More on Black Holes Perceiving the Dark Dimension

In the last two years the dark dimension scenario has emerged as focal point of many research interests. In particular, it functions as a stepping stone to address the cosmological hierarchy problem and provides a colosseum for dark matter contenders. We reexamine the possibility that primordial black holes (PBHs) perceiving the dark dimension could constitute all of the dark matter in the universe. We re-assess limits on the abundance of PBHs as dark matter candidates from γ -ray emission resulting from Hawking evaporation. We re-evaluate constraints from the diffuse γ -ray emission in the direction of the Galactic center which offer the best and most solid upper limits on the dark matter fraction composed of PBHs. The revised mass range which allows PBHs to assemble all cosmological dark matter is estimated to be 10 15 ≲ M BH / g ≲ 10 21 . We demonstrate that due to the constraints from γ -ray emission, quantum corrections due to the speculative memory burden effect do not modify this mass range. We also investigate the main characteristics of PBHs which are localized in the bulk. We show that PBHs localized in the bulk


I. INTRODUCTION
The Swampland program seeks to distinguish effective theories which can be completed into quantum gravity in the ultraviolet from those which cannot [1].In theory space, the swampland frontier is drawn by a family of conjectures classifying the properties that an effective field theory should have/avoid to enable a consistent completion into quantum gravity.These conjectures deliver a bridge from quantum gravity to astrophysics, cosmology, and particle physics [2][3][4].
For example, the distance conjecture (DC) asserts that along infinite distance geodesics there is an infinite tower of states which become exponentially light asymptotically [5].
Connected to the DC is the anti-de Sitter (AdS) distance conjecture [6], which ties in the dark energy density to the mass scale m characterizing the infinite tower of states, m ∼ |Λ| α , in the limit of a small negative AdS vacuum energy with α a positive constant of O(1).In addition, under the hypothesis that this scaling behavior remains valid in dS (or quasi dS) space, a large number of light modes also would pop up in the limit Λ → 0, with Λ being positive.
The AdS-DC in de Sitter space provides a pathway, called the Dark Dimension scenario [7], to elucidate the origin of the cosmological hierarchy Λ/M 4 p ∼ 10 −122 , because it connects the size of the compact space R ⊥ to the dark energy scale Λ −1/4 via where the proportionality factor is estimated to be within the range 10 −1 < λ < 10 −4 .
Actually, (1) derives from constraints by theory and experiment.On the one hand, since the associated Kaluza-Klein (KK) tower contains massive spin-2 bosons, the Higuchi bound [8] sets an absolute upper limit to the exponent of Λ α , whereas explicit string calculations of the vacuum energy (see e.g.[9][10][11][12]) yield a lower bound on α.All in all, the theoretical constraints lead to 1/4 ≤ α ≤ 1/2.On the other hand, experimental arguments (e.g.constraints on deviations from Newton's gravitational inverse-square law [13] and neutron star heating [14]) lead to the conclusion encapsulated in (1): The cosmological hierarchy problem can be addressed if there is one extra dimension of radius R ⊥ in the micron range, and the lower bound for α = 1/4 is basically saturated [7].The Standard Model (SM) should then be localized on a D-brane transverse to the dark dimension [15].A theoretical amendment on the connection between the cosmological and KK mass scales confirms α = 1/4 [16].Assembling all this together, we can further conclude that the KK tower of the new (dark) dimension opens up at the mass scale m KK ∼ 1/R ⊥ at the eV range.Within this setup, the 5-dimensional Planck scale (or species scale where gravity becomes strong [17][18][19][20]) is given by where M p = 1/ √ G N is the Planck mass, leading to 10 9 ≲ M * /GeV ≲ 10 10 .
The dark dimension accumulates interesting phenomenology [21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39].For example, it was noted by us [22] that primordial black holes (PBHs) with Schwarzschild radius smaller than a micron could be good dark matter candidates.A generalization to near-extremal black holes has been carried out in [38].Complementary to the PBHs, it was observed in [25] 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 contender.The cosmic evolution of the dark graviton gas is primarily dominated by "dark-to-dark" decays, yielding a specific realization of the dynamical dark matter framework [40].An interesting close relation between PBHs and the dark gravitons has been pointed out in [24].
