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Home , False vacuum decay, Micro black hole

Bounds on extra from micro black holes in the context of the metastable Higgs

Katherine J. Mack∗ North Carolina State University, Department of Physics, Raleigh, NC 27695-8202, USA

Robert McNees† Loyola University Chicago, Department of Physics, Chicago, IL 60660, USA.

We estimate the rate at which collisions between ultra-high cosmic rays can form small black holes in models with extra dimensions. If recent conjectures about catalyzed by evaporation apply, the lack of vacuum decay events in our past light cone places tight bounds on the black hole formation rate and thus on the fundamental scale of in these models. Conservatively, we find that the lower bound on the fundamental scale E∗ must be within about an order of magnitude of the energy where the spectrum begins to show suppression from the GZK effect, in order to avoid the abundant formation of semiclassical black holes with short lifetimes. Our bound, which assumes a Higgs vacuum instability scale at the low 18.8 end of the range compatible with experimental data, ranges from E∗ ≥ 10 eV for n = 1 extra 18.1 down to E∗ ≥ 10 eV for n = 6. These bounds are many orders of magnitude higher than the previous most stringent bounds, which derive from collider experiments or from estimates of Kaluza-Klein processes in neutron stars and supernovae.

I. INTRODUCTION that evaporating black holes formed in theories with ex- tra dimensions are capable of seeding vacuum decay. The decay of the false vacuum is a dramatic consequence that In models with extra dimensions, the fundamental scale presents an unmistakable (and fatal) observational sig- of gravity may be lower than the four-dimensional of microscopic black hole production. Thus, its

