Injection spectrum of ultrahigh energy cosmic rays in the top-down scenario

Abstract

We analyze the uncertainties involved in obtaining the injection spectra of UHECR particles in the top-down scenario of their origin. We show that the Dokshizer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) Q/Sup2 evolution of fragmentation functions (FF) to Q=MX (mass of the X particle) from their initial values at low Q is subject to considerable uncertainties. We therefore argue that, for x≲0.1 (the x region of interest for most large MX values of interest, x≡2E/MX being the scaled energy variable), the FF obtained from the DGLAP evolution is no more reliable than that provided, for example, by a simple Gaussian form [in the variable ln(1/x)] obtained under the coherent branching approach to a parton shower development process to lowest order in perturbative QCD. Additionally, we find that for x≳0.1, the evolution in Q/Sup 2 of the singlet FF, which determines the injection spectrum, is “minimal”—the singlet FF changes by barely a factor of 2 after evolving it over ˜14 orders of magnitude in Q˜MX. We therefore argue that as long as the measurement of the UHECR spectrum above ˜10/Sup20 eV is going to remain uncertain by a factor of 2 or larger, it is good enough for most practical purposes to directly use any one of the available initial parametrizations of the FFs in the x region x≳0.1 based on low energy data, without evolving them to the requisite Q/Sup2 value.