The effects of dark energy on the early Universe with radiation and an ad hoc potential G. A. Monerat, F. G. Alvarenga, S. V. B. Gonçalves, G. Oliveira-Neto, C. G. M. Santos and E. V. Corrêa Silva Eur. Phys. J. Plus, 137 1 (2022) 117 Published online: 13 January 2022 DOI: 10.1140/epjp/s13360-021-02316-9