On The Injection Spectrum of Ultrahigh Energy Cosmic Rays in the Top-Down Scenario

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 DGLAP Q 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 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 parton shower development process to lowest order in perturbative QCD. Additionally, we find that for x >∼ 0.1, the evolution in Q 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 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 parametrisations of the FFs in the x region x >∼ 0.1 based on low energy data, without evolving them to the requisite Q value. [email protected] [email protected]