apply this idea in the computation
of the propagator on the wall of the dark bubble introduced in [1].

##  1 Introduction and overview

The dark bubble model, first proposed in [1] and subsequently developed in [2,
3, 4, 5, 6, 7, 8, 9, 10, 11, 12], is an alternative to standard string
compactifications, which potentially circumvents the swampland obstructions
against obtaining a stringy model of dark energy. In this paper, we will focus
on presenting a complete and consistent holographic description of this model
in terms of a necessary generalization of techniques used in the AdS/CFT
correspondence [13, 14, 15]. In particular, we will sort out how the choice of
holographic boundary conditions in our model is related to the mass of the 4D
graviton imprinted on the bubble wall.

Similarly to the Randall-Sundrum scenario [16], the dark bubble model makes
use of branes embedded into a five-dimensional AdS space, even though there
are some crucial differences. The Randall-Sundrum model embeds the universe as
a hypersurface between two higher-dimensional AdS spaces. The extrinsic
curvatures on the two sides of the embedded brane have the same sign, such
that the two spaces both correspond to insides with respect to the braneworld.
It is also common to impose a $\mathbb{Z}_{2}$ symmetry, identifying the two
bulk spacetimes. The dark bubble is geometrically very different since the
hypersurface is given by the spherical shell of a bubble of true vacuum
expanding inside a decaying metastable AdS vacuum in the exterior. Clearly,
there is no $\mathbb{Z}_{2}$ symmetry, but even more important is the fact
that the bubble has an inside and an outside.

The imbalance between the inner $(-)$ and outer $(+)$ space of the bubble is
reflected in different values of the bulk cosmological constants
$\Lambda_{\pm}=-6k_{\pm}^{2}$, with $k_{-}>k_{+}$, on the two sides of the
bubble wall. The cosmological constant outside of the shell is larger than the
one inside of the shell, which induces the decay of the false vacuum to the
true one via the nucleation of a brane. The metric is of the form

$\displaystyle\mathrm{d}s^{2}=-f_{\pm}(r)\,\mathrm{d}t^{2}+\frac{1}{f_{\pm}(r)}\,\mathrm{d}r^{2}+r^{2}\,\mathrm{d}\Omega_{3}^{2}$
(1.1)

with $f_{\pm}(r)=1+k_{\pm}^{2}r^{2}$ for the interior $(-)$ or exterior $(+)$,
respectively. The dynamical expansion parameter $a(t)$ determines the location
of the shell, so that $r<a(t)$ holds for the inside and $r>a(t)$ for the
outside of the bubble. For an observer on the bubble wall, a proper time
$\tau$ can be chosen in such a way that the shell metric recovers the geometry
of a FLRW universe. The proper time evolution of the bubble radius $r=a(\tau)$
is determined by the Israel junction conditions, which require

$\displaystyle\sigma=\frac{3}{8\pi
G_{5}}\bigg{(}\sqrt{k_{-}^{2}+\frac{1+\dot{a}^{2}}{a^{2}}}-\sqrt{k_{+}^{2}+\frac{1+\dot{a}^{2}}{a^{2}}}\bigg{)},$
(1.2)

where $\sigma$ is the tension of the shell. Here,
$\dot{a}=\mathrm{d}a/\mathrm{d}\tau$. The critical value of the brane tension
is given by

$\displaystyle\sigma_{\text{crit}}:=\frac{3}{8\pi G_{5}}(k_{-}-k_{+}),$ (1.3)

and corresponds to a flat Minkowski universe with vanishing cosmological
constant. In case of the dark bubble, the tension stays slightly subcritical,
meaning $\sigma=\sigma_{\text{crit}}(1-\epsilon)$ with a small and positive
parameter $\epsilon$. With that, equation (1.2) can be expanded in powers of
$\epsilon$ yielding the Friedmann equation in four dimensions

$\displaystyle
H^{2}\equiv\frac{\dot{a}^{2}}{a^{2}}=-\frac{1}{a^{2}}+\frac{8\pi
G_{4}}{3}\Lambda_{4}+\mathcal{O}(\epsilon^{2}),$ (1.4)

where

$\Lambda_{4}=\sigma_{\text{crit}}-\sigma>0$ (1.5)

is the 4D cosmological constant, associated with a positive energy density. As
described in [1], and reviewed in [2], the 4D Newton’s constant is given by

$G_{4}=\frac{2k_{-}k_{+}}{k_{-}-k_{+}}G_{5}.$ (1.6)

One can also consider more general asymptotically AdS spacetimes, e.g. AdS-
Schwarzschild geometry, which induces an additional term to the Friedmann
equation (1.4) that can be identified as radiation on the bubble wall [1].
Similarly, a spherically symmetric cloud of strings radially stretched in the
bulk can potentially induce four-dimensional matter on the wall [2].

Contrary to RS, neither gravity nor matter is localized on the brane. The
uplift of 4D gravitational waves to 5D has been explored in detail in [10].
There, it was demonstrated how the dark bubble sustains 5D gravitational
waves, which induces the expected 4D waves in metric on the brane.
Furthermore, in [11], it was shown how electromagnetic waves, identified with
the excitations of the gauge field within the brane, source the string theory
Kalb-Ramond $B$-field in the bulk. The backreaction of these fields on the 5D
bulk metric induces the expected 4D gravitational backreaction. What these
works explicitly show, which is further reviewed in, e.g. [7], is that the
matter fields attached to the brane are governed by dynamics that is
constrained to that of 4D Einstein gravity coupled to matter. This follows
from the Gauss-Codazzi equations, which express the induced metric in terms of
the extrinsic curvature and the bulk metric, together with the 5D Einstein
equations and the junction conditions. As explained in [11], the spreading out
of matter and gravity in the fifth dimension only show up as deviations in the
Einstein equations at high energy densities.111Just as in the case of RS,
there are also modifications in the force of gravity at distances of the order
of the AdS-scale. Note that this scale is always assumed to be microscopic.
For further discussions on this see [11, 12].

The case of RS is superficially similar, but fundamentally different. The key
difference is the junction condition (1.2), where the sign of the second term
of the junction condition is changed from a minus to a plus since there are
two insides. Considering the $\mathbb{Z}_{2}$-symmetric case for simplicity,
this leads to an effective 4D cosmological constant given by $\sigma-2k$, and
the 4D Newton’s constant given by $G_{4}=kG_{5}$. The changes in the
phenomenology these differences bring about are profound, as elaborated in [4,
7]. In particular, there is now a lower bound on $\sigma$ to obtain a positive
cosmological constant. Contrary to the case of the dark bubble, extremal
branes are far from the bound, which naturally leads to an AdS braneworld
rather than a universe with positive dark energy. We will not discuss RS any
further in the present paper, but instead focus on how the phenomenologically
promising dark bubble model can be embedded, consistently, in a holographic
setting and the consequences thereof.