scale, MPl. This presents the possibility that high-energy non-observation allows us to place limits on the higher- collisions between particles, for instance in colliders or dimensional fundamental scale that are several orders of via cosmic rays, may form black holes if a high enough magnitude more stringent than those derived from ex- center-of-mass energy is achieved [1–4]. Large extra di- periments searching for production in mensions, if discovered, would constitute new physics other ways, such as via signs of Hawking evaporation in and potentially provide an explanation for the relative colliders or from nearby cosmic ray collisions. weakness of gravity in relation to the other fundamental forces. In addition to searches for microscopic black holes Our analysis relies on two main assumptions, both of formed in particle collisions, experimental constraints on which come with some important caveats that we de- extra dimensions have generally come from searches for scribe here. The first is the of the Higgs modifications of the inverse-square force law of gravity vacuum, as implied by recent measurements of the Higgs at small scales or from signatures of Kaluza-Klein gravi- boson and top masses [9]. This result is based on tons or other exotic particles. Constraints on the higher- the validity of the Standard Model of , dimensional fundamental scale depend on the number of so any beyond-Standard-Model (BSM) physics may alter extra dimensions proposed, with collider limits in the TeV the effective potential of the Higgs field in a way that range and astrophysical limits as high as O(102)-O(103) rescues our vacuum from metastability [10]. A high en- TeV [1,5]. ergy scale of inflation, were it to be confirmed, would give evidence that new physics stabilizes the vacuum in arXiv:1809.05089v1 [hep-ph] 13 Sep 2018 We present new limits from black hole creation in the some way, since high-energy-scale inflationary fluctua- context of recent work proposing that Hawking evapora- tions would likely have instigated a transition to the true tion of microscopic black holes can induce the decay of vacuum in the early [11, 12]. While we fully the standard model Higgs vacuum [6–8]. These papers recognize (and, in fact, hope) that vacuum metastability argue that the nucleation of a bubble of true vacuum in ends up being ruled out by BSM physics or a better un- general precedes the final evaporation of the black hole, derstanding of inflation, we will, for the purpose of this suggesting that any production of black holes with evap- study, rely on the great successes of the Standard Model oration less than the in our to justify the apparent metastability of the Higgs vac- past light cone should have already led to vacuum decay. uum as an observational tool for establishing constraints The most recent work in the series [8] explicitly confirms on higher-dimensional theories. Second, we are assuming that the results of [6,7] hold in a qualitatively similar way for theories with more than four dimensions, ∗ [email protected]; 7 @AstroKatie i.e., that black hole evaporation can seed vacuum decay. † [email protected]; 7 @mcnees This was explored in [8], where the authors construct an 2 approximate solution for a braneworld black the maximum center-of-momentum achieved in hole in a theory with one extra dimension and then es- collisions. As a result, bounds on the fundamental scale timate that their results extend to regimes where small of theories with extra dimensions may be even higher black holes are produced in particle collisions. We will than the values we present here. apply these results beyond one extra dimension, though earlier calculations suggest the effect may be somewhat In SectionII we discuss a method for estimating the num- suppressed [6].1 More importantly, the conclusions of [8] ber of collisions that have taken place in our past light cone between UHE cosmic rays, and review the Pierre require the instability scale ΛI of the Higgs vacuum to lie Auger Observatory’s spectrum of these particles. In Sec- below the fundamental scale E∗ of the higher-dimensional theory. Otherwise, the Standard Model calculation of the tionIII we extend this to collisions capable of forming Higgs potential no longer applies. For our analysis, we black holes in higher-dimensional theories, obtain bounds must assume that the instability scale is at the low end of on the fundamental scale of these theories in SectionIV, the range consistent with experimental limits. The most and then discuss these results in SectionV. AppendixA 19 20 considers the various criteria that must be satisfied for likely range calculated by [9] is ΛI ∼ 10 − 10 eV, with some uncertainty around that value. For our anal- a reliable semiclassical analysis of black hole formation, ysis to be fully reliable, we require scales no higher than and AppendixB discusses an analytical result for black 18 hole formation rates that supports the numerical results ΛI ∼ 10 eV for theories with one or two extra dimen- 17 used in the main text. sions and ΛI ∼ 10 eV for theories with up to six. This is an important qualifier on our main results, and will be discussed in more detail at the end of the paper. II. COLLISIONS OF ULTRA-HIGH-ENERGY The structure of our calculation is as follows. Assuming COSMIC RAYS that the Higgs vacuum is metastable and that its decay is catalyzed by black hole evaporation, we take its persis- tence as evidence against black hole evaporation in our At ultra-high energies, cosmic rays are rare enough that past light cone. While this observation can also constrain we expect interactions between them to be exceedingly the production of low mass primordial black holes in the infrequent. But on timescales comparable to the age of early universe, we apply it here to the production of mi- the Universe, even a low rate can to an apprecia- croscopic black holes in particle collisions. Specifically, ble number of collisions with center-of-momentum (CM) we consider the formation of black holes in collisions be- energies several orders of magnitude greater than any- tween ultra-high-energy (UHE) cosmic rays, in theories thing that can be achieved in existing accelerators. Let with extra dimensions and a fundamental scale well be- us quickly review the estimate of collisions between UHE low the four-dimensional Planck scale. If the instability cosmic rays with energies above 1020 eV given by Hut and scale for the Higgs vacuum is low enough, this allows us Rees in [16]. to place lower bounds on the fundamental scale of such theories which are in general much more stringent than Assuming a homogeneous and isotropic distribution, the current lower bounds from both accelerator and astro- density of UHE cosmic rays with energy greater than E physical processes. For a given value of the fundamen- is proportional to the integrated flux tal scale, we use the UHE cosmic ray spectrum from the 4π Z ∞ Auger experiment (see SectionII and ref. [13]) to make a ρ(E) = dE0 J(E0) , (1) conservative estimate of the number of black holes formed c E in particle collisions in our past light cone. We note that where J(E) is the differential flux. For constant density the measured cosmic ray spectrum includes a steep drop- ρ and cross section σ, the rate of collisions per particle is off at high energies. This is believed to be due to the ρσc, and the overall rate of collisions per unit volume is GZK effect [14, 15], which prevents the highest-energy cosmic rays from traveling unimpeded across cosmologi- R = ρ2σc . (2) cal distances. If this is the case then it is likely that even more energetic particles are plentiful in parts of the cos- The total number of collisions in our past light cone is mos where high-energy astrophysical processes accelerate given by this rate times the spacetime volume them. But without knowing more about the mechanisms 3 4 involved we restrict our analysis to cosmic rays with ener- N = R c T , (3) gies below the GZK cut-off. Thus, our calculation proba- where T is the over which these collisions have oc- bly under-estimates both black hole formation rates and curred and our assumptions hold. Hut and Rees calcu- lated the density of cosmic rays above 1020 eV using the differential flux given by Cunningham et. al. in [17]

