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arXiv:1906.07262v1 [astro-ph.HE] 17 Jun 2019 vrmn er ttePer ue bevtr [1–3], of Observatory energies Auger maximum Pierre reach the the which for observed at events years physics many (UHECR) known over ray of cosmic energy terms ultra-high in explanation factory ail pnigbakhlsobey into holes black which runs to spinning according explanation rapidly [8], an black theorem Thorne’s such Kerr with Unfortunately, rotating difficulties rapidly [7]. a process from hole Penrose energy the invoke emerged. extracting to has of is picture possibility clear in exotic no discussed more far A so amply but been known [3], have literature of waves, the acceleration shock Fermi by as such physics, ventional both events, energy ray cosmic quantitatively. and explanation energetic qualitatively simple most fairly here the a argue furnish for will principle we in As can these interacting. strongly fractionally mas- and furthermore very of are charged gas which dilute gravitinos, con- extremely stable have an could of sive we (DM) part where in least 6] dark at [5, sist that work possibility previous possible the our a raised proposals. on present previous based we is from This different Letter very Stan- this explanation the In beyond new (SM). physics new Model for dard scenario need compelling the a indicate of absence may The (He, [1]. ions energies heavier highest but , Fe, is just N, feature not puzzling of particularly occurrence A the [4]. bound GKZ the ing (where rcs,btte h aiu tanbeeeg falls required energy the attainable a of maximum a such short the that in far then acquire velocity can but the ) process, for bound elementary limit other this any that upper (or assume an to sets reasonable also seems It lution). 1 xs,i sntcerwehradhwte ol accumu- could they how and do whether clear these not assuming is it even mass exist, However, GUT as monopoles. such objects, magnetic GUT-like of annihiliation via − odt hr osntapa oeitaflysatis- fully a exist to appear not does there date To osbeaclrto ehnssrligo oecon- more on relying mechanisms acceleration Possible β ∼ etc. a 10 steuulrtto aaee fteKr so- Kerr the of parameter rotation usual the is hc pert oiaetwrstevery the towards dominate to appear which ) − atclr tese htUEReet a eudrto to understood be fo can of the essential events horizon UHECR of are GKZ that interactions matter ensues strong standard it into particular, and anti- decays mass gravitino against large via bility en The Observatory, ultra-high Auger stars. the Pierre for previous the explanation had at possible we served a that offer gravitinos can superheavy Matter the that argue We etkna vdnefrtefnaetlast oad unifi towards events ansatz these fundamental in the ions for heavy evidence of as taken presence be the explaining in role 20 .Aohrpsil xlnto ol be could explanation possible Another ). uehayGaiio n lr-ihEeg omcRays Cosmic Energy Ultra-High and Gravitinos Superheavy E 2 a-lnkIsiu ¨rGaiainpyi (Albert-Ei f¨ur Gravitationsphysik Max-Planck-Institut ∼ & 0Mc h opsto fnurnsasna hi ufc c surface their near stars neutron of composition The Mpc. 50 10 19 eiae otemmr fMra Gell-Mann Murray of memory the to Dedicated ryzo .Meissner A. Krzysztof a/M V(orsodn to (corresponding eV E ¨ hebr ,D146Ptdm Germany Potsdam, D-14476 M¨uhlenberg 1, & 1 aut fPyis nvriyo Warsaw of University Physics, of Faculty . 10 atua5 203Wra,Poland Warsaw, 02-093 5, Pasteura 19 β max Vsaturat- eV 0 = . 998 1 n emn Nicolai Hermann and olwn rpris ( properties. following dynamics. the de- of conse- without knowledge even interesting framework tailed some kinematic this derive from nevertheless quences can underlying one of symmetries the- it), understanding duality group better de- infinite-dimensional much on dynamical the a solely proper require relies would a scription far (whereas so field considerations quantum it oretic based Although space-time of theory. framework the in try oa a ute eeoe n[,1,1]i re ofully to order in pro- 13] SU(3) This 12, the [5, for 10]. account in [9, developed Gell-Mann further was to posal due orginally proposal a maximal of content h xsec fegtmsiegaiio nadto to addition in - gravitinos fundamental massive 48 eight the of existence entails framework the theoretical group this , shr o eesrl elzda a as realized necessarily not supersymmetry here emergent scale) with is (Planck unification however, for [11]; framework space-time wider on a based of simply part not thus is posal ue aia ult symmetry duality maximal tured h Mfrin,b xliigpoete ftemaximal symmetry’) the (‘R of subgroup properties compact exploiting by , SM the con o h bevdaudneo ev ions. heavy of abundance nor observed known energies, the required of for the to account acceleration particles the new) suspected explain scenarios, (or SM” plausibly the could “Beyond SM that discussed from widely nor mecha- or astrophysics