Inflationary $\alpha$-attractor models in supergravity, which provide excellent fits to the latest observational data, are based on the Poincaré disk hyperbolic geometry. We refine these models by constructing Kähler potentials with built-in inflaton shift symmetry and by making a canonical choice of the Goldstino Kähler potential. The refined models are stable with respect to all scalar fields at all $\alpha$; no additional stabilization terms are required. The scalar potential $V$ has a nearly Minkowski minimum at small values of the inflaton field $\phi$ and an infinitely long de Sitter (dS) valley of constant depth and width at large $\phi$. Because of the infinite length of this shift-symmetric valley, the initial value of the inflaton field at the Planck density is expected to be extremely large. We show that the inflaton field $\phi$ does not change much until all fields lose their energy and fall to the bottom of the dS valley at large $\phi$. This provides natural initial conditions for inflation driven by the inflaton field slowly rolling along the dS valley toward the minimum of the potential at small $\phi$. A detailed description of this process is given for $\alpha$-attractors in supergravity, but we believe that our general conclusions concerning naturalness of initial conditions for inflation are valid for a broad class of inflationary models with sufficiently flat potentials.

• Received 20 July 2015

DOI:https://doi.org/10.1103/PhysRevD.92.063519