1 The reduced branching ratio of false vacuum decay rate to the 1.14 × 10−33 1019 eV2.31 Hawking evaporation rate may be offset by the production of J(E) = . (4) large numbers of black holes. m2 · s · sr · eV E 3

Then (1) gives a density of 1.8 × 10−23 m−3. The cross section is taken to be of order the Compton wavelength squared

2π c2 σ(E) ' ~ , (5) E which at 1020 eV gives 1.5 × 10−52 m2. For their order of magnitude estimate, Hut and Rees use σ ' 10−52 m2. The per-particle rate of collision is then ' 3 × 10−67 s−1, and the rate of collisions per unit volume comes out to R ' 3 × 10−90 m−3 s−1. Taking the age of the Universe to be about T = 1010 yr, the number of collisions in the past light cone is roughly N' 8 × 105. Hut and Rees give a final estimate of N ≈ 105. FIG. 1. The Auger spectrum of UHE cosmic rays with E > For our calculations, we will use more recent results for 1018 eV, approximated as a set of power laws [13]. the differential flux of UHE cosmic rays in place of (4). The Pierre Auger Observatory is a hybrid cosmic-ray ob- servatory consisting of surface Cerenkov detectors and a homogeneous and isotropic distribution for cosmic rays air-shower observing telescopes, which allows it to collect at those energies over the full volume of our past light- large samples of cosmic rays across a wide range of ener- cone. Without knowing more about the origin of UHE gies. Its measurements of the cosmic ray energy spectrum cosmic rays, we will conservatively limit all of our cal- above 1018 eV are well described by a series of power laws culations to cosmic rays with energies below the break with free breaks between them, or else by broken power energy in the Auger spectrum. laws with an additional smooth suppression factor at the highest energies [13, 18–20]. Here we adopt the values The approximation (2) for the rate of collisions per unit given in [13], with the differential flux in each range of volume treats all particles as if they had roughly the energies taking the form same energy, with a constant cross section for collisions. We can refine the estimate by dropping these assump- J(E) ∝ E−γ . (6) tions, accounting instead for all collisions above a given CM energy and including the energy dependence of the The flux is shown in Fig.1. Below the ‘ankle’ en- cross section. Assuming once again a homogeneous and 18.61±0.01 ergy, Eankle = 10 eV, the spectral index is γ1 = isotropic distribution of UHE cosmic rays, the rate per 3.27±0.02. Above the ankle energy, but below the ‘break’ unit volume of collisions with CM energy greater than E 19.46±0.03 energy Ebreak = 10 eV, the spectral index flat- is given by tens to γ2 = 2.59 ± 0.02. Above the break energy the spectral index drops off to γ = 4.3 ± 0.2. The spectrum E 1 3 16π2 Z break Z between E and E is thought to possibly repre- R = dE0dE00 du σ(E ) ankle break c CM sent the transition to a population of extragalactic cos- Emin 0 0 00 mic rays, while the steep fall off above Ebreak is likely due J(E )J(E ) Θ(ECM − E) , (7) to the GZK effect [14, 15]. where u = (1−cos ψ)/2 is related to the angle ψ√between Repeating the estimate of Hut and Rees with the Auger the momenta of the colliding particles, E = 2 E0E00u 20 CM spectrum yields fewer collisions above 10 eV – on the is the CM energy, and the Heaviside theta function re- order of a few thousand – because of the steep drop off stricts the domain of integration to collisions with ECM > in the flux above Ebreak that is not accounted for in (4). E. The upper limit in the energy integrals is taken to be Indeed, the presence of this feature in the spectrum sug- Ebreak, which restricts the calculation to cosmic rays with gests that the upper limit in (1) should not extend to ar- energies below the GZK cut-off, while the lower limit is bitrarily high energies. The GZK effect prevents cosmic rays from traveling cosmological distances with energies E2 2 Emin = , (8) greater than Ebreak. So it seems unwarranted to assume 4 Ebreak which is the minimum energy of a particle that can par- ticipate in a collision with CM energy of at least E. As 2 It is possible that the GZK effect is only partly responsible for the before, the number of events in our past lightcone is drop-off in flux above Ebreak, which may also reflect, for instance, 3 4 10 the maximum energies that can be achieved by the sources that N = R c T . In our calculations, we will take T = 10 accelerate the particles [18]. While its origin does not impact years. This may be a conservative assumption, as the our estimates, we interpret Ebreak as the scale associated with production rate for UHE cosmic rays is likely to have the GZK effect. been higher toward the early part of that time window, 4 closer to the peak of (AGN) ac- approximation is valid. For a discussion of this tivity around a of 2. requirement, see AppendixA. In the next section we will employ (7) to estimate the rate For the collisions between UHE cosmic rays in the previ- of black hole formation in models where the fundamen- ous section, the cross section (5) was proportional to the tal scale of gravity is below the maximum CM energies square of the Compton wavelength and hence decreased accessible in collisions between UHE cosmic rays. at higher energies. But at CM energies well above E∗ the cross section for black hole formation exhibits the oppo- site behavior. The geometric cross section for black hole formation is [21]-[22] III. BLACK HOLE FORMATION VIA COSMIC RAY COLLISION 2 σBH = O(1) πrH , (11)

