Generalized two-field α-attractor models from geometrically finite hyperbolic surfaces

We consider four-dimensional gravity coupled to a non-linear sigma model whose scalar manifold is a non-compact geometrically finite surface Sigma endowed with a Riemannian metric of constant negative curvature. When the space-time is an FLRW universe, such theories produce a very wide generalization of two-field alpha-attractor models, being parameterized by a positive constant alpha, by the choice of a finitely-generated surface group Gamma subset of PSL(2, R) (which is isomorphic with the fundamental group of Sigma) and by the choice of a scalar potential defined on Sigma. The traditional two-field alpha-attractor models arise when Gamma is the trivial group, in which case Sigma is the Poincare disk. We give a general prescription for the study of such models through uniformization in the so-called "non-elementary" case and discuss some of their qualitative features in the gradient flow approximation, which we relate to Morse theory. We also discuss some aspects of the SRST approximation in these models, showing that it is generally not well-suited for studying dynamics near cusp ends. When Sigma is non-compact and the scalar potential is "well-behaved" at the ends, we show that, in the naive local one-field truncation, our generalized models have the same universal behavior as ordinary one-field alpha-attractors if inflation happens near any of the ends of Sigma where the extended potential has a local maximum, for trajectories which are well approximated by non-canonically parameterized geodesics near the ends; we also discuss spiral trajectories near the ends. Generalized two field alpha-attractors illustrate interesting consequences of nonlinear sigma models whose scalar manifold is not simply connected. They provide a large class of tractable cosmological models with non-trivial topology of the scalar field space. (C) 2018 The Author(s). Published by Elsevier B.V.