compelling physics relativistic no from be neither to nism, ago. appears long there already conclusion, decayed have In likely most , would hypothetical gauge these heavy other as or to event like applies observed objects objection GUT-like the same explain The to rates. amounts sufficient in late rtoso 6qak n etn ah rmtespin- the from each, gen- and three with 16 SM, ansatz the of of attempt erations fundamental content an fermion by motivated more the originally explain was to a work That on 6]. based [5, being different, h ih asv rvtnsaecaatrzdb the by characterized are gravitinos massive eight The h e xlnto ffrdi hsLte sentirely is Letter this in offered explanation new The h rpsdepaaint ok In work. to explanation proposed the r fcnre,tepeetrslscan results present the confirmed, If . rvtn niiaini h si’of ‘skin’ the in annihilation gravitino rycsi a UER vnsob- events (UHECR) ray cosmic ergy aino hc hyaebased. are they which on cation eprils oehrwt hi sta- their with together particles, se rgnt rmnurnsasisd a inside stars neutron from originate ypooe scniae o Dark for candidates as proposed ly nstein-Institut) 2 c × i rmtegoptertcanal- theoretic group the From ) SU(2) 2 1 udpa crucial a play ould N emos h rsn pro- present The fermions. 8sprrvt,following , =8 w × dhoc ad U(1) E K 10 ( E oafide bona utlike Just . em 10 otlts but postulates, fteconjec- the of ) sinet of assignments symme- N 8 = 2 1 2 ysis given in [12, 13] it follows that they transform as ticles. As participants in the strong interactions, they would nevertheless be in thermal equilibrium, with a very 1 1 2 2 2 3 , 3¯ , 1 , 1 , (1) small velocity dispersion of (∆v) kT/MPL. In con- 3 ⊕ −3 ⊕ 3 ⊕ −3 ∼         tradistinction to luminous matter (for which ∆v is very large), they would thus continue to slowly migrate to- under SU(3)c U(1)em [19]. Hence, unlike DM candi- wards the center of the star, especially if the latter devel- dates usually× considered (such as or WIMPs), ops a core of heavier nuclei, thereby avoiding the usual do these particles participate in strong and electromag- problem of ‘missing the center’ [15]. netic interactions, with coupling strengths of order (1). O At this stage the gravitino density is still so small (ii) All gravitinos are assumed to be supermassive with that annihilation processes can be neglected. Further- masses not too far from the (reduced) Planck mass more, because the color singlet gravitinos in (1) inter- 18 2 −9 MPL 2 10 GeV/c 4 10 kg (in a supersymmetric act only electromagnetically their annihilation cross sec- context∼ this· would correspond∼ · to Planck scale breaking tion is proportional to the inverse mass squared (σv of supersymmetry). (iii) The charge assignments (1) en- (πα2~2/(4M 2c) for small initial velocities) [21]. The sit-∼ sure that, despite their strong and electromagnetic inter- uation is entirely different for the strongly interacting actions with ordinary matter, the superheavy gravitinos gravitinos corresponding to the two color triplets in (1); are stable because there are no (confined or unconfined) as they are mainly responsible for the effect to be dis- fractionally charged final states in the SM into which they cussed here we will henceforth restrict attention to them could decay in a way compatible with SU(3) U(1) only. For strongly interacting particles the annihilation c × em cross section σ varies only very slowly with the energy and (1). Hence the only process that can lead to their √s, and can be approximated by the formula [16] disappearance is mutual annihilation, and this will be 2 the crucial effect considered here. As we will see, how- √s √s σβ 36 4 ln + 0.84 ln mb(2) ever, there will appear an essential difference between the h i ∼ " − ΛQCD ΛQCD # two kinds of gravitinos in (1), in that only the strongly      interacting gravitinos will contribute significantly to the with ΛQCD =0.4 GeV. This formula is non-perturbative annihiliation processes producing UHECR events. in the sense that it does not rely on a perturbative calcu- Both the large gravitino mass and the amount of ac- lation, but is based on fitting a general ansatz consistent cumulated mass of these particles, which is on the order with the Froissart bound with experimental data [16]. of the combined DM mass in the Universe, are neces- Putting √s = 2γmp and γ 1 we find σβ 32mb, a value that we will use below∼ (1 mb h 10i− ∼31m2 sary to understand the large energies and the rates of −2 ∼ ∼ the observed UHECR events, as we shall now explain. 