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 $Q^2$ evolution of fragmentation functions (FF) to $Q=M_X$ (mass of the X particle) from their initial values at low $Q$ is subject to considerable uncertainties. We therefore argue that, for $x\lsim 0.1$ (the $x$ region of interest for most large $M_X$ values of interest, $x\equiv 2E/M_X$ 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 Modified Leading Log Approximation (MLLA). Additionally, we find that for $x\gsim0.1$, the evolution in $Q^2$ 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 $\sim$ 14 orders of magnitude in $Q\sim M_X$. We, therefore, argue that as long as the measurement of the UHECR spectrum above $\sim10^{20}\ev$ 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 $x\gsim0.1$ based on low energy data even without evolving them to the requisite $Q^2$ value.