# Shedding light on dark bubble cosmology

Ivano Basile  Ulf Danielsson  Suvendu Giri  Daniel Panizo

###### Abstract

Dark bubble cosmology is an alternative paradigm to compactification, which
can circumvent issues of moduli stabilization and scale separation. In this
paper we investigate how electromagnetic fields can be incorporated in this
framework. Worldvolume fields backreact on the ambient universe in which the
bubble expands, which in turn affects the energy-momentum distribution and the
effective gravity induced on the brane. We compute these effects, showing that
the resulting four-dimensional cosmology consistently includes electromagnetic
waves.

##  1 Introduction

Building a realistic model of cosmology and particle physics from string
theory is the holy grail of string phenomenology. Many efforts in this
direction resulted in great progress over the last few decades, but some key
obstacles remain. Three of the most prominent issues are stabilizing moduli,
obtaining scale separation for the extra dimensions (once stabilized), and
producing a positive and small dark energy. The last of which requires
breaking supersymmetry, typically generating instabilities which can be
difficult to control.111See [1, 2, 3] for a review of the difficulties
involved in constructing stable dS vacua in string theory. Dark bubble
cosmology [4] constitutes a compelling alternative framework in which these
issues may be addressed.222See [5, 6] for a review of dark bubble cosmology,
[7, 8, 9, 10] for earlier work discussing cosmology on a 3-brane and brane
gravity localization, and [11, 12] for recent work in the context of
supergravity. To begin with, its very existence requires breaking
supersymmetry and producing a positive dark energy,333Other recent approaches
towards constructing dS vacua in string theory/M-theory include breaking
supersymmetry using negative energy sources like O-planes [13, 14] or Casimir
energy [15]. See [16] and references within, for a construction closely
related to the dark bubble scenario, and [17] for recent work studying a
3-brane mediating the decay between 4D vacua since its underlying mechanism
rests on the decay of a metastable anti-de Sitter (AdS) universe via bubble
nucleation.444See [18, 19, 20] for catalytic effects on bubble nucleariton in
the presence of black holes or a cloud of strings. Moreover, scale separation
on the bubble universe can be implemented not via small extra dimensions, but
rather via quasi-extremal tension-to-charge ratios of the bubble. This
scenario reproduces important features of gravitational interactions at long
distances [21, 22, 23, 24, 25], and it can be constructed in non-
supersymmetric settings in string theory [26, 27].

One of the most important problems in dark bubble cosmology, first introduced
in [4], is to understand the uplift of four dimensional matter fields
compatible with the standard model of particle physics into the higher
dimensions. In [4] it was argued that massless radiation is induced in 4D by
heating up the system with the help of a bulk black hole at the centre of the
bubble. Furthermore, it was shown that fundamental semi-infinite strings,
pulling on the brane from above, give rise to particles with Planckian masses
in 4D. Neither of these are promising candidates to represent known matter. As
discussed in [26, 28], to obtain masses compatible with the standard model of
particle physics, we would need fundamental strings stretching between branes
separated by distances much shorter than the AdS-scale.555In [21], it was
speculated how semi-infinite strings with a lower tension could play a similar
role. Given the stringy embedding presented in [27], a natural possibility
would be to separate the branes on the five internal compact dimensions at a
distance much smaller than the size of the dimensions.666Note that fundamental
strings stretching between opposite poles of the sphere in internal dimensions
would yield masses of order Planck scale.

In this paper we will not consider the massive, or non-Abelian sector of the
standard model of particle physics, but instead focus on pure
electromagnetism. This is already highly non-trivial, requiring an intricate
interplay between the 4D physics induced on the brane and what happens in the
bulk. As we will see, the 4D electromagnetic waves will not only involve a
gauge field trapped on the bubble, but also crucially excite oscillating Kalb-
Ramond fields in the bulk. Together, they make sure that all equations of
motion, as well as the 4D and 5D Einstein equations together with Israel’s
junction conditions across the bubble wall, are solved.

There are interesting similarities, but also differences, compared to the
uplift of gravitational waves that was done in [25]. We therefore start out
with a review of these results in section 2. In section 3 we discuss our
expectations concerning electromagnetic waves in 4D. Section 4 is where we
perform the uplift of 4D electromagnetic waves to 5D, and subsequently check
that they induce the expected 4D energy momentum tensor through the Gauss-
Codazzi equations. Furthermore, we explore the case of a constant electric
field, which clearly shows how the bulk B-field pulls on the brane, in a very
similar way to how a pulling string induces a point mass [23]. Section 5 is
devoted to provide a general proof of the consistency of the model. Finally,
results are discussed in section 6, together with an outline on further steps
to accommodate the standard model of particle physics within the dark bubble
framework.

##  2 Dark bubble cosmology: un résumé

Let us quickly summarise the key features of the dark bubble model that will
be relevant to the new results presented in this paper.

###  2.1 Hierarchies from an explicit stringy embedding

According to the dark bubble proposal, our 4D space time can be identified
with the surface of an expanding bubble of true vacuum in an unstable five
dimensional anti de Sitter ($AdS_{5}$) space [4], where, as explained in [29],
the nucleation of the bubble can be identified with the quantum cosmology of
Vilenkin. An explicit stringy realization was proposed in [27], where the
hierarchy of energy scales was fixed by the top-down construction.777Other
embeddings in non-supersymmetric string theories have been studied in [26,
28]. The type 0’B string [30, 31] is particularly promising
phenomenologically, since it contains charged D3-branes [32, 33], hence 4d
worldvolumes. We start with the 10D Planck length, which is given by the
string coupling and string length in the usual way:

$l_{10}^{4}\sim g_{s}l_{s}^{4}.$ (2.1)

In the following we will, for simplicity, not care about numerical factors.
These can be inserted back to get precise relations. Next, we reduce to 5D.
The background we consider is a deformation of $AdS_{5}\times S_{5}$. There is
no scale separation and the length $L$ sets the size of the $S_{5}$ as well as
of the curvature scale of the non-compact $AdS_{5}$. The 5D Planck scale is
obtained by a simple dimensional reduction on the $S_{5}$ and is given by

$l_{5}^{3}\sim l_{10}^{8}/L^{5}.$ (2.2)

The next step is to go down to four dimensions. In a standard dimensional
reduction on a compact dimension of size $L$, you would get $l_{4}^{2}\sim
l^{5}/L$. The same result is obtained in the Randall-Sundrum (RS) model on a
non-compact AdS-direction with curvature scale $1/L$ [34]. This is not how it
works for the dark bubble. The RS-result for the strength of 4D gravity is due
to the bubble being embedded into $AdS_{5}$ such that it has two insides. The
extrinsic curvature, appearing in the junction conditions, have equal signs on
both sides. In case of the dark bubble, which nucleates in a pre-existing
space time, there is an inside and an outside. As explained in [23], this
leads to a 4D Planck scale given by

$l_{4}^{2}\sim\frac{k_{-}k_{+}}{k_{-}-k_{+}}l_{5}^{3},$ (2.3)

where $k_{\pm}=1/L_{\pm}$ with the minus sign referring to the inside, and the
plus to the outside. In a dimensional reduction, as well as in the case of RS,
the lower dimensional Planck length is smaller than the higher dimensional
one. This means that gravity is weaker in the lower dimensional theory.
Requiring that gravity not to be too strong in the higher dimensions, sets a
limit on the size of the extra dimensions; higher the number of large extra
dimensions, stricter the limit.

This is no longer true for the dark bubble, which explains many of its unique
features. If $k_{-}$ is close to $k_{+}$, then $l_{4}$ can be larger than
$l_{5}$. In fact, the hierarchy that we will find is $l_{10}\gg l_{4}\gg
l_{5}$. Let us look at this in more detail.

As reviewed in [35], the background $AdS_{5}\times S_{5}$ is dual to a
background of $N$ D3-branes. One of these branes can nucleate and start to
expand. By studying its junction condition, this was found to correspond to
$\left(k_{-}-k_{+}\right)\sim k/N\sim\tfrac{1}{NL}$, where we have put
$k_{-}\sim k_{+}\sim k$. From this we see that:

$l_{4}^{2}\sim\frac{N}{L}l_{5}^{3}.$ (2.4)

It is a large $N$ what ensures that $l_{4}\gg l_{5}$. As is well known, the
scale $L$ of the background is given as $L^{2}\sim l_{s}^{2}\sqrt{g_{s}N}$,
which, using (2.1) and (2.2), is equivalent to

$l_{5}^{3}\sim\frac{L^{3}}{N^{2}}.$ (2.5)

There is only one remaining relation to be fixed, which will determine $N$ in
terms of the cosmological constant. The nucleated brane is supposed to be
critical and to have a tension equal to the standard value of a D3-brane:

$T_{3}=\frac{1}{(2\pi)^{3}g_{s}l_{s}^{4}}.$ (2.6)

It is worth pointing out that if all numerical coefficients are restored, it
exactly solves the condition for a critical bubble:

$\sigma_{c}=\frac{3}{8\pi G_{5}}(k_{-}-k_{+}).$ (2.7)

As argued in [27], we expect there to be corrections of order $1/N$ that will
reduce the value of the tension in line with the Weak Gravity Conjecture
(WGC)888This will be discussed and calculated in an upcoming paper [36]. along
the lines of the results found in [37, 38]. This will shift the tension from
its critical value, and generate an effective 4D cosmological constant of
magnitude

$\rho_{\Lambda}\sim\frac{1}{g_{s}l_{s}^{4}}\frac{1}{N}\sim\frac{1}{L^{4}}.$
(2.8)

From this we find that $L\sim 10^{-5}m$, which has the size of the dark
dimension discussed in [39]. This is an intriguing result, but one should keep
in mind that in our scenario the scalings do not arise from a hierarchy
between a single mesoscopic dimension and the other (scale-separated) ones.
Instead, (2.8) arises from the peculiar relation between the 5D and 4D Planck
scales given by (2.3).

