We explore the implications of a possible cosmic ray (CR) background generated during the first supernova explosions that end the brief lives of massive Population III stars. We show that such a CR background could have significantly influenced the cooling and collapse of primordial gas clouds in minihaloes around redshifts of z ~ 15 - 20, provided the CR flux was sufficient to yield an ionization rate greater than about 10^-19 s^-1 near the center of the minihalo. The presence of CRs with energies less than approximately 10^7 eV would indirectly enhance the molecular cooling in these regions, and we estimate that the resulting lower temperatures in these minihaloes would yield a characteristic stellar mass as low as ~ 10 M_sun. CRs have a less pronounced effect on the cooling and collapse of primordial gas clouds inside more massive dark matter haloes with virial masses greater than approximately 10^8 M_sun at the later stages of cosmological structure formation around z ~ 10 - 15. In these clouds, even without CR flux the molecular abundance is already sufficient to allow cooling to the floor set by the temperature of the cosmic microwave background.

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