Birth of an Isotropic and Homogeneous Universe with a Running Cosmological Constant

The present work discusses the birth of the Universe via quantum tunneling through a potential barrier, based on quantum cosmology, taking a running cosmological constant into account. We consider the Friedmann–Lemaître–Robertson–Walker (FLRW) metric with positively curved spatial sections (k=1) and the matter’s content is a dust perfect fluid. The model was quantized by the Dirac formalism, leading to a Wheeler–DeWitt equation. We solve that equation both numerically and using a WKB approximation. We study the behavior of tunneling probabilities TPWKB and TPint by varying the energy E of the dust perfect fluid, the phenomenological parameter ν, the present value of the Hubble function H0, and the constant energy density ρΛ0, with the last three parameters all being associated with the running cosmological constant. We observe that both tunneling probabilities, TPWKB and TPint, decrease as one increases ν. We also note that TPWKB and TPint grow as E increases, indicating that the Universe is more likely to be born with higher dust energy E values. The same is observed for the parameter ρΛ0, that is, TPWKB and TPint are larger for higher values of ρΛ0. Finally, the tunneling probabilities decrease as one increases the value of H0. Therefore, the best conditions for the Universe to be born, in the present model, would be to have the highest possible values for E and Λ and the lowest possible values for ν and H0.