Advances in Mathematical Physics

Volume 2018 (2018), Article ID 7323090, 24 pages

https://doi.org/10.1155/2018/7323090

## Generalized -Attractor Models from Elementary Hyperbolic Surfaces

Center for Geometry and Physics, Institute for Basic Science, Pohang 37673, Republic of Korea

Correspondence should be addressed to Elena Mirela Babalic; or.enpin.yroeht@cilababm

Received 15 October 2017; Accepted 31 January 2018; Published 19 March 2018

Academic Editor: Luigi C. Berselli

Copyright © 2018 Elena Mirela Babalic and Calin Iuliu Lazaroiu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 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 , such surfaces include the hyperbolic punctured disk and the hyperbolic annuli of modulus . 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.

#### 1. Introduction

In [1], we introduced a class of cosmological models which form a wide generalization of -attractors [2–7]. Such models are defined by cosmological solutions (with flat spatial section) of the Einstein theory of gravity coupled to two real scalar fields, the dynamics of the latter being described by a nonlinear sigma model whose scalar manifold is a noncompact, oriented, and topologically finite borderless surface endowed with a complete metric of constant Gaussian curvature . The scalar potential is given by a smooth real-valued function defined on . The rescaled scalar field metric has Gaussian curvature equal to ; hence is a geometrically finite hyperbolic surface in the sense of [8]. Using the uniformization theorem of Koebe and Poincaré [9] and the theory of surface groups (Fuchsian groups without elliptic elements), [1] discussed the basic features of cosmological dynamics and inflation in such models.

In the present paper, we consider the nongeneric situation when is elementary, focusing on special features which arise in this case. As explained in [1], an elementary hyperbolic surface is isometric with the hyperbolic disk , with the hyperbolic punctured disk or with the hyperbolic annulus of modulus , where is a real number. Each of these surfaces is planar, having the unit sphere as its end compactification. This allows one to parameterize globally well-behaved [1] scalar potentials through the coefficients of the Laplace expansion of their global extension to the end compactification, which in this case reduces to an expansion in real spherical harmonics. Both the hyperbolic metric and a fundamental polygon are known explicitly for and . In particular, one can describe explicitly the hyperbolic geometry of such surfaces and one can compute scalar field trajectories on by determining trajectories of an appropriate lift of the model to the Poincaré half plane and projecting them to or to through the uniformization map. When the scalar potential is constant, this illustrates how the different projections from to and can take the same trajectory on the Poincaré half plane into qualitatively different trajectories on the corresponding elementary hyperbolic surface. The explicit embedding of , , and into their common Kerekjarto-Stoilow compactification is also different. As a consequence, a smooth real-valued function defined on will generally induce rather different globally well-behaved scalar potentials on the disk, the punctured disk, and the annulus. When combined with the different projections from , this leads to qualitatively different dynamics on , , and . By considering some simple but natural choices of extended scalar potentials on , we use numerical methods to compute various trajectories on and , thus illustrating the rich dynamics of such models, which is not entirely visible in the gradient flow approximation.

The paper is organized as follows. Section 2 briefly recalls the definition of generalized -attractor models and the lift of their cosmological evolution equation to the Poincaré half plane. In Section 3, we consider globally well-behaved scalar potentials on topologically finite planar oriented surfaces (of which the elementary surfaces are particular cases), showing how they can be parameterized through the coefficients of the Laplace expansion of their extension to the end compactification. In Section 4, we review the classification of elementary hyperbolic surfaces and some of their basic properties. Section 5 recalls the case of the hyperbolic disk, showing how it fits into the general approach developed in [1] and how well-behaved scalar potentials can be described through the Laplace expansion of their extension to . Section 6 discusses the hyperbolic punctured disk. After giving the explicit form of the hyperbolic metric on , of a partial isometric embedding into three-dimensional Euclidean space, and of the decomposition into horn and cusp regions, we compute certain scalar field trajectories for a few globally well-behaved scalar potentials which are induced naturally from . This illustrates the rich dynamics of our models on the punctured hyperbolic disk. We also discuss inflationary regions and the number of e-folds for various trajectories and provide an explicit example of an inflationary trajectory which produces between 50 and 60 e-folds, thus showing that such models can match observational constraints. Section 7 performs a similar analysis for hyperbolic annuli, using globally well-behaved scalar potentials induced by the same choices of functions on . In particular, this illustrates how the dynamics of our models differs on and . Section 8 comments briefly on the relation of such models with observational cosmology, while Section 9 contains our conclusions and sketches a few directions for further research.