It turns out that this key difference between the Randall-Sundrum braneworld
model and our dark bubble becomes pivotal to ensure a consistent holographic
description of the latter. Since the bubble has an inside and an outside, it
can naturally define a monotonically increasing holographic direction,
contrary to the case of Randall-Sundrum, where a holographic description
requires a mirroring of the holographic direction across the brane. Therefore,
the applicability of holography is even more manifest in our dark bubble
model. This is what we aim to elaborate on in this paper.

Massive particles can be represented by strings that end on the bubble wall
and extend in the radial direction. They represent matter fields in the
expanding universe whose masses are effectively renormalized by virtue of the
holographic renormalization of the Newton constant $G_{4}$ [4]. The energy
carried by the endpoints of the strings is associated with non-normalizable
metric modes on the boundary hypersurface. However, there is no obvious reason
why non-normalizable modes should be necessary or even wanted in a holographic
description. Here, we show that this follows quite naturally from the
requirement of a massless graviton in a quantum description of gravity,
induced on the bubble wall. The present work presents a systematic and
consistent holographic approach to elucidate the pivotal role of bulk non-
normalizable modes in obtaining a massless graviton on the bubble wall. These
non-normalizable modes, in the context of the dark bubble model, are
manifestations of string sources which eventually provide the energy for the
expansion of the bubble [4]. A very similar role of the non-normalizable modes
in the context of the Karch-Randall braneworld model [17] has been discussed
in a recent work [18]. Our conclusion is very much in line with that.

In the holographic setup, variation of the gravity action in AdS5 yields a
boundary contribution of the form

$\delta
S_{5}=\frac{1}{2}\int_{\partial\mathcal{M}}d^{4}x\sqrt{-\gamma}T_{\mu\nu}^{\mathrm{(CFT)}}\delta\gamma^{\mu\nu},$
(1.7)

where $\gamma^{\mu\nu}$ is the induced metric on the cutoff hypersurface $r=a$
and $\gamma$ is the determinant of $\gamma^{\mu\nu}$ on that hypersurface.
$T_{\mu\nu}^{\mathrm{(CFT)}}$ is the bare stress-energy tensor of the dual
conformal field theory living on the $r=a$ hypersurface. Conventionally, in
this holographic setup one tends to employ Dirichlet boundary conditions on
the boundary hypersurface, which fix the boundary metric. As a consequence,
(1.7) vanishes leading to a well-defined variational principle. Nevertheless,
by construction, it does not allow for any dynamical boundary metric. On the
contrary, having a dynamical metric on the bubble wall is a necessity in our
model in order to reproduce an expanding universe. From (1.7) it is therefore
clear that in order to achieve a dynamic metric on the wall, we do need to go
for different boundary conditions that can be imposed without requiring to fix
the metric on the boundary. One possibility to obtain such a dynamic geometry
on the boundary has been investigated in the context of holographic cosmology
[19, 20], where it was shown that a “mixed” boundary condition instead of a
usual Dirichlet can potentially yield cosmological evolution on the boundary.
This mixed boundary condition, which amounts to adding a suitable local action
on the boundary, falls into the allowed class of boundary conditions in
AdS/CFT [21]. We show in this paper that this particular choice of boundary
condition is crucial to obtain a holographic realization of the dark bubble
model.

The structure of this paper is as follows: To set the stage, in section 2 we
start with a lightning review of the holographic approach to cosmology,
emphasizing the importance of choosing an appropriate boundary condition to
obtain expanding cosmology on a cutoff near the boundary of an asymptotically
AdS spacetime. This involves a holographic renormalization of the induced
Newton’s constant, and the cosmological constant on the cutoff brane. In
section 3, we illustrate the role of normalizable modes in inducing a massive
graviton in the simple setup of holographic cosmology discussed in section 2.
In this model, there is only one AdS bulk, which we generalize, in section 4,
to accommodate our dark bubble. Apart from having an inside and an outside,
this model also requires consideration of the dynamics of the brane source.
Incorporating both these features, and adopting the lessons learnt from
previous sections, we present the computation of the graviton propagator on
the bubble wall. Our computation elucidates the precise role of non-
normalizable modes towards obtaining the dynamics of a massless graviton on
the wall of the dark bubble. We conclude our short report in section 5.

##  2 Holographic Cosmology - a primer

In this section we present the central ingredients that are essential for
building up an expanding cosmological model on a holographic screen. The
holographic screen is here a UV cutoff near the boundary of an asymptotically
AdS spacetime which can as well be a codimension-1 brane with a fixed value of
tension. There had been earlier attempts to realize cosmology in the realm of
AdS/CFT, including a surge of research activities to establish a holographic
description of RS braneworld models [22, 23, 24, 25, 26, 27, 28, 29, 30, 31,
32, 33, 34, 35, 36, 37]. However, as discussed above, there was still an
outstanding puzzle regarding the choice of appropriate boundary conditions
that can potentially give rise to a dynamical cosmological spacetime on the
boundary. As discussed above, this was due to the fact that the Dirichlet
boundary condition used in standard AdS/CFT renders the boundary metric fixed
and time-independent, which was clearly not conducive for realizing expanding
cosmology at the boundary. This issue was successfully addressed in [20, 19]
which we briefly gloss over below.

Our setup consists of a $d$-dimensional de Sitter hypersurface living close to
the boundary of an $\text{AdS}_{d+1}$ spacetime, which we will later promote
to a holographic model of a dark bubble. The metric of the bulk is of the form

$\displaystyle\mathrm{d}s^{2}=-f(r)\,\mathrm{d}t^{2}+\frac{1}{f(r)}\,\mathrm{d}r^{2}+r^{2}\,\mathrm{d}\Omega_{d-1}^{2},$
(2.1)

with the radial function $f(r)$. The spacetime is cut off by a brane at a
radial position $a$ that evolves in time. This is a crucial requirement since
the boundary hypersurface in this setup corresponds to the expanding universe
and hence has to be dynamical. It is convenient to introduce a proper time
parameter $\tau$ as the brane time, and then to choose a suitable
parametrization $\\{t(\tau),a(\tau)\\}$ in order to adjust the boundary metric
appropriately. For cosmological applications, we want to imprint an FRW metric
on the boundary:

$\displaystyle\mathrm{d}s^{2}=-\mathrm{d}\tau^{2}+a^{2}(\tau)\mathrm{d}\Omega_{d-1}^{2}.$
(2.2)

