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, $H=1/(εt)$, where the principal slow roll parameter $ε$ depends on $ξ$, reaching maximally $ε=2$ (radiation era scaling) in the limit when $ξ\rightarrow -\infty$. Similar results are found in the Einstein frame (E), with $H_E=1/(ε_E t)$, but now $ε_E \rightarrow 4/3$ as $ξ\rightarrow -\infty$. Therefore in the presence of a nonminimally coupled scalar de Sitter is not any more an attractor, but instead (when $ξ<-1/2$) the Universe settles in a decelerating phase. Next we show that, when the scalar field $ϕ$ decays to matter with $ε_m>4/3$ at a rate $Γ\gg H$, the scaling changes to that of matter, $ε\rightarrow ε_m$, and the energy density in the effective cosmological becomes a fixed fraction of the matter energy density, $M_{\rm P}^2Λ_{E\rm eff}/ρ_m={\rm constant}$, 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.

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