# Landscape, Swampland, and Extra Dimensions

Luis A. Anchordoqui  Ignatios Antoniadis  Dieter Lüst

###### Abstract

By combining swampland conjectures with observational data, it was recently
suggested that the cosmological hierarchy problem (i.e. the smallness of the
dark energy in Planck units) could be understood as an asymptotic limit in
field space, corresponding to a decompactification of one extra (dark)
dimension of a size in the micron range. In these Proceedings we examine the
fundamental setting of this framework and discuss general aspects of the
effective low energy theory inherited from properties of the overarching
string theory. We then explore some novel phenomenology encompassing the dark
dimension by looking at potential dark matter candidates, decoding neutrino
masses, and digging into new cosmological phenomena.

##  1 Introduction

The challenge for a fundamental theory of nature is to describe both particle
physics and cosmology. Accelerator experiments and cosmological observations
provide complementary information to constrain the same theory. We have long
known that only about 4% of the content of the universe is ordinary baryonic
matter; the remainder is dark matter ($\sim 22\%$) and dark energy ($\sim
74\%$). The $\Lambda$CDM model, in which the expansion of the universe today
is dominated by the cosmological constant $\Lambda$ and cold dark matter
(CDM), is the simplest model that provides a reasonably good account of all
astronomical and cosmological observations [1].

The cosmological evolution is described by Einstein’s field equation,

${\cal R}_{\mu\nu}-\frac{1}{2}\ g_{\mu\nu}\ {\cal R}+g_{\mu\nu}\
\Lambda=\frac{8\pi G}{c^{4}}\ T_{\mu\nu}\,,$ (1)

where ${\cal R}_{\mu\nu}$ and ${\cal R}$ are respectively the Ricci tensor and
scalar, $g_{\mu\nu}$ is the metric tensor, $T_{\mu\nu}$ is the energy momentum
tensor, Greek subscripts run from 0 to 3, and $G$ is Newton’s gravitational
constant. The cosmological constant encapsulates two length scales: the size
of the observable Universe $[\Lambda]=L^{-2}$ and of the dark energy
$[\Lambda/G\times c^{3}/\hbar]=L^{-4}$. The observed value of the cosmological
constant $\Lambda_{\rm obs}\simeq 0.74\times 3H_{0}^{2}/c^{2}\simeq
1.4\times(10^{26}~{}{\rm m})^{-2}$ gives a characteristic length of dark
energy $\simeq 85~{}\mu{\rm m}$, where we have adopted the recent measurement
of the Hubble constant $H_{0}\simeq 73~{}{\rm km/s/Mpc}$ by the SH0ES team [2,
3].

The $SU(3)_{C}\otimes SU(2)_{L}\otimes U(1)_{Y}$ Standard Model (SM) of strong
and electroweak interactions encapsulates our current best understanding of
physics at the smallest distances ($<10^{-21}~{}{\rm m}$) and highest energies
(center-of-mass energies $\sqrt{s}\sim 14~{}{\rm TeV}$). Even though the SM
continues to survive all experimental tests at accelerators [1], it is widely
considered to be an effective theory. Besides missing gravity, the SM does not
include a mechanism for giving neutrinos their masses, and does not
incorporate dark matter or dark energy. Moreover, if we compare the strength
of gravity to that of the SM interactions we find that

$GM_{H}^{2}/(\hslash c)=(M_{H}/M_{p})^{2}\approx 10^{-34}\ll 1\,,$ (2)

where $M_{H}$ is the Higgs mass and $M_{p}$ is the Planck mass.

Leaving aside for the moment the SM downsides, a question to ask ourselves is:
why is the gravitational interaction between SM particles so much weaker than
the other SM interactions? On the flip side, why is the Planck mass so huge
relative to the SM or dark energy scales?

A way to explain hierarchies in fundamental physics is via the size of extra
dimensions which are necessary ingredients for consistency of string theory.
Indeed, if the size of the extra dimensions is large compared to the
fundamental (string) length, the strength of gravitational interactions
becomes strong at distances larger than the actual four-dimensional (4D)
Planck length [4, 5]. As a result, the string scale is detached from the
Planck mass consistently with all experimental bounds if the observable
universe is localized in the large compact space [5]. By combining swampland
conjectures with observational data, it was recently suggested [6] that the
cosmological hierarchy problem (i.e. the smallness of the dark energy in
Planck units) could be understood as an asymptotic limit in field space,
corresponding to a decompactification of one extra (dark) dimension of a size
in the micron range. In addition in cosmology, such an extra fifth dimension
also provides a nice explanation for the 60 e-foldings in the course of cosmic
inflation [7, 8].

In these Proceedings we summarize the state-of-the-art in this subject area,
and discuss future research directions. We begin by reviewing the emergence of
a new paradigm of quantum gravity.

##  2 The Landscape and the Swampland

The Swampland Program seeks to amplify our understanding of the fundamental
constraints accompanying a consistent theory of quantum gravity (QG) [9]. The
basic thought is that an effective field theory (EFT) might seem consistent as
a stand-alone theory in the IR (anomaly free), but coupling the theory to
gravity in the UV may uproot its consistency. Actually, given the scarcity of
consistent EFTs coupled to QG, it is easier to rule inconsistent theories out.
The goal is therefore to circumscribe the subset of 4D EFTs that can be UV-
completed to a QG theory and are said to belong in the landscape from the
complementary subset of theories that do not admit such a completion and are
relegated to the swampland. This is done by enumerating criteria classifying
the properties that an EFT must satisfy in order to enable a consistent
completion into QG. As the energy increases and we get closer to the QG scale
the swampland criteria provide stronger constraints on the boundary that
separates the swampland from the landscape. This implies that the space of UV-
consistent EFTs encircles a conical-shape structure that is cartooned in the
left panel of Fig. 1.

Figure 1: Left. The space of EFTs that are a low energy limit of quantum
gravity forms a cone in theory space, because the swampland constraints become
stronger at high energies. Right. Moduli space $\mathscr{M}$ with an infinite
distance limit between points $P$ and $Q$. From Ref. [10].

The swampland criteria have evolved to some set of conjectures, which can be
used as new guiding principles to construct compelling UV-completions of the
SM. There are many swampland conjectures in the literature; actually, too many
to be listed here and so readers are referred to comprehensive reviews [11,
12, 13, 14]. Conjectures linked to towers of states, however, deserve some
attention because they are closely related to the topics we will explore in
this proceedings.

Consider a gravitational EFT with a moduli space $\mathscr{M}$ (i.e. a space
parameterized by the massless scalar fields in the theory) and whose metric is
given by the kinetic terms of the scalar fields.111In the presence of a
mechanism of moduli stabilization, the moduli space is discretized
corresponding to a finite landscape of vacua which is assumed to be large
enough for the following arguments to hold. The distance conjecture (DC)
states [15] that infinite distance limits $\Delta\phi\to\infty$, with
reference to geodesic field distance $\Delta\phi\equiv d(P,Q)$ illustrated in
Fig. 1, are accompanied by an infinite tower of exponentially light states

$m(Q)\sim m(P)\ e^{-\lambda\Delta\phi}\,,$ (3)

where distance and masses are measured in Planck units, and $\lambda$ is an
order one positive constant.

The archetypal example to gain intuition about the DC is a theory compactified
on a circle. Taking $R$ to be the modulus controlling the radius (or radion),
it is straightforward to see that the Kaluza-Klein (KK) modes have masses that
scale as $m_{n}^{2}=n^{2}/R^{2}$, with $n\in\mathbb{Z}$. Now, after
dimensional reduction of the gravitational piece of the action and the
corresponding field redefinition to go into the Einstein-frame, the kinetic
term of the Lagrangian for the radion field takes the form $\mathscr{L}_{{\rm
kin}}\supset(\partial R)^{2}/R^{2}$. The distance in field space between
points $R_{\rm i}$ and $R_{\rm f}$ is measured by the field space metric,
given by $1/R^{2}$, and is found to be

$d(R_{\rm i},R_{\rm f})=\int_{R_{\rm i}}^{R_{\rm f}}\sqrt{\dfrac{1}{R^{2}}}\
dR=\left|\ln(R_{\rm f}/R_{\rm i})\right|.$ (4)

As predicted by the DC, Taking the decompactification limit $R\to\infty$ we
see that the KK tower becomes light, $m_{n}\sim e^{-d(R_{\rm i},R_{\rm
f}\rightarrow\infty)}$. In the opposite limit at infinite distance $R\to 0$
the KK tower does not satisfy the DC. However, in this limit there is another
infinite tower that is becoming light; namely, the winding modes (wrapping
strings), which present the same behaviour as the KK modes in the
decompactification limit. All in all, we arrive at a staggering conclusion: to
satisfy the DC in KK compactifications we need extended objects that can wrap
some compact directions and become light in the limit in which they shrink to
zero size.

Before moving on, we bring up a refinement of the DC, which is known as the
emergent string conjecture (ESC) and states that any infinite distance limit
is either a decompactification limit or a limit in which there is a weakly
coupled string becoming tensionless [16]. Recently it was argued [17, 18] that
bottom-up arguments from black hole properties provide general evidence for
the ESC.

The DC and the ESC can be generalized to other field space configurations
beyond the moduli space. For example, we can define a notion of distance
between different metric and flux configurations of anti-de Sitter spacetimes
[19, 20, 21, 22], such that the flat space limit $\Lambda\to 0$ is located at
infinite distance in this metric configuration space. The related distance in
the space of AdS vacua is then given as

$d(\Lambda_{\rm i},\Lambda_{\rm f})=\left|\ln(\Lambda_{\rm f}/\Lambda_{\rm
i})\right|,$ (5)

and the anti-de Sitter distance conjecture (AdS-DC) states [19] that any AdS
vacuum has an infinite tower of states that becomes light in the flat space
limit $\Lambda\to 0$, satisfying

$m\sim|\Lambda|^{\alpha}\,,$ (6)

with $\alpha$ a positive constant of order one. Besides, under the hypothesis
that this scaling behavior holds in de Sitter (dS) space, an unbounded number
of massless modes also pop up in the limit $\Lambda\to 0$ [19]. A point worth
noting at this juncture is that the AdS-DC modifies the EFT expectation
$\Lambda\sim\Lambda_{0}+m^{1/\alpha}$, in which $\Lambda_{0}$ stands for the
contribution of the heavy modes. The vanishing of $\Lambda_{0}$ is argued to
be intrinsically related to the modular invariance of string theory (see also
the discussion in [23]). Furthermore one can formulate a conformal field
theory (CFT) distance conjecture about the spectrum of certain
(holographically dual) conformal and world sheet field theories [24, 25, 26,
27], and one can also generalize it to topology change and non-geometric
spaces at infinite distance [28].

