Primordial black holes PBHs hidden in the incredible bulk of the dark dimension could escape constraints from nonobservation of their Hawking radiation. Since these fivedimensional 5D PBHs are bigger colder and longerlived than usual 4D PBHs of the same mass mmlmath altimgsi139.svg displayinline idd1e358mmlmiMmmlmimmlmath they could make all cosmological dark matter if mmlmath altimgsi2.svg displayinline idd1e363mmlmrowmmlmn1mmlmnmmlmsupmmlmrowmmlmn0mmlmnmmlmrowmmlmrowmmlmn11mmlmnmmlmrowmmlmsupmmlmo linebreakgoodbreak linebreakstyleaftermmlmommlmiMmmlmimmlmommlmommlmi mathvariantnormalgmmlmimmlmo linebreakgoodbreak linebreakstyleaftermmlmommlmn1mmlmnmmlmsupmmlmrowmmlmn0mmlmnmmlmrowmmlmrowmmlmn21mmlmnmmlmrowmmlmsupmmlmrowmmlmath i.e. extending the 4D allowed region far down the asteroidmass window. We show that these evasive PBHs could be search for by measuring their mmlmath altimgsi140.svg displayinline idd1e395mmlmiXmmlmimmlmathray microlensing events from faraway pulsars. We also show that future mmlmath altimgsi140.svg displayinline idd1e401mmlmiXmmlmimmlmathray microlensing experiments will be able to probe the interesting range mmlmath altimgsi5.svg displayinline idd1e407mmlmrowmmlmn1mmlmnmmlmsupmmlmrowmmlmn0mmlmnmmlmrowmmlmrowmmlmn16mmlmnmmlmo.mmlmommlmn5mmlmnmmlmrowmmlmsupmmlmo linebreakgoodbreak linebreakstyleaftermmlmommlmiMmmlmimmlmommlmommlmi mathvariantnormalgmmlmimmlmo linebreakgoodbreak linebreakstyleaftermmlmommlmn1mmlmnmmlmsupmmlmrowmmlmn0mmlmnmmlmrowmmlmrowmmlmn17mmlmnmmlmo.mmlmommlmn5mmlmnmmlmrowmmlmsupmmlmspace classnbsp width1emmmlmspacemmlmi mathvariantnormalgmmlmimmlmrowmmlmath where an all dark matter interpretation in terms of 4D Schwarzschild PBHs is excluded by the nonobservation of their Hawking radiation.
['Bulk black hole dark matter', 'Microlensing of X-ray pulsars', 'High Energy Physics - Phenomenology']
Through the looking glass into the dark dimension Searching for bulk black hole dark matter with microlensing of mmlmath altimgsi6.svg displayinline idd1e142mmlmi mathvariantbolditalicXmmlmimmlmathray pulsars
{'Through the Looking Glass into the Dark Dimension: Searching for Bulk Black Hole Dark Matter with Microlensing of X -ray Pulsars': "Luis A. Anchordoqui, 1, 2, 3 Ignatios Antoniadis, 4, 5 Dieter Lust, 6, 7 and Karem Pe˜nal'o Castillo 1  \n1 Department of Physics and Astronomy, Lehman College, City University of New York, NY 10468, USA 2 Department of Physics, Graduate Center, City University of New York, NY 10016, USA 3 Department of Astrophysics, American Museum of Natural History, NY 10024, USA 4 High Energy Physics Research Unit, Faculty of Science, Chulalongkorn University, Bangkok 1030, Thailand 5 Laboratoire de Physique Th'eorique et Hautes ' Energies - LPTHE Sorbonne Universit'e, CNRS, 4 Place Jussieu, 75005 Paris, France 6 Max-Planck-Institut fur Physik, Werner-Heisenberg-Institut, 80805 Munchen, Germany 7 Arnold Sommerfeld Center for Theoretical Physics, Ludwig-Maximilians-Universitat Munchen, 80333 Munchen, Germany  \n(Dated: September 2024)  \nPrimordial black holes (PBHs) hidden in the incredible bulk of the dark dimension could escape constraints from non-observation of their Hawking radiation. Since these five-dimensional (5D) PBHs are bigger, colder, and longer-lived than usual 4D PBHs of the same mass M , they could make all cosmological dark matter if 10 11 ≲ M/ g ≲ 10 21 , i.e., extending the 4D allowed region far down the asteroid-mass window. We show that these evasive PBHs could be search for by measuring their X -ray microlensing events from faraway pulsars. We also show that future X -ray microlensing experiments will be able to probe the interesting range (10 16 . 5 ≲ M/ g ≲ 10 17 . 