Using the previous relations, we can now express the 5D and 10D Planck scales,
4D Hubble length $R_{H}$, and length scale $L$ of $AdS_{5}$ in terms of $N$
and the 4D Planck length as:999If we instead consider corrections to be of
order $1/g_{s}N$, the first relation is changed to $R_{H}\sim
g_{s}^{1/2}Nl_{4}$.

$R_{H}\sim Nl_{4},\quad L\sim N^{1/2}l_{4},\quad l_{10}\sim N^{1/4}l_{4},\quad
l_{5}\sim N^{-1/6}l_{4}.$ (2.9)

Furthermore, we have $l_{s}\sim l_{10}/g_{s}^{1/4}$. This finally fixes $N\sim
10^{60}$. From there we find the non-trivial prediction that the 10D Planck
scale should sit at around $10\>\text{TeV}$, with the string scale just below
and a string coupling $g_{s}$ that is less than one but not too small.

The scalings we have found may be rather general for an embedding meeting the
requirements of the dark bubble. The input needed, include the standard
relations between scales for a compactification (without scale separation)
like $AdS_{5}\times S_{5}$ or some similar warped throat. This, together with
the junction conditions for a brane with an inside and an outside, and the
tension shifted away from the critical one through a $1/N$-effect, is what
leads to the hierarchy.

###  2.2 Gravitational waves in dark bubble cosmology

Let us now quickly summarise how gravitational perturbations in the bulk
induce gravitational waves (GW) on the bubble’s boundary, as studied in [25].
The four dimensional energy momentum tensor on the brane is induced due to the
difference between bulk metrics across the brane, which is captured by
Israel’s second junction condition,101010Israel’s first junction condition
requires that the induced metric is continuous across the brane:
$h_{ab}^{+}=h_{ab}^{-}\coloneqq h_{ab}$. given by:

$\kappa_{5}S_{ab}=\left.\left[K_{ab}-Kh_{ab}\right]\right|^{-}_{+},$ (2.10)

where $[A]_{+}^{-}$ stands for the difference between the object $A$ evaluated
inside ($-$) and outside $(+)$ the bubble wall.
$K_{ab}=\nabla_{\beta}n_{\alpha}\>e^{\alpha}_{a}\>e^{\beta}_{b}$ (with trace
$K$) represents the extrinsic curvature and carries information about the
bubble’s embedding in the bulk geometry.111111The unit normal vector, denoted
by $n_{\alpha}$, points in the direction in which the bubble’s volume
increases. The tangent vector to the hyper-surface is $e_{a}^{\alpha}$.
$h_{ab}$ is the induced metric on the wall. Throughout, bulk coordinates are
represented by Greek letter indices, while coordinates on the brane are
represented by Latin letter indices.

We would like to rewrite previous expression in terms of the energy-momentum
in the bulk. This can be achieved by inserting the Gauss-Codazzi equation,

$R^{(5)}_{\alpha\beta\gamma\delta}e^{\alpha}_{a}e^{\beta}_{b}e^{\gamma}_{c}e^{\delta}_{d}=R^{(4)}_{abcd}+K_{ad}K_{bc}-K_{ac}K_{bd},$
(2.11)

into the junction condition (2.10), to find:

$\displaystyle G_{ab}^{(4)}$
$\displaystyle=\left(\kappa_{4}\sigma-3k_{+}k_{-}\right)h_{ab}-\kappa_{4}\left(T_{\textrm{brane}}\right)_{ab}+\frac{k_{+}k_{-}}{k_{-}-k_{+}}\left[\left(\frac{\mathscr{J}_{ab}^{+}}{k_{+}}-\frac{\mathscr{J}_{ab}^{-}}{k_{-}}\right)-\frac{1}{2}\left(\frac{\mathscr{J}^{+}}{k_{+}}-\frac{\mathscr{J}^{-}}{k_{-}}\right)h_{ab}\right]$
(2.12) $\displaystyle\quad+\mathscr{O}\left((\Lambda_{4}/k^{2})^{2}\right),$

where $\mathscr{J}_{ab}$ is a tensor defined by:

$\mathscr{J}_{ab}=R^{(5)}_{\alpha\beta\gamma\delta}e^{\alpha}_{a}e^{\beta}_{b}e^{\gamma}_{c}e^{\delta}_{d}h^{cd}.$
(2.13)

and $\mathscr{J}=h^{ab}\mathscr{J}_{ab}$, its trace. $T_{\textrm{b}rane}$
above corresponds to matter on the brane, and contributes with a negative sign
to the energy-momentum tensor in 4D.121212Note that our conventions with
metric signature $(-++++)$ and $h_{00}<0$, implies $T_{00}>0$ and
$T_{0}^{0}<0$ for positive energy. We can see from expression (2.12) that it
is $\mathscr{J}_{ab}$, the energy-momentum of the bulk, that is in charge of
sourcing the four-dimensional geometry. It is important to notice, that even
in pure $\text{AdS}_{5}$, $\mathscr{J}_{ab}$ contributes with $-3k^{2}h_{ab}$,
which provides a net cosmological constant given by

$\Lambda_{4}=6k_{+}k_{-}-\kappa_{4}\sigma=\kappa_{4}\left(\frac{3}{\kappa_{5}}(k_{-}-k_{+})-\sigma\right).$
(2.14)

Here we see how the tension is matched against the difference in energy
between the outside and the inside to yield an effective cosmological
constant, always positive. One can then go further and study how different
bulk geometries source different energy densities (i.e. dust or radiation) in
the four dimensional geometry [21].

Gravitational waves are vacuum solutions of Einstein’s equations of motion.
They can be described as transverse and traceless (TT) perturbations to a FLRW
metric, as:

$ds_{\rm{FLRW}}^{2}=a^{2}(\eta)\left(-d\eta^{2}+\left(\gamma_{ij}+\zeta\>h_{ij}(\eta,x)\right)dx^{i}dx^{j}\right),$
(2.15)

where $\eta$ is the conformal time, $\gamma_{ij}$ is the metric for spatial
slices, $h_{ij}$ corresponds to the traceless and transverse perturbations,
and $\zeta$ is the perturbation parameter. Assuming a single mode of the
perturbation, as $h_{ij}(\eta,x)=h_{4D}(\eta)Y_{ij}(x)$, where $Y_{ij}(x)$
represents spherical harmonics over the closed spatial sections, one finds the
time dependent solution to be:

$h_{\rm
4D}\left(\cos(\eta)\right)=\frac{1}{n+1}\cos\left((n+1)\eta\right)+\sin\eta\sin(n\eta),$
(2.16)

with $n$ labelling the wave number.

The notion of energy of a gravitational wave is a bit tricky. Since it solves
the vacuum equation of motion, one could wrongly conclude that they do not
carry any energy. The right procedure is to solve the Einstein equations to
first order, and then to identify the energy of the first order waves with the
second order piece representing the failure of the first order waves to solve
the vacuum equations. In [25] we used this procedure in 4D as well as in 5D.
The isotropic, wavelength averaged energy momentum in 4D becomes:131313With
coordinates $x^{a}\equiv\left\\{t,\theta,\phi,\psi\right\\}$.

$\langle\tensor{T}{{}^{a}_{b}}\rangle_{\rm
iso}=\frac{7}{8\kappa_{4}}\frac{1}{a^{2}}\begin{pmatrix}1&0&0&0\\\
0&1/3&0&0\\\ 0&0&1/3&0\\\
0&0&0&1/3\end{pmatrix}+\frac{q^{2}}{4\kappa_{4}H^{2}}\frac{1}{a^{4}}\begin{pmatrix}-1&0&0&0\\\
0&1/3&0&0\\\ 0&0&1/3&0\\\ 0&0&0&1/3\end{pmatrix}.$ (2.17)

Observe how the first term dilutes as $\sim 1/a^{2}$, behaving like curvature
in the Friedmann equations, while the second one, with equation of state
$p=-\rho/3$ and dilution $\sim 1/a^{4}$, has a radiation-like behaviour. When
the wavelength of the waves is larger than the horizon, it freezes and the
only remaining contribution is curvature.