In order to describe the dynamics of the hypersurface, we now need to impose
suitable boundary conditions. As argued above, in order to ensure real-time
dynamics on the cutoff surface, which can as well be thought of as an end-of-
the-world brane with fixed tension, we need to dump the Dirichlet conditions,
and instead resort to mixed boundary conditions. In order to implement this
boundary condition, we need an appropriate local stress-energy tensor
$T_{ij}^{\mathrm{local}}$ along with the induced CFT stress-energy tensor
$T_{ij}^{\mathrm{CFT}}$ on the brane. $T_{ij}^{\mathrm{CFT}}$ is obtained by
varying the full bulk action supplemented by the Gibbons-Hawking-York counter
term [38]. Furthermore, to render the full stress-energy tensor finite, even
in the limit when the expansion parameter goes to infinity
$a(\tau)\rightarrow\infty$, we need to add, additionally, a counterterm
$T_{ij}^{\mathrm{ct}}$ which is obtained by varying the counterterm action

$\displaystyle
S_{\mathrm{ct}}=-\frac{1}{2}\int\limits_{\partial\mathcal{M}}\mathrm{d}^{d}x\sqrt{-\gamma}(\kappa_{1}\mathcal{R}[\gamma]+\kappa_{2}),$
(2.3)

that appears in the bulk action following holographic renormalization[39].
Compiling all these ingredients, the mixed boundary condition reads [21]

$\displaystyle
T_{ij}^{\mathrm{CFT}}+T_{ij}^{\mathrm{local}}+T_{ij}^{\mathrm{ct}}=0.$ (2.4)

The different components appearing in (2.4) take the following explicit
expressions,

$\displaystyle T_{ij}^{\mathrm{CFT}}$ $\displaystyle=\frac{1}{8\pi
G_{d+1}}(K_{ij}-K\gamma_{ij}),$ (2.5) $\displaystyle T_{ij}^{\mathrm{local}}$
$\displaystyle=-\frac{1}{8\pi G_{d}}\bigg{(}{\cal R}_{ij}-\frac{1}{2}{\cal
R}\gamma_{ij}+\Lambda_{d}\gamma_{ij}\bigg{)},$ (2.6) $\displaystyle
T_{ij}^{\mathrm{ct}}$ $\displaystyle=-\kappa_{1}\bigg{(}{\cal
R}_{ij}-\frac{1}{2}{\cal R}\gamma_{ij}\bigg{)}-\kappa_{2}\gamma_{ij}.$ (2.7)

Here, the Ricci tensor ${\cal R}_{ij}$ and the Ricci scalar $\cal R$ are
defined on the brane, so is the extrinsic curvature
$K_{ij}=1/2\,n^{k}\partial_{k}\gamma_{ij}$. The parameters
$\kappa_{1},\kappa_{2}$ are determined through holographic renormalization
[39]. Technically, these parameters are chosen such that the energy density
$\epsilon$ and the pressure $p$ on the cutoff brane are finite. In a 4+1
dimensional asymptotically AdS spacetime, the energy density, pressure and the
trace of the stress-energy tensor on the brane are given by [19]

$\displaystyle\epsilon$
$\displaystyle=T_{\tau\tau}^{\mathrm{CFT}}+T_{\tau\tau}^{\mathrm{ct}}=\kappa_{2}+\kappa_{1}\bigg{(}H^{2}+\frac{\tilde{k}}{a^{2}}\bigg{)}-\frac{3}{8\pi
G_{5}a}\sqrt{\dot{a}^{2}+f(r)},$ $\displaystyle p$
$\displaystyle=T_{i}^{i,\,\mathrm{CFT}}+T_{i}^{i,\,\mathrm{ct}}=-\kappa_{2}-\kappa_{1}\bigg{(}H^{2}+\frac{\tilde{k}}{a^{2}}+\frac{2\ddot{a}}{a}\bigg{)}+\frac{1}{16\pi
G_{5}}\frac{af^{\prime}+2(a\ddot{a}+2\dot{a}^{2})+4f}{a\sqrt{\dot{a}^{2}+f}},$
$\displaystyle{\rm Tr}\,T$ $\displaystyle=-\frac{3L^{3}}{16\pi
G_{5}}\bigg{(}H^{2}+\frac{1}{a^{2}}\bigg{)}\frac{\ddot{a}}{a}+\mathcal{O}(H^{4}),$
(2.8)

with $H=\dot{a}/a$ being the Hubble parameter and $\tilde{k}=0,\pm 1$,
depending on whether the asymptotic boundary is flat, spherical or hyperbolic,
respectively. In order to render (2) finite in the limit $a\rightarrow\infty$,
one needs to choose

$\displaystyle\kappa_{1}=\frac{3L}{16\pi
G_{5}}\,\,\,\,\text{and}\,\,\,\,\kappa_{2}=\frac{3}{8\pi G_{5}L},$ (2.9)

which can be thought of as renormalizing the bare 4-dimensional Newton’s
constant and the 4-dimensional cosmological constant, respectively.

Having a non-trivial dynamic metric on the boundary will nevertheless break
the conformal invariance on the boundary, which can be tracked,
systematically, by computing the Weyl anomaly. To demonstrate this, let us
consider a 4+1 dimensional AdS Schwarzschild black hole with the blackening
factor $f(r)={\tilde{k}}+\frac{r^{2}}{L^{2}}-\frac{M}{r^{2}}$ with a flat
static boundary so that $H={\tilde{k}}=0$. In this case, the renormalized
energy density and pressure on the brane simplifies as an expansion in large
$a$ as

$\displaystyle\epsilon=\frac{3}{8\pi G_{5}L}\bigg{(}\frac{(\pi
LT)^{4}}{2}-\frac{7(\pi LT)^{8}}{8}+...\bigg{)},\,\,\,p=\frac{1}{8\pi
G_{5}L}\bigg{(}\frac{(\pi LT)^{4}}{2}-\frac{3(\pi LT)^{8}}{8}+...\bigg{)},$
(2.10)

with $T=T_{\mathrm{Hawking}}/\sqrt{\dot{a}^{2}+f(a)}$ being the red-shifted
Hawking temperature on the cutoff brane. Clearly, $\epsilon=3p\propto T^{4}$
holds at the leading order of $a$-expansion which is thermodynamically
expected for a conformal fluid. This conformality is broken when the
subleading contributions are considered, i.e. when we move a finite distance
away from the boundary and is broken further when the full dynamics of the
brane with $H\neq 0$ is considered. The latter is evident from the non-
vanishing trace term $\text{Tr}\,T=\epsilon-3p$ as in (2). Nevertheless, (2)
still satisfies the first law of thermodynamics at the leading order of
$a$-expansion,

$\displaystyle\mathrm{d}E=T\mathrm{d}S-p\mathrm{d}V,$ (2.11)

$S=(\pi LT)^{3}V^{(3)}/(4G_{5})$ being the entropy of the black hole,
$E=\epsilon V$ and $V=a^{3}V^{(3)}$ where $V^{(3)}$ is the three-dimensional
transverse volume. Finally, plugging (2) and (2.9) back in the $\tau\tau$
component of (2.4) yields the cosmological evolution in the form of a
Friedmann equation on the brane, in the leading order of expansion in $1/a$.