In this paper we provide new insights on PBHs as dark matter candidates.The layout is as follows.In Sec.II we provide a concise overview of black hole evaporation within Hawking's semiclassical approximation.In Sec.III we explore the impact of quantum effects, possibly associated to the memory burden, which could take the evaporation process out of the semiclassical regime by half-decay time [41,42,44].In Sec.IV we re-assess limits on the abundance of PBHs as dark matter candidates, focusing on holes localized on the brane during the evaporation process.In Sec.V we examine the possibility that bulk PBHs make the dark matter in the universe.In Sec.VI we consider a space with two boundaries for the dark dimension [45].We study general phenomenological aspects of this construct, and in particular, we examine the possibility that the dark matter in the universe is made of tubular-pancake shape black holes localized on a brane, which is place at the space boundary.
The paper wraps up in Sec.VII with some conclusions.

II. SEMICLASSICAL BLACK HOLE EVAPORATION
In the mid-70's Hawking pointed out that a black hole emits thermal radiation as if it were a black body, with a temperature inversely proportional to its mass [46,47].In this section we first review the main properties of Hawking evaporation of Schwarzschild black holes and of its generalization to dimension d.After that we review the evaporation of near extremal black holes within the semiclassical approximation.

A. Schwarzschild Black Holes
It is well known that inertial observers in Minkowski space perceive the vacuum (the absence of particles), whereas observers moving with uniform proper acceleration a measure a thermal bath (thermal distribution of particles) of temperature T = a/(2π) [48][49][50].In special relativity, an observer moving with uniform proper acceleration through Minkowski spacetime is conveniently described with Rindler coordinates [51], which are related to the standard (Cartesian) Minkowski coordinates by x = ρ cosh σ and t = ρ sinh σ.The line element in Rindler space is found to be where ρ = 1/a and where σ is related to the observer's proper time τ by σ = aτ .Note that the local acceleration diverges as ρ → 0.
By the same token, what inertial Schwarzschild observers measure as vacuum, the uniformly accelerated ones identify as thermal bath.The group of inertial Schwarzschild observers are the ones falling towards the black hole, while the uniformly accelerated observers are the ones who manage to keep constant distance from the event horizon, the acceleration being there to prevent the gravitational pull.This implies that black holes should have an approximate Rindler region near the horizon.More indicatively, consider the Schwarzschild metric where t is the universal time coordinate, r is the circumferential radius (defined so that the circumference of a sphere of radius r is 2πr), dΩ 2 is an interval of spherical solid angle, and r s = 2M BH is the horizon radius (in Planck units), with M BH the black hole mass.The proper distance from the horizon is given by Substituting ( 5) into (4) the Schwarzschild metric can be recast as For δ/r s ≪ 1, in the near horizon limit, r = r s + δ, it follows that ρ = 2 √ r s δ.Bearing this in mind, the local metric to lowest order in δ is found to be The (t, ρ) piece of this metric is Rindler space; we can rescale t = 2r s σ to make it look exactly like (3).
In analogy with Rindler space, the near-horizon observer must see the field excited at a local temperature The gravitational redshift is given by the square root of the time component of the metric.
So for the field theory state to consistently extend, there must be a thermal background everywhere with the local temperature redshift-matched to the near horizon temperature The inverse temperature redshifted to r ′ at infinity is found to be lim For small δ, in the near-horizon limit lim All in all, a field theory defined on Schwarzschild background is in a thermal state whose Hawking temperature at infinity is given by Armed with the Hawking temperature, we can now calculate the entropy of the black hole [52].By adding a quantity of heat dQ the change in the black hole entropy is given by where we assumed that the heat energy that enters serves to increase the total mass.The black hole entropy is then found to be Next, in line with our stated plan, we generalize the previous discussion to dimension d.Following [53], we conveniently work with the mass scale where [15].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.
This is an adequate approximation provided M BH is well above the brane tension, which is presumably of the order of but smaller than M d .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 d-dimensional Schwarzschild radius where Γ(x) is the Gamma function [54][55][56].Using (15) it is easily seen that for dimension d, (12) and ( 14) generalize to and respectively [57].
