Primordial Pairing and Binding of Superheavy Charged Particles in the Early Universe¦ V
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JETP Letters, Vol. 77, No. 7, 2003, pp. 335–338. From Pis’ma v Zhurnal Éksperimental’noÏ i TeoreticheskoÏ Fiziki, Vol. 77, No. 7, 2003, pp. 403–406. Original English Text Copyright © 2003 by Dubrovich, Khlopov. Primordial Pairing and Binding of Superheavy Charged Particles in the Early Universe¶ V. K. Dubrovich*, 1 and M. Yu. Khlopov**, ***, **** * Special Astrophysical Observatory, Russian Academy of Sciences, St. Petersburg, 196140 Russia 1 e-mail:[email protected] ** Keldysh Institute of Applied Mathematics, Russian Academy of Sciences, Moscow, 125047 Russia *** Moscow Engineering Physics Institute, Moscow, 115409 Russia **** Physics Department, Università degli studi “La Sapienza,” I 00185 Roma, Italy Received July 10, 2002; in final form, February 18, 2003 Primordial superheavy particles, considered as the source of Ultra High Energy Cosmic Rays (UHECR) and produced in local processes in the early Universe, should bear some strictly or approximately conserved charge to be sufficiently stable to survive to the present time. Charge conservation dictates that they be produced in pairs, and the estimated separation of particle and antiparticle in such a pair is shown to be in some cases much smaller than the average separation determined by the averaged number density of considered particles. If the new U(1) charge is the source of a long-range field similar to the electromagnetic field, the particle and antipar- ticle, possessing that charge, can form a primordial bound system with an annihilation timescale, which can satisfy the conditions assumed for this type of UHECR source. These conditions severely constrain the possible properties of the considered particles. © 2003 MAIK “Nauka/Interperiodica”. PACS numbers: 98.70.Sa; 98.80.Cq The origin of cosmic rays with energies exceeding If the considered particles are absolutely stable, the the GZK cutoff energy [1] is widely discussed, and one source of UHECRs is related with their annihilation in of the popular approaches is related to decays or anni- the Galaxy. But their averaged number density, con- hilation in the Galaxy of primordial superheavy parti- strained by the upper limit on their total density, is so cles [2–5] (see [5] for review and references therein). low that a strongly inhomogeneous distribution is The mass of such particles is assumed to be higher than needed to enhance the effect of annihilation to the level the reheating temperature of the inflationary Universe, desired to explain the origin of UHECR by this mecha- so it is assumed that the particles are created in some nism. nonequilibrium processes, such as inflation decay [6] at In the present note, we offer a new approach to the the stage of preheating after inflation [7]. solution of the latter problem. If superheavy particles possess new U(1) gauge charge, related to the hidden The problems related to this approach are as fol- sector of particle theory, they are created in pairs. The lows. If the source of ultrahigh-energy cosmic rays Coulomb-like attraction (mediated by the massless (UHECR) is related with particle decay in the Galaxy, U(1) gauge boson) between particles and antiparticles the timescale of this decay necessary to reproduce the in these pairs can lead to their primordial binding, so UHECR data needs a special nontrivial explanation. that the annihilation in the bound system provides the Indeed, the relic unstable particle should survive to the mechanism for UHECR origin. present time and, having mass m on the order of Note, first of all, that in quantum theory particle sta- 1014 GeV or larger, should have the lifetime τ, exceed- bility reflects the conservation law, which according to ing the age of the Universe. On the other hand, even if Noether’s theorem is related with the existence of a particle decay is due to gravitational interaction and its conserved charge possessed by the particle under con- τ 4 probability is on the order of 1/ = (m/mPl) m, where sideration. Charge conservation implies that a particle 19 mPl = 10 GeV is the Planck mass, the estimated life- should be created together with its antiparticle. This time would be many orders of magnitude smaller. This means that, being stable, the considered superheavy implies some specific additional suppression factors in particles should bear a conserved charge, and such the probability of decay, which need a rather nontrivial charged particles should be created in pairs with their physical realization [2, 5], e.g., in the model of cryptons antiparticles at the stage of preheating. [8] (see [9] for review). Being created in the local process of inflation field decay, the pair is localized within the cosmological ¶This article was submitted by the authors in English. horizon in the period of creation. If the