$\displaystyle
H^{2}\equiv\frac{\dot{a}^{2}}{a^{2}}=-\frac{\tilde{k}}{a^{2}}+\frac{8\pi
G_{4}}{3}\epsilon+\frac{1}{3}\Lambda_{4}.$ (2.12)

##  3 Non-normalizable modes and the graviton mass in a braneworld
construction

In the simple holographic setup discussed above, in this section, we will
derive a formula for graviton mass induced on the cutoff brane. For this
purpose, we start with a Fefferman-Graham expansion of an asymptotically
$\text{AdS}_{d+1}$ bulk metric (2.1) which assumes a generic form [40]

$\displaystyle\mathrm{d}s^{2}=\frac{L^{2}}{4\rho^{2}}\mathrm{d}\rho^{2}+\frac{L^{2}}{\rho}g_{ij}(x,\rho)\mathrm{d}x^{i}\mathrm{d}x^{j},$
(3.1)

where $g_{ij}(x,\rho)$ assumes a power series expansion near the boundary,

$\displaystyle g(x,\rho)=g_{(0)}(x)+\cdots
g_{(d)}(x)\rho^{\frac{d}{2}}+{\bar{g}}_{d}(x)\,\rho^{\frac{d}{2}}\log\rho,$
(3.2)

$g_{(0)}$ being the boundary metric. The last term is an anomaly term which
contributes only for even boundary dimensions. $L$ is the AdS radius and
$\rho$ is the new radial coordinate with $\rho\rightarrow 0$ being the
asymptotic boundary. As described in the previous section, we introduce a
brane at the hypersurface close to the boundary of AdS at $\rho=\epsilon$
where $\epsilon$ serves as a UV cutoff from the perspective of the dual field
theory. In order to have a fully consistent boundary theory without any UV
divergence, one would naively expect the modes of the induced metric
$\gamma_{ij}=L^{2}/\epsilon\,g_{ij}(x,\epsilon)$ on the brane to be
normalizable in the asymptotic limit. In what follows we will now show that
this contradicts the physical requirement of a massless graviton.

In order to evaluate the relation between normalizable modes and the graviton
mass, let us consider a small metric fluctuation

$\displaystyle\delta\gamma_{ij}=\gamma_{ij}-\bar{\gamma}_{ij},$ (3.3)

where $\bar{\gamma}_{ij}$ is the background metric on the cutoff surface. The
dynamics of this graviton excitation is then determined by the linearized
Einstein’s field equations on the hypersurface, which in turn is encoded
automatically in the mixed boundary condition (2.4). At the linearized order,
the equations of motion of the graviton excitation read

$\displaystyle-\frac{1}{2}\bigg{(}1+\frac{(d-1)(d-2)}{2}\bigg{)}\,\Box\,\delta\gamma_{ij}+\bigg{(}\underbrace{\frac{(d-1)(d-2)}{L^{2}}+\Lambda_{d}}_{\bar{\Lambda}_{d}}\bigg{)}\delta\gamma_{ij}=8\pi
G_{d}\delta T_{ij}^{\mathrm{CFT}},$ (3.4)

where we have plugged in the corresponding values of the parameters in general
dimensions $\kappa_{1}=\frac{(d-1)L}{16\pi G_{d+1}}$,
$\kappa_{2}=\frac{d-1}{8\pi G_{d+1}L}$, and used the relation between the
Newton constants of different dimensions $G_{d}L=(d-2)G_{d+1}$ [15]. As such,
we can identify a new effective cosmological constant $\bar{\Lambda}_{d}$ as a
combination of the renormalization term governed by $\kappa_{2}$ and the bare
$d$-dimensional cosmological constant, the latter being a part of the local
stress-energy tensor on the brane. We are also free to set ${\Lambda}_{d}=0$
which yields an effective de Sitter cosmological constant.

The term including the perturbation of the stress-energy tensor $\delta
T_{ij}^{\mathrm{CFT}}$ appearing on the right-hand side of (3.4) yields the
mass term for the metric fluctuation. If we demand the graviton to be
massless, this term has to vanish. We will show that this only happens when
the fluctuation contains non-normalizable modes only.

In order to demonstrate this let us further linearize the right-hand side of
(3.4) using its expression given in (2.5). This yields

$\displaystyle\begin{split}8\pi G_{\mathrm{N}}^{(d+1)}\delta
T_{ij}^{\mathrm{CFT}}&=\frac{1}{2}n^{k}\partial_{k}\delta\gamma_{ij}\\\
&-\frac{1}{2}n^{k}(\partial_{k}\bar{\gamma}_{nm})\delta\gamma^{nm}\bar{\gamma}_{ij}+\frac{1}{2}n^{k}(\partial_{k}\delta\gamma_{nm})\bar{\gamma}^{nm}\bar{\gamma}_{ij}+\frac{1}{2}n^{k}(\partial_{k}\bar{\gamma}_{nm})\bar{\gamma}^{nm}\delta\gamma_{ij}.\end{split}$
(3.5)

Since the graviton on the brane should be a traceless, symmetric rank-2
tensor, the second and third term of (3.5) cancel by virtue of the fact that
$\delta\gamma_{nm}\bar{\gamma}^{nm}=0$. This is a very important point to
emphasize. This is the same property that we will use explicitly to derive the
propagator for traceless excitations on the brane. The trace part, as we will
discuss in the upcoming section, is related to the number of degrees of
freedom on the brane. The last term of (3.5) drops out due to the fact that
the trace of the induced metric is coordinate independent. Evaluating the
extrinsic curvature on the hypersurface $S=\rho-\epsilon=\text{const.}$, the
remaining terms read

$\displaystyle 8\pi G_{d}\,\delta
T_{ij}^{\mathrm{CFT}}=\frac{d-2}{L^{2}}\left[\partial_{\rho}(\rho\delta\gamma_{ij})+\rho(\partial_{\rho}\delta\gamma_{nm})\bar{\gamma}^{nm}\bar{\gamma}_{ij}-\delta\gamma_{ij}\right],$
(3.6)

The last term only provides a new contribution to the cosmological constant
term of Einstein’s equations. Bringing this to the left-hand side of (3.4),
this redefines the cosmological constant further as

$\displaystyle\Lambda^{\rm
eff}_{d}=\bar{\Lambda}_{d}+\frac{d-2}{L^{2}}=\frac{(d-1)(d-2)}{L^{2}}+\Lambda_{d}+\frac{d-2}{L^{2}}.$
(3.7)

Again, using the freedom to choose $\Lambda_{d}$, we can fix $\Lambda^{\rm
eff}_{d}=\frac{(d-1)(d-2)}{L^{2}}$, the de Sitter value.

