Effect of Finite-Temperature $\beta$-Decay Rates on the Rapid Neutron Capture Process

$\beta$-decay is known to play an essential role in the rapid neutron capture process ($r$-process) during $(n, \gamma) \leftrightarrow (\gamma, n)$ equilibrium and freeze-out when the neutron-rich nuclei decay back to stability. Recent systematic theoretical studies on $\beta$-decay at finite temperature indicated that under hot conditions ($T\sim10$~GK), a significant acceleration of $\beta$-decay rates is expected, especially for nuclei near stability. This corresponds to the early stage of the $r$-process. In this study, we investigate the effect of the $\beta$-decays in finite temperature using the rates calculated with the finite-temperature proton-neutron relativistic quasiparticle random-phase approximation (FT-PNRQRPA). We explore a variety of astrophysical conditions and find that the effect on the abundance pattern is significant in hot and moderately neutron-rich conditions such as are expected in magnetorotational supernovae. Accelerated $\beta$-decay rates also increase the heating rate in the early phase, resulting in an additional modification of the final abundance pattern.