Tracking extended quintessence

Abstract

We investigate the cosmological role of a tracking field $\ensuremath{\varphi}$ in extended quintessence scenarios, where the dynamical vacuum energy driving the acceleration of the universe today possesses an explicit coupling with the Ricci scalar R of the form $F(\ensuremath{\varphi})R/2,$ where $F(\ensuremath{\varphi})$ mimics general relativity today, $F({\ensuremath{\varphi}}_{0})=1/8\ensuremath{\pi}G.$ We analyze explicit nonminimally coupled (NMC) models where $F(\ensuremath{\varphi})=1/8\ensuremath{\pi}G+\ensuremath{\xi}({\ensuremath{\varphi}}^{2}\ensuremath{-}{\ensuremath{\varphi}}_{0}^{2}),$ with $\ensuremath{\xi}$ is the coupling constant and ${\ensuremath{\varphi}}_{0}$ is the Q value today. Tracker solutions for these NMC models, with inverse power-law potentials, possess an initial enhancement of the scalar field dynamics, named the R-boost, caused by the effective potential generated by the Ricci scalar in the Klein-Gordon equation. During this phase the field performs a ``gravitational'' slow rolling until the true potential becomes important. We give accurate analytic formulas describing the R-boost, showing that the quintessence energy in this phase scales with the redshift z as ${(1+z)}^{2}.$ When the R-boost ends, the field trajectory matches the tracker solution in minimally coupled theories. We compute perturbations in these tracking extended quintessence models, by integrating the full set of equations for the evolution of linear fluctuations in scalar-tensor theories of gravity, and assuming Gaussian scale-invariant initial perturbations. The integrated Sachs-Wolfe (ISW) effect on the cosmic microwave background (CMB) angular spectrum causes a change $\ensuremath{\delta}{C}_{l}{/C}_{l}\ensuremath{\simeq}6[1\ensuremath{-}8\ensuremath{\pi}GF({\ensuremath{\varphi}}_{\mathrm{dec}})]$ at $l\ensuremath{\lesssim}10,$ where ``dec'' stands for decoupling. Similarly, the CMB acoustic peak multipoles shift compared to ordinary tracking quintessence models by roughly an amount $\ensuremath{\delta}l/l\ensuremath{\simeq}[8\ensuremath{\pi}GF({\ensuremath{\varphi}}_{\mathrm{dec}})\ensuremath{-}1]/8.$ The turnover wave number ${k}_{\mathrm{turn}}$ in the matter power spectrum shifts by an amount $\ensuremath{\delta}{k}_{\mathrm{turn}}{/k}_{\mathrm{turn}}\ensuremath{\simeq}[1\ensuremath{-}8\ensuremath{\pi}GF({\ensuremath{\varphi}}_{\mathrm{eq}})]/2,$ where ``eq'' stands for matter-radiation equivalence. All these corrections may assume positive as well as negative values, depending on the sign of the NMC parameter $\ensuremath{\xi}.$ We show that the above effects can be as large as 10--30 % with respect to equivalent cosmological constant and ordinary tracking quintessence models, respecting all the existing experimental constraints on scalar-tensor theories of gravity. These results demonstrate that the playground where the data of the next decade will have their impact includes the nature of the dark energy in the Universe, as well as the structure of the theory of gravity.