5 g) where an all dark matter interpretation in terms of 4D Schwarzschild PBHs is excluded by the non-observation of their Hawking radiation.", 'I. INTRODUCTION': 'The dark dimension, an innovative five-dimensional (5D) scenario that has a compact space with characteristic length-scale in the micron range, provides a stepping stone to address the cosmological hierarchy problem [1]. This is because the anti-de Sitter distance conjecture in de Sitter space [2] connects the size of the compact space R ⊥ to the dark energy scale Λ 1 / 4 via R ⊥ ∼ λ Λ 1 / 4 , where Λ ∼ 10 -122 M 4 p is the cosmological constant, M p the Planck mass, and the proportionality factor is estimated to be within the range 10 -1 < λ < 10 -4 [1].  \nPrimordial black holes (PBHs), presumably formed through the collapse of sufficiently sizable overdensities originated by an enhancement in the comoving curvature perturbation power spectrum at small scales during inflation [3-5], may be one of the most interesting candidates of dark matter [6-8]. These intriguing dark matter candidates are particular important in the dark dimension scenario, because 5D PBH are bigger, colder, and longer-lived than usual 4D PBHs of the same mass M [911]. 1 Furthermore, we have recently shown that a back reaction of Hawking evaporation process is to kick 5D Schwarzschild PBHs out of the brane [18]. As a consequence, 5D PBHs could make all cosmological dark  \nmatter if 10 11 ≲ M/ g ≲ 10 21 , i.e., well down into the asteroid-mass window. In this paper we investigate a potential signal that may allow us to unmask these as yet evasive 5D PBHs.  \nAn astonishing coincidence is that R ⊥ corresponds approximately to the wavelength of visible light. This implies that the Schwarzschild radius of black holes perceiving the dark dimension 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 [19-21], obstructing the sensitivity to 5D PBH microlensing signals. Nevertheless, it was pointed out in [22] that X -ray pulsars, with photon energies above 1 keV, are good candidate sources to search for microlensing of PBHs with M ≲ 10 21 g. In this paper we reexamine this idea focussing attention on 5D PBHs perceiving the dark dimension.  \nThe layout is as follows. In Sec. II we review the generalities of gravitational microlensing. In Sec. III we focus attention on the particulars of microlensing of X -ray pulsars, which is relevant to unmask PBHs perceiving the dark dimension. The paper wraps up in Sec. IV with some conclusions.', 'II. POINT-LENS MICROLENSING BASICS': "It has long been known that gravitational lensing becomes observable when a massive object is located between a light-emitting source and an observer [23]. This  \nFIG. 1: The geometry of the microlensing setup for a pointlike lens, seen to produce two images after deflection [25].  \n<!-- image -->  \nis because the gravitational potential of the central object acts as a lens that bends the path of the light rays coming from the source, by bending the spacetime through which the photon travels. If the central object is a Schwarzschild black hole, described by the line element  \nds 2 = -f ( r ) dt 2 + f -1 ( r ) dr 2 + r 2 d Ω 2 2 , (1)  \nthe deflection angle of a light ray is estimated to be  \nα = 4 GM r 0 +4 ( GM r 0 ) 2 ( 15 π 16 -1 ) + O ( ϵ 3 ) , (2)  \nwhere f ( r ) = 1 -2 GM/r is the blackening function, M is the mass of the black hole (i.e. the lens), G = M -2 p is Newton's gravitational constant, d Ω 2 2 is the metric of a 2-dimensional unit sphere, and r 0 is the closest distance of approach to the lens in the lens-Earth-source plane, with ϵ = GM/r 0 , and where all measurements are taken in the observer's reference frame [24]. To develop some sense for the orders of magnitude involved, we consider the set up shown in Fig. 1 for a point-like lens. In the absence of the lens, we would then observe the source at an angular position β on the source plane, but because of the deflection, we actually observe it at θ . Since the observer distances to the lens D OL and source D OS are very large in relation to the deflection angle, the distances of the source and image from the optical axis in the source plane are estimated using the small angle approximation  \nθD OS = βD OS + αD LS , (3)  \nwhere D LS is the distance from the lens to the source [19].  \nThe impact parameter of the unperturbed light ray is found to be ξ = r 0 (1 -2 GM/r 0 ) -1 / 2 . In the point-lens approximation, however, ξ ≡ θD OL ∼ r 0 , and so (3) can be rewritten as  \nθ 2 = βθ + θ 2 E , (4)  \nwhere  \nθ E = √ 4 GM D LS D OL D OS (5)  \nis the angular size of the Einstein ring which is formed when the source is perfectly aligned behind the lens, i.e., when β = 0, and where we have neglected terms O ( ϵ 2 ). θ E in turn defines the Einstein radius r E = D OL θ E on the lens plane. Note that for β ∼ 0, the impact parameter ξ ∼ r E is the characteristic scale of the source-lensobserver system. All in all, the isolated Schwarzschild black hole will split a point source into two images with angular positions defined implicitly by  \nθ 1 , 2 = 1 2 ( β ± √ β 2 +4 θ 2 E ) . (6)  \nThe two images are on opposite sides of the source, with one image inside the Einstein ring and the other outside. As the source moves away from the lens (i.e. as β increases), one of the images approaches the lens and becomes very faint, whereas the other image approaches the true position of the source and asymptotes to its unlensed flux.  \nA point worth noting at this juncture is that if a source is closer than θ E in separation from a lensing PBH on the sky, the source is multiply imaged by its lensing. Nevertheless, the separation between multiple images is too small to be resolved by optical telescopes. What is observed instead in the so-called 'microlensing phenomenon' is the temporary magnification of the total flux of two images relative to that of the unlensed source. Now, if a gravitational field were to deflect a light ray this would produce a change in the cross-section of a bundle of rays. However, according to Liouville's theorem the phase space density of photons must be conserved, which implies that gravitational lensing should preserve the surface brightness of the source and should only change its apparent surface area. In other words, the magnification of an image becomes the ratio of the solid angles of the image and of the unlensed source (at the observer position). If the central object is spherically symmetric, the magnification factor is found to be  \nµ = sin θ dθ sin β dβ ≃ θ β dθ dβ , (7)  \nwhere the sign of the magnification gives the parity of the particular image. Substituting β from the point-lens equation (4) into (7), it follows the magnifications of the two images,  \nµ 1 , 2 = [ 1 -( θ E θ 1 , 2 ) 4 ] -1 = u 2 +2 2 u √ u 2 +4 ± 1 2 , (8)  \nwhere u is the angular separation of the source from the point mass in units of the Einstein angle, u = β/θ E . Note that θ 2 < θ E implies µ 2 < 0, and so the magnification of the image which is inside the Einstein ring is negative implying that this image has its parity flipped with respect to the source. The net magnification of flux in the two images is obtained by adding the absolute magnifi-  \nns,  \nµ tot ( u ) = µ 1 + µ 2 = u 2 +2 u √ u 2 +4 . (9)  \nNote that when the source lies on the Einstein radius, we have β = θ E and u = 1, so that the total magnification becomes µ tot = 1 . 17 + 0 . 17 = 1 . 34.  \nNow, the source and the lens have a relative motion with respect to the observer. As a consequence, the observed flux of a source varies with time, yielding a characteristic light curve of the observed source flux. Such a unique light curve allows a microlensing event to be identified from the observation of other variable sources. A typical timescale of the microlensing light curve can be estimated from a crossing time of the Einstein radius for a lensing PBH with respect to a distant source,  \nt E = r E /v , (10)  \nwhere v is the relative velocity for a observer-lens-source system [26].  \nMicrolensing surveys are typically sensitive to stars that are 10 ≲ D OS / kpc ≲ 1000 away and to transit times of minutes to years. For example, the Subaru-HSC instrument was sensitive to the short transit times of light PBHs reaching sensitivities to constrain an all dark matter PBH interpretation for masses M ≳ 10 22 g [19-21]. However the Subaru-HSC survey was limited by two effects as the PBH mass is reduced:  \n- · the finite apparent size of the source stars in its sky target, M31, being larger than the apparent Einstein radius, yielding poorly focused light;\n- · transition from geometric to wave optics as the PBH Schwarzschild radius becomes comparable to or smaller than the wavelength of optical light.  \nIndeed, the wave effects characterized by  \nw = 8 πGM λ = 0 . 3 ( M 10 22 g )( λ 0 621 nm ) -1 , (11)  \nbecome important when w ≲ 1, where λ 0 is the characteristic wavelength of light in an observation [27]; the default choice in the normalization of (11) corresponds to a central wavelength of r -band in the Subaru telescope. As it was first pointed out in [22], consideration of sources emitting in the X -ray spectrum could allow us to search deep into the PBH low-mass window. It is this that we now turn to study.", 'III. MICROLENSING OF BLACK HOLES PERCEIVING THE DARK DIMENSION': 'It is a known fact that M p is the natural cutoff of quantum gravity. However, if there were light species of particles in the theory, then the consistency of black hole entropy with the effective field theory description would  \ndemand a breakdown of the classical picture at a lower scale,  \nM ∗ = m ( d -4) / ( d -2) KK M 2 / ( d -2) p , (12)  \ndubbed the species scale, where d is the spacetime dimension and m KK ∼ R -1 ⊥ [28, 29]. Now, despite the fact that M ∗ is motivated by the emergence of the tower of light states, curiously the mass scale of this tower, m KK , does not seem to be directly captured by M ∗ [30].  \nOne way the lower-dimensional theory can find out about the m KK scale is through the study of black holes [10, 11, 30]. When black holes get smaller than R ⊥ , the black hole becomes thermodynamically unstable because of the Gregory-Laflamme transition [31]. Black holes with r s ≫ R ⊥ are actually black branes wrapped around the extra dimensions, but the ones with r s ≪ R ⊥ are localized in the extra dimensions. This transition to a new black hole solution marks a new scale Λ BH in the lower-dimensional theory [30]. In this section we show that Earth-based microlensing experiments are insensitive to the Λ BH scale, independently of the black hole mass M .  \nThroughout we assume that the Standard Model fields are confined to a D-brane and only gravity spills into the compact space of dimension ( d -4) [32].', 'A. Microlensing of Bulk Black Holes': "The Gregory-Laflamme transition induces a change in the scaling of the black hole's Schwarzschild radius r s and its Hawking temperature T H [10]. If the black hole is spherically symmetric and r s ≪ R ⊥ , then it can be treated as a flat d -dimensional object with line element given by  \nds 2 = -U ( r ) dt 2 + U -1 ( r ) dr 2 + r 2 d Ω 2 d -2 , (13)  \nwhere U ( r ) = 1 -( r s /r ) d -3 is the blackening function,  \nd Ω 2 d -2 = dχ 2 2 + d -2 ∏ i =2 sin 2 χ i dχ 2 i +1 (14)  \nis the metric of a ( d -2)-dimensional unit sphere, and  \nr s = 1 M ∗ [ M M ∗ ] 1 / ( d -3) [ 8 Γ ( d -1 2 ) ( d -2) π ( d -3) / 2 ] 1 / ( d -3) , (15)  \nand where Γ( x ) is the Gamma function [33-35]. The d -dimensional black hole behaves like a thermodynamic system [36], with temperature T H ∼ ( d -3) / (4 πr s ) and entropy S = (4 πMr s ) / ( d -2) [37]. If the black hole is localized on the brane, Hawking evaporation [38, 39] proceeds dominantly through emission of Standard Model fields [40]. However, we have recently shown that the recoil effect due to graviton emission imparts the black hole  \na relative kick velocity with respect to the brane, allowing Schwarzschild black holes to escape into the bulk [18]. Because the escape from the brane is almost instantaneous 5D Schwarzschild black holes evaporate almost entirely into gravitons in the bulk, and so can evade constraints from the non-observation of Hawking radiation in: (i) the extragalactic γ -ray background [41], (ii) the cosmic microwave background [42], (iii) the 511 keV γ -ray line [43-46], (iv) the EDGES 21-cm signal [47, 48], (v) Lymanα forest [49], and (vi) the MeV Galactic diffuse emission [50-52].  \nThe deflection angle of a light ray expected to be induced by a d -dimensional Schwarzschild black hole localized on the brane is estimated to be  \nα d = 2( d -2) M M d -2 ∗ r d -3 0 2 F 1 ( 1 2 , κ ; 3 2 ; 1 ) + O ( M 2 M d ∗ r d -2 0 ) , (16)  \nwhere 2 F 1 ( a, b, ; c ; z ) is the Gaussian hypergeometric function, with κ = 1 / 2 -( d -3) / 2 [53]. A straightforward calculation shows that for d = 4, (16) reduces to the first term in (2). For d = 5,  \nα 5 = 3 2 π M 3 ∗ M r 2 0 + O ( M 2 M 5 ∗ r 3 0 ) , (17)  \nwhich implies that α 5 < α 4 for fixed M and r 0 , with r s < r 0 < R ⊥ . This was interpreted in [53] as the need for advanced sensitive detection devices to observe lensed images influenced by the dark dimension.  \nThe previous statement should be revised with caution. It is clear that while the source is on the brane, in the dark dimension scenario the lens has an extension into the bulk. Therefore, the light emitted by the source could be a micron away from the lens 4D position. The scale that counts in addressing whether the 5D geometry of the lens would bring important corrections to the source magnification µ tot is actually the size ξ of the lens involved in the deflection. As we have stressed in Sec. II, in the point-lens approximation of Fig. 1, the characteristic scale of the source-lens-observer system is ξ ≳ r E .  \nAt this point a reality check is in order. A source could be a good object to undergo microlensing if it is a long distance away from the telescopes (viz., the Earth), because this would increase the optical depth and/or the number of possible lensing events. Following [22], we consider the X -ray pulsar SMC X-1 in the Small Magellanic Cloud at a distance D OS = 65 kpc [54]. For such a distance, the Einstein radius of a M ∼ 10 17 g black hole is given by  \nr E = √ 4 GMx (1 -x ) D OS = 12 km [( M 10 17 g )( D OS 65 kpc )( x (1 -x ) 1 / 4 )] 1 / 2 (18)  \nwhere x = D OL /D OS . Since r E ≫ R ⊥ is the closest distance that source photons get to the lens when it lies directly along the line of sight, we conclude that corrections to µ tot due to the 5D geometry of the lens can be  \nsafely neglected. In other words, for r ≫ R ⊥ the gravitational potential of a 5D black hole falls as 1 /r and therefore is indistinguishable from that of a 4D black hole of the same mass. This implies that for Earth-based experiments the microlensing signal of a 5D black hole would be indistinguishable from that of a 4D black hole of the same mass, and this is independent on whether the black hole is localized on the brane or propagates through the bulk.  \nSince line-of-sight distances of interest are usually much larger than transverse distances, X -ray microlensing events can be pictured as projections on the lenscontaining transverse plane, viz. the lens plane. The source radius in the lens plane is estimated to be  \na s ( x ) = xR s r E ( x ) ∼ 0 . 8 × ( x √ x (1 -x ) ) ( R S 20 km ) × ( D OS 65 kpc ) -1 ( M 10 17 g ) -1 / 2 , (19)  \nwhere following [22] we have taken a fiducial source size 10 ≲ R S / km ≲ 20. From (19) we infer that X -ray pulsars could overcome finite source size effects for M ∼ 10 17 g if x ≲ 0 . 6.  \nNow, wave effects becomes important when w ≲ 1 [27] or equivalently for photon energies  \nE γ ≲ 1 4 GM = 660 keV ( M 10 17 g ) -1 . (20)  \nThus, for a source emitting primarily with X -ray energy of 1 ≲ E γ / keV ≲ 10 [54], wave effects must be taken into account in order to probe a lower PBH mass of about 10 17 g.  \nThe magnification, including the wave optics effect, is given by  \nµ tot ( w,u ) = πw 1 -e -πw ∣ ∣ ∣ ∣ 1 F 1 ( i 2 w, 1; i 2 wu 2 )∣ ∣ ∣ ∣ 2 (21)  \nwhere 1 F 1 ( a ; b ; z ) is the confluent hypergeometric function [55]. For u = 0 the magnification has a maximum  \nµ tot ( w,u ) ∣ ∣ ∣ ∣ max = µ w ( w, 0) = πw 1 -e -πw . (22)  \nIt is easily seen from (22) that for microlensing events with w ≪ 1, the maximum magnification  \nµ tot ( w,u ) ∣ ∣ max = 1 + πw 2 , (23)  \nwould be significantly reduced as compared to the maximum magnification µ tot ( u ) →∞ in the geometric optics approximation. 