In higher-dimensional theories the fundamental scale of where rH is the horizon radius of a black hole of mass gravity may be lower than MPl. We will consider a 2 MBH = ECM/c , and an overall factor of order 1 reflects generic 4 + n-dimensional theory with fundamental scale various corrections. Collisions at higher energies produce 2 M∗ = E∗/c related to the Newton’s constant by black holes with larger mass, and hence larger horizon

n+5 n+1 radius, resulting in a cross section that grows as a positive (4+n) c ~ power of ECM. GN = 2+n . (9) E∗ Assuming the collision forms a Schwarzschild black hole, If the scale E∗ is low enough, collisions between UHE the horizon radius in 4 + n dimensions is [23] cosmic rays at sufficiently high CM energy are expected 2 to form black holes of mass MBH ∼ ECM/c . 1 1  2  n+1  3+n  n+1 ~ c MBHc 8π Γ( 2 ) For our estimates of black hole formation via scattering rH = 3+n . (12) E E n + 2 2 to make sense, the black holes should be large enough ∗ ∗ π that a semiclassical treatment is appropriate. We enforce this by considering only black holes with entropy above Then, up to the O(1) factor in Eq. (11), the cross section 2 for a collision forming a black hole with MBH = ECM/c some minimum value Smin (see AppendixA for a brief is discussion). This implies that the ratio of MBH/M∗ must be greater than 2 2 2     n+1 8 Γ( 3+n ) n+1 1 (4+n) ~ c ECM 2 n+3 ! n+2 σBH = . (13) 2 n+1 n + 2 π  n+2 E∗ E∗ n + 2 λ = n+3 Smin . (10) 4π 2 Γ( 2 ) Using this cross section in (7), we can estimate the rate 2 We will typically take Smin = 10 , which implies that at which black holes are formed by collisions between

MBH must be greater than M∗ by a factor that is O(10) UHE cosmic rays in a higher-dimensional theory with 2 for n = 1 and increases to O(10 ) for n = 6. Neglecting fundamental scale E∗. energy loss during the formation process, M = E /c2 BH CM As an example, consider a theory with one extra di- and collisions with ECM ≥ λ E∗ are considered to form mension (n = 1). Using (10), black holes with entropy semiclassical black holes. 2 SBH ≥ 10 have mass MBH ≥ 8.76 M∗. The number of We also require that the black holes be small enough such black holes formed in our past light cone by colli- compared with the compactification scale L that a flat- sions between UHE cosmic rays is approximately

2 √ 2   Z Ebreak Z 1 0 00 √ 3 4 128π ~ c 0 00 E E u 0 00 0 00  N = c T dE dE du J(E )J(E )Θ 2 E E u − 8.76 E∗ . (14) 3 c E∗ Emin 0 E∗

Fixing a fundamental scale allows us to evaluate this ex- T = 1010 yr, and a fundamental scale of 1018.5 eV. The pression explicitly to determine a number of black hole- numerical evaluation of this integral gives N' 1.6×1011. producing events above that scale. As an illustrative cal- Raising the fundamental scale to 1018.8 eV lowers the culation, we use the Auger results for the differential flux, number of events to N' 2.6 × 106. 5

