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

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

2004-03-31

arxiv.org/abs/hep-ph/0403299v2

Phys.Rev. D70 (2004) 023510

Physics

High Energy Physics

High Energy Physics - Phenomenology

Minor changes, added a reference, version to appear in Phys. Rev. D

Scientific paper

10.1103/PhysRevD.70.023510

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.

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