*Notations and Conventions*. All manifolds considered are smooth, connected, oriented, and paracompact (hence also second-countable). All homeomorphisms and diffeomorphisms considered are orientation-preserving. By definition, a Lorentzian four-dimensional manifold has “mostly plus” signature. The symbol denotes the imaginary unit. The Poincaré half plane is the upper half plane with complex coordinate : endowed with its unique complete metric of Gaussian curvature , which is given by The real coordinates on are denoted by and . The complex coordinate on the hyperbolic disk, the hyperbolic punctured disk, or an annulus is denoted by . We define the* rescaled Planck mass* through where is the (reduced) Planck mass.

#### 2. Generalized -Attractor Models

##### 2.1. Definition of the Models

Let be a noncompact oriented, connected, and complete two-dimensional Riemannian manifold without boundary (called the* scalar manifold*) and be a smooth function (called the* scalar potential*). We say that is* hyperbolic* if the metric has constant Gaussian curvature equal to . We say that is* topologically finite* and that is* geometrically finite* if the fundamental group is finitely generated. Let , where is a fixed positive real number.

Let be any four-dimensional manifold which can support Lorentzian metrics. The Einstein-Scalar theory defined by the triplet includes four-dimensional gravity (described by a Lorentzian metric defined on ) and a smooth map , with action [1]: Here is the volume form of , is the scalar curvature of , and is the (reduced) Planck mass.

When is diffeomorphic with and is a FLRW metric with flat spatial section, solutions of the equations of motion for action (4) for which depends only on the cosmological time define the class of so-called* generalized **-attractor models* [1]. In this case, the map (where is a real interval) defines a curve in which obeys an invariantly defined nonlinear second-order ordinary differential equation (which is locally equivalent with a system of two second-order equations). We refer the reader to [1] for a general discussion of such models.

##### 2.2. Lift to the Poincaré Half Plane

As explained in [1], the cosmological equations of motion can be lifted from to the Poincaré half plane by using the covering map which uniformizes to . This allows one to determine the cosmological trajectories by projecting to the trajectories of a “lifted” model defined on . The lifted model is governed by the following system of second-order nonlinear ordinary differential equations [1, eq. ]: where and is the cosmological time while , are the Cartesian coordinates on the Poincaré half plane with complex coordinate and is the* lifted potential*. Let be any point of and let be chosen such that . An initial velocity vector defined at on and its unique lift through the differential of at are related through Notice that the differential of at is a bijective linear map because is a covering map and hence a local diffeomorphism. Writing and , we have with real , . As shown in [1], a cosmological trajectory on with initial condition can be written as , where is the solution of lifted system (5) with initial conditions:

*Eliminating the Planck Mass*. Let and , where is the rescaled Planck mass (3). Then (5) becomes showing that we can eliminate the Planck mass from the equations provided that we measure both and (and hence also ) in units of . The numerical solutions extracted in latter sections of this paper were obtained using system (8), after performing such a rescaling of and .

#### 3. Laplace Expansion of Globally Well-Behaved Scalar Potentials

Let denote the Kerekjarto-Stoilow (a.k.a. end) compactification of (see [10, 11]) and identify with its image in through the embedding map . A smooth scalar potential is called* globally well-behaved* [1] if there exists a smooth function whose restriction to equals , in which case is uniquely determined by through continuity. Since all elementary hyperbolic surfaces are planar, their end compactification is diffeomorphic with the unit sphere , so in this case globally well-defined scalar potentials on correspond bijectively to smooth functions .

Let and be spherical coordinates on ; thus where and . Any smooth map is square-integrable with respect to the round Lebesgue measure on and admits the Laplace-Fourier series expansion: where are the complex spherical harmonics and The series in (10) converges* uniformly* to on since is smooth (see [12]). Recall that , where are the associated Legendre functions. Since is real-valued, expansion (10) reduces to where and are real constants. Equivalently, we have where and are the real (a.k.a. tesseral) spherical harmonics, which correspond to orbitals. This expansion is again uniformly convergent and gives a systematic way to approximate by truncating away the contributions with greater than some cutoff value.

*Some Particular Choices for *. If only the modes with and ( and orbitals) are present, then we have where are real constants and we used the expressions The constant term in (14) corresponds to the orbital ) while the terms with prefactors , , and correspond to the orbitals , , and .

For , two simple choices are and , where is the rescaled Planck mass (3). These give the following -independent potentials, which involve only the orbitals and and are shown in Figure 1: Notice that has a maximum at (north pole) and a minimum at (south pole) while has a minimum at (north pole) and a maximum at (south pole).