Since we restrict our attention to UHE cosmic rays with with energies above the GZK cut-off, and as explained in 19.46 energy below the GZK cut-off at Ebreak = 10 eV, the previous section, our assumption of a homogeneous the maximum possible CM energy in our analysis is and isotropic distribution seems questionable for that 19.76 2 Ebreak = 10 eV. This places an upper limit on the population of UHE cosmic rays. mass of the black hole, so the requirement MBH ≥ λ M∗ implies that our analysis is valid only for E∗ < 2 Ebreak/λ. In this example with n = 1, the largest fundamental scale 18.82 we can consider is 10 eV, and N plunges to zero as E∗ approaches this value. The drop-off is steep enough that IV. BOUNDS ON E∗ FROM BLACK HOLE 6 3 FORMATION the difference between E∗ for N ∼ 10 and N ∼ 10 is much less than the uncertainties in Ebreak and other factors. If vacuum decay triggered by black hole evapo- Now we calculate the number of black holes formed via ration is as likely as the claims of [6–8], then essentially collisions between UHE cosmic rays for different numbers any value of E∗ up to 2 Ebreak/λ results in too many black of extra dimensions n, and use this to establish lower holes being formed. bounds on the fundamental scale E∗. Note that by raising the maximum cosmic ray energy in As in the previous section, we consider only black holes (7), E∗ could be greater than 2 Ebreak/λ by a factor of with entropy above a minimum value Smin that justifies ∼5 and still allow a significant number of black holes to the use of semiclassical methods. This implies M ≥ form and evaporate over the history of our universe. But BH λ M∗, where λ is given in (10). Then the number of the majority of those collisions would involve particles black holes formed is approximately

2 Z Ebreak Z 1 3 4 16π 0 00 (4 + n) 0 00  N = c T dE dE du σBH (ECM)J(E )J(E )Θ ECM − λ E∗ , (15) c Emin 0

√ 0 00 where ECM = 2 E E u. Since we only consider UHE for the Higgs vacuum. A more conservative condition 3 cosmic rays with energies below Ebreak, these collisions SBH ≥ 10 would require larger black holes and may not can only form semiclassical black holes when the funda- be consistent with the assumed instability scale. Fig.4 2 mental scale satisfies E∗ ≤ 2Ebreak/λ. Because the Auger translates the lower bound on E∗ (with SBH ≥ 10 ) into spectrum is defined piecewise for different values of the an upper bound on the size of the extra dimensions (A7) energy, (15) is most easily evaluated numerically. The with the typical size of extra dimensions for TeV-scale number of black holes formed as a function of E∗ is shown gravity [24] included for comparison. The values for E∗ in Fig.2, for 1 ≤ n ≤ 6. and L are summarized in TableI.

As in the n = 1 example, N  1 for all values of E∗ up 2 n λ (Smin = 10 ) log (E∗/eV) log (L/m) to the maximum value 2Ebreak/λ that can be probed us- 10 10 ing our method. Thus, avoiding vacuum decay catalyzed 1 8.8 18.8 −7.8 by the evaporation of black holes formed in collisions be- tween UHE cosmic rays requires a fundamental scale 2 16.1 18.6 −16.5 3 23.9 18.4 −19.4 2 E 1019.76 eV E ≥ break = , (16) ∗ λ λ 4 31.5 18.3 −20.9 5 38.9 18.2 −21.7 where λ is given in (10). For smaller values of E∗, colli- sions between UHE cosmic rays that form rapidly evap- 6 46.0 18.1 −22.3 orating black holes are plentiful within our past light cone. Although they cannot be reliably estimated with TABLE I. Lower bounds on the fundamental scale E∗ and the method used here, collisions between cosmic rays at upper bounds on the size L of extra dimensions (under the even higher energies are of course possible, and would assumption of a toroidal compactification). raise the lower bound on E∗.

The lower bound on E∗ for different values of n is shown The values of N used in Fig.2 were calculated by nu- 2 in Fig.3. For SBH ≥ 10 , the lower bound on E∗ is more merically evaluating (15). However, it is easy to show or less within an order of magnitude of the Auger break that once E∗ is larger than about 0.8 Ebreak/λ, both of energy, and safely above our assumed instability scale the UHE cosmic rays participating in the collision must 6