Let us now focus on single modes of TT gravitational perturbations in the five
dimensional bulk that will induce the aforementioned four dimensional ones.
The partial differential equation to be satisfied, from Einstein’s equations
is:

$\frac{\partial^{2}h}{\partial t^{2}}-f^{2}\frac{\partial^{2}h}{\partial
r^{2}}-\frac{f}{r}\left(2+4k^{2}r^{2}+f\right)\frac{\partial h}{\partial
r}+\frac{n^{2}-1}{r^{2}}fh=0,$ (2.18)

with $f\equiv f(r)$, the AdS function in global coordinates, and $h\equiv
h_{\textrm{5D}}(t,r)$, the wave to solve for. This wave is required to behave
like the induced four dimensional one (2.16), up to supressed $H/k$
corrections, and to be sourceless ($h_{\textrm{5D}}(t,r)=0$ when $r\rightarrow
0$). Imposing these boundary conditions, the resulting bulk wave is:

$h_{\rm
5D}(t,r)=\frac{(kr)^{n-1}}{\left(1+k^{2}r^{2}\right)^{\frac{n-1}{2}}}\left[\frac{\frac{1}{2}(1+n)(2-n)+k^{2}r^{2}}{(n+1)\left(1+k^{2}r^{2}\right)}\cos\left((n+1)kt\right)+\sin(kt)\sin(nkt)\right]\,.$
(2.19)

with a five dimensional energy momentum:141414With coordinates
$x^{\alpha}\equiv\left\\{x^{a},r\right\\}$.

$\displaystyle\begin{aligned} \langle\tensor{T}{{}^{\mu}_{\nu}}\rangle_{\rm
rad}=\frac{k^{2}n^{2}t^{2}}{4\kappa_{5}r^{2}}\begin{pmatrix}-1&0&0&0&0\\\
0&1/3&0&0&0\\\ 0&0&1/3&0&0\\\ 0&0&0&1/3&0\\\ 0&0&0&0&0\end{pmatrix},&&\langle
T^{\mu}{}_{\nu}\rangle_{\rm
flux}=\frac{n^{2}}{8\kappa_{5}r^{2}}\begin{pmatrix}0&0&0&0&-2tk^{-2}r^{-3}\\\
0&0&0&0&0\\\ 0&0&0&0&0\\\ 0&0&0&0&0\\\
2k^{2}tr&0&0&0&0\end{pmatrix},\end{aligned}$ (2.20)
$\displaystyle\langle\tensor{T}{{}^{\mu}_{\nu}}\rangle_{\rm
curv}=\frac{1}{8\kappa_{5}r^{2}}\left(7-\frac{n^{4}}{2k^{4}r^{4}}\right)\begin{pmatrix}1&0&0&0&0\\\
0&1/3+\mathscr{O}\left(\frac{n^{2}}{k^{2}r^{2}}\right)&0&0&0\\\
0&0&1/3+\mathscr{O}\left(\frac{n^{2}}{k^{2}r^{2}}\right)&0&0\\\
0&0&0&1/3+\mathscr{O}\left(\frac{n^{2}}{k^{2}r^{2}}\right)&0\\\
0&0&0&0&1+\mathscr{O}\left(\frac{n^{2}}{k^{2}r^{2}}\right)\end{pmatrix}.$

Here, the energy momentum tensor has been averaged and made isotropic, in the
same spirit as in the four dimensional case (2.17). The first line corresponds
to gravitational radiation. Note in particular that the non-zero $\\{t,r\\}$
components of the flux indicate a flow of energy in the direction in which the
bubble expands, as expected. The second line corresponds to waves that are
frozen as they become larger than the horizon scale.

The very nontrivial check that the dark bubble construction makes sense, is to
use the backreacted 5D bulk metric caused by the above energy momentum (2.20)
and calculate its contribution to the extrinsic curvature on the bubble. This
metric is given by:

$\mathrm{d}s^{2}_{\rm
back}\approx-\left[1+k^{2}r^{2}+\epsilon^{2}\>\left(q_{1}-q_{2}k^{2}t^{2}\right)\right]\mathrm{d}t^{2}+\left[1+k^{2}r^{2}+\epsilon^{2}\>\left(q_{1}-q_{2}k^{2}t^{2}\right)\right]^{-1}\mathrm{d}r^{2}+r^{2}\mathrm{d}\Omega_{3}^{2},$
(2.21)

with $|\epsilon|\ll 1$ and $\\{q_{i}\\}$ coefficients:

$q_{1}=-\frac{7}{24},\quad\quad q_{2}=-\frac{q^{2}}{6},\quad\quad
q_{3}=\frac{q_{2}}{2}=-\frac{q^{2}}{12}.$ (2.22)

This, in turn, gives rise to an effective energy momentum tensor in 4D which,
remarkably, is exactly the energy momentum corresponding to the 4D
gravitational waves (2.17).

Let us now see how a similar approach can give rise to electromagnetic waves.
Once again, it is crucial to show that the backreaction of the electromagnetic
field on the brane induces the correct energy-momentum tensor in the 4D
cosmology, effectively changing the sign of the bare contribution
$T_{\text{brane}}$. In the following, we discuss in detail how this is
achieved.

##  3 Electromagnetic waves in curved spacetime

In this section, we will review concepts of electromagnetism in four
dimensional curved space, as a required foundation to understand its higher
dimensional embedding.

###  3.1 Maxwell’s equations in disguise

Let us begin by recalling the nature of electromagnetism in a minimally
coupled 4D Einstein-Maxwell theory:

$S=\int\mathrm{d}^{4}x\sqrt{-g}\left(R-\frac{1}{4g^{2}}F_{\mu\nu}F^{\mu\nu}\right)\,,$
(3.1)

where $F^{\mu\nu}$ is the field strength associated to the $U(1)$ gauge field
$A^{\mu}$, and $g$ is the gauge coupling constant. Electromagnetic wave
solutions in this background are solutions to sourceless Maxwell’s equations,
which can be formulated in three equivalent ways:

  1. 1.

The first is the standard Maxwell equations that follow from varying the
action

$*\mathrm{d}*F=0=\nabla_{\beta}F^{\alpha\beta},\quad\mathrm{d}F=0=\nabla_{[\mu}F_{\nu\rho]}\,.$
(3.2)

  2. 2.

Maxwell’s equations above can be combined by taking a covariant derivative of
the second equation and using the first. This gives a wave equation for the
electromagnetic tensor

$\nabla^{\alpha}\nabla_{\alpha}F_{ab}\equiv\nabla^{2}F_{ab}=-2R_{abcd}F^{cd}+R_{a}{}^{c}F_{cb}+F_{a}{}^{c}R_{cb}\,.$
(3.3)

This highlights an interesting feature of curved spacetime: electromagnetic
waves couple to the spacetime curvature. Consequently, curvature can act as a
source for electromagnetic waves. Recall that $F_{ab}$ can be decomposed along
the world line of an observer (see [40, 41] for further details)

$F_{ab}=2u_{[a]}E_{b]}+\epsilon_{abc}B^{c}\,,$ (3.4)

where $u^{a}\coloneqq\mathrm{d}x^{a}/\mathrm{d}\tau$ is the 4-velocity tangent
to the observer’s worldline. This implies that the electric and magnetic
fields experienced by the observer are given by the projections
$E_{a}=F_{ab}u^{b}$, and $H_{a}=\epsilon_{abc}F^{bc}/2$. Projecting (3.3)
along $u^{a}$ gives the wave equation for the electric field $E^{a}$, and its
dual gives the wave equation for the magnetic field.

  3. 3.

Yet another way to reformulate Maxwell’s equations is to write them for the
gauge field instead of the field strength

$\nabla_{\beta}\nabla^{\beta}A^{\alpha}-\nabla^{\alpha}\nabla_{\beta}A^{\beta}-R^{\alpha}{}_{\beta}A^{\beta}=0\,.$
(3.5)

The curvature term comes from commuting the covariant derivatives. One can use
the gauge freedom in the system to choose Lorenz gauge:
$\nabla_{\alpha}A^{\alpha}=0$. With this choice, the left hand side becomes
the de Rham operator $\Delta_{\textrm{(dR)}}$, which is the ordinary Laplacian
supplemented with the Ricci curvature. The resulting wave equation is also
referred to as the de Rham equation (see for example [42])

$\Delta_{\textrm{(dR)}}A^{\alpha}\coloneqq\nabla_{\beta}\nabla^{\beta}A^{\alpha}-R^{\alpha}{}_{\beta}A^{\beta}=0\,.$
(3.6)

The three formulations above are equivalent in content, and one can choose to
solve either of them. The second and third formulations are inherently second
order in the field strength and the gauge field respectively, while the first
formulation in (3.2) involves only a single derivative on $F^{\mu\nu}$.
Additionally, the third formulation is simple only in the Lorenz gauge, which
will need to be generalized for higher form fields that we will be interested
in solving wave equations for in the next section. The best bet is therefore
to use the first formulation, which is what we will do.