In a way, similar to the AdS-DC, a distance conjecture for black hole
solutions in the EFT was put forward [29]. Concretely the black hole entropy
distance conjecture (BHEDC) states that the limit of infinite black entropy
$S_{\rm BH}\rightarrow\infty$ is at infinite distance in the space of BH
metrics and the associated BH entropy distance can be expressed as

$d(S_{\rm BH,i},S_{\rm BH,f})=\left|\ln(S_{\rm BH,f}/S_{\rm BH,i})\right|\,.$
(7)

Following the infinite distance conjecture, there must be a corresponding mass
scale with a tower of “states", whose masses in Planck units are given as

$m\sim({S_{\rm BH}})^{-\beta}\;,$ (8)

where $\beta$ is a positive constant. Further discussion and more evidence for
the BHEDC for extremal and non-extremal black holes was subsequently given in
[30, 31, 32, 33]. In addition, the BHEDC can be also applied for so called
minimal black holes. Their minimal horizon size $L_{\rm min}$ serves as a
working definition for the species scale $\Lambda_{\rm sp}$ [34, 35, 36, 37,
38] and the corresponding species length $l_{\rm sp}$ in the EFT (see also the
discussion in section 3.1 and section 8.):222Alternatively, $\Lambda_{\rm
sp}=M_{p}/\sqrt{F_{1}}$ can be identified with the scale at which $R^{2}$
corrections to the Einstein action become important, with $F_{1}\simeq N$
being the one-loop topological string free energy [39]. In [40] it was shown
that this definition of the species agree with the one from minimal black
holes for a specific type of charged black holes. Further recent discussion
about the species scale can be found in [41, 42, 43, 44, 45, 46, 47, 48, 49,
50, 51] and recently further arguments for the agreement of these two
definitions of the species scale were provided in [18].

$l_{\rm sp}\equiv\Lambda_{\rm sp}^{-1}\equiv L_{\rm min}.$ (9)

It follows that the species scale is related to the entropy of the minimal
black hole in the following way:

$\Lambda_{\rm sp}={M_{p}\over(S_{\rm BH,{\rm min}})^{1\over d-2}}\,.$ (10)

Employing the BHEDC to minimal black holes one can then infer that the limit
of vanishing species scale $\Lambda_{\rm sp}\rightarrow 0$ is at infinite
distance in the space of EFTs, with a distance given as

$d(\Lambda_{\rm sp,i},\Lambda_{\rm sp,f})=\left|\ln(\Lambda_{\rm
sp,f}/\Lambda_{\rm sp,i})\right|,$ (11)

and with a tower of massless states, in which masses scale as

$m\sim(\Lambda_{\rm sp})^{\gamma}\;,$ (12)

where $\gamma$ is a positive constant. Combing this with the AdS-DC it follows
that the cosmological constant $\Lambda$ and the species scale, i.e. the UV
cut-off of the EFT, are parametrically related at the boundary of the moduli
space as

$|\Lambda|\simeq(\Lambda_{\rm sp})^{\gamma/\alpha}\,.$ (13)

So the limit of a small cosmological constant goes along with the limit of a
small species scale and vice versa. Furthermore minimal black holes and the
species scale can also be used to introduce the notion of species
thermodynamics [43, 51], which connects the thermodynamic properties of the KK
and string species to an entropy and temperature functional over the moduli
space of vacua.

The DC, the AdS-DC, and the BHEDC can be further generalized for more general
manifolds using geometric flow equations [52, 53, 54, 55, 56, 57, 58].
Specifically, the following Ricci flow distance conjecture was formulated
[52]: For a $d$-dimensional Riemannian manifold, the distance in the field
space of the background metrics along the Ricci-flow is determined by the
scalar curvature ${\cal R}(g)$:

$d({\cal R}_{\rm i},{\cal R}_{\rm f})=\left|\ln({\cal R}_{\rm f}/{\cal R}_{\rm
i})\right|.$ (14)

Here ${\cal R}_{i}$ and ${\cal R}_{f}$ are the corresponding initial and final
values of the scalar curvature. Furthermore it is stated that at ${{\cal
R}}=0$ there is an infinite tower of additional massless states in quantum
gravity.

One more related distance conjecture, the large D distance conjecture [59,
60], states that the limit of large space-time dimension,
$d\rightarrow\infty$, is also at infinite distance in the space of quantum
gravity vacua.

Another interesting limit arises with the gravitino mass going to zero. This
is because the gravitino mass is generally related to the scale of spontaneous
supersymmetry (SUSY) breaking in non-supersymmetric vacua (like the one we
happen to live in) [61, 62]. The gravitino conjecture states [63, 64] that in
a supersymmetric theory with a non-vanishing gravitino mass $m_{3/2}$, a tower
of states becomes light in the limit $m_{3/2}\to 0$ according to

$m\sim\left(\frac{m_{3/2}}{\bar{M}_{p}}\right)^{\delta}\ \bar{M}_{p}$ (15)

where $\delta$ is an order-one positive parameter and
$\bar{M}_{p}=M_{p}/\sqrt{8\pi}$ is the reduced Planck mass.

In closing, we review one last conjecture, which is not linked to a tower of
states, but will be relevant for our discussion. The weak gravity conjecture
(WGC) states [65, 66] that for given a gauge theory, weakly coupled to
Einstein gravity, there exists a charged state with

$\frac{Q}{m}\geq\left.\frac{{\cal Q}_{\rm BH}}{M_{\rm BH}}\right|_{\rm
extremal}={\cal O}(1)$ (16)

in Planck units, where ${\cal Q}_{\rm BH}$ and $M_{\rm BH}$ are the charge and
mass of an extremal black hole, $Q=qg$, $q$ is the quantized charge of the
state, and $g$ is the gauge coupling.

The WGC can be seen as a kinematic requirement that allows extremal black
holes to decay. As can be inferred from the weak cosmic censorship [67],
charged black holes must satisfy an extremality bound in order to avoid the
presence of naked singularities. Intuition about the extremality bound (16)
arises spontaneously from the the Reissner-Nordström (RN) metric that
describes the simplest extremal black hole, which has its mass $M_{\rm BH}$
equal to its charge ${\cal Q}_{\rm BH}$ in appropriate units. Note that if
${\cal Q}_{\rm BH}=M_{\rm BH}$ the single option for the black hole to decay
is if there exists a particle whose charge to mass ratio is at least one. If
$M_{\rm BH}>{\cal Q}_{\rm BH}$ the black hole has inner (Cauchy) and outer
(event) horizons, whereas if $M_{\rm BH}<{\cal Q}_{\rm BH}$, the RN metric
describes a naked singularity.

##  3 Foundations of the Dark Dimension

###  3.1 In which region of the landscape do we live?

Applying the AdS-DC to dS (or quasi dS) space provides one possible pathway to
answer this question. Moving forward, we will walk along this pathway.

Many asymptotic limits are expected to have towers of KK modes. According to
the dS-DC, the vacuum energy scales as $\Lambda\sim m^{1/\alpha}$, with $m$
the mass scale of the leading tower [19]. Since the KK tower contains massive
spin-2 bosons, there is a strong constraint from fundamental physics,
unitarity, which is expressed in the form of the Higuchi bound and imposes an
absolute upper limit $\alpha=1/2$ [68]. Besides, $\alpha$ has a lower limit
set by contributions of the Casimir energy; in four dimensions $\alpha\geq
1/4$ [6]. Another property of asymptotic limits is that both $m$ and $\Lambda$
are very small in Planck units. Therefore, since the observed amount of dark
energy in our world is also very small, $\Lambda\sim 10^{-122}M_{p}^{4}$, it
is tempting to speculate whether we could be living near an asymptotic limit.
We will then assume that we live within sight of the space boundary in some
infinite distance limit. This assumption automatically leads to the prediction
[6] of a tower of light fields at the energy scale

$m_{\rm KK}\lesssim\Lambda^{1/4}\sim 2.31~{}{\rm meV}\,.$ (17)

Since we have not seen yet experimental evidence of such a tower, it must
couple very weakly (if at all) to SM fields.

The ESC connects infinite distance limits with the decompactification of $n$
extra dimensions. Now, consistency of large-distance black hole physics in the
presence of a tower of $N$ light fields imposes a bound on the gravitational
cutoff of the EFT, and the fundamental length is no longer $l_{p}=M_{p}^{-1}$,
but rather the species length [34, 35]

$l_{\rm sp}=\sqrt{N}\ l_{p}\,.$ (18)

The origin of (18) can be traced back using different arguments, we follow
here the reasoning given in [69] based on quantum information storage [70].
Consider a pixel of size $L$ containing $N$ species storing information. The
minimal energy required to localize $N$ wave functions is of order $N/L$. This
energy can be associated to a Schwarzschild radius $r_{s}=N/(LM_{p}^{2})$,
which must be smaller than the pixel size if we want to avoid the system to
collapse into a black hole. Now, $r_{s}\leq L$, implies there is a minimum
size $l_{\rm sp}\equiv\Lambda_{\rm sp}^{-1}\equiv L_{\rm
min}=\sqrt{N}\,M_{p}^{-1}$ associated to the scale where gravity becomes
strongly coupled and the EFT necessarily breaks down. Since $l_{\rm sp}$
constitutes the smallest black hole size described by the EFT (involving only
the Einstein term), $\Lambda_{\rm sp}$ codifies the “number of light degrees
of freedom” (i.e., the number of KK excitations lighter than the cut-off),
given by $N\sim R_{\perp}^{n}l_{\rm sp}^{-n}$, up to energies of order

$M_{*}\sim\Lambda_{\rm sp}=m_{\rm KK}^{n/(n+2)}M_{p}^{2/(2+n)}\,,$ (19)

where $n$ is the number of decompactifying dimensions of radius $R_{\perp}\sim
m_{\rm KK}^{-1}$ and where the species scale $\Lambda_{\rm sp}$ corresponds to
the Planck scale $M_{*}$ of the higher dimensional theory.