FIG. 4. Upper bound on the size of extra dimensions (blue), with typical values for TeV-scale gravity (red), for different FIG. 2. Number of collisions N forming a black hole with values of the number of extra dimensions, n. The blue shaded 2 10 entropy SBH ≥ 10 over the past T = 10 yr, as a function region of the plot is excluded by our constraints. of the fundamental scale. The colored lines are for different values of n, the number of extra dimensions. N drops off rapidly as E∗ approaches 2Ebreak/λ. catalyzed by microscopic black holes. This scenario is based on the mechanism outlined in [6–8], in which black holes seed vacuum decay before their evaporation is com- plete. It assumes the meta-stability of the Higgs vacuum, supported by recent measurements of the mass of the and , at a scale below the funda- mental scale of the higher-dimensional theory. While this concept has been used to place limits on the production of primordial black holes [6,7], ours is the first analysis to place quantitative limits on the fundamental scale of extra dimensional theories based on this method. TableII summarizes current limits on the fundamental scale and/or size of extra dimensions from a range of methods. Comparing with TableI, limits from vacuum decay catalyzed by black hole evaporation are many or- ders of magnitude more constraining for 1 ≤ n ≤ 6 extra FIG. 3. Lower bound on the fundamental scale for 1 ≤ n ≤ 6 2 dimensions. for SBH ≥ 10 , where n is the number of extra dimensions and SBH is the entropy of the black hole, set to ensure the validity Our analysis is limited by the assumptions underlying of a semiclassical treatement. Note that the scales of existing 2 3 the conclusions of [6–8]. Given the values for the lower constraints (O(10 )-O(10 ) TeV) are all below the plot range bound on E in TableI, the most important of these as- shown here. The blue shaded region of the plot is excluded ∗ sumptions is the requirement ΛI < E∗. The authors of by our constraints. 17 [8] quote ΛI ∼ 10 eV as the lowest value consistent with experimental limits on the top quark mass, which is well below our lower bounds on E∗ for 1 ≤ n ≤ 6. They esti- have E > Eankle. In that case, the relevant part of the 17 mate that for n = 1 extra dimension, with ΛI ∼ 10 eV Auger differential flux is given by a single power law and 18 and E∗ ∼ 10 eV, vacuum decay would be caused by N can be evaluated analytically as a function of E∗ and 20 18.8 n. This is described in more detail in AppendixB. black holes with MBH ∼ 10 eV. For E∗ < 10 eV, we find that a significant number of black holes with 20 MBH ∼ 10 eV are produced. Even if the instability 18 scale is as high as ΛI ∼ 10 eV, our n = 1 (and pos- V. DISCUSSION sibly n = 2) value for E∗ seems high enough to justify concerns about vacuum decay. However, the most likely range of values for the instability scale ΛI appears to be We have presented constraints on the fundamental scale around 1019-1020 eV. In that case the fundamental scale 20 21 and the size of extra dimensions in higher-dimensional E∗ would have to be greater than 10 -10 eV, which theories, based on the non-observation of vacuum decay is outside the regime that can be probed with this con- 7

relies on qualitatively different physics than the tests re- Method Reference n log (E∗/eV) log (L/m) 10 10 sponsible for the strongest existing constraints on extra Grav force [25] 2 12.5 −4.36 dimensions. It does not rely on assumptions about gravi- SN1987A [26] 2 13.4 −6.18 tons or other BSM particle physics, and is therefore an interesting complement to existing methods. 3 12.4 −9.10 NS cooling [27] 1 −4.35 We expect that our analysis can be made more robust with improved inferences about the instability scale from 2 −9.81 collider data, along with a more complete accounting of 3 −11.6 the full range of high-energy particle interactions in our past light cone. In particular, collisions in regions where 4 −12.5 UHE cosmic rays are accelerated to energies above the 5 −13.0 GZK cut-off may achieve higher CM energies than we 6 −13.4 considered. Such collisions could form black holes for even larger values of the fundamental scale E∗, extending CMS [28] 2 13.0 our analysis to more likely values of the Higgs instability 3 12.9 scale. 4 12.8 5 12.8 6 12.7 VI. ACKNOWLEDGMENTS