In the rest frame of the Schwarzschild black hole, both the average number [46,47] and the probability distribution of the number [58][59][60] 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" σ s , which controls the black hole absorption cross section, depends upon the spin of the emitted particles s, their energy Q, and M BH [61][62][63][64][65].The prevailing energies of the emitted particles are ∼ T H ∼ 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 [66,67].Thereby, the process of evaporation is driven by the large number of SM brane modes.
The emission rate per degree of particle freedom i of particles of spin s with initial total energy between (Q, Q + dQ) is found to be The average total emission rate for particle species i is then where ζ(x) is the Riemann zeta function and f = 1 (f = 3/4) for bosons (fermions) [69].
For the different spins, the emission rate can be parametrized by and The rate of change of the black hole mass in the evaporation process is estimated to be: and (ii) where c i (T H ) counts the number of internal degrees of freedom of particle species i of mass ) for bosons (fermions), and where for d = 5, we have taken M * ∼ 10 10 GeV as expected in the dark dimension scenario [7].A direct comparison of ( 24) and (25) shows that for equal masses, five-dimensional (5D) Schwarzschild black holes evaporate slower than their 4D cousins.This is because 5D black holes are bigger and colder.Indeed, using (24) a straightforward calculation shows that 4D black holes with M BH ≳ 5 × 10 14 g can survive the age of the universe, whereas for the dark dimension scenario we obtain A discussion of a 4D black hole evaporation entering the 5D regime together with details of the 4D to 5D transition is provided in the Appendix.

B. Near-Extremal Black Holes
Next, we consider the Reissner-Nordström black hole in Einstein-Maxwell gravity, which is specified by its ADM mass M BH and electric charge Q BH .The Einstein-Maxwell action in dimension d is given by, where F = dA and The d-dimensional Reissner-Nordström metric can be written as where is the blackening function and is the metric of a (d − 2)-dimensional unit sphere, and where m is related to the ADM mass M BH and q to the charge Q BH of the black hole by and with the volume of the unit (d − 2)-sphere [70].The black hole has two horizons located at the zeroes of the blackening function, u(r ± ) = 0, yielding where following [71] we conveniently introduce the extremality parameter .To avoid naked singularities, we must require . Now, we define the temperature, where β is a factor of order-one that controls the differences between M BH and Q BH [72].
By inspection of ( 36) and (37) we can see that near-extremal black holes are extremely cold due to the fact that c/M BH , which quantifies the near-extremality, is extremely small because of the large entropy [38].

III. EFFECTS OF MEMORY BURDEN ON BLACK HOLE EVAPORATION
It was recently suggested [41,42,44] that quantum effects (e.g.memory burden) would take the evaporation process out of the semiclassical regime after about half-decay time has elapsed.It is this that we now turn to study.Actually, the black hole evaporation estimates presented in the previous section, based on the semiclassical approximation, rely on the assumption of self-similarity.Namely, we have assumed that in the course of evaporation, a black hole gradually shrinks in size while maintaining the standard semi-classical relations between its parameters, such as its mass, the Schwarzschild radius, and the Hawking temperature.However, there have been some objections to this viewpoint, and actually arguments have been put forward suggesting the self-similarity assumption is inconsistent over the long time-scales of the evolution.
The breakdown of self-similarity is deep-rooted on the nature of entanglement.For a cavity emitting black body radiation, early on photon emission is entangled with atomic states in the cavity.However, once half the energy in the cavity is emitted subsequent radiation emitted is entangled with radiation emitted earlier.As a result the entanglement entropy increases to some maximum, at about half the energy emitted, and then declines.
A black hole may experience a similar behavior, because Hawking radiation is emitted from an entangled pair of photons or electron positron pairs.The Page time is defined by the condition that the mass of a black hole has decreased to half its original value via Hawking radiation, τ half ∼ M BH /2 [73,74].Within this time r s → r s /2 and S BH → S BH /4.It is at time ∼ τ half when an observer that has configured a black hole with a set of known states on the horizon might find that they have been randomized beyond what can be recovered.