Decompactification limits are tightly constrained by observations. Indeed,
astrophysical bounds from the requirement that neutron stars are not
excessively heated by KK decays lead to a very restrictive limit on the mass
scale of the KK tower, which depends on the number of dimensions that are
decompactifying: $R_{\perp}\leq 44~{}\mu{\rm m}$ for $n=1$, and $R_{\perp}\leq
10^{-4}~{}\mu{\rm m}$ for $n=2$ [71]. The limit on $R_{\perp}$ becomes more
restrictive with rising $n$. For $n=1$, deviations from Newton’s law impose a
more restrictive constraint, $R_{\perp}\leq 30~{}\mu{\rm m}$ [72]. Remember
that the starting point here is the hypothesis that associates the
cosmological constant to a tower of states whose mass scale satisfies

$m_{\rm KK}\simeq\Lambda^{1/4}/\lambda\,,$ (20)

where $\lambda$ is an order one parameter (more on this below). Now, we have
seen that $\Lambda^{-1/4}\sim 85\mu{\rm m}$ and so from (20) we conclude that
if $n=2$ the hypothesis is excluded by experiment, but if $n=1$ the hypothesis
can be made compatible with the experiment adjusting the proportionality
factor, which is estimated to be within the range
$10^{-4}\lesssim\lambda\lesssim 10^{-2}$. Substituting $n=1$ into (19) we have
$M_{*}\sim m_{\rm KK}^{1/3}M_{p}^{2/3}$, and so $10^{9}\lesssim M_{*}/{\rm
GeV}\lesssim 10^{10}$.333Auger data of highest energy cosmic rays favor
$M_{*}\sim 10^{10}~{}{\rm GeV}$ [73, 74]. We can combine this with the
relation between $m_{\rm KK}$ and $\Lambda_{\rm sp}$ to derive the following
relation

$M_{*}\sim\Lambda_{\rm sp}=\Lambda^{1/12}\ \lambda^{-1/3}\ M_{p}^{2/3}\,.$
(21)

In summary, when astrophysical bounds and gravitational tests of Newton’s law
are combined with the requirement that the size of the extra dimension is
related to the cosmological constant, we arrive at the conclusion encapsulated
in (20); namely, that there is one extra dimension of radius $R_{\perp}$ in
the micron range, and that the lower bound for $\alpha=1/4$ is basically
saturated [6]. Because of its connection to the observed dark energy, this
dimension has been nicknamed the dark dimension.

In closing we note that explicit string calculations of the vacuum energy (see
e.g. [75, 76, 77, 78]) show that the lower limit on $\alpha$ is saturated. In
particular, a strongly warped throat with its redshifted KK tower provides a
natural string theoretic mechanism that realizes the scaling
$m\sim\Lambda^{1/4}$, with the dark dimension being the one along the throat
[79]. Actually, a theoretical amendment on the connection between the
cosmological and KK mass scales confirms $\alpha=1/4$ [23].444Unsubstantiated
criticisms raised in [80, 81] have been addressed in [23]. We reiterate herein
that quantum gravity and string theory are different from field theory and
lead to a finite result relating $\Lambda$ to the KK scale via (20). Two
arguments support this statement: (i) the AdS-dC stressing that all infinities
cancel out (for AdS, this conjecture holds in many examples) and (ii) the
explicit one-loop string computation, which is finite due to modular
invariance and confirms $\alpha=1/4$.

###  3.2 The scale of SUSY breaking

It is of course interesting to explore whether there is a relation between the
SUSY breaking scale and the measured value of the dark energy density
$\Lambda$. Such a relation can be derived by combining two quantum gravity
consistency swampland constraints, which tie $\Lambda$ and the gravitino mass
$m_{3/2}$, to the mass scale of a light KK tower and, therefore, to the UV
cut-off of the EFT [61, 63, 64]. One can then use the constraint on $m_{3/2}$
to infer the implications of the dark dimension scenario for the scale of
supersymmetry breaking. In general, one can distinguish two situations. In the
first case, the gravitino mass and the cosmological constant are related to
the same tower of states. This is arguably the simplest scenario, in which the
natural scale for SUSY signatures is of order $\Lambda^{1/8}\sim{\rm TeV}$,
and therefore is within reach of LHC and/or of the next generation of hadron
colliders [82]. In the second case, $m_{3/2}$ and $\Lambda$ are related to
different towers. This scenario requires a decoupling of the gravitino mass
from the cosmological constant and is thus more difficult to realize in
concrete models.

Possible string theory and effective supergravity realizations of the dark
dimension scenario with broken supersymmetry are discussed in [82].

###  3.3 The dark dimension as a space with two boundaries

It was recently conjectured that the dark dimension can be viewed as a line
interval with end-of-the-world 9-branes (EW9-brane) attached at each end [83].
This construction derives from the 10D $E_{8}\times E_{8}$ heterotic string
theory in the strong coupling limit that has an orbifold
$S^{1}/\mathbb{Z}_{2}$ eleventh dimension (the dark dimension), which is not
visible in perturbation theory. The gravitational field propagates in the bulk
$\mathbb{R}^{10}\times S^{1}/\mathbb{Z}_{2}$, whereas the $E_{8}\times E_{8}$
fields propagate only at the $\mathbb{Z}_{2}$ fixed points on the two
EW9-branes.

The 10D coupling of the $E_{8}$ is a fixed number in 11D supergravity
(M-theory) units. When six out of ten space-time dimensions are compactified
on a Calabi-Yau (CY6) manifold, the tree-level couplings of the effective 4D
theory are simply given by the volume $V$ of CY6 (always in $M_{11}$ units)
[84, 85]. The SM gauge couplings $(g)$ are observed to be ${\cal O}(1)$, and
this forces $V$ on one of the branes to be ${\cal O}(1)$ in $M_{11}$ units.

For smooth compactifications, the volume of CY6 at given position of the
coordinate $x^{11}$,

$v(x^{11})=\int_{{\rm CY}6}\sqrt{g}\ d^{6}x\,,$ (22)

becomes an approximate linear function of the extra coordinate, and decreases
from one $E_{8}$ to the other $E_{8}$ [84]. This implies that $v$ has
different values at the two fixed points $x^{11}=0$ and $x^{11}=\pi\rho$.
Herein we identify $V=v(0)$ and $V^{\prime}=v(\pi\rho)$. Note that when the
theory in one $E_{8}$ is perturbative the theory on the other $E_{8}$ becomes
non-perturbative when the radius $\rho$ of the dark dimension is large. This
forces an upper bound on the size of the 11th dimension to avoid $V^{\prime}$
to become negative. If the anomaly coefficient is ${\cal O}(1)$, then
$\rho\sim 1~{}\mu{\rm m}$ is not allowed. This is a general phenomenon in
orientifold compactifications in which couplings of localized interactions
acquire a linear dependence in the extra dimension if one extra dimension
(transverse to the brane) is large [86]. The absence of such divergences
requires local tadpole cancellation (between branes and orientifolds).

However, there are some particular non-geometric compactifications where the
correction to the other $E_{8}$ coupling vanishes and there is no constraint
on $\rho$ [87]. In such a particular case, we have the following connections
between the 11D Planck mass, $M_{11}$ (defined in terms of the coefficient of
the Einstein Lagrangian in 11D supergravity, as $M_{11}=\kappa^{-2/9}$), the
radius of the dark dimension $\rho$, and the compactification radius
$R=V^{1/6}$,

$\kappa=(2\alpha)^{3/4}R^{9/2}\,,$ (23)

and

$\rho=(\alpha/2)^{3/2}\ M_{p}^{2}\ R^{3}\,$ (24)

where $V$ is the CY6 volume on the SM boundary and $\alpha\equiv
g^{2}/(4\pi)$. For $\rho\sim 1~{}\mu{\rm m}$ and phenomenological value
$\alpha\sim 1/25$, we obtain $M_{\rm KK}\sim R^{-1}\sim 7\times 10^{8}{\rm
GeV}$ and $M_{11}\sim 10^{9}{\rm GeV}$.

In summary, when six dimensions ($x^{5},\cdots,x^{10}$) are compactified on a
CY manifold, the eleven-dimensional bulk of the world becomes 5D while the
9-branes at its boundaries become 3-branes. The entire SM lives on one of
those 3-branes and is oblivious to the bulk of the 5D world or its other
boundary. The threshold structure of the $\rho\gg R\gtrsim l_{\rm sp}\sim
M_{11}^{-1}$ regime of the M-theory can be summarized as follows:

  * •

Gravity has a threshold at a rather low energy scale $m_{\rm KK}\sim
1/\rho\sim{\rm eV}$ above which it becomes 5D. However, this threshold does
not affect any gauge, Yukawa or scalar forces of the SM, which remains 4D at
distances shorter than $\rho$.

  * •

The next threshold arises at the KK scale $M_{\rm KK}\sim 1/R\sim 7\times
10^{8}~{}{\rm GeV}$, where six more dimensions open up for both gravity and
gauge interactions.

  * •

Almost immediately above this scale (around $\Lambda_{\rm sp}$), the effective
field theory description breaks down and the fully quantized M-theory
(whatever that is) takes over.

##  4 Dark Matter Candidates

The dark dimension provides a colosseum for dark matter contenders. In this
section we review the general properties of the various dark matter
candidates.