###  3.2 Electromagnetic waves in a conformally flat de Sitter universe

Let us solve for EM waves in a homogeneous isotropic conformally flat universe
whose metric is given by

$\mathrm{d}s^{2}=a(\eta)^{2}(-\mathrm{d}\eta^{2}+\mathrm{d}x_{i}^{2})\,,$
(3.7)

with a scale factor $a(\eta)=\left(H\eta\right)^{-1}$, this is a patch of
conformally flat de Sitter spacetime. Decomposing $F^{\mu\nu}$ in a basis of
plane waves, let us make the ansatz
$F^{\mu\nu}=e^{in_{\lambda}x^{\lambda}}f^{\mu\nu}(\eta)$, where $n_{\lambda}$
represents the wave vector. To further simplify the problem, we will consider
waves propagating along one of the three isotropic space directions (say $z$).
The full isotropic wave can be obtained by averaging the solution over all
space directions. Equations (3.2) imply (for $i=x,y$)

$f^{t,z}=\eta\dot{f}^{\eta,i}-4f^{\eta,i}-in\,\eta
f^{i,z}=-\eta\dot{f}^{i,z}+4f^{i,z}+in\,\eta f^{\eta,i}=0\,,$ (3.8)

where a dot denotes a conformal time derivative. This has a very simple
solution

$\displaystyle F^{\eta,x}$
$\displaystyle=\frac{e^{inz}}{a^{4}}\left(c_{1}\cos(n\,\eta)+c_{2}\sin(n\,\eta)\right),\quad
F^{x,z}=\frac{e^{inz}}{a^{4}}i\left(c_{2}\cos(n\,\eta)-c_{1}\sin(n\,\eta)\right)\,$
(3.9) $\displaystyle F^{\eta,y}$
$\displaystyle=\frac{e^{inz}}{a^{4}}\left(c_{3}\cos(n\,\eta)+c_{4}\sin(n\,\eta)\right),\quad
F^{y,z}=\frac{e^{inz}}{a^{4}}i\left(c_{4}\cos(n\,\eta)-c_{3}\sin(n\,\eta)\right).$

As a demonstration of another way to solve the problem, let us use the third
formulation above (3.5) and solve the wave equation for the gauge field
$A^{\mu}$. Making an ansatz for the gauge field
$A^{\mu}=\left(0,A^{i}(\eta)e^{inz}\right)$, the Lorenz gauge condition gives
$\nabla_{\mu}A^{\mu}=ine^{ikz}A^{3}=0\Rightarrow A^{3}=0$. With this, de
Rham’s equation gives

$\eta^{2}\left(\ddot{A}^{i}+n^{2}A^{i}\right)-4\eta\dot{A}^{i}+6A^{i}=0,\quad\textrm{for
}i\in{x,y}\textrm{ and 0 otherwise}.$ (3.10)

With the redefinition $A^{i}(\eta)\coloneqq B^{i}(\eta)\,\eta^{2}$, this
becomes a simple wave equation $\ddot{B}^{i}+n^{2}B^{i}=0$, which gives the
same solutions as (3.9) above.

The energy momentum tensor carried by the electromagnetic wave is given by

$T^{a}{}_{b}=F^{ac}F_{bc}-\frac{1}{4}\delta^{a}{}_{b}F^{cd}F_{cd}\,.$ (3.11)

A uniform background of electromagnetic radiation requires to average overall
possible phases and directions of the waves with different polarizations. This
results in an isotropic energy momentum tensor of the form:

$\langle T^{a}{}_{b}\rangle_{\rm
iso}=\frac{\mathscr{E}^{2}}{2a^{4}}\begin{pmatrix}-1&0&0&0\\\
0&\frac{1}{3}&0&0\\\ 0&0&\frac{1}{3}&0\\\ 0&0&0&\frac{1}{3}\\\ \end{pmatrix},$
(3.12)

where $\mathscr{E}$ accounts for the amplitude of the wave, encoded by the
$c_{i}$ in eq (3.9). This section foreshadows the next, where we will solve
for vacuum waves of the Kalb-Ramond 2-form field $B^{\mu\nu}$. If we were to
use the third formulation to solve for the waves there, we would first need to
find an analog of the Lorenz gauge for the problem. A simple guess would be to
use $\nabla_{\mu}B^{\mu\nu}=0$. However, this leaves behind extra curvature
terms that do not result in a de Rham wave equation, and makes it very
complicated to solve. Another complication arises when one tries to find a
suitable ansatz for the $B^{\mu\nu}$ field. A natural guess is to make an
ansatz that reduces on the shell to the 4D $A^{\mu}$ found above. However,
there are multiple ways of doing this, which complicates the problem. One
could choose a better gauge, or make a better ansatz, but even then, one has
to solve a cumbersome second order PDE, which carries its own set of
challenges. For the problem that we are interested in, we will therefore use
the first formulation to solve directly for the 3-form field strength
$H_{\mu\nu\rho}$ instead.

##  4 Fiat lux in obscura bulla151515In English, Let there be light in the
dark bubble.

In this section, we are going to study how gauge fields in the $AdS_{5}$ bulk,
which also enjoy a ten dimensional supergravity interpretation, can give rise
to their four dimensional counterparts on the boundary of the bubble.

###  4.1 The uplift of electromagnetic waves

We will start by exploring the Dirac-Born-Infeld (DBI) action for a D3-brane,
given by171717Note the change in notation in this section where Greek letters
are used to label 4D indices on the brane.

$S_{D3}=-T_{3}\int\mathrm{d}^{4}x\sqrt{-\det(g_{4}+\tau\mathscr{F})},$ (4.1)

with $\tau=2\pi\alpha^{\prime}$, where $\alpha^{\prime}\equiv l_{s}^{2}$ is
the string tension. Here and in the following we suppress indices in the
determinants. In string theory there is also a coupling to bulk RR-flux, but
we will not consider this here. There is an immediate and obvious candidate
for the EM-waves: the gauge field $\cal{F}_{\mu\nu}$. Let us now explicitly
expand (4.1) up to second order for small $\alpha^{\prime}$, which results in:

$S_{D3}=-\int\mathrm{d}^{4}x\sqrt{-\det
g_{4}}\left(T_{3}+\underbrace{T_{3}\,\tfrac{\tau^{2}}{4}}_{1/\left(4g^{2}\right)}\mathscr{F}_{\mu\nu}\mathscr{F}^{\mu\nu}+\mathscr{O}\left(\mathscr{F}\mathscr{F}\right)^{2}\right),$
(4.2)

where $g^{2}=2\pi g_{s}$ is the gauge coupling. The second piece in eq (4.2)
can then be identified with the action of electromagnetism in four dimensions
(3.1). However, contrary to the case of other kinds of brane world
constructions, this is far from the whole story. The first puzzle to solve is
that the energy density of such a gauge field would seem to add to the energy
of the tension, and therefore, recalling, e.g. [21], it would give a negative
contribution to the 4D energy momentum. Nevertheless, as explained in [25], we
also need to take into account the back reaction from the bulk. We will come
back to this at the end of the section.

As it is well known, the gauge field on the brane can be written as:

$\tau\>{\cal F}_{\mu\nu}=\tau\>F_{\mu\nu}+B_{\mu\nu}.$ (4.3)

It is only ${\cal F}$ that is gauge invariant, while $F$ needs to shift when
there is a change of gauge for the B-field. The bulk field strength $H=dB$ is
of course also gauge invariant. By varying the DBI action (4.1) with respect
to $B$, we get a source term for $H$ given simply by ${\cal F}$. Since we are
working with a shell with co-dimension one, this means that there is a jump in
$H$ across the shell. We assume that $H$ vanishes inside the shell and that
the non-vanishing value outside is purely due to the presence of the
electromagnetic field on the brane. To be more precise, the bulk equations in
$AdS_{5}$ that we need to solve, are obtained from the following bulk action,
with the DBI-contribution included:

$S_{5}=\frac{1}{2\kappa_{5}}\int\mathrm{d}^{5}x\sqrt{-\det\left(g_{5}\right)}\left(R-\frac{1}{12g_{s}}H^{2}\right)-T_{3}\int\mathrm{d}^{5}x\>\delta(r-a[\eta])\>\sqrt{-\det\left(g_{4}+\tau\cal{F}\right)}.$
(4.4)

Here we work in the 5D Einstein frame, where the dilaton coupling of the
B-field is fixed to $e^{-\phi}=g_{s}^{-1}$, and
$\sqrt{-\det\left(g_{5}\right)}=\frac{1}{kr}\sqrt{-\det\left(g_{4}\right)}$,
with the bulk metric of the form:

$\mathrm{d}s_{5}^{2}=k^{2}r^{2}\left(-\mathrm{d}t^{2}+\mathrm{d}x^{2}+\mathrm{d}y^{2}+\mathrm{d}z^{2}\right)+\frac{\mathrm{d}r^{2}}{k^{2}r^{2}}.$
(4.5)

