Nonminimal coupling and the cosmological constant problem

We consider a universe with a positive effective cosmological constant and a nonminimally coupled scalar field. When the coupling constant is negative, the scalar field exhibits linear growth at asymptotically late times, resulting in a decaying effective cosmological constant. The Hubble rate in the Jordan frame reaches a self-similar solution, $H=1/(\epsilon t)$, where the principal slow roll parameter $\epsilon$ depends on $\xi$, reaching maximally $\epsilon=2$ (radiation era scaling) in the limit when $\xi\rightarrow -\infty$. Similar results are found in the Einstein frame (E), with $H_E=1/(\epsilon_E t)$, but now $\epsilon_E \rightarrow 4/3$ as $\xi\rightarrow -\infty$. Therefore in the presence of a nonminimally coupled scalar de Sitter is not any more an attractor, but instead (when $\xi<-1/2$) the Universe settles in a decelerating phase. Next we show that, when the scalar field $\phi$ decays to matter with $\epsilon_m>4/3$ at a rate $\Gamma\gg H$, the scaling changes to that of matter, $\epsilon\rightarrow \epsilon_m$, and the energy density in the effective cosmological becomes a fixed fraction of the matter energy density, $M_{\rm P}^2\Lambda_{E\rm eff}/\rho_m={\rm constant}$, exhibiting thus an attractor behavior. While this may solve the (old) cosmological constant problem, it does not explain dark energy. Provided one accepts tuning at the $1\%$ level, the vacuum energy of neutrinos can explain the observed dark energy.