Waterfall-modulated $\alpha$-attractors

Abstract

Hybrid $\alpha$-attractor models \cite{Kallosh:2022ggf} can have significantly greater values of $n_{s}$ and smaller $r$, while preserving the relation $r\cong 3\alpha (1-n_s)^2$, which is valid for exponential T- and E-models at large values of the inflaton field. Here we study single-field $\alpha$-attractors with features inspired by hybrid models: one can uplift the potential, and one can also have a waterfall regime that leads to a premature termination of inflation near the critical point $\varphi_c$. This allows one to increase the effective number of e-foldings $N_c$ in formulas like $n_s\simeq 1-{2\over N_c}$, $r\simeq {12 \alpha\over N_c^2}$. By changing the waterfall's steepness and location, one can continuously move the predictions along the curves with $r\cong 3\alpha (1-n_s)^2$ as $n_s$ increases and $r$ decreases. We also study the effect of waterfall insertions and uplift on $n_s$ in quintessential $\alpha$-attractors that describe inflation and dynamical dark energy.