Note that the conventions are such that ${\tau\cal F}$ and $B$ are
dimensionless. We assume that internal (geometric) moduli and the dilaton are
stabilized. If we expand for small $\alpha^{\prime}$, as we did in expression
(4.2) and then vary the action with respect to $B$, we find the equations of
motion

$\partial_{r}H^{r\mu\nu}=\frac{2\kappa_{5}\>kr}{\alpha^{\prime}\>\pi^{2}}\>{\cal
F}^{\mu\nu}\>\delta\left(r-a[\eta]\right),$ (4.6)

Integrating across the brane at $r=a(\eta)$ gives

$\Delta
H^{r\mu\nu}\big{|}_{r=a}=H^{r\mu\nu}\big{|}_{r=a}=\frac{2\kappa_{5}\>ka}{\alpha^{\prime}\>\pi^{2}}\>\>{\cal
F}^{\mu\nu}\big{|}_{r=a}\,.$ (4.7)

Let us now understand how the four and five dimensional field strengths relate
to one another. We first connect the field strength $\mathscr{F}$ in the DBI
piece of expression (4.4) and that from Maxwell in eq (3.1). To be specific,
we focus on electromagnetic waves propagating along $z$, with the electric
field polarized along $x$, and the magnetic field polarized along $y$. Given a
wave propagating in the $z$ direction, and $\mathbf{E,B}$ respectively
polarised along $\\{x,y\\}$ directions, we have

${\cal F}^{tx}=\frac{{\cal E}(t,z)}{k^{2}a^{4}},\quad{\cal F}^{xz}=\frac{{\cal
E}(t,z)}{k^{2}a^{4}},$ (4.8)

where $\cal E$ is a dimensional function that determines the amplitude of the
wave.181818Here we have used conformal coordinates with ${\cal F}^{\mu\nu}\sim
1/k^{2}a^{4}$ and ${\cal F}_{\mu\nu}\sim k^{2}$ as in section 3.1. With proper
time and length given by $\mathrm{d}x^{\mu}_{p}=ka\mathrm{d}x^{\mu}$ we would
instead have ${\cal F}^{\mu\nu}\sim{\cal F}_{\mu\nu}\sim 1/a^{2}$. With this
information, we can then identify the behaviour of the $H$ field strength on
the brane, assuming there is no $H$-field turned on inside of the bubble.
Imposing this boundary condition on expression (4.7), gives us:

$\displaystyle H^{rtx}\big{|}_{r=a}$ $\displaystyle\equiv
h_{1}(t,a,z)=\frac{2\kappa_{5}\>ka}{\alpha^{\prime}\>\pi^{2}}{\cal
F}^{tx}=\frac{2\kappa_{5}}{\alpha^{\prime}\>\pi^{2}k}\frac{{\cal
E}(t,z)}{a^{3}},$ (4.9) $\displaystyle H^{rxz}\big{|}_{r=a}$
$\displaystyle\equiv
h_{2}(t,a,z)=\frac{2\kappa_{5}\>ka}{\alpha^{\prime}\>\pi^{2}}{\cal
F}^{xz}=\frac{2\kappa_{5}}{\alpha^{\prime}\>\pi^{2}k}\frac{{\cal
E}(t,z)}{a^{3}},$

In figure 1 we schematically see how the electromagnetic waves on the brane
source the H-field in the bulk above the brane.

Figure 1: A sketch of the dark bubble carrying electromagnetic waves sourcing
the B-field in the bulk. The junction conditions imply a jump in the B-field
encoded in eq (4.7).

The $H$ field must solve the equation of motion, as well as the Bianchi
identities away from the brane.191919Note that we assume that the dilaton as
well as any other moduli of the background are fixed. We get:

$\displaystyle*\mathrm{d}*H=0$
$\displaystyle\Rightarrow\nabla_{\alpha}H^{\alpha\beta\gamma}=0\,,$ (4.10)
$\displaystyle*\mathrm{d}H=0$
$\displaystyle\Rightarrow\epsilon^{\alpha\mu\nu\lambda}_{\quad\quad\kappa}\partial_{\alpha}H_{\mu\nu\lambda}=0\,,$

where $*$ is the 5D Hodge star and $H$ is the 3-form corresponding to $H$
above. As we will see, it is not enough to turn on these components of the
bulk field. Any non-trivial time dependence also requires the presence of
$H^{txz}\equiv h_{3}(t,r,z)$.

For the AdS-background, in the limit of flat 4D space time, the equations of
motion become:

$\displaystyle\left(\partial_{r}+\frac{3}{r}\right)h_{1}+\partial_{z}h_{3}$
$\displaystyle=0,$ (4.11)
$\displaystyle\left(\partial_{r}+\frac{3}{r}\right)h_{2}+\partial_{t}h_{3}$
$\displaystyle=0,$ $\displaystyle\partial_{z}h_{2}-\partial_{t}h_{1}$
$\displaystyle=0,$

while the Bianchi identity becomes

$r^{2}(\partial_{t}h_{2}-\partial_{z}h_{1})+\>k^{4}\>\partial_{r}\left(r^{6}h_{3}\right)=0$
(4.12)

A trivial solution is $h_{1}=h_{2}=(kr)^{-3}\sin(kn(t+z))$ with $h_{3}=0$, and
the momentum of the wave given by $n/r$. Using eq (4.21), this trivial
solution gives an energy momentum tensor of the form:

$T^{\mu}{}_{\nu}\sim\frac{1}{r^{4}}\>\text{diag}\left(-1,1,1,1,0\right),$
(4.13)

which does not correspond to the expected field strength in the bulk. To be
more precise, with such an energy momentum tensor the back reacted bulk metric
will contain a logarithmic dependence on the scale factor. If the brane were
to move in this background, the 4D observer would note an unphysical change in
the energy density on the brane, in addition to the familiar $1/a^{4}$. To
compensate for this, we must have an explicit time dependence of the bulk
metric, corresponding to time dependent flux of energy in the radial
direction, which corresponds to a non vanishing $T^{t}_{\>\>r}$ component.

Let us now think of a more general ansatz for the $H$ field strength,
accounting for the aforementioned $H^{txz}$ component that represents such a
flux. We make the ansatz:

$h_{1}=f_{1}(t,z,x)\,k^{-2}r^{-3},\,\qquad
h_{2}=f_{2}(t,z,x)k^{-2}r^{-3},\,\qquad h_{3}=f_{3}(t,z,x)\,k^{-3}r^{-4},\,$
(4.14)

with $x=2k^{3}tr^{2}n^{-1}$ and

$f_{i}(t,z,x)\equiv\alpha_{i}(t,x)\sin\left(kn\left(t+z\right)\right)+\beta_{i}(t,x)\cos\left(kn\left(t+z\right)\right),$
(4.15)

where the functions $\alpha_{i}$ and $\beta_{i}$ are dimensionless. The reason
for an ansatz of this form is the following. We note that on the brane (i.e.
$\eta=kt=-\left(Hr\right)^{-1}$):

$\frac{1}{knt}\sim\frac{rH}{n}\sim\frac{H}{p}\,,$ (4.16)

where $p\sim n/r$ is the energy of the quanta. Usually, $p\gg H$ and hence
$1/\left(knt\right)\ll 1$, is a natural variable for expansion. We also note
that, again on the brane:

$x=\frac{n}{2k^{3}tr^{2}}\sim\frac{nH}{k^{2}r}\sim\frac{pH}{k^{2}}\sim\frac{H}{k}\cdot\frac{p}{k}\sim\frac{p}{M_{pl}}\,,$
(4.17)

where in the last step, we have used relations from [27], where $1/H\sim
Nl_{4}$ and $1/k\sim N^{1/2}l_{4}$ and $M_{pl}$ is the Planck mass in 4D. It
is amusing to see how these specific results guarantee that the corrections
remain small until $p$ reaches the Planck scale. To summarize, we conclude
that the corrections are small, provided that $H\ll p\ll M_{pl}$, which is
also the relevant regime for us.

With this ansatz, we find the solution to the equations of motion and the
Bianchi identity, to order $(knt)^{-1}$ but all orders in $x$, given by:

$\displaystyle
h_{1}(t,z,x)=\frac{\alpha}{k^{2}r^{3}}\sin\left(kn\left(t+z\right)\right)+\frac{2\beta}{k^{2}r^{3}}\left[\sin\left(\tfrac{x}{2}+kn\left(t+z\right)\right)-\sin\left(kn\left(t+z\right)\right)\right],$
(4.18) $\displaystyle h_{2}(t,z,x)=h_{1}(t,z,x)-\frac{\beta
x}{k^{3}ntr^{3}}\cos\tfrac{x}{2}\sin\left(kn\left(t+z\right)\right),$ (4.19)
$\displaystyle h_{3}(t,z,x)=\frac{2\beta
x}{nk^{3}r^{4}}\sin\left(\tfrac{x}{2}+kn\left(t+z\right)\right),$ (4.20)

where $\alpha$ and $\beta$ are now constants. We note that it is only $h_{2}$
that needs a correction at $(knt)^{-1}$.