TABLE II. Current bounds on extra dimensions from: gravi- RM was supported in part by the National Science tational force law tests [25]; constraints on the production of Kaluza-Klein gravitons from the 1987A [26]; con- Foundation under Grant No. NSF PHY-1748958 through straints based on the expectation that Kaluza-Klein gravitons the KITP Scholars program, and by Loyola University would decay into and heat neutron stars [27]; and col- Chicago through a Summer Research Stipend. KJM and lider searches, the most stringent of which currently provided RM acknowledge the hospitality of Caltech and the Burke by the CMS collaboration [28]. We provide values for both E∗ Institute, especially for the workshop “Unifying Tests of and L when provided in the cited references. In other cases, ” (July 2016) during which this project it is possible to deduce the corresponding value via equation was conceived. KJM also acknowledges the Australian A7 for the toroidal compactifications considered here. It is of Research Council and Melbourne University for support note that the most stringent constraints (SN1987A and NS during early stages of this project, and RM thanks the cooling) require some assumptions about Kaluza-Klein gravi- Kavli Institute for and the KITP tons. Scholars Program for hospitality and a productive work environment. We are grateful to Leo Stein and Walter Tangarife for helpful comments on an earlier draft of this servative calculation. On the other hand, UHE cosmic paper. ray observatories have detected particles with energies as high as E = 3×1020 eV.The propagation of such particles on cosmological scales is suppressed by the GZK effect, so our method for estimating the number of collisions form- Appendix A: Black Hole Formation and ing black holes is not applicable. But if collisions between Semiclassical Methods particles at these energies occur in regions where UHE 21 cosmic rays are produced, then CM energies ECM ∼ 10 may be achieved. In that case rapidly evaporating black Our analysis is based on the formation of black holes in holes may have been formed for fundamental scales as collisions between UHE cosmic rays, which rapidly evap- 20 high as E∗ ∼ 10 eV, potentially inducing vacuum de- orate via . Thus, we must consider 19 cay even if the instability scale is ΛI ∼ 10 eV, which is three questions. First, under what conditions can we in the most likely range of values [9]. For larger values say that a collision has formed a black hole? Second, of the instability scale it seems unlikely that black hole when can those black holes be described semiclassically? formation via UHE cosmic ray collisions could be used to And third, since the extra dimensions of spacetime are constrain E∗. assumed to be small and compact, when can the black holes be described using results that assume an asymp- The approach we have taken here comes with important totically flat space time? caveats described above and in the introduction. Nev- ertheless, the possibility of establishing bounds on extra A basic criteria for saying that a black hole has formed is dimensions that are many orders of magnitude stronger that the decay time should be very long compared to the than existing constraints makes this is a promising di- time scale associated with the formation process. For rection for continued research. Additionally, our method black holes formed via collision, we take that to mean 8 that the decay time should be much longer than the time needed for the particles to cross a region of linear size rH. In 4 + n dimensions the decay time is of order

n+3   n+1 ~ MBH τD ∼ 2 , (A1) M∗c M∗ while the crossing time τC ' rH/c is

1 1 1 8 Γ( 3+n ) n+1 M  n+1 ~ √ 2 BH τC = 2 . (A2) M∗c π n + 2 M∗

In all cases of interest, the n-dependent factors in τC are

O(1), so the condition τD  τC is equivalent to

n+2   n+1 M FIG. 5. The ratio λ = MBH/M∗ for a black hole with entropy BH  1 . (A3) Smin. M∗ The power on the left-hand side of this inequality is al- ways greater than 1, so black holes with MBH  M∗ with length scale L. The formulas above assume that satisfy τD  τC. spacetime is asymptotically flat, but we may regard them as approximately true when the black hole radius To justify a semiclassical treatment, the entropy of the (12) is much smaller than the compactification scale: black hole should satisfy SBH  1. In any dimension rH  L. In the case of toroidal extra dimensions, the the entropy is given by one-quarter of the horizon area in four-dimensional Planck mass is related to the compact- units of the fundamental length scale. For a non-rotating ification scale of the higher dimensional theory by black hole this is: n 3 2+n 3 2 n 2+n c AHc ω2+n rH c M = (2πL) (M ) . (A7) S = = , (A4) Pl ∗ n BH (4+n) (4+n) ~ 4 ~ GN 4 ~ GN