The contributions to the graviton mass then come from the first two terms of
(3.6). Let us now expand these terms in terms of normalizable and non-
normalizable modes decomposed through the Feffermann-Graham expansion of the
graviton fluctuation,

$\displaystyle\delta\gamma_{ij}(\rho,x)=(\alpha\,\rho^{-1}+\cdots+\beta\,\rho^{d/2-1})\,h_{ij}(x).$
(3.8)

Clearly, $\alpha$ is the non-normalizable mode and $\beta$ is the normalizable
mode in this asymptotic expansion. This expansion implies that the
tracelessness condition of the graviton holds only if
$h_{ij}\bar{\gamma}^{ij}=0$. However, since in this expansion $h_{ij}$ only
depends on $x$, it is obvious that
$(\partial_{\rho}\delta\gamma_{nm})\bar{\gamma}^{nm}=0$ in our case and hence
the second term of (3.6) vanishes identically. Inserting the mode
decomposition in the single non-vanishing term of (3.6) yields

$\displaystyle\frac{d-2}{L^{2}}\partial_{\rho}(\rho\delta\gamma_{ij})=\frac{d-2}{L^{2}}\partial_{\rho}(\alpha+\beta\,\rho^{d/2})\,h_{ij}(x)=\frac{d(d-2)}{2L^{2}}\beta\,\rho^{d/2}\,\rho^{-1}\,h_{ij}(x)\sim\frac{d(d-2)}{2L^{2}}\frac{\beta}{\alpha}\rho^{d/2}\delta\gamma_{ij}.$
(3.9)

In the last step, we used the fact that the non-normalizable part of the
graviton fluctuation dominates near the position of brane at $\rho=\epsilon$
close to the boundary, namely, $\delta\gamma_{ij}\sim\alpha\rho^{-1}h_{ij}$.
In the near boundary expansion $\epsilon\rightarrow 0$ one can read off the
graviton mass directly from (3.9) as

$\displaystyle m_{\rm
grav}^{2}=-\frac{\beta}{\alpha}\frac{2\,d(d-2)}{(d-1)(d-2)+2}\frac{\epsilon^{d/2}}{L^{2}}.$
(3.10)

As is evident from (3.10), in order to get a massless graviton, one does need
to set $\beta=0$, i.e. we need to exclude the normalizable mode from the field
expansion. In other words, in order to get a consistent holographic
description of such a cosmological model, one needs to consider only non-
normalizable boundary conditions. This conclusion is very much in line with
that of [18].

As mentioned in the previous section, there should also be additional trace
contributions to the CFT stress-energy tensor. By construction, such
contributions are normalizable, as can be verified directly by performing a
near boundary expansion $a\rightarrow\infty$ of (2). However, this anomalous
part of the stress-energy tensor does not contribute to a mass term, rather it
provides the number of degrees of freedom imprinted on the cutoff brane. We
will come back to this, once again, in the context of our dark bubble model in
the upcoming section.

##  4 From braneworld to bubbleworld: The graviton propagator on the shell

So far we have discussed a simple one-sided case with an $\text{AdS}_{d+1}$
bulk geometry with a cutoff brane near the boundary. Now we extend this
configuration, replacing the cutoff with a spherical shell having an interior
as well as an exterior region. This is the dark bubble model set up and
developed in [2, 3, 4, 5, 6, 7, 8, 9, 10, 11]. In this model, the AdS spaces
in the inside $(-)$ and outside $(+)$ have different length scales with
$\Lambda_{+}>\Lambda_{-}$ which ensures the expansion of the dark bubble. The
junction across the shell, separating the two regions, determines the tension
of the bubble wall. As discussed before, a subcritical tension of the wall
leads to a positive cosmological constant on it, corresponding to the desired
de Sitter universe. The non-identical AdS spaces in the interior and exterior
region render the gravitational constant finite, but for this one does need to
allow non-normalizable modes in these regions, which in turn imprint massless
graviton modes on the bubble wall. In the dark bubble model such non-
normalizable modes are physically realized using hanging strings as sources.
These provide the required kinetic energy for the bubble to expand in the
presence of particles of dust represented by the end points of the strings
[4].

Our goal in this subsection is to fit the dark bubble model into the
holographic formalism discussed in the previous sections. The major technical
difference is how to incorporate the junction condition across the bubble wall
that separates the two holographic bulk spacetimes. The location of the bubble
wall coincides with the near boundary regions of both the bulk regions. In
this setup the graviton propagator on the bubble was computed in [3]. We will
revisit this computation in this section, but, this time, in the light of the
holographic formalism developed in the previous sections.

Unlike the previous section, where the brane is merely a cutoff near the
boundary, the dynamics of the bubble wall plays a more crucial role in the
dark bubble model. As discussed above, in order to implement the junction
between interior and exterior with a view to get an expanding universe on the
bubble, we require a subcritical positive tension spherical brane to be
realized as the bubble wall. The latter modifies the bulk action as

$\displaystyle
S=S_{\mathrm{\rm{bulk}}}+S_{\mathrm{\rm{brane}}}=S_{\mathrm{\rm{bulk}}}+\sigma\int\mathrm{d}^{d}x\sqrt{-\gamma}\,,$
(4.1)

where $\sigma$ is the brane tension.