###  4.1 Primordial black holes

It has long been speculated that black holes could be produced from the
collapse of large amplitude fluctuations in the early universe [88, 89, 90,
91]. For an order of magnitude estimate of the black hole mass $M_{\rm BH}$,
we first note that the cosmological energy density scales with time $t$ as
$\rho\sim 1/(Gt^{2})$ and the density needed for a region of mass $M_{\rm BH}$
to collapse within its Schwarzschild radius is $\rho\sim c^{6}/(G^{3}M_{\rm
BH}^{2})$, so that primordial black holes (PBHs) would initially have around
the cosmological horizon mass [92]

$M_{\rm BH}\sim\frac{c^{3}t}{G}\sim 10^{15}\left(\frac{t}{10^{-23}~{}{\rm
s}}\right)~{}{\rm g}\,.$ (25)

This means that a black hole would have the reduced Planck mass
($\bar{M}_{p}\sim 10^{-5}~{}{\rm g}$) if they formed at the Planck time
($10^{-43}~{}{\rm s}$), $1~{}M_{\odot}$ if they formed at the QCD epoch
($10^{-5}~{}{\rm s}$), and $10^{5}M_{\odot}$ if they formed at $t\sim 1~{}{\rm
s}$, comparable to the mass of the holes thought to reside in galactic nuclei.
This back-of-the-envelope calculation suggests that PBHs could span an
enormous mass range. Despite the fact that the mass spectrum of these PBHs is
yet to be shaped, on cosmological scales they would behave like a typical cold
dark matter particle.

However, an all-dark-matter interpretation in terms of PBHs is severely
constrained by observations [92, 93, 94, 95]. To be specific, the
extragalactic $\gamma$-ray background [96], the cosmic microwave background
(CMB) [97], the 511 keV $\gamma$-ray line [98, 99, 100, 101], EDGES 21-cm
signal [102], and the MeV Galactic diffuse emission [103, 104, 105] constrain
evaporation of black holes with masses $\lesssim 10^{17}~{}{\rm g}$, whereas
the non-observation of microlensing events by MACHO [106], EROS [107], Kepler
[108], Icarus [109], OGLE [110] and Subaru-HSC [111] set an upper limit on the
black hole abundance for masses $M_{\rm BH}\gtrsim 10^{21}~{}{\rm g}$.

Before proceeding, we pause and call attention to a captivating coincidence:

${\tt size\ of\ the\ dark\ dimension}\sim{\tt wavelength\ of\ visible\
light}\,,$ (26)

which implies that the Schwarzschild radius of 5D black holes is well below
the wavelength of light. For point-like lenses, this is precisely the critical
length where geometric optics breaks down and the effects of wave optics
suppress the magnification, obstructing the sensitivity to 5D PBH microlensing
signals [111]. So 5D PBHs escape these microlensing constraints; at the same
time, as pointed out in [112], they are: bigger, colder, and longer-lived than
a usual 4D black hole of the same mass.

Throughout we rely on the probe brane approximation, which ensures that the
only effect of the brane field is to bind the black hole to the brane [113].
This is an adequate approximation provided $M_{\rm BH}$ is well above the
brane tension, which is presumably of the order of but smaller than
$M_{\star}$. We also assume that the black hole can be treated as a flat $d$
dimensional object. This assumption is valid for extra dimensions that are
larger than the 5D Schwarzschild radius, which is given by [114, 115, 116]

$r_{s}(M_{\rm BH})\sim\frac{1}{M_{*}}\left[\frac{2}{3\,\pi}\ \frac{M_{\rm
BH}}{M_{*}}\right]^{1/2}\,.$ (27)

Schwarzschild black holes radiate all particle species lighter than or
comparable to their Hawking temperature, which in four dimensions is related
to the mass of the black hole by

$T_{H}=\frac{\bar{M}_{p}^{2}}{8\pi M_{\rm BH}}\sim\,\bigg{(}\frac{M_{\rm
BH}}{10^{16}\,{\rm g}}\bigg{)}^{-1}~{}{\rm MeV}\,,$ (28)

whereas for 5D black holes the Hawking temperature mass relation is found to
be [117]

$T_{H}\sim\frac{1}{r_{s}}\sim\left(\frac{M_{\rm BH}}{10^{12}~{}{\rm
g}}\right)^{-1/2}~{}{\rm MeV}\,.$ (29)

The numerical estimate of (29) applies to the dark dimension scenario with
$M_{*}\sim 10^{10}~{}{\rm GeV}$.555We have taken the highest possible value of
$M_{*}$ to remain conservative in the estimated bound on the fraction of dark
matter composed of primordial black holes $f_{\rm PBH}$. It is evident that 5D
black holes are colder than 4D black holes of the same mass.

Armed with the Hawking temperature, we can now calculate the entropy of the 5D
black hole [118]

$S_{\rm BH}=\frac{4}{3}\pi\ M_{\rm BH}\ r_{s}\,.$ (30)

In the rest frame of the Schwarzschild black hole, both the average number
[119, 120] and the probability distribution of the number [121, 122, 123] of
outgoing particles in each mode obey a thermal spectrum. However, in the
neighborhood of the horizon the black hole produces an effective potential
barrier that backscatters part of the emitted radiation, modifying the thermal
spectrum. The so-called “greybody factor”, which controls the black hole
absorption cross section, depends upon the spin of the emitted particles $s$,
their energy $Q$, and $M_{\rm BH}$ [124, 125, 126, 127, 128]. The prevailing
energies of the emitted particles are $\sim T_{\rm H}\sim 1/r_{s}$, resulting
in $s$-wave dominance of the final state. This implies that the black hole
evaporates with equal probability to a particle on the brane and in the
compact space [129, 130]. Thereby, the process of evaporation is driven by the
large number of SM brane modes.

The Hawking radiation causes a 4D black hole to lose mass at the following
rate [101]

$\displaystyle\left.\frac{dM_{\rm BH}}{dt}\right|_{\rm evap}$ $\displaystyle=$
$\displaystyle-\frac{\bar{M}_{p}^{2}}{30720\ \pi\ M_{\rm BH}^{2}}\
\sum_{i}c_{i}(T_{H})\ \tilde{f}\ \Gamma_{s}$ (31) $\displaystyle\sim$
$\displaystyle-7.5\times 10^{-8}\ \left(\frac{M_{\rm BH}}{10^{16}~{}{\rm
g}}\right)^{-2}\ \sum_{i}c_{i}(T_{H})\ \tilde{f}\ \Gamma_{s}~{}{\rm g/s}\,,$

whereas a 5D black hole has an evaporation rate of [117]

$\displaystyle\left.\frac{dM_{\rm BH}}{dt}\right|_{\rm evap}$
$\displaystyle\sim$ $\displaystyle-9\ \pi^{5/4}\zeta(4)T_{H}^{2}\
\sum_{i}c_{i}(T_{H})\ \tilde{f}\ \Gamma_{s}$ (32) $\displaystyle\sim$
$\displaystyle-1.3\times 10^{-12}\ \left(\frac{M_{\rm BH}}{10^{16}~{}{\rm
g}}\right)^{-1}\ \sum_{i}c_{i}(T_{H})\ \tilde{f}\ \Gamma_{s}~{}{\rm g/s}\,,$

where $c_{i}(T_{H})$ counts the number of internal degrees of freedom of
particle species $i$ of mass $m_{i}$ satisfying $m_{i}\ll T_{H}$,
$\tilde{f}=1$ $(\tilde{f}=7/8)$ for bosons (fermions), and where
$\Gamma_{s=1/2}\approx 2/3$ and $\Gamma_{s=1}\approx 1/4$ are the (spin-
weighted) dimensionless greybody factors normalized to the black hole surface
area [131]. Now, comparing (31) and (32) it is easily seen that 5D black holes
live longer than 4D black holes of the same mass.

Integrating (32) we can parametrize the 5D black hole lifetime as a function
of its mass and temperature,

$\tau_{s}\sim 13.8\ \Bigg{(}\frac{M_{\rm BH}}{10^{12}~{}{\rm g}}\Bigg{)}^{2}\
\left(\frac{6}{\sum_{i}c_{i}(T_{s})\ \tilde{f}\ \Gamma_{s}}\right)~{}{\rm
Gyr}\,,$ (33)

where we have used (29) to estimate that $T_{H}\sim 1~{}{\rm MeV}$ and
therefore $c_{i}(T_{H})$ receives a contribution of 6 from neutrinos, 4 for
electrons, and 2 from photons, yielding $\sum_{i}c_{i}(T_{H})\ \tilde{f}\
\Gamma_{s}=6$. Armed with (33) we can estimate the bound on the 5D PBH
abundance by a simple rescaling procedure of the $d=4$ bounds on the fraction
of dark matter composed of primordial black holes $f_{\rm PBH}$. The key point
for such a rescaling is that for a given photon energy, or equivalently a
given Hawking temperature, we expect a comparable limit on $f_{\rm PBH}$ for
both $d=4$ and $d=5$. For example, from (28) and (29) we see that the
constraint of $f_{\rm PBH}\lesssim 5\times 10^{-5}$ for 4D black holes with
$M_{\rm BH}\sim 10^{16}~{}{\rm g}$ [105], should be roughly the same for the
abundance of 5D black holes with $M_{\rm BH}\sim 10^{12}~{}{\rm g}$. Now,
since in $d=4$ for $M_{\rm BH}\sim 4\times 10^{17}~{}{\rm g}$ we have $f_{\rm
PBH}\sim 1$ [105], this implies the same abundance for 5D black holes of
$M_{\rm BH}\sim 10^{15}~{}{\rm g}$. By duplicating this procedure for heavier
black holes we conclude that for a species scale of ${\cal O}(10^{10}~{}{\rm
GeV})$, an all-dark-matter interpretation in terms of 5D black holes must be
feasible for masses in the range [117]

$10^{15}\lesssim M_{\rm BH}/{\rm g}\lesssim 10^{21}\,.$ (34)

This range is extended compared to that in the 4D theory by more than two
orders of magnitude in the low mass region.