2.5 GeV ). In addition we need to make one important further as- To estimate the present density ρ0 of strongly inter- sumption concerning the local distribution of DM in stel- acting (color triplet) gravitinos, we observe that with a lar systems. The average density of DM within a typical gravitino mass close to MPL, the usual requirement of 6 −3 thermal equilibrium reads galaxy is commonly given as ρDM 0.3 10 GeV m ∼ × · 2 (corresponding to one proton per cubic centimeter) [14]. π(kB T ) Extrapolating this number to Planck mass particles we Γ= ρ σv > H = 2 (3) h i 3√5~c MPL would get 10−13 gravitinos per cubic meter [20], hence an extremely∼ dilute gas of DM particles. Most of these Adopting from now on the usual unit conventions ~ = c = k = 1 (hence 2 10−7 eV m = 1), this translates will be color singlet gravitinos, whereas the strongly in- B · · teracting non-singlets make up only a small fraction of into an equation for the relic abundance ρT these, see below. Now in [6], we have already raised the 3/2 2 mT −m/T T possibility that DM, while more or less uniformly dis- (32 mb) ρT (32 mb) g e = (4) ≡ 2π 2MPL tributed in interstellar space, might be subject to larger   local variations near stars. This could happen if the DM (g = 4 for a massive gravitino), or co-rotates with the stars around the center of the galaxy, m 59 −3 but not relative to them, unlike the dust that gives rise 90 ρ 3 10 m (5) T T to planets and ends up rotating around (and not being ∼ ⇒ ∼ · 16 absorbed by) the star. Then, a typical star could eat The temperature T 2 10 GeV corresponds to cosmic 2 ∼ · −39 up much of the surrounding DM in its vicinity over its time tT = MPL/T 3 10 s. The present density ρ0 ∼ · lifetime, depleting a ball of diameter of a few lightyears is obtained from ρT by the well known formula around it. For a pre-supernova star and a volume of (two 3 3 49 3 aT lightyears) 10 m this would yield a total amount of ρ0 = ρT (6) ∼ 27 a0 DM within the star of . 10 kg, corresponding to a frac-   tion of . 10−3 of its total mass. Although seemingly non- 22 (since superheavy gravitinos are non-relativistic). Taking negligible it corresponds to only one gravitino per 10 the end of the radiation dominated era as 1012 s we get protons or helium nuclei. This number is far too small −39 1/2 12 2/3 to affect standard stellar processes in any significant way, aT 3 10 10 −29 = · 12 17 10 (7) especially since our gravitinos cannot decay into SM par- a0 10 3 10 ∼    ·  3 where the two factors correspond to the radiation domi- Here the strong coupling αs is to be evaluated at Λ 6 ∼ nated and matter dominated eras, respectively. Thus 0.35√s. Plugging in √s MPL gives (10 ) particles ∼ O −28 −3 −9 −3 per annihilation [22]. The total energy MPL will be ρ0 5 10 m 10 GeV m (8) distributed over all these particles, with∼ an average en- ∼ · ∼ · 13 22 Assuming now (as before) that the star ‘swallows’ all ergy of 10 GeV 10 eV per particle. This happens to  ∼ gravitinos within a radius of two lightyears we get for the be of the same order of magnitude as the maximum en- total number of color triplet gravitinos inside the star ergy observed in high energy cosmic rays! Nevertheless, because of their strong interactions and the large density N 2 1022 (9) g ∼ · inside the neutron star the annihilation products cannot As we argued above, the gravitinos inside the star are escape because they will either lose too much energy or in thermal, but not mechanical equilibrium, so for a pre- be stopped altogether on their way out from the core of supernova star we expect them to cluster more towards the neutron star. For this reason, we expect the main the iron core (where no nuclear reactions take place any- contribution to UHECR particles to come from the out- more). As already pointed out, the number (9) is too ermost shell of the neutron star of width d . (100 m) i.e. 3% of the volume. There the density dropsO down small to produce any significant effects in the star – even ∼ −5 in the iron core the lifetime of a gravitino still exceeds by a factor 10 relative to the core density [17], and is given by 1013 kg m−3 such that ρ(R d)=1040 m−3. the lifetime of the Universe, see below. ∼ − The situation changes dramatically if the star collapses (The width d is here determined by the requirement that to a neutron star. In that case, as explained above, most the number of collisions times the loss of energy per col- lision, here assumed to be (1GeV), should be much of the gravitinos will be contained in its iron core even ∼ O 12 prior to the supernova collapse, and the gravitinos will lower than the total energy of the proton of 10 GeV, which gives ρ(R d)σd < 1012, where σ 10−30 m2). collapse with the core due to the sudden increase of grav- − ∼ ititional pull towards the center. As a consequence, they Importantly, this outer shell is thought to be rich in will get squeezed into a ball of radius (10 km) [17], and heavier nuclei, and also iron nuclei [17], so the high en- their density increases to O ergy particles that can escape will ‘sweep up’ not just and , but also heavier ions before ex- 9 −3 ρNS 5 10 m . (10) iting the neutron star. Independently of subtleties of ∼ · strong interaction dynamics the ‘proto-nuclei’ generated This ‘compactification’ is absolutely