Non-linear Laplace equation, de Sitter vacua and information geometry

Three exact solutions say ${\varphi }_0$ of massless scalar theories on Euclidean space, i.e. $D=6{\varphi }^3$, $D=4{\varphi }^4$ and $D=3{\varphi }^6$ models are obtained which share similar properties. The information geometry of their moduli spaces coincide with the Euclidean ${AdS}_7$, ${AdS}_5$ and ${AdS}_4$ respectively on which ${\varphi }_0$ can be described as a stable tachyon. In D=4 we recognize that the SU(2) instanton density is proportional to ${\varphi }_0^4$. The original action $S[\varphi ]$ written in terms of new scalars $\tilde{\varphi }=\varphi -{\varphi }_0$ is shown to be equivalent to an interacting scalar theory on $D$-dimensional de Sitter background.