Two-Field Cosmological Models and the Uniformization Theorem

Abstract

We propose a class of two-field cosmological models derived from gravity coupled to non-linear sigma models whose target space is a noncompact and geometrically-finite hyperbolic surface, which provide a wide generalization of so-called α-attractor models and can be studied using uniformization theory. We illustrate cosmological dynamics in such models for the case of the hyperbolic triply-punctured sphere.

Cosmological flows on hyperbolic surfaces

“…The cosmological flow of such models can already be very intricate, especially when (Σ, G) has finite hyperbolic area. Numerical studies as well as arguments based on the gradient flow approximation indicate [11,12,15] that such models can be compatible with current observational constraints.…”

Two-field cosmological α-attractors with Noether symmetry

“…Most of the time, the equations of motion of two-field cosmological models are solved numerically in the literature. See, in particular, [15,16,17] for such numerical investigations in two-field α-attractor models. Our goal here will be to find exact solutions by imposing the requirement that the model possesses a Noether symmetry.…”

“…that the universal properties of the original one-field α-attractors persist under certain conditions. Specific examples of generalized two-field α-attractors were explored in more detail in [15,16,17]. In particular, [15] studied α-attractors whose scalar manifold is an elementary hyperbolic surface, i.e.…”

Hidden symmetries of two-field cosmological models

“…When G is rotationally invariant 2 , we show that the model is weakly Hessian iff Σ is diffeomorphic to a disk, a punctured disk or an annulus and G is a metric of constant Gaussian curvature K = − 3 8 . In particular, weakly-Hessian two-field cosmological models coincide with those elementary two-field α-attractor models (in the sense of reference [22]) for which α = 8 9 (here and in the following we use the convention K = − 1 3α ), being special examples of the much wider class of two-field generalized α-attractors introduced and studied in [21][22][23][24] 3 . We show that such weakly-Hessian models are Hessian iff their scalar potential has a specific form which we determine explicitly in all cases, thereby classifying all Hessian two-field models with rotationally invariant scalar manifold metric.…”

“…Notice that the Hesse symmetry generating function Λ is not assumed to be rotationally symmetric, even though we assume that the scalar manifold metric is.3 Generalized two-field α-attractors extend ordinary two-field α-attractor models[25][26][27][28][29][30][31][32] (whose target space is the hyperbolic disk) to models whose target space is allowed to be an arbitrary complete hyperbolic surface. As explained in[21][22][23][24], such models can be approached through uniformization theory, which relates them to the framework of modular inflation developed and studied in[33][34][35][36][37].…”