crucial since the in this process will subsequently decay to stable isotopes, gravitinos need to be packed sufficiently closely to enable and Fe nuclei in particular, and thus end up as ultra-high them to annihilate in any appreciable rate. The inverse energy stable ions of the type observed by [1]. lifetime of the gravitino as a function of the neutron star Due to various uncertainties, however, it is not possible time from its birth is to give more precise estimates at this point. For instance, t ′ ′ the density of gravitinos near the skin may actually be ΓNS(t)= ρNS exp ΓNS(t )dt σv (11) − 0 h i enhanced by a ‘centrifuge effect’ for rapidly spinning neu-  Z  tron stars. For this reason we shall simply take the value which gives (10) to hold also near the skin of the neutron star, and ΓNS(0) assume that 3% of the neutron star volume is effectively ΓNS(t)= (12) available for this process. A young neutron star would 1+ΓNS(0)t thus continuously ‘spray’ high energy protons or heavy with the initial value (and σβ 32mb) ions at a rate h i∼ 9 −31 8 −1 −12 −1 22 −12 6 −1 16 −1 ΓNS(0) 5 10 (32 10 ) 3 10 s 5 10 s 6 0.03 (2 10 ) (5 10 ) 10 s 2 10 s (15) ∼ · · · · · ∼ · (13) ∼ · · · · · · ∼ · Therefore the actual annihilation rate depends on the age from its surface into outer space (the factor of 6 comes of the neutron star. We also see that before the collapse from the number of the strongly interacting gravitino for a pre-supernova star with an iron core of (1000 km) species). To calculate how many of these will eventually diameter the rate would be lower by a factorO of 10−12. reach Earth, we recall that, with an estimated average 8 7 Having derived the approximate annihilation∼ rate in- number of neutron stars per galaxy of 10 and 10 ∼ ∼ side the neutron stars we can now estimate the number galaxies within a GKZ horizon of 50 Mpc [3], we have 15 of UHECR particles coming from the annihilation. Be- a total number of 10 UHECR emitters. Denoting the cause the superheavy gravitinos interact strongly, each density of neutron stars in the universe by ρN (x) (where single annihilation will result in a violent burst of Planck x = 0 corresponds to the position of Earth), the total scale energy, producing a multitude of (mostly hadronic) rate arriving at Earth is thus particles. We can roughly estimate their multiplicity by 3 − ρ (x)d x extrapolating to Planckian energies the formula [18] N 2 1016s 1 N (16) E ∼ · × 4π x 2 Z | |  2.26 For a rough estimate of the total flux we neglect density multiplicity 0.27 αs(Λ) exp (14) ∼ αs(Λ)! variations, taking ρN = const, in which case the integral p 4 is easily evaluated to be rate that is not too far from the one observed. Evidently, there remain many uncertainties in our calculation, quite 16 −1 NE ρN Rmax 2 10 s (17) apart from questions concerning the viability of the uni- ∼ × · fication scenario proposed in [5]. Some of these are due Putting R 50Mpc as a cutoff we arrive at the flux max ∼ to our ignorance of the detailed dynamics, others are due of UHECR arriving on Earth as to untested extrapolations and yet others are due to our 15 16 insufficient knowledge of astrophysical data and difficult 10 2 10 −2 −1 −18 −2 −1 issues with strong interaction dynamics in neutron stars NE · 24· 2 m s 5 10 m s (18) ∼ 4(10 ) ∼ · (and their composition, density and age profile, in par- ticular). Nevertheless, we find it remarkable that the which is not too far off the observed rate of one UHECR present proposal could tie in with the scheme proposed event per month and per 3000 km2 [1]. To be sure, the in [5] to explain the fermion content of the SM, with three UHECR emitters are not evenly distributed throughout generations of quarks and leptons. the universe, and we therefore expect an increased num- ber of events to originate from superclusters of galaxies rich in neutron stars (the supergalactic plane, in partic- Acknowledgments: We are most grateful to Masaru ular, as also suggested by the data [1]). In particular Shibata and Kacper Zalewski for enlightening discus- the integral in (16) may receive its dominant contribu- sions. K.A. Meissner thanks AEI for hospitality and tion from a disk rather than the full ball. We also note support; he was partially supported by the Polish Na- that with a maximum available energy of (1022 eV) our tional Science Center grant DEC-2017/25/B/ST2/00165. proposal can also explain the existenceO of (very rare) The work of H. Nicolai has received funding from the UHECR events exceeding the GKZ bound, if these origi- European Research Council (ERC) under the Euro- nate from neutron stars within the Milky Way or nearby pean Union’s Horizon 2020 research and innovation pro- galaxies. gramme (grant agreement No 740209). We thus arrive at an explanation which agrees qual- itatively with observations, and at an estimated event

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