Over the last few years , a large family of cosmological attractor models has been discovered , which can successfully match the latest inflation-related observational data .
Many of these models can also describe a small cosmological constant \Lambda , which provides the most natural description of the present stage of the cosmological acceleration .
In this paper , we study \alpha -attractor models with dynamical dark energy , including the cosmological constant \Lambda as a free parameter .
Predominantly , the models with \Lambda > 0 converge to the asymptotic regime with the equation of state w = -1 .
However , there are some models with w \neq - 1 , which are compatible with the current observations .
In the simplest models with \Lambda = 0 , one has the tensor to scalar ratio r = \frac { 12 \alpha } { N ^ { 2 } } and the asymptotic equation of state w = -1 + \frac { 2 } { 9 \alpha } ( which in general differs from its present value ) .
For example , in the seven disk M-theory related model with \alpha = 7 / 3 one finds r \sim 10 ^ { -2 } and the asymptotic equation of state is w \sim - 0.9 .
Future observations , including large-scale structure surveys as well as B-mode detectors will test these , as well as more general models presented here .
We also discuss gravitational reheating in models of quintessential inflation and argue that its investigation may be interesting from the point of view of inflationary cosmology .
Such models require a much greater number of e -folds , and therefore predict a spectral index n _ { s } that can exceed the value in more conventional models by about 0.006 .
This suggests a way to distinguish the conventional inflationary models from the models of quintessential inflation , even if they predict w = -1 .