In other words, information may remain encoded inside the black hole because the emitted radiation is thermal in character.However, after τ half , the remaining black hole has only 1/4 of its initial entropy and so much less information storage capacity.A far reaching proposal that can accommodate these ideas and describe the black hole evaporation for times > τ half is the so called memory burden effect, which insinuates that Hawking evaporation is slowed down by further n powers of the entropy S BH , after the Page time has elapsed [41][42][43][44].
It is not clear whether the memory burden effect still needs further justification, and whether the black hole lifetime already gets increased after Page time, but herein we will explore its spectacular phenomenological consequences.Before proceeding, we pause to note that memory burden holds in any dimension and herein we assume that the behavior of t half is roughly universal for any dimension.
Bearing this in mind, a comparison of the scaling behavior of the different rates of particle emission is of interest.The scaling behavior of the semiclassical decay rate ( Ṅi ≡ Γ {0} H , ∀i) of a Schwarzschild black hole in dimension d follows from (20).Neglecting the effect of grebody factors and mass thresholds we have The quantum decay rate has an additional suppression (compared to the Hawking decay rate), and the suppression factor, 3) , is dimension-dependent.Throughout, n is a non-negative integer parametrizing the quantum suppression when the black hole enters the memory burden phase.
For near-extremal black holes, the semiclassical decay rate in dimension d is suppressed compared to the Schwarzschild decay rate by a factor 1/S The quantum decay rate of a near-extremal black hole in dimension d gets again an additional suppression compared to the near-extremal Hawking decay rate by a factor 1/S n BH , In Fig. 1 we show a comparison of the various decay rates.For large masses (or equivalently large entropies), the quantum decay rates are suppressed by several orders of magnitude.
The difference, of course, is reduced with decreasing entropy, and all the decay rates become equal at the minimum black hole mass M BH,min ∼ M d .
In summary, Hawking's semiclassical approximation applied to 5D black holes perceiving the dark dimension scenario gives very robust predictions and lead to reliable conclusions.
More spectacular results can be obtained adopting the black hole portrait endowed with the memory burden.However, the theoretical underpinning and the precise implementation of the memory burden is rather speculative.In particular via memory burden the black hole decay rate gets dramatically changed already after Page time, whereas it also could be conceivable that quantum effects affect the black hole decay rate at much later time scales.

IV. PRIMORDIAL BLACK HOLES AS DARK MATTER
Since the late 60's there have been speculation suggesting that black holes could be formed from the collapse of large amplitude fluctuations in the early universe [86][87][88][89].An order of magnitude estimate of M BH can be obtained by equating the scaling of the cosmological energy density with time t in the radiation dominated epoch, to the required density in a region of mass M BH which is able to collapse within its Schwarzschild radius Linkening ( 42) and (43) gives the idea that at production PBHs would have roughly the cosmological horizon mass [90] M BH ∼ tM 2 p ∼ 10 15 t 10 −23 s g .
Now, it is straightforward to see that a black hole would have the Planck mass (M p ∼ 10 −5 g) if it formed at the Planck time (10 −43 s), 1 M ⊙ if it formed at the QCD epoch (10 −5 s), and 10 5 M ⊙ if it formed at t ∼ 1 s, comparable to the mass of the holes thought to reside in galactic nuclei.The back-of-the-envelope calculation yielding (44) hints that PBHs could span an enormous mass range.Even though the spectrum of masses of these PBHs is yet to see the light of day, on cosmological scales they would behave like a typical cold dark matter particle.
Before proceeding, we pause and call attention to a captivating coincidence: size of the dark dimension ∼ wavelength of visible light , 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 [109].So 5D PBHs escape these microlensing constraints; at the same time, as pointed out in [22], they are longer lived than their 4D counter parts, and this makes them more promising all-dark-matter candidates.
In light of the stunning coincidence (45), hereafter we will now focus on the bounds imposed by the extragalactic γ-ray background [94], the CMB spectrum [95], the 511 keV γ-ray line [96,97], and the MeV Galactic diffuse emission [101][102][103] which directly constrain the black hole decay rate.As previously noted, these constraints place an upper limit on the dark matter fraction f PBH composed of PBHs.
