# Generalized $α$-attractor models from geometrically finite hyperbolic surfaces [CL]

We consider four-dimensional gravity coupled to a non-linear sigma model whose scalar manifold is a geometrically finite hyperbolic surface $\Sigma$, which may be non-compact and may have finite or infinite area. 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 $\Gamma\subset \mathrm{PSL}(2,\mathbb{R})$ (which is isomorphic with the fundamental group of $\Sigma$) and by the choice of a scalar potential defined on $\Sigma$. The traditional $\alpha$-attractor models arise when $\Gamma$ is the trivial group, in which case $\Sigma$ is the Poincar\'{e} disk. When $\Sigma$ is non-compact, we show that our generalized models have the same universal behavior as ordinary $\alpha$-attractors if inflation happens near any of the Freudenthal ends of $\Sigma$, for trajectories which are well approximated by non-canonically parameterized geodesics near the ends. We also discuss some aspects of these models in the SRST approximation and give a general prescription for their study through uniformization in the non-elementary case. Our generalized models can sustain multipath inflation starting near any of the ends of $\Sigma$ and proceeding toward the compact core. They illustrate interesting consequences of nonlinear sigma models whose scalar manifold is not simply connected and provide a large class of tractable cosmological models with non-trivial topology of the scalar field space.

C. Lazaroiu and C. Shahbazi
Wed, 22 Feb 17
35/37