Generalizedα-Attractor Models from Elementary Hyperbolic Surfaces

Abstract

We consider generalized -attractor models whose scalar potentials are globally well-behaved and whose scalar manifolds are elementary hyperbolic surfaces. Beyond the Poincaré disk D, such surfaces include the hyperbolic punctured disk D * and the hyperbolic annuli A( ) of modulus = 2 log > 0. For each elementary surface, we discuss its decomposition into canonical end regions and give an explicit construction of the embedding into the Kerekjarto-Stoilow compactification (which in all three cases is the unit sphere), showing how this embedding allows for a universal treatment of globally well-behaved scalar potentials upon expanding their extension in real spherical harmonics. For certain simple but natural choices of extended potentials, we compute scalar field trajectories by projecting numerical solutions of the lifted equations of motion from the Poincaré half plane through the uniformization map, thus illustrating the rich cosmological dynamics of such models.

Hidden symmetries of two-field cosmological models

“…When the scalar manifold metric G is rotationally invariant, we showed that the two-field model admits a Hessian symmetry iff Σ is a disk, a punctured disk or an annulus and G is a complete metric of Gaussian curvature K = − 3 8 , i.e. iff the model is an elementary two-field α-attractor in the sense of reference [22], for the particular value α = 8/9 of the α-parameter. In all cases, we determined the explicit general form of the scalar potential V which is compatible with a given Hessian symmetry.…”

“…that it is isometric with the Poincaré disk D, with the hyperbolic punctured disk D * or with a hyperbolic annulus A(R) of modulus µ = 2 log R (where R > 1). We refer the reader to Appendix D and to reference [22] for the description of elementary hyperbolic surfaces. We will use the notations D β , D * β and A β (R) for the disk, punctured disk and annulus endowed with the metric G = 1 β 2 G of Gaussian curvature equal to −β 2 .…”

“…Section 4 gives the classification of weakly-Hessian models with rotationally-invariant scalar manifold metric, summarizing the results proved in Appendix C. This shows that the only scalar manifolds which are allowed in such models are the disk, punctured disk and annuli of constant Gaussian curvature K = − 3 8 . It follows that weakly-Hessian models with rotationally-invariant scalar manifold metric are elementary two-field α-attractors in the sense of reference [22], for the specific value α = 8 9 . We next proceed to determine the general form of the scalar potential of the corresponding Hessian models in each of the three cases.…”

“…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.…”

“…Finally, A(R) (where R > 1) indicates the Euclidean annulus of inner radius 1/R and outer radius R. The notations D and D * indicate the surfaces D andḊ when endowed with their unique hyperbolic metric, while A(R) indicates the annulus A(R), when the latter is endowed with its unique hyperbolic metric of modulus µ = 2 log R. The isometry group of an oriented Riemannian 2-manifold (Σ, G) is denoted by Iso(Σ, G), while its subgroup of orientation-preserving isometries is denoted by Iso + (Σ, G). We refer the reader to the Appendices and to references [21,22] for a summary of some relevant information about elementary hyperbolic surfaces.…”