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General limit on the relation between abundances of D and $^7$Li in big bang nucleosynthesis with nucleon injections

The injections of energetic hadrons could have occurred in the early universe by decays of hypothetical long-lived exotic particles. The injections induce the showers of nonthermal hadrons via nuclear scattering. Neutrons generated at these events can react with $^7$Be nuclei and reduce $^7$Be abundance solving a problem of the primordial $^7$Li abundance. We suggest that thermal neutron injection is a way to derive a model independent conservative limit on the relation between abundances of D and $^7$Li in a hadronic energy injection model. We emphasize that an uncertainty in cross sections of inelastic $n+p$ scattering affects the total number of induced neutrons, which determines final abundances of D and $^7$Li. In addition, the annihilations of antinucleons with $^4$He result in higher D abundance and trigger nonthermal $^6$Li production. It is concluded that a reduction of $^7$Li abundance from a value in the standard big bang nucleosynthesis (BBN) model down to an observational two $σ$ upper limit is necessarily accompanied by an undesirable increase of D abundance up to at least an observational 12 $σ$ upper limit from observations of quasi-stellar object absorption line systems. The effects of antinucleons and secondary particles produced in the hadronic showers always lead to a severer constraint. The BBN models involving any injections of extra neutrons are thus unlikely to reproduce a small $^7$Li abundance consistent with observations.