3+n Ignoring factors of O(1), the condition rH  L becomes 2 3+n where ω2+n = 2π /Γ( 2 ) is the area of a unit 2 + n- sphere. Using (9) and (12), the entropy can be expressed 1 2 M  n+1 M  n as BH  Pl (A8) n+2 1 n+2 M∗ M∗   n+1   n+1   n+1 4π 4 MBH SBH = . (A5) n + 2 ωn+2 M∗ Thus, MBH/M∗ should be large enough to justify a semi- classical calculation, but not so large that the black hole The first two factors give a number greater than 1 for begins to notice the extent L of the extra dimensions. 1 ≤ n ≤ 9, and of O(1) out to n ∼ 35. So the condition In the text we consider theories with fundamental scale 19 SBH  1 is essentially the same as the previous condition, as large as E∗ ∼ 10 eV, and limit ourselves to UHE τD  τC, in all cases of interest. cosmic ray collisions with CM energy no greater than E ∼ 1020 eV. In that case, for collisions forming black For a black hole of mass M = 10 M , the entropy CM BH ∗ holes with entropy greater than S , the ratio r /L al- ranges from S ' 120 when n = 1, down to S ' 20 for min H BH BH ways satisfies n = 6. Since SBH decreases with n for fixed MBH/M∗, we will always set a minimum entropy Smin that is sufficient 1 1  2  n rH − (16π) 1 to justify semiclassical calculations, and then restrict our 17 n − n ≤ 2π 1.79 × 10 2 (Smin) . attention to black holes with entropy at or above this L (n + 2) ωn+2 cut-off. Using (A5), this fixes the minimum value λ of (A9) the ratio MBH/M∗ for a semiclassical black hole in our 2 −18 analysis as For Smin = 10 , this is of order 10 for n = 1, and of −3 1 order 10 for n = 6. In these cases, the black holes we n + 2 wn+2  n+2 n+1 λ = (S ) n+2 . (A6) consider are all much smaller than the size of the extra 4π 4 min dimensions and the physics should be well-described by This is shown for 1 ≤ n ≤ 6 in Fig.5. Black holes with formulas that assume an asymptotically flat spacetime. A quick calculation shows that the size of extra dimen- MBH ≥ λ M∗ have SBH ≥ Smin. sions is also much larger than the fundamental length For the higher dimensional theories considered in this scale L  `∗, so that corrections may paper we assume that n dimensions are compactified safely be neglected. 9

2 Thus, for black holes with entropy SBH ≥ 10 , the pro- formation rates, N is well approximated by cess of formation via collision and subsequent evapora- tion should be well described using semiclassical meth- 4(n + 2)2 N' 2.38 × 105 (S )2 χ3× ods and asymptotically flat-space results for 1 ≤ n ≤ 6. min 3 The case n = 7 is borderline, with the conditions de-  4n + 5  scribed above and the assumptions outlined elsewhere in 1 + γ2 χ + χ . (B4) 2(n + 1) the paper beginning to break down. 2 18.8 For Smin = 10 and E∗ = 10 eV (corresponding to χ = 0.042) this approximation gives N = 2.55 × 106, which is within about 2% of the result obtained directly Appendix B: Analytic expression for N from (B3).

Since we consider cosmic rays with energies below the

GZK cut-off at Ebreak, the minimum energy of a cosmic ray that can participate in a collision with CM energy above λE∗ is

2 (λE∗) Emin = . (B1) 4 Ebreak

If we express the fundamental scale as a fraction of the maximum value that we can probe, E∗ = (1−χ) 2Ebreak/λ with 0 ≤ χ < 1, then both cosmic rays must have energy greater than Eankle when

r E 1 − χ > ankle = 0.38 . (B2) Ebreak

In this regime the differential flux in (15) is described by a single power law, and the integral can be evaluated analytically.

Expressing the particle energies in units of Ebreak, the number of black holes with SBH ≥ Smin formed over the past T = 1010 yr, in a theory with fundamental scale

E∗ = (1 − χ) 2Ebreak/λ, is

2(n+2)  1  n+1 N = 2.38 × 105 (n + 2)2 (S )2 min 1 − χ 1 Z 1 1 0 00 0 00 −γ2 0 00 2 de de du u n+1 (e e ) n+1 Θ e e u − (1 − χ) (1−χ)2 (B3) where γ2 = 2.59 is the spectral index given by Auger for cosmic rays with energies between Eankle and Ebreak. Notice that N grows with the minimum entropy for semi- 2 classical calculations as (Smin) . This is due to the fact that as Smin goes up, the fundamental scales we probe go down like 1/λ, resulting in a larger cross-section (13). The full expression obtained from evaluating the integral in (B3) is not especially illuminating, but was used to verify the numerical results presented in sectionIV. For χ  1, the regime where E∗ is extremely close to the maximum value for which we can estimate black hole 10

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