Quantum mechanics forbids singularities in a closed radiation-dominated FRW universe

The existence of singularities in a closed FRW universe depends on the assu mption that general relativity is valid for distances less than the Planck length. However, the stationary state wave functions of the Schrodinger equ ation for a closed radiation-dominated FRW universe derived by Elbaz et al. are zero at zero radius of curvature. Thus, even if general relativity is assumed valid at distances less than the Planck length, quantum mechanics s eems to forbid singularities in a closed radiation-dominated FRW universe.