Quantum Creation of a FRW Universe: applying the Riesz fractional derivative

Abstract

In this work, we apply fractional calculus to study quantum cosmology. Specifically, our Wheeler-DeWitt equation includes a FRW geometry, a radiation fluid, a positive cosmological constant, and an ad hoc potential; we employ the Riesz fractional derivative, which brings a parameter $\alpha$, where $1 < \alpha \leq 2$, appearing explicitly in the mentioned equation. We investigate numerically the tunnelling probability for the Universe to emerge using a suitable WKB approximation. Our findings are as follows. When we decrease the value for $\alpha$, the tunnelling probability also decreases, suggesting that if fractional features could be considered to ascertain among different early universe scenarios, then the value $\alpha=2$ (meaning strict locality and standard cosmology) would be the most likely. Finally, our results also allow for an interesting discussion between selecting values for $\Lambda$ (in a non-fractional conventional set-up) versus balancing, e.g., both $\Lambda$ and $\alpha$ in the fractional framework. Concretely, the probability transition in the former if, e.g., $\Lambda=0.7$, is very close to the value computed if in the latter we employ instead, e.g., $\Lambda=1.5$ and $\alpha=1.9397961$.