The constant-roll inflation model in Barrow entropy

In this work, we study a constant-roll inflation model embedded in the Barrow entropy scenario. In this regards, we derive the modified of the Friedmann–Robertson–Walker (FRW) universe from the Barrow entropy using the first law of thermodynamics for the apparent horizon of the universe. We consider the inflation dynamics of the early universe under the constant-roll condition where the inflation is driven by a power-law scalar potential field, $V_{(\mathrm{\Phi })}={\mathrm{\Phi }}^n$. We calculated the tensor-to-scalar ratio $r$ and scalar spectral index $n_s$ by applying the constant-roll condition with some other parameters and compared them with the Planck 2020 observable data. To reveal the effect of the Barrow parameter $\mathrm{\Delta }$ on the inflation, we fixed the constant-roll inflation parameter as $\gamma =0.014$ and focused on the exponent of potential in the range $0<n<1$. We observed that as the value of the Barrow parameter approaches zero in the range $0\le \mathrm{\Delta }\le 1$, a long and sufficient inflation occurs, consistent with the observation data. This strengthens the claim that the observable values of the Barrow parameter occur at very small values. In addition, the obtained results were also examined numerically.