2 This is because the gravitational potential induced by extremely light PBHs is too weak to bent the path of the photons.  \nFIG. 2: Compilation of projected sensitivities on the PBH dark matter fraction f PBH as a function of M for future X -ray telescopes, assuming a monochromatic mass function and including wave optics effects. The sensitivity curves have been taken from [22, 57]. For comparison, the shaded region shows contraints on the 4D f PBH from the non-observation of Hawking radiation in: (i) the extragalactic γ -ray background [41], (ii) the cosmic microwave background [42], (iii) the 511 keV γ -ray line [43-46], (iv) the EDGES 21-cm signal [47, 48], (v) Lymanα forest [49], and (vi) the MeV Galactic diffuse emission [50-52]. .  \n<!-- image -->  \nThe mass distribution of PBHs is usually characterized by the mass function  \nψ ( M ) = M ρ CDM dn PBH dM , (24)  \nwhere dn PBH is the number density of PBHs within the mass range ( M,M + dM ), and ρ CDM is the energy density of cold dark matter [56]. Integrating ψ ( M ) gives the total fraction of dark matter in PBHs,  \nf PBH ≡ ρ PBH ρ CDM = ∫ ψ ( M ) dM , (25)  \nwhere ρ PBH = ∫ Mdn PBH is the energy density of PBHs. If all of the dark matter were made of PBH, we would have f PBH = 1. In Fig. 2 we show a compilation of the projected sensitivity to f PBH of future X -ray telescopes, including Athena [58], Lynx [59], and eXTP for a 300day observation [60], as well as STROBE-X [61], and Xµ for a 30-day observation [57]. We can see that these instruments will break down into the mass range relevant for bulk black hole dark matter, including the interesting region in which a 4D PBH all dark matter interpretation has been excluded due to the non-observation of Hawking radiation.", 'B. Microlensing of Near-Extremal Black Holes': "According to the no-hair theorem [62], 4D black hole geometries in asymptotically flat spacetimes eventually settled down to Kerr-Newman solutions [63, 64], which are characterized by three measurable parameters: the mass M , the angular momentum ⃗ J and the electric charge Q . An important feature of Kerr-Newman black holes is that the three parameters are not all independent from each other. Actually, for a given set of parameters there is a minimal extremal mass M e satisfying  \nM 2 ≥ M 2 e = ( M 2 p J M ) 2 +( M p Q ) 2 , (26)  \nwhere Q is measured in units in which the Coulomb force between two charges separated by a distance d has magnitude F = Q 2 /d 2 and J = | ⃗ J | [65]. If the parameters saturate this bound the black hole is dubbed extremal, and if the parameters are close to saturating it near-extremal. Before proceeding we pause to note that:  \n- · When the black hole mass is tuned below M ext , the event horizon disappears leaving behind a naked singularity, which violates the cosmic censorship conjecture [66].\n- · When a black hole reaches its extreme limit, the thermal description breaks down, and it cannot continue to evaporate by emitting (uncharged) elementary particles.\n- · When black holes are near-extremal their evaporation temperature decreases, and consequently so does their luminosity. Thus, near-extremal black holes can also evade constraints from the nonobservation of Hawking radiation [67, 68].  \nIt has long been suspected that any electromagnetic charge or spin would be lost very quickly by any 4D black hole population of primordial origin. On the one hand, the electromagnetic charge of a black hole is spoiled by the Schwinger effect [69], which allows pair-production of electron-positron pairs in the strong electric field outside the black hole, leading to the discharge of the black hole and subsequent evaporation [70, 71]. On the other hand, a rapidly rotating black hole spins down to a nearly nonrotating state before most of its mass has been given up, and therefore it does not approach to extremal when it evaporates [72]. All in all, near-extremal primordial Kerr-Newman black holes are not expected to prevail in the universe we live in.  \nAn alternative interesting possibility is to envision a scenario where the black hole is charged under a generic unbroken U (1) symmetry (dark photon), whose carriers (dark electrons with a mass m ' e and a gauge coupling e ' ) are always much heavier than the temperature of the black hole [73]. This implies that the charge Q does not get evaporated away from the black hole and remains therefore constant. Strictly speaking, the pair production rate per unit volume from the Schwinger effect can be slowed down by arbitrarily decreasing e ' , whereas the  \nweak gravity conjecture (WGC) imposes a constraint on the charge per unit mass; namely, for each conserved gauge charge there must be a sufficiently light charge carrier such that  \ne ' q/m e ' ≥ √ 4 π √ ( d -3) / ( d -2) ¯ M -( d -2) / 2 p , (27)  \nwhere q is the integer-quantized electric charge of the particle and ¯ M p = M p / (8 π ) is the reduced Planck mass [74, 75]. Setting e = e ' = √ 4 πα the (4D) Schwinger effect together with the WGC lead to a bound on the minimum black hole mass of near extremal black holes with evaporation time longer than the age of Universe, M ne ≳ 5 × 10 15 g( m e ' / 10 9 GeV) -2 [73].  \nSince we have seen that microlensing experiments cannot distinguish 4D from 5D black holes, in what follows we consider the line element (1) with blackening function  \nf ( r ) = 1 -2 M M 2 p r + Q 2 M 2 p r 2 . (28)  \nThe deflection angle of a light ray expected to be induced by the geometry given in (28) has been computed in [76] and is given by  \nα Q = 4 M M 2 p r 0 + ( 15 16 π -1 ) 4 M 2 M 4 p r 2 0 -3 4 π Q 2 M 2 p r 2 0 + · · · . (29)  \nWe can see that the effect of Q is to slightly reduce the second order correction to the deflection angle. Therefore, we can conclude that the sensitivity of future X -ray microforlensing experiments to near-extremal black holes is actually shown in Fig. 2.  \nIn summary, evaporation constraints on f PBH can be substantially altered when moving away from the Schwarzschild picture. (Other PBH models where Hawking radiation can be slowed down have been recently discussed in [77, 78].) Microlensing experiments of X -ray pulsars are well positioned to uncover these models.", 'IV. CONCLUSIONS': "Wehave investigated the potential of searching for bulk black hole dark matter through microlensing events us-  \n- [1] M. Montero, C. Vafa and I. Valenzuela, The dark dimension and the Swampland, JHEP 02 , 022 (2023) doi:10.1007/JHEP02(2023)022 [arXiv:2205.12293 [hepth]].\n- [2] D. Lust, E. Palti and C. Vafa, AdS and the Swampland, Phys. Lett. B 797 , 134867 (2019) doi:10.1016/j.physletb.2019.134867 [arXiv:1906.05225 [hep-th]].\n- [3] Y. B. Zel'dovich and I. D. Novikov, The hypothesis of cores retarded during expansion and the hot cosmological  \ning future X -ray telescopes. We have shown that for these telescopes the microlensing signal of a 5D black hole would be indistinguishable from that of a 4D black hole of the same mass, and this is independent on whether the black hole is localized on the brane or propagates through the bulk. We have demonstrated that future instruments observing microlensing events from X -ray pulsars will probe the mass range relevant for bulk black hole dark matter, including the interesting region in which a 4D PBH all dark matter interpretation has been excluded due to the non-observation of Hawking radiation.  \nWe end with an observation. PBHs may experience a memory burden effect, which splits the evaporation process into two distinct phases: semiclassical and quantum [79, 80]. If this were the case, then the minimum black hole mass allowing a PBH all-dark-matter interpretation would also be relaxed [81-84]. The quantum decay rate has an additional suppression factor compared to the Hawking decay rate, which in the most realistic scenario scales with the inverse of the black hole entropy. For d = 4 and a quantum decay rate Γ ∼ T H /S , the memory burden effect opens a new window in the mass range 10 9 ≲ M/ g ≲ 10 10 [82]. 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