Before proceeding, we pause to note that femtolensing of cosmological gamma-ray bursts (GRBs) could be used to search for PBHs in the mass range 10 17 ≲ M BH /g ≲ 10 20 [110].
The lack of femtolensing detection by the Fermi Gamma-ray Burst Monitor was used to constrain PBHs with mass in the range 10 17.7 < M BH /g ≲ 10 20 to contribute no more than 10% to the total dark matter abundance [111].However, the validity of this GRB constraint has been called into question, because it is based on the assumption that the gamma-ray source is point-like in the lens plane [112].The non-pointlike nature of GRBs imply that most of them are too big when projected onto the lens plane to yield meaningful femtolensing limits.Indeed, only a small GRB population with very fast variability might be suitable for PBH searches.When this systemtic effect is taken into consideration to decontaminate the Fermi sample, the GRB constraint on PBHs becomes obsolete.A sample of 100 GRBs with transverse size 10 9 cm would be needed to probe the 5D PBHs [112].
Black hole evaporation has two main channels contributing to the (diffuse) isotropic γ-ray flux: (i) direct photon emission and (ii) positron emission (these positrons annihilate with thermal electrons producing an X-ray background at energies around and below 511 keV).
The diffuse photon emission consists of a contribution from PBH from extragalactic structures at all redshifts.Limits on f PBH are obtained by demanding that the emission from PBH evaporation does not exceed the X-ray and soft γ-ray isotropic fluxes, neither the Galactic intensity associated with the line-of-sight integrated Navarro-Frenk-White dark matter density profile [113].Observations of the diffuse γ-ray emission in the direction of the Galactic center offer the best and most solid constraints of f PBH [103].
The upper panel of Fig. 2 shows the constraints on the PBH fraction in the mass range of interest for d = 4.The concomitant constraints for d = 5, can be estimated by a simple rescaling procedure.The central idea to decipher the change of scale is that for a given photon energy, or equivalently a given Hawking temperature, it is reasonable to expect a comparable limit on f PBH for both d = 4 and d = 5, i.e., Note that at given T H , the required PBH number density in 5D is larger than the one in 4D but this is automatically compensated by the fact that at fixed Hawking temperature the 5D black holes have smaller M BH than those in 4D, i.e., yielding equal density.
For example, in d = 4, the Hawking temperature is related to the mass of the black hole by [99] T whereas in d = 5 the Hawking temperature mass relation is found to be where we have taken M d ∼ 10 10 GeV. 1 This means that since f PBH ≲ 5 × 10 −5 in d = 4 for M BH ∼ 10 16 g, our reasoning indicates that f PBH ≲ 5 × 10 −5 for M BH ∼ 10 12 g in d = 5.
The complete rescaling of f PBH for the M BH range of interest is displayed in the lower panel of Fig. 2.
In Sec.III we have seen that near-extremal black holes are extremely cold.For example, a black hole of M ∼ 10 5 g has a temperature T H ∼ 4 GeV.Substituting these figures into (24) we find that the Hawking lifetime of a 10 5 g Schwarzschild black hole is τ H ∼ 4 × 10 −5 yr.
For a near extremal black hole of the same mass, using (37) we find that its temperature would be T ne ∼ 10 −5 √ β eV, and so using (26) its lifetime τ ne ∼ 15/ √ β Gyr.Note that the energy of the emitted particles by the near-extremal black hole is well below the peak of the CMB photons.We can conclude that if there were 5D primordial near-extremal black holes in nature, then a PBH all-dark-matter interpretation would be possible in the mass range Of course, by tuning the β parameter we can have a PBH all-dark-matter interpretation with very light 5D black holes. 2ext, we explore how the memory burden effect impacts the mass range of a PBH alldark-matter interpretation.Memory burden dramatically enhances the black hole life time and implies that a-priori the bounds from the age of the universe are much milder than compared to those from the semiclassical black hole decay.However, photon emission will put additional constraints, which we will discuss in the following.To be specific, we assume only two phases in the evaporation process: semiclassical evaporation up to τ half followed by a quantum regime characterized by n = 1.We can duplicate the procedure adopted above to derive the allowed mass range for a PBH all-dark-matter interpretation.For d = 4, the memory burden effect opens a new window in the mass range 10 mass range for the same temperature interval is 10 Imposing the S BH enhancement on (26), we find that i.e., black holes could survive the age of the universe if M BH ≳ 5 g.Bounds on the black hole abundance as a function of the initial black hole mass are encapsulated in Fig. 3.We conclude that within the dark dimension scenario the memory burden effect with n = 1 does not open any new window that could allow an all-dark-matter interpretation composed of primordial black holes.

