It has been recently shown that the competition between unscreened Coulomb and Fr{o}hlich electron-phonon interactions can be described in terms of a short-range spin exchange $J_p$ and an effective on-site interaction $tilde{U}$ in the framework of the polaronic $t$-$J_p$-$tilde{U}$ model. This model, that provides an explanation for high temperature superconductivity in terms of Bose-Einstein condensation (BEC) of small and light bipolarons, is now studied as a charged Bose-Fermi mixture. Within this approximation, we show that a gap between bipolaron and unpaired polaron bands results in a strong suppression of low-temperature spin susceptibility, specific heat and tunneling conductance, signaling the presence of normal state pseudogap without any assumptions on preexisting orders or broken symmetries in the normal state of the model.
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