momentum dis- 0021-3640/03/7707-0335 $24.00 © 2003 MAIK “Nauka/Interperiodica” 336 DUBROVICH, KHLOPOV tribution of created particles peaks below p ~ mc, they If the considered charge is the source of a long- do not spread beyond the proper region of their original range field similar to the electromagnetic field, which localization, being in the period of creation l ~ c/H, can bind a particle and antiparticle into an atomlike sys- where the Hubble constant H at the preheating stage is tem analogous to positronium, it may have important ≤ ≤ in the range Hr H Hend. Here, Hend is the Hubble con- practical implications for the UHECR problem. The stant at the end of inflation and Hr is the Hubble con- annihilation timescale of such a bound system can pro- stant in the period of reheating. For relativistic pairs, vide a rate of UHE particle sources corresponding to the region of localization is determined by the size of UHECR data. the cosmological horizon in the period of their derela- tivization. In the course of successive expansion, the A pair of a particle and antiparticle with opposite distance l between particles and antiparticles grows gauge charges forms a bound system when, in the course of expansion, the absolute magnitude of the with the scale factor, so that, after reheating at the tem- α perature T, it is equal to (here and further, if not indi- potential energy of the pair V = y/l exceeds the kinetic 2 cated otherwise, we use the units ប = c = k = 1) energy of the particle’s relative motion Tk = p /2m. The mechanism is similar to that proposed in [12] for the m 1/2 1 lT()= -------Pl- --- . (1) binding of magnetic monopole–antimonopole pairs. It H T is not a recombination one. The binding of two oppo- sitely charged particles is caused just by their Cou- The averaged number density of superheavy parti- lomb-like attraction, once it exceeds the kinetic energy cles n is constrained by the upper limit on their modern of their relative motion. density. If we take their maximum possible contribu- Ω tion in units of critical density, X, not to exceed 0.3, In case plasma interactions do not heat superheavy the modern cosmological average number density particles created with relative momentum p ≤ mc in the ≥ should be period corresponding to Hubble constant H Hs, their 14 Ω initial separation, being on the order of –2210 GeV 3 n = 410× ----------------------- -------X T . m 0.3 p lH()= --------- , (3) This corresponds at the temperature T to a mean dis- mH –1/3 tance (ls ~ n ) equal to 1/3 1/3 experiences only the effect of general expansion, pro- 7m 0.3 1 l ≈ 1.6× 10 ----------------------- ------- --- . (2) portional to the inverse first power of the scale factor, s 14 Ω 10 GeV X T while the initial kinetic energy decreases as the square of the scale factor. Thus, the binding condition is ful- 1/3 1/3 6m 0.3 (l ≈ 4.6× 10 ----------------------- ------- cm filled in the period corresponding to the Hubble con- s 14 Ω 10 GeV X stant Hc, determined by the equation in ordinary units). H 1/2 p3 One finds that superheavy nonrelativistic particles ------ = --------------------2 , (4) Hc 2m α H created just after the end of inflation, when H ~ Hend ~ y 1013 GeV, are separated from their antiparticles by dis- where H is the Hubble constant in the period of particle tances more than 4 orders of magnitude smaller than the α average distance between these pairs. On the other creation and y is the “running constant” of the long hand, if the nonequilibrium processes of superheavy range U(1) interaction possessed by the superheavy particle creation (such as decay of inflation) take place particles. If the local process of pair creation does not at the end of the preheating stage and the reheating tem- involve nonzero orbital momentum, due to primordial perature is as low as it is constrained from the effects of pairing the bound system is formed in the state with gravitino decays on 6Li abundance (T < 4 × 106 GeV zero orbital momentum. The size of the bound system r strongly depends on the initial momentum distribution [10, 11]), the primordial separation of pairs given by ≤ Eq. (1) can exceed the value given by Eq. (2). This and for p mc is equal to means that the separation between particles and anti- 4 α p y particles can be determined in this case by their aver- l ==-----------------------2----------, (5) aged density, if they were created at c α 3 2 β2 2 ym H m 14 2/3 Ω 2/3 ≤ ∼ –15 10 GeV X HHs 10 mPl ----------------------- ------- where m 0.3 Ω 2/3 α 4 2 ymH ∼ 10 -------X GeV. β = ------------------ . (6) 0.3 p2 JETP LETTERS Vol. 77 No. 7 2003 PRIMORDIAL PAIRING AND BINDING OF SUPERHEAVY CHARGED PARTICLES 337 The annihilation timescale of this bound system can be results of the classical problem of massive particles estimated from the annihilation rate given by with opposite electric charges falling down the center due to radiation in the bound system.