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 $w eq -1$, which are compatible with the current observations. In the simplest models with $Lambda = 0$, one has the tensor to scalar ratio $r=frac{12alpha}{N^2}$ and the asymptotic equation of state $w=-1+frac{2}{9alpha}$ (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 $w = -1$.
Download