V. BLACK HOLES LOCALIZED IN THE BULK
Hitherto, we have assumed that 5D black holes stay attached to the brane during the evaporation process.In this section 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 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 proposal introduced elsewhere [26,35], in which we postulated 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 this section we then assume that PBHs are localized or propagating 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 ⊥ ) d−4 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 [66].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 where we have remained conservative, and following [68] 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.
We note that within the black hole memory burden an all dark-matter interpretation is almost unconstrained, and so the allowed mass range estimate given in ( 53) can be extended down to 1 ≲ M BH /g ≲ 10 21 .Actually, the minimum black hole mass can be further reduced by advocating a multiple-phase black hole evolution, such that in each phase there is a 1/S BH increase in the memory burden suppression factor of the decay rate.

VI. THE DARK DIMENSION AS A SPACE WITH TWO BOUNDARIES
Very recently, it was conjectured that the dark dimension can also be viewed as a line interval with end-of-the-world 9-branes attached at each end [45].In this section, we briefly comment on the impact that this thought-provoking viewpoint could have on phenomenological aspects of the dark dimension scenario.
Actually, the line interval along the dark dimension y can also be understood as a semicircular dimension endowed with S 1 /Z 2 symmetry.This symmetry has radical consequences for models in which neutrino masses originate in three 5D Dirac fermions Ψ α , which are singlets under the SM gauge symmetries and interact in our brane with the three active lefthanded neutrinos ν αL in a way that conserves lepton number, where the indices α = e, µ, τ denote the generation [116][117][118].In the Weyl basis each Dirac field can be decomposed into two two-component spinors Ψ α ≡ (ψ αL , ψ αR ) T .Now, the Z 2 symmetry of S 1 /Z 2 contains y to −y which acts as chirality (γ 5 ) on spinors.This implies that one of the two-component Weyl spinors, say ψ αR , would be even under Z 2 , while the other spinor ψ αL would be odd.
If ν αL are restricted to the brane located at the fixed point y = 0, then ψ αL vanishes at this point and so the coupling is between ν αL and ψ αR .As we have shown elsewhere [37], the model remains consistent with experiment because only the ψ αR degrees of freedom contribute to the effective number of equivalent neutrino species N eff .
Two different types of PBHs can be produced if the dark dimension is seen as space with two boundaries: (i) 5D Schwarzschild black holes that propagate freely (or else are localized) in the bulk and (ii) tubular-pancake shape black holes, which are bound to a brane localized at the boundary of space.The second type of black holes would resemble the black cigars described in [119], and therefore far away from the space boundary the black hole metric on the end-of-the-world brane would be approximately Schwarzschild.In the spirit of [120], we argue that the phenomenology of these black cigars is similar to that of the 5D Schwarzschild black holes discussed throughout this paper.This implies that primordial black cigars could provide an all-dark-matter interpretation for masses in the range 10 15 < M BH /g < 10 21 ; see Fig. 2 for details.We have estimated in the previous section that to accommodate the observed dark matter density Schwarzschild black holes living in the bulk should have masses in the range 10 11 ≲ M BH /g < 10 21 .