###  4.2 Energy-momentum tensor in the bulk

Varying the bulk part of the action (4.4) with respect to $g_{\mu\nu}$, yields
the energy-momentum tensor for the electromagnetic fields as:

$T_{\tau\omega}=\frac{1}{\kappa_{5}}\left(-\frac{1}{2}g_{\tau\omega}H^{2}+3\>H_{\tau\alpha\beta}H_{\omega}^{\>\>\>\alpha\beta}\right).$
(4.21)

So far, our previous computations have been for a single electromagnetic wave
traveling in the $z$ direction, with electric and magnetic fields accordingly
polarised along $\\{x,y\\}$ directions. A uniform background of
electromagnetic waves can be realised by accounting for all ingoing and
outgoing waves along some given direction. Furthermore, one should average
over waves traveling isotropically (along $\\{x,y,z\\}$) with any possible
polarisation. This results in the following isotropic energy-momentum tensor:

$\langle\tensor{T}{{}^{\mu}_{\nu}}\rangle_{\rm
iso}=\langle\tensor{T}{{}^{\mu}_{\nu}}\rangle_{\rm
rad}+\langle\tensor{T}{{}^{\mu}_{\nu}}\rangle_{\rm flux},$ (4.22)

where the radiation and flux contributions to the 5D energy-momentum tensor
are given by

$\langle\tensor{T}{{}^{\mu}_{\nu}}\rangle_{\rm
rad}=\frac{3\alpha^{2}}{\kappa_{5}\,k^{2}r^{4}}\begin{pmatrix}-1&0&0&0&0\\\
0&\frac{1}{3}&0&0&0\\\ 0&0&\frac{1}{3}&0&0\\\ 0&0&0&\frac{1}{3}&0\\\
0&0&0&0&0\end{pmatrix},\qquad\quad\langle\tensor{T}{{}^{\mu}_{\nu}}\rangle_{\rm
flux}=\frac{3\alpha^{2}}{\kappa_{5}\,k^{2}r^{4}\,t}\begin{pmatrix}0&0&0&0&\frac{-\beta}{\alpha\,k^{4}r^{3}}\\\
0&0&0&0&0\\\ 0&0&0&0&0\\\ 0&0&0&0&0\\\
\frac{\beta\,r}{\alpha}&0&0&0&0\end{pmatrix}.$ (4.23)

Off-diagonal terms in the flux correspond to the energy flow that we
anticipated,202020If we were to lower the upper index, the energy momentum
tensor is of course symmetric. analogous to energy flow along the radial
direction in eq. (2.20). Covariant conservation of this energy momentum tensor
is guaranteed, as each component $h_{i}$ of the bulk field strength $H$ solve
both their equation of motion (4.11) and the Bianchi identity (4.12).

In the next subsection we will calculate the backreaction and move on to
matching parameters with the 4D dark bubble cosmology.

###  4.3 Backreaction

As already hinted in eq (4.13), in the case where there is no net flux along
$tr$ direction, the metric will have the general form of an AdS-Schwarzschild-
like one, where $f(r)=1+k^{2}r^{2}-1/r^{2}$ should be replaced by
$\hat{f}(r)=1+k^{2}r^{2}-\log r/r^{2}$. In 4D, this would correspond to a
logarithmic increase of energy with time on the brane, compared to the
expected $1/a^{4}$ decay as the universe expands. What happens is that the
expanding bubble scoops up matter from bulk that is added to the brane. To
avoid this, we need the bulk matter to expand up along the throat of AdS, just
as it does in the case of GW [25]. This is precisely what we have achieved
above.

The energy momentum tensor in the bulk that does the job has $\left(\log
r\right)/r^{2}\rightarrow\left(\log tr\right)/r^{2}$. Since
$kt=-1/\left(Hr\right)$ on the bubble this give the correct behaviour in
4D.212121It is amusing to compare with the case of gravitational waves. There,
$1/r^{2}\rightarrow t^{2}r^{2}/r^{2}$. Presumably this has to do with the fact
that electromagnetism is scale invariant and renormalizable, while this is not
the case for gravity. The cosmological expansion mirrors the behaviour of
renormalization group flows, given the holographic nature of the direction
along the throat of the AdS. To be more precise, the 5D metric (in the flat 4D
limit) is given by

$\mathrm{d}s^{2}_{\rm back}\approx-
f_{1}(t,r)\mathrm{d}t^{2}+f_{2}(t,r)^{-1}\mathrm{d}r^{2}+r^{2}\mathrm{d}\Omega_{3}^{2},$
(4.24)

where

$f_{i}(t,r)\equiv k^{2}r^{2}-\frac{\epsilon^{2}\left(\log(-\xi
k^{2}tr)+q_{i}\right)}{k^{2}r^{2}}$ (4.25)

with $\\{q_{i}\\}$ undetermined numbers. This line invariant yields an energy
momentum tensor of the form:

$\langle\tensor{T}{{}^{\mu}_{\nu}}\rangle_{\rm
back}=\frac{3\>\epsilon^{2}}{2\,\kappa_{5}\,k^{2}\,r^{4}}\begin{pmatrix}-1&0&0&0&\frac{-1}{k^{4}r^{3}t}\\\
0&\frac{1}{3}&0&0&0\\\ 0&0&\frac{1}{3}&0&0\\\ 0&0&0&\frac{1}{3}&0\\\
\frac{r}{t}&0&0&0&0\end{pmatrix},$ (4.26)

provided that $q_{1}-q_{2}=1/4$. Note that $\xi$ is a free parameter that does
not affect the energy momentum tensor. Tuning it corresponds to changing a
piece of the metric that is the vacuum AdS-Schwarzschild background. Fixing it
will play an important role later on.

Let us now compare both energy momentum tensor (4.22) and (4.26) to relate the
parameters $\alpha,\beta,\epsilon$. It is easy to see that:

$\alpha=\beta=\frac{\epsilon}{\sqrt{2}}.$ (4.27)

We can also express these coefficients in terms of the amplitude of the 4D
electromagnetic wave. Comparing to eq (4.9) we can read off that:

$\alpha=\frac{2\kappa_{5}k}{\alpha^{\prime}\pi^{2}}\mathscr{E}.$ (4.28)

Finally, the covariant conservation of the energy momentum tensor (4.26) is
still respected, as it coincides with that derived for the $H$ field in eq
(4.22) and it obeys the Einstein equation.

###  4.4 Checking the induced 4D energy momentum tensor on the bubble

Let us now study how the backreaction of the bulk geometry, due to the
presence of $H$ field strength, changes the motion of the bubble wall, and
hence the induced energy momentum tensor. One can equivalently arrive to the
same result by computing Israel’s second junction condition (2.10) or the
Gauss-Codazzi equation (2.12). There will two main contributions to the 4D
energy momentum tensor:

  1. 1.

A positive contribution coming from the extrinsic curvature.

  2. 2.

A piece that comes from the gauge-field on the brane, which adds to the
tension and thus gives a negative contribution.

Israel’s second junction condition (2.10) yields an induced Friedmann equation
of the form:

$\begin{split}\left(\frac{\dot{a}}{a}\right)^{2}\delta^{a}_{b}=\frac{\Lambda_{4}}{3}\delta^{a}_{b}+\>&\epsilon^{2}\,\frac{\kappa_{4}}{6\kappa_{5}k^{3}a^{4}}\left(3\delta^{a}_{0}\delta^{0}_{b}-\delta^{a}_{i}\delta^{i}_{b}\right)\left(q_{2}+\log\left[-\xi\frac{k_{+}}{H}\right]\right)\\\
&+\frac{\kappa_{4}}{3}\left(\mathscr{F}^{ac}\mathscr{F}_{bc}-\frac{1}{4}\,\delta^{a}_{b}\mathscr{F}_{ij}\mathscr{F}^{ij}\right).\end{split}$
(4.29)

Note that the left hand side (LHS) is the usual geometrical evolution of the
scale factor for late time cosmologies (i.e. there is no contribution of the
curvature term) in Friedmann equations, while the right hand side (RHS)
corresponds to the energy momentum tensor associated with the induced
cosmology. For indices $a=b=0$ we obtain the first Friedmann equation, and one
notes that the gauge field on the brane, in the second line, gives a negative
contribution to the energy density, just like the brane tension does. Recall
that our conventions are such that $T^{0}{}_{0}=-\rho$. This needs to be
corrected to a positive contribution by the first term due to the extrinsic
curvature.

