The possibility of a connection between dark energy and gravity through a direct coupling in the Lagrangian of the underlying theory has acquired an increasing interest due to the recently discovered capability of the extended quintessence model to encompass the fine-tuning problem of the cosmological constant. The gravity induced R-boost mechanism is indeed responsible for an early, enhanced scalar field dynamics, by virtue of which the residual imprint of a wide set of initial field values is cancelled out. The initial conditions problem is particularly relevant, as the most recent observations indicate that the dark energy equation of state approaches, at the present time, the cosmological constant value, wDE = -1; if confirmed, such observational evidence would cancel the advantage of a standard, minimally coupled scalar field as a Dark Energy candidate instead of the cosmological constant, because of the huge fine tuning it would require. We give here a general classification of the scalar-tensor gravity theories admitting R-boost solutions scaling as a power of the cosmological redshift, outlining those behaving as an attractor for the quintessence field. In particular, we show that all the R-boost solutions with the dark energy density scaling as the relativistic matter or shallower represent attractors. This analysis is exhaustive as for the classification of the couplings which admit R-boost and the subsequent enlargement of the basin of attraction enclosing the initial scalar field values.
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