Cosmological Attractors and Initial Conditions for Inflation
  Inflationary $\alpha$-attractor models in supergravity, which provide
excellent fits to the latest observational data, are based on the Poincare disk
hyperbolic geometry. We refine these models by constructing Kahler potentials
with built-in inflaton shift symmetry and by making a canonical choice of the
goldstino Kahler 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 $\varphi$, and an infinitely long dS valley of constant depth
and width at large $\varphi$. 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
$\varphi$ does not change much until all fields lose their energy and fall to
the bottom of the dS valley at large $\varphi$. This provides natural initial
conditions for inflation driven by the inflaton field slowly rolling along the
dS valley towards the minimum of the potential at small $\varphi$. 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.