We still have not determined the free parameters $\xi$ and $q_{2}$, which are
degenerate and can be shifted into each other. It corresponds to the fact that
we can always add a piece of dark radiation by adding a 5D black hole in the
background. This was already explored in [25]. Let us now impose the absence
of such contribution, which restricts the RHS of eq (4.29) above to be that of
four dimensional electromagnetism in an expanding cosmology, as given in eq
(3.12). Furthermore, for simplicity, we choose the free parameter $\xi$ so
that $-\xi\frac{k}{H}=1$. This implies:

$\epsilon^{2}\left(\frac{q_{2}}{2a^{4}k_{+}^{3}\kappa_{5}}\right)\left(3\delta^{a}_{0}\delta^{0}_{b}-\delta^{a}_{i}\delta^{i}_{b}\right)+\underbrace{\left(\mathscr{F}^{ac}\mathscr{F}_{bc}-\tfrac{1}{4}\,\delta^{a}_{b}\mathscr{F}_{ij}\mathscr{F}^{ij}\right)}_{\langle
T^{a}_{\>\>b}\rangle_{\rm iso}}\equiv-\langle T^{a}_{\>\>b}\rangle_{\rm iso}.$
(4.30)

Making use of relations (4.27), (4.28) and (2.4) one can fix $q_{2}$ to be:

$q_{2}=\frac{k_{+}^{2}M_{pl}^{2}}{2^{7}\,T_{D3}\,g_{s}}\,N\,\pi,$ (4.31)

where $N$ is the number of $D3$ branes in the 10D background, as explained in
section 2.222222Using the scaling relations, one finds that $q_{2}\sim
N/g_{s}$.

This is all analogous to the case where we only have dark radiation associated
with a black hole with mass $M$ in the bulk, instead of the electromagnetic
waves. As already discussed in [6], the 4D energy momentum tensor has two
contributions yielding

$\rho=\frac{M_{+}}{k_{+}a^{4}}-\frac{M_{-}}{k_{-}a^{4}},$ (4.32)

with $M_{-}=M_{+}=M$. If we now move the mass from the center of the bubble
and deposit it evenly onto the brane, without changing the exterior metric
this becomes

$\rho=\frac{M_{+}}{k_{+}a^{4}}-\rho_{\mathrm{brane}}\,.$ (4.33)

So, the matter on the brane actually does give a negative contribution, but it
is overcompensated by the contribution from the extrinsic curvature of the
exterior. This is nothing but the gravitational back reaction from the bulk.
This is exactly what happens for the electromagnetic field. As we have seen,
the back reaction happens through the $H$-field that sources the term $q_{2}$
in the metric that plays the role of $M_{+}$. We will come back to a more
general discussion of this structure in the next section.

It is important to note that electromagnetic radiation is realized in a very
different way from the dark radiation. It is accompanied by a non-trivial
$H$-field, with a non-vanishing energy density in the bulk. This yields a
different bulk geometry. In order to further highlight the role of the B-field
in determining the correct energy-momentum tensor on the brane, we will now
consider electrostatic field configurations.

###  4.5 Un pliage électrique 232323En anglais, an electric bending.

Let us consider a constant electric field $\cal E$, pointing in the
$z$-direction. For simplicity, we ignore the expansion of the universe. Also,
such a constant electric field cannot be supported over an extended region. In
practice, it could be approximately realized as a piece of an electric field
far away from a localized source. In this case, the solution of eqs (4.11) are
very simple, and we find

$H^{rtx}\big{|}_{r=a}\equiv h_{1}=\frac{\gamma}{k^{2}r^{3}},\quad
H^{rxz}\big{|}_{r=a}\equiv h_{2}=0,\quad H^{txz}\big{|}_{r=a}\equiv h_{3}=0,$
(4.34)

where $\gamma\sim{\cal E}$ is a dimensionless constant. This sources a 5D
energy-momentum tensor

$\langle
T^{\mu}{}_{\nu}\rangle\approx\frac{\gamma^{2}}{k^{2}\kappa_{5}}\begin{pmatrix}-1/r^{4}&0&0&0&0\\\
0&1/r^{4}&0&0&0\\\ 0&0&1/r^{4}&0&0\\\ 0&0&0&1/r^{4}&0\\\
0&0&0&0&-1/r^{4}\end{pmatrix}\,,$ (4.35)

which leads to the following back reaction in the bulk:

$\mathrm{d}s^{2}_{\rm back}\approx
A(r)\left(-\mathrm{d}t^{2}+\mathrm{d}z^{2}\right)+\frac{1}{A(r)^{2}}\mathrm{d}r^{2}+B(r)(\mathrm{d}x^{2}+\mathrm{d}y^{2}),$
(4.36)

with

$\displaystyle A(r)$
$\displaystyle=(kr)^{2}-\frac{2}{3}\gamma^{2}\frac{\log(k\zeta r)}{(kr)^{2}},$
(4.37) $\displaystyle B(r)$
$\displaystyle=(kr)^{2}+\gamma^{2}\frac{\log(k\zeta r)}{(kr)^{2}}\,,$ (4.38)

where $\zeta$ is a constant. The next step is to find the embedding of the
brane that gives the correct induced metric in 4D corresponding to a constant
electric field. To do this, we must construct the constant electric field as a
local approximation within a setup containing an explicit, physical source.
Let us use the 4D metric of a point source, where we are far enough from the
source that the electric field can be considered constant in a big enough
region. We will not take into account any gravitational forces, so $1\gg
Q^{2}/r^{2}\gg M/r$. (This is the relevant situation for any currently
realistic experiment.) The induced 4D metric we need on our embedded brane is
therefore of the form

$\displaystyle\mathrm{d}s^{2}$
$\displaystyle=-\left(1+\frac{Q^{2}}{\rho^{2}}\right)\mathrm{d}t^{2}+\left(1+\frac{Q^{2}}{\rho^{2}}\right)^{-1}\mathrm{d}\rho^{2}+\rho^{2}\mathrm{d}\Omega_{2}$
$\displaystyle\rightarrow-\left(1+\frac{Q^{2}}{\rho^{2}}\right)\mathrm{d}t^{2}+\left(1+\frac{Q^{2}}{\rho^{2}}\right)\mathrm{d}\tilde{\rho}^{2}+\rho^{2}\mathrm{d}\Omega_{2}$
$\displaystyle\rightarrow-\left(1+\frac{Q^{2}}{z^{2}}\right)\mathrm{d}t^{2}+\left(1+\frac{Q^{2}}{z^{2}}\right)\mathrm{d}\tilde{z}^{2}+\mathrm{d}x^{2}+\mathrm{d}y^{2}$
$\displaystyle\sim-\left(1+\frac{Q^{2}}{\tilde{z}^{2}}\right)\mathrm{d}t^{2}+\left(1+\frac{Q^{2}}{\tilde{z}^{2}}\right)\mathrm{d}\tilde{z}^{2}+\mathrm{d}x^{2}+\mathrm{d}y^{2}.$
(4.39)

The first arrow represents a coordinate transformation in the $\rho$-direction
introducing the new coordinate $\tilde{\rho}$, through
$\mathrm{d}\tilde{\rho}/\mathrm{d}\rho=\left(1+Q^{2}/\rho^{2}\right)^{-1}$.
For convenience, we express the result using $\rho(\tilde{\rho})$. The second
arrow means that we zoom in on $x=\rho\phi$ and $y=\rho\theta$ small, with the
angles $\phi$ and $\theta$ small. We can then express the metric using
Cartesian coordinates where we write $\tilde{z}=\tilde{\rho}$. Similarly, it
is natural to write $z=\rho$. Finally, since $Q^{2}/z^{2}$ is small, we can
put $z\sim\tilde{z}$. Now let us see if we can match this to the induced
metric on a brane embedded into (4.36).

Since the electric field giving rise to the metric in (4.36) is assumed to be
constant (for simplicity), we can only hope to match the induced metric in
(4.5) over a short interval. We choose to study the embedding close to
$\tilde{z}=\tilde{z}_{1}$, where $\tilde{z}_{1}$ is far away from the source
such that $Q/\tilde{z}$ is small. This means that
$r(\tilde{z})=r_{1}+\alpha(\tilde{z}-\tilde{z}_{1})$ is almost constant with
$\alpha$ small. We set the asymptotic value of $r$ at infinity to $r_{0}$.
Noting that the $\mathrm{d}r^{2}$-piece of the bulk metric only contributes at
quadratic order in $\alpha$ to the induced metric, we see that the induced
metric can be put in the form (4.5) provided that

$\begin{split}k^{2}r^{2}-\frac{2\gamma^{2}\log(k\zeta
r)}{3k^{2}r^{2}}&=k^{2}r_{0}^{2}\left(1+\frac{Q^{2}}{z^{2}}\right),\\\
k^{2}r^{2}+\frac{\gamma^{2}\log(k\zeta
r)}{3k^{2}r^{2}}&=k^{2}r_{0}^{2}.\end{split}$ (4.40)

The first equation comes from matching $\mathrm{d}t^{2}$ and
$\mathrm{d}z^{2}$, while the second comes from matching $\mathrm{d}x^{2}$ and
$\mathrm{d}y^{2}$. Here, we have already made a conformal rescaling of $x$,
$y$ and $z$ to match the equations at zeroth order. This gives

$\begin{split}\frac{r^{2}}{r_{0}^{2}}&=1+\frac{Q^{2}}{3z^{2}},\\\
\frac{\gamma^{2}\log(k\zeta
r)}{3k^{2}r^{2}}&=-\frac{k^{2}r_{0}^{2}Q^{2}}{3z^{2}}.\end{split}$ (4.41)

Due to our approximation for the bulk metric, using a constant electric field,
these equations are only valid close to a given point such as $z=z_{1}$. The
first of the two equations give us $r$, for a given $z$, while the second
equation determines $\zeta$.252525Changing $z_{1}$, leads to a different
$\zeta$ as a consequence of the approximation. We immediately see from the
first equation that $r$ grows in the direction towards the charge sourcing the
field. The interpretation is simple. In order to capture the blue shift of the
induced metric as we approach the source, the brane must bend upwards to make
use of the bulk blue shift due to AdS. In this way the induced metric comes
out right. We expect the result to be similar if we were to construct the full
uplift of an electric field decaying away from a point source.