Extremal black holes trace the boundary between black-hole configurations and
horizonless naked singularities. As put forward by the WGC, they are
characterized by the minimally allowed mass (radius) for a given amount of
black-hole charge ${\cal Q}_{\rm BH}$ (or angular momentum). Near-extremal
black holes are characterized by a finite (non-zero) mass gap of the first
excited state above the extremal (zero-temperature) black-hole configuration
[132]. The temperature of such near-extremal black holes is found to be

$T_{ne}\sim\frac{\beta^{1/2}T_{H}}{S_{\rm BH}^{1/2}}\,,$ (35)

where $\beta$ is a factor of order-one that controls the differences between
$M_{\rm BH}$ and ${\cal Q}_{\rm BH}$; for details see [31, 51]. If there were
5D primordial near-extremal black holes in nature, then an all-dark-matter
interpretation would be possible in the mass range [133]

$10^{5}\sqrt{\beta}\lesssim M_{\rm BH}/{\rm g}\lesssim 10^{21}\,.$ (36)

Up until now we have assumed that the 5D black holes stay attached to the
brane during the evaporation process. Hereafter we relax this assumption and
allow them wander off into the bulk. Without knowing more details of the bulk
and brane theory it is not worth considering to calculate the probability of
such wandering in detail. However, we can assume that the black holes are out
of the brane-world and study the evaporation effects of these bulk PBHs.
Furthermore, it is always possible that the PBHs are produced in the bulk to
start with. This situation will be more appealing within the model discussed
in Sec. 6, in which we theorize that the dark dimension may have undergone a
uniform rapid expansion, together with the three-dimensional non-compact
space, by regular exponential inflation driven by an (approximate) higher
dimensional cosmological constant. If this were the case, then primordial
fluctuations during inflation of the compact space could lead to the
production of black holes in the bulk. In what follows, we then assume that
PBHs are localized or propagate in the bulk.

Bulk black holes live longer than those attached to the brane. This is because
KK modes are excitations in the full transverse space and so their overlap
with small (higher dimensional) black holes is suppressed by the geometric
factor $(r_{s}/R_{\perp})$ relative to the brane fields. This geometric
suppression precisely compensates for the enormous number of modes and the
total KK contribution is only of same order as that from a single brane field
[129]. Actually, greybody factors suppress graviton emission when compared to
fermions and gauge bosons, and hence bulk black holes which do not have access
to the brane degrees of freedom are expected to live longer. In addition,
since there is no emission on the brane the bounds due to photon evaporation
can be avoided. This implies that PBHs localized in the bulk can provide an
all-dark-matter interpretation if

$10^{11}\lesssim M_{\rm BH}/{\rm g}<10^{21}\,,$ (37)

where we have remained conservative, and following [134] we assumed that the
ratio of the emitted flux into a single brane field over a single bulk field
is roughly a factor of two [117].

###  4.2 KK gravitons

It was observed in [135] that the universal coupling of the SM fields to the
massive spin-2 KK excitations of the graviton in the dark dimension provides
an alternative dark matter candidate. Within this model the cosmic evolution
of the hidden sector is primarily dominated by “dark-to-dark” decays, yielding
a specific realization of the dynamical dark matter framework [136]. Consider
a tower of equally spaced dark gravitons, indexed by an integer $l$, and with
mass $m_{l}=l\ m_{\rm KK}$. The partial decay width of KK graviton $l$ to SM
fields is found to be,

$\Gamma^{l}_{{\rm SM}}=\frac{\tilde{\lambda}^{2}\ m^{3}_{\rm KK}\
l^{3}}{80\pi\bar{M}_{p}^{2}}\,,$ (38)

where $\tilde{\lambda}$ takes into account all the available decay channels
and is a function of time [137].

In the absence of isometries in the dark dimension, which is the common
expectation, the KK momentum of the dark tower is not conserved. This means
that a dark graviton of KK quantum $n$ can decay to two other ones, with
quantum numbers $n_{1}$ and $n_{2}$. If the KK quantum violation can go up to
$\delta n$, the number of available channels is roughly $l\,\delta n$. In
addition, because the decay is almost at threshold, the phase space factor is
roughly the velocity of decay products, $v_{\rm r.m.s.}\sim\sqrt{m_{\rm KK}\
\delta n/m_{l}}$. Putting all this together we obtain the total decay width,

$\displaystyle\Gamma^{l}_{\rm tot}$ $\displaystyle\sim$
$\displaystyle\sum_{l^{\prime}<l}\ \
\sum_{0<l^{\prime\prime}<l-l^{\prime}}\Gamma^{l}_{l^{\prime}l^{\prime\prime}}\sim\beta^{2}\frac{m_{l}^{3}}{\bar{M}_{p}^{2}}\times\frac{m_{l}}{m_{\rm
KK}}\ \delta n\times\sqrt{\frac{m_{\rm KK}\delta_{n}}{m_{l}}}$ (39)
$\displaystyle\sim$ $\displaystyle\beta^{2}\ \delta
n^{3/2}\frac{m_{l}^{7/2}}{\bar{M}_{p}^{2}m_{\rm KK}^{1/2}}\,,$

where $\beta$ parametrizes our ignorance of decays in the dark dimension
[135].

To estimate the time evolution of the dark matter mass assume that for times
larger than $1/\Gamma^{l}_{\rm tot}$ dark matter which is heavier than the
corresponding $m_{l}$ has already decayed, and so it follows that

$m_{l}\sim\left(\frac{\bar{M}_{p}^{4}\ m_{\rm KK}}{\beta^{4}\ \delta
n^{3}}\right)^{1/7}t^{-2/7}\,,$ (40)

where $t$ indicates the time elapsed since the big bang [135].

Consistency with CMB anisotropies requires $\Gamma^{l}_{\gamma\gamma}<5\times
10^{-25}~{}{\rm s}^{-1}$ between the last scattering surface and reionization
[138]. Taking $\tilde{\lambda}=1$ (to set out the decay into photons) and
using (38) it follows that the CMB requirement is satisfied for $l\lesssim
10^{8}$ at the time $t_{\rm MR}\sim 6\times 10^{4}~{}{\rm yr}$ of matter-
radiation equality. In other words, by setting $\tilde{\lambda}\sim 1$ and
$m_{l}(t_{\rm MR})\lesssim 1~{}{\rm MeV}$, the evolution of $m_{l}$ with
cosmic time given in (40) is such that at the last scattering surface the
dominant KK state in the dynamical dark matter ensemble has the correct decay
width to accommodate the CMB constraints [139].

Now, we have seen that dark matter decay gives the daughter particles a
velocity kick. Self-gravitating dark-matter halos that have a virial velocity
smaller than this velocity kick may be disrupted by these particle decays.
Consistency with existing data requires roughly $\delta n\sim 1$, and
$\beta\sim 635$ [140, 141]. For selected fiducial parameters, the cosmic
evolution of the incredible bulk predicts via (40) a dominant particle mass of
$\sim 900~{}{\rm keV}$ at CMB, of $\sim 500~{}{\rm keV}$ in the Dark Ages, of
$\sim 150~{}{\rm keV}$ at Cosmic Dawn, and of $\sim 50~{}{\rm keV}$ in the
local universe. This is in sharp contrast to typical dark matter decay
scenarios with one unstable particle (such as sterile neutrinos [142]).
Simultaneous observations of signals at Cosmic Dawn and in the local universe
could constitute the smoking gun of the incredible bulk [7].

For many purposes, a black hole can be replaced by a bound state of gravitons
[143]. As a matter of fact, a correspondence between 5D PBHs and massive KK
gravitons as dark matter candidates has been conjectured in [144].

###  4.3 A fuzzy radion

The radion stabilizing the dark dimension could be yet another dark matter
contender [145]. This is because in principle the radion could be ultralight,
and if this were the case it would serve as a fuzzy dark matter candidate. A
simple cosmological production mechanism brings into play unstable KK graviton
towers which are fueled by the decay of the inflaton. As in the previous
model, the cosmic evolution of the dark sector is mostly driven by “dark-to-
dark” decay processes that regulate the decay of KK gravitons within the dark
tower, conveying another realization of the dynamical dark matter framework
[136]. In the spirit of [146], within this model it is assumed that the intra-
KK decays in the bulk carry a spontaneous breakdown of the translational
invariance in the compact space, such that the 5D momenta are not conserved
(but now $\delta n\gg 1$). Armed with these two reasonable assumptions it is
straightforward to see that the energy the inflaton deposited in the KK tower
should have collapsed all into the radion well before BBN.

##  5 Neutrino Masses and Mixing

The dark dimension scenario provides a profitable arena to realize an old idea
for explaining the smallness of neutrino masses by introducing the right-
handed neutrinos as 5D bulk states with Yukawa couplings to the left-handed
lepton and Higgs doublets that are localized states on the SM brane stack
[147, 148, 149]. The neutrino masses are then suppressed due to the wave
function of the bulk states.

More indicatively, we introduce three 5D Dirac fermions $\Psi_{\alpha}$, which
are singlets under the SM gauge symmetries and interact in our brane with the
three active left-handed neutrinos in a way that conserves lepton number. The
$S^{1}/\mathbb{Z}_{2}$ symmetry contains $x^{11}$ to $-x^{11}$ which acts as
chirality ($\gamma_{5}$) on spinors. In the Weyl basis each Dirac field can be
decomposed into two two-component spinors $\Psi_{\alpha}\equiv(\psi_{\alpha
L},\psi_{\alpha R})^{T}$.

The generation of neutrino masses originates in 5D bulk-brane interactions of
the form

$\mathscr{L}\supset h_{ij}\ \overline{L}_{i}\ \tilde{H}\
\psi_{jR}(x^{11}=0)\,,$ (41)

where $\tilde{H}=-i\sigma_{2}H^{*}$, $L_{i}$ denotes the lepton doublets
(localized on the SM brane), $\psi_{jR}$ stands for the 3 bulk (right-handed)
$R$-neutrinos evaluated at the position of the SM brane, $x^{11}=0$ in the
dark-dimension coordinate $x^{11}$, and $h_{ij}$ are coupling constants. This
gives a coupling with the $L$-neutrinos of the form $\langle H\rangle\
\overline{\nu}_{L_{i}}\ \psi_{jR}(x^{11}=0)$, where $\langle
H\rangle=175~{}{\rm GeV}$ is the Higgs vacuum expectation value. Expanding
$\psi_{jR}$ into modes canonically normalized leads for each of them to a
Yukawa $3\times 3$ matrix suppressed by the square root of the volume of the
bulk $\sqrt{\pi R_{\perp}M_{s}}$, i.e.,

$Y_{ij}=\frac{h_{ij}}{\sqrt{\pi R_{\perp}M_{s}}}\sim
h_{ij}\frac{M_{s}}{M_{p}}\,,$ (42)

where $M_{s}\lesssim M_{*}$ is the string scale, and where in the second
rendition we have dropped factors of $\pi$’s and of the string coupling.

