Quantum cosmologies with varying speed of light and the $\mathrm{\Lambda }$ 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 $\mathit{\text{P}}\sim \exp (-{\alpha }^2\mathrm{/}\mathrm{\Lambda })$ where $\alpha$=const and $\mathrm{\Lambda }$ is the cosmological constant, so once it nucleates, the universe immediately starts the de Sitter inflationary expansion. The probability $\mathit{\text{P}}$ will be large for values of $\mathrm{\Lambda }$ that are large enough, whereas $\mathrm{\Lambda }$ of our Universe is definitely small. Of course, for the early universe filled with radiation or another ‘‘matter’’ the mentioned probability is large nevertheless ($\mathit{\text{P}}\sim 1$) 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 $\mathit{\text{P}}\sim 1$, ${\rho }_{{}_{\mathrm{\Lambda }}}\mathrm{/}{\rho }_c\sim 0.7$ ($\mathrm{\Lambda }$-problem) and both horizon and flatness problems are solvable without inflation.