Topological densities in Einstein-scalar-Gauss-Bonnet gravity

The present work is devoted to studying the background dynamical evolution of a scalar field in Einstein-Gauss-Bonnet gravity in maximally symmetric space-time. This study is useful for giving meaning to the presence of two Gauss-Bonnet vacua, instead of using the spherically symmetric bubbles of the "true" vacuum expand in the "false" vacuum. The theory admits two possible effective cosmological constants, which lead to two maximally symmetric vacuum solutions. The first solution corresponds to the dynamics of dark energy. When there is matter, the second solution describes dark matter. In Einstein-Gauss-Bonnet gravity, we establish the expression of the topological mass spectrum which depends on the golden ratio and its inverse. In the Schwarzschild limit, the topological density corresponds to the standard model radiation energy density. We find the mass loss rate which gives the evolution of mass over time.