The Flatness Problem and Λ

By way of a complete integration of the Friedmann equations, in terms of observables, it is shown that for the cosmological constant Λ>0 there exist nonflat Friedmann-Lemaître-Robertson-Walker models for which the total density parameter Ω remains ∼1 throughout the entire history of the Universe. Further, it is shown that in a precise quantitative sense these models are not finely tuned. When observations are brought to bear on the theory, and, in particular, the Wilkinson microwave anisotropy probe observations, they confirm that we live in just such a universe. The conclusion holds when the classical notion of Λ is extended to dark energy.