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Quantum cosmologies with varying speed of light and the $Λ$ problem

In quantum cosmology the closed universe can spontaneously nucleate out of the state with no classical space and time. For the universe filled with a vacuum of constant energy density the semiclassical tunneling nucleation probability can be estimated as $\emph{P}\sim\exp(-α^2/Λ)$ where $α$=const and $Λ$ is the cosmological constant, so once it nucleates, the universe immediately starts the de Sitter inflationary expansion. The probability $\emph{P} $ will be large for values of $Λ$ that are large enough, whereas $Λ$ of our Universe is definitely small. Of course, for the early universe filled with radiation or another ''matter'' the mentioned probability is large nevertheless ($\emph{P}\sim 1$) but in this case we have no inflation which is a standard solution for the flatness and horizon problems. In the other hand, the alternative solution of these problems can be obtained in framework of cosmologies with varying speed of light $c(t)$ (VSL). We show that, as a matter of principle, such quantum VSL cosmologies exist that $\emph{P}\sim 1$, $ρ_{_Λ}/ρ_c\sim 0.7$ ($Λ$-problem) and both horizon and flatness problems are solvable without inflation.