Quantum Creation of a Universe with varying speed of light: $\Lambda$-problem and Instantons

This article is dedicated to cosmologies with variable speed of light (VSL) - models, which one can consider as a particular case of models of a modified gravitation. In quantum cosmology the closed universe can spontaneously nucleate out of the state with no classical space and time. As known, in case of classical cosmology with varying speed of light $c(t)$ it is possible to solve the horizon problem, the flatness problem and the $\Lambda$-problem if $c=sa^n$ with $s$=const and $n<-2$. We show that in VSL quantum cosmology with $n<-2$ the semiclassical tunneling nucleation probability is $\emph{P}\sim\exp(-\beta^2\Lambda^k)$ with $\beta$=const and $k>0$. Thus, the semiclassical tunneling nucleation probability in VSL quantum cosmology is very different from that in quantum cosmology with $c$=const. In particular, it can be strongly suppressed for large values of $\Lambda$. In addition, we propose two instantons that describe the nucleation of closed universes in VSL models. These solutions are akin to the Hawking-Turok instanton in sense of O(4) invariance but, unlike to it, are both non-singular. Moreover, using those solutions we can obtain the probability of nucleation which is suppressed for large value of $\Lambda$ too. We also discuss some unusual properties of models with $n>0$.