Quantum cosmologies with varying speed of light and the $\Lambda$   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 $\emph{P}\sim\exp(-\alpha^2/\Lambda)$ where $\alpha$=const and $\Lambda$ 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 $\Lambda$ that are large enough, whereas $\Lambda$ 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 $\emph{P}\sim 1$, $\rho_{_\Lambda}/\rho_c\sim 0.7$ ($\Lambda$-problem) and both horizon and flatness problems are solvable without inflation. 
A.V. Yurov, V.A. Yurov