The extra contribution due to the brane only depends on the determinant of the
induced metric and is of the same form as that of the constant term
proportional to $\kappa_{2}$ in the counterterm action (2.3) in the bulk.
Therefore, in the equation of motion, this will provide a new contribution to
the cosmological constant term. The derivation is similar to that leading to
(3.7), however, there are a couple of interesting catch points, particularly
for the dark bubble model, which has an interior and an exterior. The latter
is manifest in the difference of the AdS scales appearing in the denominator
of the RHS of (1.6). This has an interesting consequence in terms of a
different hierarchy between the four-dimensional and the five-dimensional
Newton’s constants given by [9, 12]

$\frac{G_{4}}{G_{5}}=-\frac{3}{L}\frac{N}{\Delta N},$ (4.2)

where $\Delta N$ is a negative integer signifying the number of nucleated D$3$
branes from a stack of N D$3$ branes in the ten-dimensional realization of the
dark bubble model in type IIB supergravity [9]. This new scaling behaviour
follows precisely due to the difference in scales appearing in the denominator
of (1.6), rather than the sum, as usually for RS-like braneworld models. The
second important point is that we need to run the holographic renormalization
scheme from either side of the brane consistent with the junction condition.
This would fix the renormalization parameters as

$\kappa_{1}=\frac{3\left(L_{+}-L_{-}\right)}{16\pi G_{5}}\ \ ,\ \ \ \
\kappa_{2}=\frac{3}{8\pi G_{5}}\left(\frac{1}{L_{-}}-\frac{1}{L_{+}}\right),$
(4.3)

for the two-sided geometry. Taking these two special features of the dark
bubble model into account, along with the contribution of the brane tension,
we end up in getting the effective four-dimensional cosmological constant
as222We fix the contribution to the cosmological constant coming from the
local term exactly as before, namely, by choosing $\Lambda_{4}$ in such a way
that it cancels the contribution coming from the CFT stress-energy tensor.

$\displaystyle\Lambda^{\rm eff}_{4}=8\pi
G_{4}\kappa_{2}-\sigma=\frac{6}{L^{2}}-\sigma.$ (4.4)

It is interesting to note that, at the end of the day, the effective
cosmological constant retains its form as in the standard one-sided
holography, as in (3.7). Nevertheless, the scale of hierarchy, $N$ appearing
in (4.2), makes our dark bubble significantly different from the inside-inside
braneworld models, including the RS. This difference provides a reversal of
hierarchy beteen the five-dimensional and four-dimensional Planck scales,
resulting in interesting phenomenological implications. Interested readers are
referred to [12] for a detailed account of the latter.

Note, that this is precisely the desired cosmological constant (1.5) on the
brane, with the critical tension of the brane holographically identified with
the parameter of holographic renormalization, $8\pi G_{4}\kappa_{2}$. Varying
the full action (4.1) yields the equation of motion

$\displaystyle\mathcal{G}_{\mu\nu}=\frac{1}{2M^{d-1}}[T_{\mu\nu}-\Lambda
g_{\mu\nu}-\sigma(g_{\mu\nu}-n_{\mu}n_{\nu})\delta(X^{d+1}-X^{d+1}(x))],$
(4.5)

where $\mathcal{G}_{\mu\nu}$ is the Einstein’s tensor, and $n_{\mu}$ is the
normal vector on the brane. The delta function appearing in the field equation
selects the bulk position of the shell parametrized by its world-volume
coordinates $x^{a}$.

Our strategy is first to linearize the equation of motion in terms of the
leading graviton fluctuation as before. For this, it turns out to be
convenient to use the Gauss normal coordinates, in which the AdS metric
assumes the form [3]

$\displaystyle
ds^{2}=d\xi^{2}+a^{2}(\xi)(\eta_{ab}+h_{ab}\left(\xi,x^{a})\right)dx^{a}dx^{b}.$
(4.6)

One can obtain this metric from the Fefferman-Graham metric (3.8) simply by
rescaling the radial coordinate as

$\displaystyle\rho\propto e^{-2k\xi}=a^{-2}(\xi),$ (4.7)

with $k=\frac{1}{L}$ being the inverse of the AdS length scale.

The main algebraic difference we have here, as compared to the simpler one-
sided braneworld construction presented in the previous section, is the last
term in (4.5). It contains products of normal vectors on the brane which give
additional derivative terms with respect to the normal coordinate when we
linearize the equations of motion in the metric fluctuation $h_{ab}$. At the
leading order of the metric perturbation in $h_{ab}$, the transverse
components of (4.5) yields [41]

$\displaystyle\begin{split}\Box\bar{h}_{ab}=e^{2k\xi}(&-\eta_{ab}\partial^{n}\partial^{m}\bar{h}_{nm}+\partial^{n}\partial_{a}\bar{h}_{bn}+\partial^{n}\partial_{b}\bar{h}_{an})\\\
&+\frac{\eta_{ab}}{2}e^{k\xi d}\partial_{\xi}(e^{-k\xi
d}\partial_{\xi}{\bar{h}})-16\pi G_{d+1}e^{2k\xi}T_{ab},\end{split}$ (4.8)

where we have defined the trace-reversed fluctuations

$\displaystyle\bar{h}_{ab}=h_{ab}-\frac{1}{2}\eta_{ab}h$ (4.9)

satisfying the transverse gauge condition $\partial^{a}\bar{h}_{ab}=0$
everywhere outside the location of the source [41]. Absorbing the scale factor
by defining $\gamma_{ab}:=a^{2}(\xi)h_{ab}$ and taking a trace, one obtains
from (4.8) a trace equation in $d=4$ of the form

$\displaystyle
a^{-2}\partial^{2}\bar{\gamma}+3(\partial^{2}_{\xi}-4k^{2})\bar{\gamma}=-16\pi
G_{5}T,$ (4.10)

where $\bar{\gamma}_{ab}$ is the trace reversed $\gamma_{ab}$.

We need to subtract (4.10) from the original linearized equation (4.8), in
order to select the traceless graviton mode. With this, finally, the equations
of motion for the graviton on the dark bubble assume the form [3]

$\displaystyle\Box\chi_{ab}+(\partial_{\xi}^{2}-4k^{2})\chi_{ab}=-16\pi
G_{5}\Sigma_{ab},$ (4.11)

where $\chi_{ab}$ and $\Sigma_{ab}$ represent the traceless metric excitation
and stress tensor, respectively.

Before we solve the equations of motion and continue with the derivation of
the propagator, let us pause a bit to check the role of normalizable and non-
normalizable modes in generating the mass of these transverse traceless modes.
We follow the same strategy as discussed in section 3, namely to study the
equations of motion of these modes near the boundary to extract the mass term.
In particular, it would be interesting to see whether the additional piece of
the dark bubble equations of motion (4.11), arising due to the shell tension,
alters the previous conclusion in any way. Following this motivation, we will
concentrate on the second term in (4.11), since the stress tensor contribution
is the same in both cases.

