Attractor Cosmology from non-minimally Coupled Gravity
  By using a bottom-up reconstruction technique for non-minimally coupled
scalar-tensor theories, we realize the Einstein frame attractor cosmologies in
the $\Omega (\phi)$-Jordan frame. For our approach, what is needed for the
reconstruction method to work is the functional form of the non-minimal
coupling $\Omega(\phi)$ and of the scalar-to-tensor ratio, and also the
assumption of the slow-roll inflation in the $\Omega (\phi)$-Jordan frame. By
appropriately choosing the scalar-to-tensor ratio, we demonstrate that the
observational indices of the attractor cosmologies can be realized directly in
the $\Omega (\phi)$-Jordan frame. We investigate the special conditions that
are required to hold true in for this realization to occur, and we provide the
analytic form of the potential in the $\Omega (\phi)$-Jordan frame. Also, by
performing a conformal transformation, we find the corresponding Einstein frame
canonical scalar-tensor theory, and we calculate in detail the corresponding
observational indices. The result indicates that although the spectral index of
the primordial curvature perturbations is the same in the Jordan and Einstein
frames, at leading order in the $e$-foldings number, the scalar-to-tensor ratio
differs. We discuss the possible reasons behind this discrepancy, and we argue
that the difference is due to some approximation we performed to the functional
form of the potential in the Einstein frame, in order to obtain analytical
results, and also due to the difference in the definition of the $e$-foldings
number in the two frames, which is also pointed out in the related literature.
Finally, we find the $F(R)$ gravity corresponding to the Einstein frame
canonical scalar-tensor theory.