Now, neutrino oscillation data can be well-fitted in terms of two nonzero
differences $\Delta m^{2}_{ij}=m^{2}_{i}-m^{2}_{j}$ between the squares of the
masses of the three mass eigenstates; namely, $\Delta m_{21}^{2}=(7.53\pm
0.18)\times 10^{-5}~{}{\rm eV}^{2}$ and $\Delta m^{2}_{32}=(2.453\pm
0.033)\times 10^{-3}~{}{\rm eV}^{2}$ or $\Delta m^{2}_{32}=-(2.536\pm
0.034)\times 10^{-3}~{}{\rm eV}^{2}$ [1]. It is easily seen that to obtain the
correct order of magnitude of neutrino masses the coupling $h_{ij}$ should be
of order $10^{-4}$ to $10^{-5}$ for $10^{9}\lesssim M_{s}/{\rm GeV}\lesssim
10^{10}$.

Note that KK modes of the 5D $R$-neutrino fields behave as an infinite tower
of sterile neutrinos, with masses proportional to $m_{\rm KK}$. However, only
the lower mass states of the tower mix with the active SM neutrinos in a
pertinent fashion. The non-observation of neutrino disappearance from
oscillations into sterile neutrinos at long- and short-baseline experiments
places a 90% CL upper limit $R_{\perp}<0.4~{}\mu{\rm m}$ for the normal
neutrino ordering, and $R_{\perp}<0.2~{}\mu{\rm m}$ for the inverted neutrino
ordering [150, 151].666We arrived at these upper bounds by looking at the low
mass limit of the lightest neutrino state in Fig. 6 of [151] and rounding the
numbers to one significant figure. This set of parameters corresponds to
$\lambda\lesssim 10^{-3}$ and so $m_{\rm KK}\gtrsim 2.5~{}{\rm eV}$ [7].

Before proceeding, it is important to stress that the upper bounds on
$R_{\perp}$ discussed in the previous paragraph are sensitive to assumptions
of the $5^{\rm th}$ dimension geometry. Moreover, in the presence of bulk
masses [152, 153], the mixing of the first KK modes to active neutrinos can be
suppressed, and therefore the aforementioned bounds on $R_{\perp}$ can be
avoided [154, 155]. It is also worth mentioning that such bulk masses have the
potential to increase the relative importance of the higher KK modes, yielding
distinct oscillation signatures via neutrino disappearance/appearance effects.

Non-minimal extensions of the dark dimension, in which $m_{3/2}$ and $\Lambda$
have different KK towers, allow a high-scale SUSY breaking and can therefore
host a rather heavy gravitino together with a modulino with a mass of about 50
eV [156]. For a particular example, we note that the modulino could be the
fermionic partner of the radion.777In the standard moduli stabilization by
fluxes, all complex structure moduli and the dilaton are stabilized in a
supersymmetric way while Kähler class moduli need an input from SUSY breaking.
The radion is Kähler class and exists in a model independent fashion within
the dark dimension scenario. These models with high-scale SUSY breaking are
fully predictive through neutrino-modulino oscillations [157] which can be
confronted with data to be collected by experiments at the Forward Physics
Facility [158, 159].

A seemingly different, but in fact closely related subject is the the
sharpened version of the WGC forbidding the presence of non-SUSY AdS vacua
supported by fluxes in a consistent quantum gravity theory [160]. This is
because (unless the gravitino is very light, with mass in the meV range)
neutrinos have to be Dirac with right-handed states propagating in the bulk so
that the KK neutrino towers compensate for the graviton tower to maintain
stable dS vacua [155].

##  6 Mesoscopic Extra Dimension from 5D Inflation

It is unnatural to entertain that the size of the dark dimension would remain
fixed during the evolution of the Universe right at the species scale, since
the Higuchi bound implies a very low inflation scale. One possible mechanism
to accommodate this hierarchy is to inflate the size of the dark dimension.
The required inflationary phase can be described by a 5D dS (or approximate)
solution of Einstein equations, with cosmological constant and a 5D Planck
scale $M_{*}\sim 10^{9}~{}{\rm GeV}$ [7]. All dimensions (compact and non-
compact) expand exponentially in terms of the 5D proper time. It is
straightforward to see that this set-up requires about 42 e-folds to expand
the 5th dimension from the fundamental length ${\cal O}(M_{*}^{-1})$ to the
micron size ${\cal O}(R_{\perp})$. At the end of 5D inflation, or at any given
moment, one can interpret the solution in terms of 4D fields using 4D Planck
units from the relation $M_{p}^{2}=M_{*}^{3}R$, which amounts going to the 4D
Einstein frame. This implies that if $M_{*}^{-1}\leq R\leq R_{\perp}$ expands
$N$ e-folds, then the 3D space would expand $3N/2$ e-folds as a result of a
uniform 5D inflation. Altogether, the 3D space has expanded by about 60
e-folds to solve the horizon problem, while connecting this particular
solution to the generation of large size extra dimension.

Besides solving the horizon problem, 4D slow-roll inflation predicts an
approximate scale-invariant Harrison-Zel’dovich power spectrum of primordial
density perturbations [161, 162] consistent with CMB observations [163]. This
is due to the fact that the 2-point function of a massless minimally coupled
scalar field in dS space behaves logarithmically at distances larger than the
cosmological horizon, a property which is though valid for any spacetime
dimensionality [164]. When some dimensions are however compact, this behaviour
is expected to hold for distances smaller than the compactification length,
while deviating from scale invariance at larger distances, potentially
conflicting with observations at large angles. Remarkably, consistency of 5D
inflation with CMB observations is maintained if the size of the dark
dimension is larger than about a micron, implying a change of behaviour in the
power spectrum at angles larger than 10 degrees, corresponding to multiple
moments $l\lesssim 30$, where experimental errors are getting large [8].
Actually, the scale invariance of the power spectrum is obtained upon
summation over the contribution of the inflaton KK-modes’ fluctuations that
correspond to a tower of scalars from the 4D point of view. Spectral indices
dependence on slow-roll parameters and tensor perturbations have been computed
in [165]. The tensor-to-scalar ratio is found to be $r=24\epsilon_{V}$, and so
the 95% CL upper limit $r<0.032$ (derived using a combination of BICEP/Keck
2018 and Planck data) [167, 168] places an experimental constraint on the
potential slow-roll parameter: $\epsilon_{V}<0.0013$.

Another interesting feature of 5D inflation is that the radion can be
stabilized in a local (metastable) dS vacuum [8], using the contributions of
bulk field gradients [169] or of the Casimir energy, assuming a mass for the
bulk $R$-handed neutrinos of the same order of magnitude [170]. Consider 5D
Einstein-de Sitter gravity compactified on a circle $S^{1}$ endowed with
$S^{1}/\mathbb{Z}_{2}$ symmetry, and assume that the SM is localized on a
D-brane transverse to the compact dimension, whereas gravity spills into the
compact space. The effective 4D potential of the radion field $R$ is found to
be

$V(R)=\frac{2\pi\ \Lambda_{5}\ r^{2}}{R}+\left(\frac{r}{R}\right)^{2}\
T_{4}+V_{C}(R)\,,$ (43)

where $\Lambda_{5}$ is the 5D cosmological constant, $r\equiv\langle R\rangle$
is the vacuum expectation value of the radion, $T_{4}$ is the total 3-brane
tension, and $V_{C}$ stands for the quantum corrections to the vacuum energy
due to Casimir forces. These corrections are expected to become important in
the deep infrared region, because the Casimir contribution to the potential
falls off exponentially at large $R$ compared to the particle wavelength.
Indeed, as $R$ decreases different particle thresholds open up,

$V_{C}(R)=\sum_{i}\frac{\pi r^{2}}{32\pi^{7}R^{6}}\ (N_{F}-N_{B})\
\Theta(R_{i}-R)\,,$ (44)

where $m_{i}=R_{i}^{-1}$ are the masses of the 5D fields, $\Theta$ is a step
function, and $N_{F}-N_{B}$ stands for the difference between the number of
light fermionic and bosonic degrees of freedom. At the classical level, i.e.
considering only the first two terms in (43), it is straightforward to see
that the potential develops a maximum at

$R_{\rm max}=-T_{4}/(\pi\Lambda_{5})\,,$ (45)

requiring a negative tension $T_{4}$. Note that if the fermionic degrees of
freedom overwhelm the bosonic contribution, they would give rise to possible
minima, as long as $R_{i}<R_{\rm max}$. This could be the case if $N_{F}=12$
takes for the three 5D Dirac neutrino fields and $N_{B}=5$ accounts for the 5D
graviton. In Fig. 2 we show an illustrative example.

Figure 2: Left. The potential $V(R)$ for $(\Lambda_{5})^{1/5}=22.6~{}{\rm
meV}$ and $|T_{4}|^{1/4}=24.2~{}{\rm meV}$, considering $N_{F}-N_{B}=6$ (AdS)
and $N_{F}-N_{B}=7$ (dS). From Ref. [166]. Right. Schematic form of the real
scalar potential $V(\phi)$.

