Quantum Creation of a Friedmann-Robertson-Walker Universe: Riesz Fractional Derivative Applied

In this work, we apply fractional calculus to study quantum cosmology. Specifically, our Wheeler-DeWitt (WDW) equation includes a Friedman-Robertson-Walker (FRW) geometry, a radiation fluid, a positive cosmological constant (Λ), and an ad-hoc potential. We employ the Riesz fractional derivative, which introduces a parameter α, where 1<α≤2, in the WDW equation. We investigate numerically the tunneling probability for the Universe to emerge using a suitable WKB approximation. Our findings are as follows. When we decrease the value of α, the tunneling probability also decreases, suggesting that if fractional features could be considered to ascertain among different early universe scenarios, then the value α=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 Λ (in a non-fractional conventional set-up) versus balancing, e.g., both Λ and α in the fractional framework.