To accomplish our goal, we need to first go back to the Fefferman-Graham
radial coordinates through (4.7). In this coordinate, the relevant piece of
the equations of motion transforms into the form

$\displaystyle(\partial_{\xi}^{2}-4k^{2})\chi_{ab}=4k^{2}[-\partial_{\rho}(\rho\chi_{ab})+\rho\partial_{\rho}^{2}(\rho\chi_{ab})].$
(4.12)

Once again, we plug in this the mode decomposition of the metric fluctuation
(3.8) for $d=4$. It yields

$\displaystyle(\partial_{\xi}^{2}-4k^{2})\chi_{ab}=4k^{2}(-2\beta\rho+2\beta\rho)h_{ab}=0.$
(4.13)

Thus, taking the limit $\rho\rightarrow a_{s}^{-2}$, with
$a_{s}\rightarrow\infty$ being the location of the brane, we obtain a
vanishing correction to the mass term at the leading order. Therefore, in the
$d=4$ holographic dark bubble, we end up getting, exactly, the same mass term
of the graviton (3.10) on the bubble wall 333The mass terms get modified in
higher dimensions. Nevertheless, they remain proportional to the ratio between
the normalizable and the non-normalizable modes. Consequently, the conclusion
remains unaltered.. Accordingly, the condition for having a massless graviton
on the brane remains the same as well, i.e., in the holographic model of dark
bubble, we do need to turn off the normalizable fluctuation modes and should
only allow non-normalizable modes instead. As discussed in the introduction,
physically, this condition fits perfectly with our dark bubble model, as the
non-normalizable modes in this model are carried by hanging strings present in
this model. A detailed discussion on this can be found in [4].

We now carry forward this finding towards determining the real dynamics of the
graviton on the bubble wall. For deriving the propagator on the wall, it is
easier to work in momentum space. We therefore apply a transverse Fourier
transformation to (4.11) that yields [3]

$\displaystyle\bigg{(}-\frac{p^{2}}{a^{2}}+\partial_{\xi}^{2}-4k^{2}\bigg{)}\tilde{\chi}_{ab}(p,\xi)=-16\pi
G_{5}\tilde{\Sigma}_{ab}.$ (4.14)

$\tilde{\chi}_{ab}$ and $\tilde{\Sigma}_{ab}$ are the transverse Fourier
transforms of the trace removed metric and stress tensor, respectively. The
shell is assumed to be localized at $a_{s}\equiv a_{+}=a_{-}$, where $a_{\pm}$
describes the scale factor in the interior or exterior region. It is
instructive to first solve for a massless minimally coupled scalar propagator
in AdS. The latter essentially satisfies the same equation of motion as
(4.14), and reproduces all the relevant qualitative features of the solutions
of (4.14). Finally, of course, we will reinstate the indices properly to write
down the solution for the graviton mode explicitly. The full computation was
done in [3]. We will here only show the main steps and make it aligned with
the holographic lessons we learnt above.

We start with an ansatz for a solution of the Green’s function for the
exterior and interior regions

$\displaystyle\Delta^{+}_{\tilde{\chi}}(p;a_{+},a_{-})$
$\displaystyle=A(p,a_{-})K_{2}\bigg{(}\frac{p}{k_{+}a_{+}}\bigg{)}+B(p,a_{-})I_{2}\bigg{(}\frac{p}{k_{+}a_{+}}\bigg{)}$
(4.15) $\displaystyle\Delta^{-}_{\tilde{\chi}}(p;a_{+},a_{-})$
$\displaystyle=C(p,a_{+})K_{2}\bigg{(}\frac{p}{k_{-}a_{-}}\bigg{)},$ (4.16)

valid outside the source, which in this case is the brane at the junction.
$K_{2}$ and $I_{2}$ denote the respective modified Bessel functions, and
represent non-normalizable and normalizable metric modes in the large $a$
limit. The normalizable piece proportional to $I_{2}$ does not appear in
(4.16) because it is divergent in the limit $a\rightarrow 0$ where we expect
our solution to be regular. We fix two of the coefficients $A,B,C$ using
junction conditions on the bubble wall. There are two matching conditions, one
for the propagator function and the other involving its derivative, namely

$\displaystyle\Delta^{-}_{\tilde{\chi}}(p;a_{+},a_{s})=\Delta^{+}_{\tilde{\chi}}(p;a_{s},a_{-}),$
(4.17) $\displaystyle\frac{1}{16\pi
G_{5}}\bigg{[}\frac{\partial}{\partial\xi_{i}}\Delta^{i}_{\tilde{\chi}}(p;a_{+},a_{-})|_{a_{i}\rightarrow
a_{s}}\bigg{]}^{i=+}_{i=-}+\frac{\sigma}{3}\Delta^{+}_{\tilde{\chi}}(p;a_{s},a_{-})=\frac{1}{16\pi
G_{5}}.$ (4.18)

Solving this, we express $A$ and $C$ in terms of $B$, which leads to a Green’s
function of the form

$\displaystyle\Delta^{+}_{\tilde{\chi}}(p;a_{+},a_{-})=$
$\displaystyle-\bigg{[}\frac{1}{g_{K}(p,a_{s})}-B(p,a_{-})\frac{g_{I}(p,a_{s})}{g_{K}(p,a_{s})}\bigg{]}K_{2}\bigg{(}\frac{p}{k_{+}a_{+}}\bigg{)}$
(4.19) $\displaystyle+B(p,a_{-})I_{2}\bigg{(}\frac{p}{k_{+}a_{+}}\bigg{)},$
(4.20)

with

$\displaystyle g_{K}(p,a_{s})$
$\displaystyle=\frac{p}{a_{s}}\frac{K_{2}\bigg{(}\frac{p}{k_{+}a_{s}}\bigg{)}K_{1}\bigg{(}\frac{p}{k_{-}a_{s}}\bigg{)}-K_{2}\bigg{(}\frac{p}{k_{-}a_{s}}\bigg{)}K_{1}\bigg{(}\frac{p}{k_{+}a_{s}}\bigg{)}}{K_{2}\bigg{(}\frac{p}{k_{-}a_{s}}\bigg{)}},$
(4.21) $\displaystyle g_{I}(p,a_{s})$
$\displaystyle=\frac{p}{a_{s}}\frac{I_{2}\bigg{(}\frac{p}{k_{+}a_{s}}\bigg{)}K_{1}\bigg{(}\frac{p}{k_{-}a_{s}}\bigg{)}+K_{2}\bigg{(}\frac{p}{k_{-}a_{s}}\bigg{)}I_{1}\bigg{(}\frac{p}{k_{+}a_{s}}\bigg{)}}{K_{2}\bigg{(}\frac{p}{k_{-}a_{s}}\bigg{)}}.$
(4.22)

This is the time to use our holographic understanding of boundary conditions
as we now aim to fix the last remaining function, $B(p,a_{s})$ on the brane.
We recall the requirement for having a massless graviton on the wall, in the
limit when the brane is close to the asymptotic boundaries of either AdS
spacetime. In the context of the present model, this demands setting the
normalizable mode to be set to zero in the near boundary expansion
$a_{s}\rightarrow 0$ of (4.19). This in turn fixes $B(p,a_{s})$ as