##  7 Living on the Edge: Cosmology within sight of the space boundary

###  7.1 Cosmic Discrepancies

Over the last few years, low- and high-redshift observations set off tensions
in the measurement of the present-day expansion rate $H_{0}$ and in the
determination of the amplitude of the matter clustering in the late Universe
(parameterized by $S_{8}$). More concretely, the values $H_{0}=67.4\pm
0.5~{}{\rm km/s/Mpc}$ and $S_{8}=0.834\pm 0.016$ inferred from Planck’s CMB
data assuming $\Lambda$CDM [163] are in $\sim 5\sigma$ tension with
$H_{0}=73.04\pm 1.04~{}{\rm km/s/Mpc}$ from the SH0ES distance ladder
measurement (using Cepheid-calibrated type-Ia supernovae) [2, 3] and in $\sim
3\sigma$ tension with $S_{8}=0.766^{+0.020}_{-0.014}$ from the cosmic shear
data of the Kilo-Degree Survey (KiDS-1000) [171], respectively. These
statistically significant discrepancies have become a new cornerstone of
theoretical physics, and many beyond SM setups are rising to the challenge
[172, 173, 174, 175].

$\Lambda_{s}$CDM [176, 177, 178, 179] is one of the many beyond SM physics
models that have been proposed to simultaneously resolve the $H_{0}$ and
$S_{8}$ tensions; see Appendix for details.888An alternative model that
accommodates the data has been presented in [180]. The model relies on an
empirical conjecture which postulates that $\Lambda$ may have switched sign
(from negative to positive) at critical redshift $z_{c}\sim 2$;

$\Lambda\quad\rightarrow\quad\Lambda_{\rm s}\equiv\Lambda_{0}\ {\rm
sgn}[z_{c}-z],$ (46)

with $\Lambda_{0}>0$, and where ${\rm sgn}[x]=-1,0,1$ for $x<0$, $x=0$ and
$x>0$, respectively. Apart from resolving the three major cosmological
tensions, $\Lambda_{s}$CDM achieves quite a good fit to Lyman-$\alpha$ data
provided $z_{c}\lesssim 2.3$ [176], and it is in agreement with the otherwise
puzzling JWST observations [181, 182].

Despite the remarkable success of $\Lambda_{s}$CDM to accommodate the
experimental data, the model is theoretically unsatisfactory because it
postulates that the Universe experienced a rapid transition from an AdS vacuum
to a dS vacuum, and this hods out against the AdS-DC conjecture, which posits
that flat space limit is at infinite distance in the space of metric
configurations and therefore these two vacua are an infinite distance appart
in metric space [19]. However, it is important to stress that this no-go
theorem is valid at zero temperature, where the number of light particles is
(in general) constant. At finite temperature, particles can decay and hence
the number of light particles can change. In this way the minima of the
potential can be lifted [166].

###  7.2 AdS $\to$ dS transition driven by Casimir forces of bulk fields

A possible explanation for the required AdS $\to$ dS crossover transition in
the vacuum energy can be obtained using the Casimir forces of fields
inhabiting the dark dimension [166]. We assume that the 5D spectrum contains a
light real scalar field $\phi$, in addition to the graviton and the three
neutrino generations. We further assume that the real scalar has a potential
with two local minima with very small difference in vacuum energy and bigger
curvature (mass) of the lower one, see Fig. 2. At $z_{c}$ the false vacuum
“tunnels” to its true vacuum state. After the quantum tunneling $\phi$ becomes
more massive and its contribution to the Casimir energy becomes exponentially
suppressed. The idea here is that for $z\gtrsim z_{c}$, we have $N_{B}=6$,
whereas for $z\lesssim z_{c}$, we have $N_{B}=5$. Taking $N_{F}=12$ to account
for the three Dirac neutrino fields we can use (43) in combination with (44)
to obtain an expression for the effective 4D radion potential. In Fig. 2 we
show an illustrative example of the AdS $\to$ dS transition produced by
$N_{F}-N_{B}=6$ for $z\gtrsim z_{c}$, and $N_{F}-N_{B}=5$ for $z\lesssim
z_{c}$. It is important to note that the 5D vacuum transition creates a
$\delta V$ contribution to $\Lambda_{5}$, where $\delta V=V(\phi_{\rm
fv})-V(\phi_{\rm tv})$ corresponding to the vacuum energies of the upper
(false vacuum) and the lower (true vacuum) minima. We have taken $\delta
V\ll\Lambda_{5}$ so that it does not perturb the analysis producing the curves
shown if Fig. 2.

Now, the AdS $\to$ dS transition shown in Fig. 2 slightly deviates from the
model analyzed in [179], because the fields characterizing the deep infrared
region of the dark sector contribute to the effective number of relativistic
neutrino-like species $N_{\rm eff}$ [183]. Using conservation of entropy,
fully thermalized relics with $g_{*}$ degrees of freedom contribute

$\Delta N_{\rm
eff}=g_{*}\left(\frac{43}{4g_{s}}\right)^{4/3}\left\\{\begin{array}[]{ll}4/7&{\rm
for\ bosons}\\\ 1/2&{\rm for\ fermions}\end{array}\right.\,,$ (47)

where $g_{s}$ denotes the effective degrees of freedom for the entropy of the
other thermalized relativistic species that are present when they decouple
[184]. The 5D graviton has 5 helicities, but the spin-1 helicities do not have
zero modes, because we assume the compactification has $S^{1}/\mathbb{Z}_{2}$
symmetry and so the $\pm 1$ helicities are projected out. The spin-0 is the
radion and the spin-2 helicities form the massless (zero mode) graviton. This
means that for the 5D graviton, $g_{*}=3$. The scalar field $\phi$ contributes
with $g_{*}=1$. The (bulk) left-handed neutrinos are odd, but the right-handed
neutrinos are even and so each counts as a Weyl neutrino, for a total
$g_{*}=2\times 3$. Assuming that the dark sector decouples from the SM sector
before the electroweak phase transition we have $g_{s}=106.75$. This gives
$\Delta N_{\rm eff}=0.25$.999It was recently noted that if the QCD axion is
localized on the SM brane, a combination of theoretical and observational
constraints forces it to have decay constant in a narrow range $10^{9}\lesssim
f/{\rm GeV}\lesssim 10^{10}$ [185]. This corresponds to a mass for the QCD
axion of $1\lesssim m_{a}/{\rm meV}\lesssim 10$. Although the axion would not
affect the Casimir corrections to the potential, it would contributes to
$\Delta N_{\rm eff}$ [186, 187, 188]. A numerical study shows that the
addition of extra relativistic degrees of freedom does not spoil the
resolution of the $H_{0}$ and $S_{8}$ tensions [189].

We end with an observation: the argument to understand the transition is
essentially the same than the one in finite temperature models, because the
number of light degrees of freedom changes due to a different transition of
the 5D scalar field. In plain English, the model avoids finite temperature
requirements and relies on an ordinary vacuum decay in five dimensions. This
obviously implies that the AdS vacuum is not a true vacuum. The vacuum in the
radius modulus is determined by the contribution to the Casimir potential of
the number of light degrees of freedom. This number changes discontinuously
due to an ordinary vacuum decay of a 5D scalar field which satisfies the AdS-
DC conjecture. This change drives the AdS to dS transition in the radius
modulus, which is therefore discontinuous as in first order transitions.

##  8 The Black Hole Transition Conjecture

In Sec. 4.1, we have argued that if $r_{s}<R_{\perp}$, then black holes are
5D, where $r_{s}$ is the Schwarzschild radius and $R_{\perp}$ the radius of
the dark dimension. This implies that if the horizon size of a 4D black hole
that is evaporating shrinks below the micron scale, then the black hole must
undergo a 4D $\to$ 5D transition. As noted in [117], the black hole transition
is instantly visible by analyzing the black hole entropy of a $d$-dimensional
black hole,

$S_{\rm BH}=\frac{4\pi\ M_{\rm BH}\ r_{s}}{d-2}\sim\left(\frac{M_{\rm
BH}}{M_{d}}\right)^{(d-2)/(d-3)}\,,$ (48)

where $M_{\rm BH}$ is the black hole mass and $M_{d}$ is the $d$-dimensional
Planck scale; note that for $d=4$ we have $M_{4}={M}_{p}$. In Fig. 3 we show a
comparison of the 4D and 5D scaling behavior of the black hole entropy as
given by (48). By adding species in a higher dimensional theory, it follows
from (48) that the scaling behavior of the entropy changes, and for the black
hole it is more convenient to be in the 5D configuration because for given
black hole mass its entropy is larger than the one in the 4D configuration. As
can be seen in Fig. 3, for $R_{\perp}\sim 1~{}\mu{\rm m}$, the transition
takes place at $M_{\rm BH}\sim 10^{21}~{}{\rm g}$. Note that the 4D and 5D
entropies as given by (48) are equal at the 5D-4D transition point where
$M_{\rm BH}\sim M_{p}^{2}R_{\perp}$. Moreover, the entropy crosses the
horizontal axis where the black hole masses are the same as the 4D or 5D
Planck masses. Then, the associated lengths are the 4D or 5D Planck lengths,
where the two entropies are equal to one.

Figure 3: Scaling of the Schwarzschild radius (left) and black hole entropy
(right) in $d=4$ and $d=5$ dimensions. From Ref. [117].

The transition between 4D and 5D black holes corresponds to Gregory-Laflamme
phase transition [190] and is also visible using the free energy in terms of
the temperature $T_{H}$ [191]. This can be seen in the left plot in Fig. 3,
where the black hole mass is approximately the same as the free energy, and
the black hole radius corresponds to the inverse temperature. So the 5D
configuration for given black hole temperature has smaller free energy than
the 4D configuration.

More generally, this has lead in [191] to the black hole transition conjecture
which states that any consistent EFT description of $d$-dimensional QG must
exhibit three scales across its moduli space $\mathscr{M}_{\rm QG}$: (i) the
$d$-dimensional Planck scale $M_{d}$, which controls the strength of the
Einstein term; (ii) the species scale $\Lambda_{\rm sp}$, where the higher
order gravitational corrections become important; and (iii) the black hole
scale $\Lambda_{\rm BH}$, where at this temperature, the black hole predicted
by EFT undergoes a phase transition to a more stable solution. Furthermore,
$\Lambda_{\rm BH}\lesssim\Lambda_{\rm sp}\lesssim M_{d}$ everywhere in
$\mathscr{M}_{\rm QG}$ and $\Lambda_{\rm BH}$ approaches the mass scale of the
lightest tower at large distances in field space. So for a KK tower one
asymptotically has that $\Lambda_{\rm BH}=m_{\rm KK}$. For the line interval
with EW9-branes attached at each end [83], the three scales are $m_{\rm KK}\ll
M_{\rm KK}\lesssim M_{11}$ [87].101010Using universality of black hole
thermodynamics and properties of gravitational scattering amplitudes it has
been shown in [18] that some intrinsic features of the density of one-particle
states above the minimum black hole mass are an indicator for the existence of
large extra dimensions, and cannot be reproduced by any lower-dimensional
field theory with finitely many fields satisfying the WEC.