We have stressed in the Introduction that the dark graviton gas proposal requires KK excitations which are unstable [25].Actually, there are strong bounds on the changing dark matter density which imply that the decays to lighter gravitons of the KK tower must not lose much mass to kinetic energy [36].A typical violation of KK quantum number δ n cannot be too large and requires δ n ∼ O(1).The proposal entertained in [45] provides a profitable arena to accommodate the dark graviton gas cascade, because S 1 /Z 2 automatically breaks the U (1) isometry associated to the KK momentum conservation.Indeed, for S 1 /Z 2 there are two level sources of KK momentum violation: • KK momenta are conserved modulo 2; • KK states decay to brane states with a coupling that falls off exponentially at large KK number n > M/m KK with M the brane width.This of course feeds back to a momentum violation among KK modes by a loop of brane states.Now, the KK momentum violation can leave a discrete Z 2 symmetry acting as the KK number parity.This symmetry can be, in principle, implemented to brane states, opening the possibility for alternative dark matter candidates, such as the radion or the lightest odd KK in the presence of KK parity symmetry.For example, in [34] we assume that no such discrete symmetry survives and KK modes decay to the radion.In this model we can avoid the velocity kick constraints on KK momentum violation because the graviton tower decays into the radion before primordial nucleosynthesis.It is of interest to see whether it is possible to translate the KK momentum violation effects into an effective δ n .We plan to elaborate on this idea in future publications.

VII. CONCLUSIONS
We have shown that 5D black holes perceiving the dark dimension are bigger, colder, and longer-lived than usual 4D black holes of the same mass.Adopting the robust Hawking's semiclassical approximation we have demonstrated that a PBH all-dark-matter interpretation would be possible for the following mass ranges: • Schwarzschild black holes localized on the brane ⇒ 10 15 < M BH /g < 10 21 ; • Schwarzschild black holes localized in the bulk ⇒ 10 11 ≲ M BH /g < 10 21 ; • near-extremal black holes localized on the brane ⇒ 10 5 √ β < M BH /g < 10 21 ; where β measures the near-extremality.The more speculative memory burden effect, with decay rate suppress by 1/S BH , could only extend this mass range if black holes are localized on the bulk.Finally, we have argued that the mass ranges given above for a PBH dark matter interpretation would not be modified if one assumes a space with two boundaries for the dark dimension. of a 4D effective theory with N species but without compact dimensions.Setting d = 5 and using M 2 p ∼ M 3 d R ⊥ we recast (55) as In Fig. 4 we show a comparison of the 4D and 5D scaling behavior of the black hole entropy as given by ( 59) and (60).We can conclude that when considering a 4D theory with addition of N species, the black hole formulas do not change in essence, but there are more channels for the evaporation process.For example, we can compare the evaporation of a black hole into SM fields and its minimal extension when the theory is extended by three right-handed Dirac neutrinos.If the black hole can emit right-handed neutrinos it would have a shorter lifetime than if it only emits SM degrees of freedom.The minimum black hole mass available in the SM extension is increased, because the black hole must store extra information, this is all.Now, by adding species in a higher dimensional theory, it follows from ( 59) and ( 60) that the scaling behavior of the entropy changes and for the black hole it is more convenient to be in the 5D configuration because it has more entropy than the 4D configuration.As can be seen in Fig. 4, for R ⊥ ∼ 1 µm, the transition takes place at M BH ∼ 10 21 g.Note that the two entropies in ( 59) and ( 60) are equal at the 5D-4D transition point where M BH = M 2 p R ⊥ .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 and where we have been cavalier on unimportant numerical factors taking m ∼ 2M BH /M d−2 d and q ∼ Q BH /M d−2 d

9 ≲FIG. 2 :
FIG.2: Contraints on f PBH as a function of the PBH mass M BH , assuming a monochromatic mass function.The yellow area indicates the region where black holes do not survive the age of the universe.The region shaded in magenta corresponds to the upper limits from the extragalactic γ-ray background[94], the CMB spectrum[95], the 511 keV γ-ray line[96,97], and the MeV Galactic diffuse emission[101][102][103].The upper panel shows the results for d = 4, and the lower panel for d = 5.

FIG. 3 :
FIG.3: Contraints on f PBH as a function of the PBH mass M BH , assuming a monochromatic mass function.The yellow area indicates the region where black holes do not survive the age of the universe, i.e. τ {1} H < 13.8 Gyr.The region shaded in magenta corresponds to the upper limits from the extragalactic γ-ray background[94], the CMB spectrum[95], the 511 keV γ-ray line[96,97], and the MeV Galactic diffuse emission[101][102][103].The upper panel shows the results for d = 4, and the lower panel for d = 5.