$\displaystyle B(p,a_{s})==-\frac{\eta}{\eta g_{I}(p,a_{s})-4g_{K}(p,a_{s})}\
\ \text{with}\ \ \eta:=3-4\gamma+4\ln{2}.$ (4.23)

Plugging (4.23) in (4.19) in the low momentum limit yields the propagator on
the bubble wall as

$\displaystyle\Delta^{s}_{\tilde{\chi}}(p;a_{s},a_{s})=\frac{a_{s}^{2}}{p^{2}}\bigg{(}\frac{2k_{-}k_{+}}{k_{-}-k_{+}}\bigg{)}+\mathcal{O}(p^{0}).$
(4.24)

Finally, after convoluting the Green’s function with the source term which
corresponds to a simple multiplication in momentum space, the Fourier
component of the graviton mode on the bubble wall is

$\displaystyle\tilde{\chi}^{s}_{ab}(p;a_{s})=-16\pi
G_{5}\frac{a_{s}^{2}}{p^{2}}\bigg{(}\frac{2k_{-}k_{+}}{k_{-}-k_{+}}\bigg{)}\tilde{\Sigma}_{ab}.$
(4.25)

This recovers, in the small momentum limit, the expected four-dimensional
graviton mode as imprinted on the bubble wall with the correct four-
dimensional Newton’s constant [1]. The key point we want to emphasize in this
paper, through the computation leading to (4.25), is the role of holography in
reproducing the correct propagator on the bubble wall. We use the requirement
of having a massless graviton on the brane, which automatically dictates the
proper boundary condition. Surprisingly enough, this was already an energetic
requirement in our dark bubble model to ensure an expanding universe on the
wall [1, 2]. The present work reestablishes it in terms of a proper
holographic requirement.

We will conclude this section with a pending discussion on the degrees of
freedom on the brane. As outlined in section 3, the CFT stress-energy tensor
generally has a trace piece due to the conformal Weyl anomaly that we
subtracted from the equations of motion in order to take into account the
tracelessness of the graviton imprinted on the bubble wall. We now want to
emphasize the physical connection of the trace piece to the number of degrees
of freedom on the wall. Similarly to section 3, we need to consider the
stress-energy tensor of the CFT which is renormalized by means of the
holographic renormalization techniques [40]. In the two-sided case of the dark
bubble, the choice of the renormalization parameters (4.3) renders finiteness
of the brane stress-energy tensor in the limit $a_{s}\to\infty$.

We now compute the trace of the regularized stress energy tensor, and compare
it to the expected Weyl anomaly for the conformal field theory on the brane,
namely, [42, 38]

$\text{Tr}\,T=-\frac{N_{\text{eff}}^{2}}{32\pi^{2}}\left(-{\cal R}_{ij}{\cal
R}^{ij}+\frac{1}{3}{\cal R}^{2}\right),$ (4.26)

which yields, for our case,

$N_{\text{eff}}^{2}=\frac{\pi}{2G_{5}}\left(L_{+}^{3}-L_{-}^{3}\right).$
(4.27)

This matches perfectly with the thermodynamical derivation presented in [1].
Usually, when we simply have one term of type $\frac{\pi L^{3}}{2G_{5}}$, this
translates into $N^{2}$, which is the number of adjoint fields. For the dark
bubble, we find, due to the subtraction, $N$, which is the dimension of the
fundamental representation. This was also used in [9], when discussing
corrections to the string tension. The physical picture is that of the
background $AdS_{5}\times S^{5}$, dual to $N$ branes. In this background a
single brane nucleates, the dark bubble. The counting is that of open strings
connecting the brane to the background. It is interesting to note that the
number of degrees of freedom is always $\sim L^{2}/G_{4}$.

##  5 A brief summary of the main claim

The aim of this work is to fit the dark bubble model consistently within a
holographic framework [2, 3, 4, 5, 6, 7, 8, 9, 10, 11]. We manage to achieve
this through a careful and thorough investigation of a broader class of
boundary conditions allowed in AdS/CFT [21]. While such constructions were
already done in the context of holographic cosmology [20, 19], the challenge
we faced here was to incorporate the two-sided bulk geometry, living on either
side of the bubble wall, which, geometrically, are the inside and the outside
of the bubble. A second challenge was to make the dark bubble model
phenomenologically reasonable to produce the correct dynamics of the graviton
on the brane, along with a finite four-dimensional Newton’s constant. We
noticed the necessity of having extended objects like strings carrying
momentum, which in turn imprints matter on the bubble wall. Not only that,
these strings also ensure proper temporal evolution of the bubble, which from
the perspective of the physics on the wall can be realized through a process
of mass renormalization (see [4] for a detailed discussion on it). The latter
ensures finite masses of the particles induced on the wall.

While we needed the strings to ensure the consistency of the model, it was a
challenge to fit this aspect in the context of holography. In this paper, we
showed that this requirement can be converted into the requirement of having a
massless induced graviton on the bubble wall. For this we need to discard the
normalizable components of the asymptotic expansion of bulk graviton
fluctuation keeping only the non-normalizable piece. A similar conclusion was
drawn recently in [18] in the context of a one-sided Karch-Randall braneworld
scenario. In our two-sided case, this requirement fits nicely, in the sense
that these non-normalizable modes are nothing but an artifact of the momentum
carrying strings that makes the dark bubble a suitable model of the expanding
universe, while being perfectly consistent with the swampland conjectures
avoiding the usual no-go theorems that prevent generating a de Sitter universe
imprinted on a conventional positive tension braneworld [6]. This is also
where our construction significantly differs from the Randall-Sundrum
braneworld story [7]. We revisited the computation of the graviton propagator
on the brane [3], but now endowed with the complete holographic understanding
of the conditions to be imposed to fix the relevant functions on the brane. It
yields the correct massless graviton propagator in the low energy limit, with
the correct four-dimensional Newton’s constant.

Last but not least, in this process of rediscovering our model in the light of
holography, we interpret the critical tension of the brane in terms of a
parameter of holographic renormalization and also compute the effective
degrees of freedom on the bubble wall from the conformal Weyl anomaly of the
holographically renormalized stress-energy tensor. The degrees of freedom
matches exactly with its thermodynamical estimation presented in [1].

## Acknowledgments

We would like to thank Johanna Erdmenger, Suvendu Giri, Andreas Karch, Dominik
Neuenfeld, Daniel Panizo and Thomas Van Riet for useful discussions and
communications. We would also like to thank Dominik Neuenfeld for comments on
an earlier version of the manuscript.

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