Let us also add a remark about the definition of the species scale by the
minimal black hole in view of the phase transition from the $d$-dimensional,
e.g. 4-dimensional, black hole to the higher-dimensional, e.g. 5-dimensional,
black hole solution. Recall that the species length $l_{\rm sp}=1/\Lambda_{\rm
sp}$ corresponds to the minimal possible Schwarzschild radius of the BH
solution in the EFT. First, from the higher dimensional perspective, the
species length corresponds to the higher-dimensional Planck length, and this
is the scale, where the entropy of the higher dimensional black hole becomes
one. However, also the lower dimensional perspective makes sense for the
definition of the species length, i.e. the species scale formula can be seen
entirely from a $d$-dimensional perspective. Namely $l_{\rm sp}$ corresponds
to the Schwarzschild radius, where the entropy of the corresponding minimal
$d$-dimensional black hole is equal to $N$, the number of species. Using (10)
one then obtains $l_{\rm sp}={N^{1\over d-2}M_{d}^{-1}}$. For the species
being the KK modes of an $n$-dimensional compact space one then obtains

$l_{\rm sp}=M_{d}^{{2-d\over d+n-2}}m_{\rm KK}^{-{n\over d+n-2}}\,.$ (49)

Replacing $m_{\rm KK}$ by $\Lambda_{\rm BH}$, one obtains a definition of the
species length entirely in terms of $n$ and $\Lambda_{\rm BH}$:

$l_{\rm sp}=M_{d}^{{2-d\over d+n-2}}\Lambda_{\rm BH}^{-{n\over d+n-2}}\,.$
(50)

It is not difficult to show that this $d$-dimensional definition of the
species scale agrees with the higher dimensional one, using the specific form
of the higher dimensional entropy formula and requiring the entropy in higher
dimensions to be one.

Finally, we discuss the correspondence (conjectured in [144]) between the
graviton-boundstate interpretation of 5D black holes and massive KK gravitons
as dark matter candidates. We first consider a 4D black hole with entropy
$S_{\rm BH,4d}=N_{\rm tot}$, with $N_{\rm tot}=N_{g,4}N_{\rm sp}$, where
$N_{g,4}$ is the number of 4D gravitons in the black hole boundstate and
$N_{\rm sp}$ is the number of additional species in the black hole boundstate
[51]. These can be the KK gravitons, i.e. $N_{\rm sp}=N_{\rm KK}$. The radius
of the 4D black hole is $r_{s,4}=N_{\rm tot}^{1/2}l_{p}$, where
$l_{p}=1/M_{p}$. For the minimal black hole, whose radius sets the species
scale, $N_{g,4}=1$, leading to $S_{{\rm BH},4d,{\rm min}}=N_{\rm KK}$. The
corresponding minimal radius is $r_{s,4,{\rm min}}=N_{\rm KK}^{1/2}l_{p}$.
This is indeed the species length, being just the 5D Planck length.

Now, we consider a 5D black hole after the transition. There are no KK
gravitons anymore, since they are part of normal 5D gravitons. The black hole
is now made entirely of $N_{g,5}$ 5D gravitons and its entropy becomes
$S_{{\rm BH},5d}=N_{g,5}$. The radius of the 5D black hole is
$r_{s,5}=N_{g,5}^{1/3}l_{5}$, where $l_{5}=1/M_{5}$ is the 5D Planck length.
For the minimal black hole, whose radius again sets the species scale,
$N_{g,5}=1$, leading to $S_{{\rm BH},5d,{\rm min}}=1$. The corresponding
minimal radius is $r_{s,5,{\rm min}}=l_{5}$. This is again the species length
$l_{\rm sp}$, so we get in a consistent way that $r_{s,5,{\rm
min}}=r_{s,4,{\rm min}}$.

In summary, the 4D/5D phase transition for the KK gravitons can be understood
assuming that above the KK scale these states are part of the 5D graviton and
therefore the effective description of space time becomes 5D. So in this case
we indeed would have a 5D (or $d$-dimensional) black hole graviton boundstate
description, just like in the original black hole $N$-portrait picture of
[143]. This implies that before the phase transition, the correspondence is
that the 4D black hole is a bound state of the $N_{\rm KK}$ particles, whereas
after the phase transition, there is the correspondence between the 5D black
hole and the bound state of the 5D gravitons.

##  9 Concluding Thoughts

We have seen that the dark dimension scenario provides one possible
explanation of the cosmological hierarchy problem and carries with it a rich
phenomenology:

  * •

It provides a profitable arena to accommodate a very light gravitino.

  * •

It encompasses a framework for primordial black holes, KK gravitons, and a
fuzzy radion to emerge as viable candidates to comprise some or all of the
dark matter.

  * •

It also encompasses an interesting framework for studying cosmology and
astroparticle physics.

  * •

It provides a natural set up for $R$-neutrinos propagating in the bulk to
accommodate neutrino masses in the range $10^{-4}<m_{\nu}/{\rm eV}<10^{-1}$,
despite the lack of any fundamental scale higher than $M_{*}$. The suppressed
neutrino masses are not the result of a see-saw mechanism, but rather because
the bulk modes have couplings suppressed by the volume of the dark dimension
(akin of the weakness of gravity at long distances).

We have also seen that uniform 5D inflation can relate the causal size of the
observable universe to the present weakness of gravitational interactions by
blowing up an extra compact dimension from the microscopic fundamental length
of gravity to a large size in the micron range, as required by the dark
dimension scenario. Moreover, uniform 5D inflation can lead to an approximate
scale invariant power spectrum of primordial density perturbations. The
predicted small-angle ($<10^{\circ}$) CMB power spectrum is compatible with
observations. Such an angle corresponds to a distance $\sim 2.3~{}{\rm Mpc}$
and multipole moment $\ell\simeq 30$. For smaller $\ell$ multipoles (larger
angles), one obtains more power spectrum than standard 4D inflation,
corresponding to a nearly vanishing spectral index, that the present data
cannot distinguish due to large errors. One caveat here is that the power
spectrum is cosmic variance limited [192]. However, even though cosmic
variance prevents identification at low $\ell$, the transition region could
provide a signal for experiments in the near future. To determine $\ell$
multipoles in the transition region, an exhaustive transfer-function analysis
(numerically solving the linearized Einstein-Boltzmann equations) would be
required. The tensor-to-scalar ratio is also consistent with observations. An
estimate of the magnitude of isocurvature perturbations based on entropy
perturbations indicates that they are suppressed [165]. A dedicated
investigation along these lines is obviously important to be done.

On a separate track, it was noted in [193] that the cosmic scale factor $a$
describing the evolution of $\Lambda_{s}$CDM is continuous and non-zero at
$t=t_{c}$, but its first derivative $\dot{a}$ is discontinuous, and its second
derivative $\ddot{a}$ diverges. In the spirit of [194], it would be
interesting to investigate the evolution of (43) during the phase transition
induced by the Casimir forces. This would allow a complete description of the
background and perturbation evolution at all redshifts.

## Acknowledgements

We thank Eleonora Di Valentino and Andriana Makridou for permission to
reproduce Figs. 1 and 4. The work of L.A.A. is supported by the U.S. National
Science Foundation (NSF Grant PHY-2112527). I.A. is supported by the Second
Century Fund (C2F), Chulalongkorn University. The work of D.L. is supported by
the Origins Excellence Cluster and by the German-Israel-Project (DIP) on
Holography and the Swampland.

## Appendix

Figure 4: 2D contours at 68% and 95% CL in the $H_{0}$-$\Omega_{m}$ plane
(left) for the $\Lambda$CDM and $\Lambda_{s}$CDM models, and in the
$\Omega_{m}$-$S_{8}$ plane for the $\Lambda$CDM (middle) and $\Lambda_{s}$CDM
(right) models. In the left pannel the model predictions have been extracted
fitting data from the Planck satellite and transversal baryon acoustic
oscillations (BAOtr). In the middle and right panels the model predictions
have its origins in Planck and KiDS-1000 data. Note that the Planck and BAOtr
contours intersect exactly at the vertical band of the SH0ES measurement. From
Ref. [179].

In this Appendix we briefly discuss the $\Lambda_{s}$CDM solution of the
cosmic discrepancies, paying special attention to the role played by data from
baryon acoustic oscillations (BAO). The most recent analysis presented in
[179] is based on: the Planck CMB data [195], the Pantheon+ supernovae type Ia
sample [196], the data release of KiDS-1000 [197], and the (angular)
transversal 2D BAO data on the shell [198], which are less model dependent
than the 3D BAO data used in previous studies of $\Lambda_{s}$CDM. The
outcome, which is displayed in Fig. 4, shows that the $\Lambda_{s}$CDM model
can simultaneously resolve both the $H_{0}$ and $S_{8}$ tensions. It is
important to stress that the BAO 3D data sample assumes $\Lambda$CDM to
determine the distance to the spherical shell, and hence could potentially
introduce a bias when analyzing beyond $\Lambda$CDM models [199].

As shown in [200], 3D BAO data leave no room for low-$z$ solutions to the
$H_{0}$ tension if the absolute magnitude of supernova $M$ is constant. This
is generally phrased as a no-go theorem which states that if $M$ is constant,
to address the $H_{0}$ tension one needs to consider some sort of new physics
at $z>1000$. If one uses 2D BAO (instead of 3D BAO) data, though, it is
possible to solve the Hubble tension without requiring new physics before
recombination. However, in this case the effective dark energy density needs
to be negative at $z\gtrsim 2$ in order to produce the correct angular
diameter distance to the last scattering surface.

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