Numerical Study of Wheeler-Dewitt Equation beyond Slow-roll approximation

The Wheeler-DeWitt (WDW) equation is analyzed using two boundary proposals: the Hartle-Hawking no-boundary condition and tunneling condition. By compactifying the scale factor $a$ into $x = a/(1+a)$, we reformulate the WDW equation to find stable numerical solutions with clearer boundary conditions. The no-boundary wave function peaks at the horizon scale, indicating quantum nucleation of classical spacetime, while the tunneling solution shows exponential decay, reflecting vacuum decay from a classically forbidden state. These dynamics are explored under slow-roll and non-slow-roll regimes of a periodic potential, separately, with non-slow-roll scenarios amplifying quantum effects that delay the classical behavior. The results emphasize the role of boundary conditions in quantum cosmology, offering insights into the universe's origin and the interplay between quantum gravity and observable cosmology.