To better understand what is going on, let us recall the case of a point mass
in 4D as discussed in [23], caused by a string pulling on the brane. The brane
bends upwards due to the pulling. The bending in itself causes a change in the
vacuum solution of the bulk, with the induced metric on the brane becoming
that of Schwarzschild. Here, we have a similar story even when ignoring the
presence of any pulling string. The brane bends upwards in the direction of
increasing electric field but now it is caused by the presence of the bulk
B-field. This gives rise to a backreaction in the bulk that makes the brane
bend upwards.

##  5 A dark bubble consistency proof

The dark bubble construction is based on two sets equations: Einstein’s
equations in the bulk (including the junction conditions) together with the
Gauss-Codazzi equations. The 4D Einstein equation is obtained from the Gauss-
Codazzi equation making use of the junction conditions. Schematically, the 4D
Einstein tensor is:

$G_{4D}=T_{\rm extrinsic}-T_{\rm brane}\,.$ (5.1)

When setting up the equations for a particular physical situation, we have
seen how certain choices need to be made. In particular, we have studied two
examples involving electromagnetic fields in this paper. We had to choose
initial conditions that were such that the 4D effective theory did not involve
any additional fields beyond those that were represented on the brane itself.
Note that in general such dark components can be present in the bulk,
affecting the 4D physics. The dark radiation due to an AdS-Schwarzschild black
hole, as already discussed in [4], is one such example. The cosmological
constant itself, can be viewed as another.

What is crucial is that this initial condition is preserved by the time
evolution. The central claim of the dark bubble proposal is, in fact, that
this is the case. The initial condition that we impose is

$G_{4D}=T_{\rm brane},$ (5.2)

implying that $T_{\rm extrinsic}=2T_{\rm brane}$ initially. The reason why
this remains true for the subsequent time evolution is twofold:

  1. 1.

$T_{\rm brane}$ is covariantly conserved on its own through its equations of
motion derived from the brane action. Therefore, there cannot be any exchange
of energy between $T_{\rm brane}$ and anything else.

  2. 2.

The total energy momentum tensor, involving $T_{\rm extrinsic}$ is also
covariantly conserved since it is equal to the 4D Einstein tensor. Hence, the
Einstein equation in the form (5.2) is preserved throughout.262626One might
worry, that there could be spontaneous creation of modes with $G_{4D}=0$, thus
not visible to the Einstein equations and constrained by our argument. In
writing down the Gauss-Codazzi equations, we have only employed them at the
level of the Ricci-tensor when obtaining the 4D Einstein equations. However,
they are valid at the level of the Riemann tensor, which then constrains such
waves. This was fully explored in [25], where the uplift of 4D gravitational
to 5D was worked out in detail.

Physically, this means the following. Let us assume matter associated with the
brane such as the electromagnetic fields studied in this paper. This could
also include other matter of the standard model represented by more elaborate
brane configurations. We will call this the visible sector. Then set up
initial conditions on the brane and in the bulk to make sure that the induced
4D energy momentum is precisely the expected one of the visible sector. Our
discussion then shows that the subsequent time evolution will be the one of
the visible sector coupled to 4D gravity. Any initial mismatch between the
energy momentum tensor of the visible sector and the effective induced 4D
energy momentum tensor, we interpret as the presence of a dark sector. This
implies that this dark sector will, from the 4D point of view, couple to 4D
gravity in the same way as the visible sector. It is natural to associate the
cosmological constant with the dark sector, but it is clear from the rich
structure of the model that other more dynamical components associated with
the extra dimensions can be turned on as well. It would be interesting to
explore whether these could be candidates for dark matter.

Note that there are corrections to the 4D Einstein equations at very high
energies. As discussed in [21], they arise through quadratic couplings of the
brane energy momentum tensor to itself.272727A simple way to see this is
through footnote 1 of [29]. Just add a contribution to the brane tension
$\sigma$. In terms of the Friedmann equation, the standard contribution is of
the form $l_{4}^{2}\rho$, where $\rho$ is the energy density. The corrections
are of the form $l_{4}^{4}L^{2}\rho^{2}\sim(l_{4}^{2}L^{2}\rho)\times
l_{4}^{2}\rho$, as indicated in equation (2.12). Using the relation between
the scales, we find that $l_{4}^{2}L^{2}\rho\sim l^{4}_{10}\rho$. Hence, these
corrections only appear at densities close to the 10D Planck scale (or string
scale).

To further understand how tightly constrained the construction is, recall how
the matter equations of motion are built into Einstein’s equations. To
illustrate, let us consider an elementary example in the form of a charged,
thin spherical shell in vacuum. The shell is uniformly charged and has some
tension. Effects due to gravity are assumed to be so small that they can be
ignored. The evolution of the shell is easily obtained using the laws of
electromagnetism. If any radial velocity of the shell is small, it is enough
to use electrostatics involving electric forces or electric potentials. There
is, however, another method that can be used. View it as a problem in general
relativity, introducing a Reissner-Nordström metric on the outside and flat
space time on the inside, together with a junction condition. Solving these
equations, and taking the limit where the gravitational effects go to zero,
you recover the elementary results obtained from electrostatics. This
demonstrates how the Einstein equations know of, and take fully into account,
the equations of motion of any matter in the system. This was also the content
of the calculation done in [25], where it was shown how the brane equations of
motion written in their Hamiltonian form were nothing other than the junction
conditions.

##  6 Discussion and outlook

In this paper we have taken the first step towards explicitly incorporating
the standard model of particle physics into the dark bubble model by
considering electromagnetism. It is interesting to see the crucial role played
by the B-field. It yields the necessary back reaction on the brane in order to
get the proper 4D gravity. In many ways, the extended structure of B-fields
into the bulk, and their back reaction on the 5D space time, play a very
similar role to the stretched strings required to sustain the dust particles
mentioned in the beginning. In the setting we have discussed in this paper,
the backreaction of the electromagnetic field on the brane is driven by the
B-field, which controls the jump in eq (4.7). However, in principle the
junction conditions could be modified even without B-fields in the theory.
This is what we expect in string constructions arising _e.g._ from type 0
orientifolds. It would be interesting to investigate such alternative
realizations of the dark bubble in future work.

Our work represents an important first step toward building particle
phenomenology within the dark bubble scenario. From a bottom-up perspective,
the next step involves realizing non-abelian gauge fields, possibly including
the Higgs mechanism. Given the results that we have presented, we expect a
similar story if we consider a non-abelian extension. The bubble wall would
then consist of several branes on top of each other, with Chan-Paton factors
furnishing the appropriate gauge degrees of freedom for quarks and other
fields. In case of the electroweak force, we would have two branes separared
by a distance proportional to the Higgs field. The simplest realizations of
this setting would not correspond to the precise Higgs mechanism that takes
place in the standard model, since the scalar fields on brane stacks typically
take values in rank-two tensor representations (the adjoint in supersymmetric
cases). The most natural way to separate the branes would be to keep them on
top of each other in AdS at the same coordinate $r$, but at different
positions on the internal $S^{5}$. Note that the mass of a string stretching
between two such branes would be given by $l\times 1/l_{s}^{2}$, where $l$ is
the length of the string, with the maximal value equal to the the Planck mass
when the branes sit at opposite poles. It would be very interesting to work
out the details of the B and H-fields, and their dependence on the $S^{5}$.
This should give rise to a qualitatively different behaviour in $AdS_{5}$,
compatible with massive particles (the W and Z) from the 4D point of view.

It is tantalizing that the string scale (and 10D Planck scale) ends up at
around $10$ TeV, which is just above what is currently accessible in
experiments. This suggests that massive excitations of open strings attached
to the brane, such as those giving rise to _e.g,_ photons, could be detectable
in future collider experiments. As we have discussed, the dark bubble model
also includes a possible candidate of dark matter through the its dark sector,
when a mismatch with the visible sector is introduced through the extrinsic
curvature of the brane embedding. We hope to return to these phenomenological
aspects in the near future.

## Acknowledgement

We thank Souvik Banerjee for collaboration at an early stage of this work, and
Thomas Van Riet for discussions. DP would like to thank the Centre for
Interdisciplinary Mathematics (CIM) for financial support. The work of SG was
conducted with funding awarded by the Swedish Research Council grant VR
2022-06157.

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