# Generalizedα-Attractor Models from Elementary Hyperbolic Surfaces

Authors: Elena Mirela Babalic; Calin Iuliu Lazaroiu
Journal: Advances in Mathematical Physics
 (2018)

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## Abstract

We consider generalizedα-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(R) of modulus μ=2log⁡R>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.


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## Body

## 1. Introduction

In [1], we introduced a class of cosmological models which form a wide generalization of α-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 Σ endowed with a complete metric G of constant Gaussian curvature K=-1/3α. The scalar potential is given by a smooth real-valued function Φ defined on Σ. The rescaled scalar field metric G=def.(1/3α)G has Gaussian curvature equal to -1; hence (Σ,G) 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(Σ,G) is elementary, focusing on special features which arise in this case. As explained in [1], an elementary hyperbolic surface (Σ,G) is isometric with the hyperbolic disk D, with the hyperbolic punctured disk D∗ or with the hyperbolic annulus A(R) of modulus μ=2log⁡R, where R>1 is a real number. Each of these surfaces is planar, having the unit sphere S2 as its end compactification. This allows one to parameterize globally well-behaved [1] scalar potentials Φ through the coefficients of the Laplace expansion of their global extension Φ^ 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 D∗ and A(R). In particular, one can describe explicitly the hyperbolic geometry of such surfaces and one can compute scalar field trajectories on (Σ,G) by determining trajectories of an appropriate lift of the model to the Poincaré half plane H and projecting them to D∗ or to A(R) through the uniformization map. When the scalar potential is constant, this illustrates how the different projections from H to D∗ and A(R) can take the same trajectory on the Poincaré half plane into qualitatively different trajectories on the corresponding elementary hyperbolic surface. The explicit embedding of D, D∗, and A(R) into their common Kerekjarto-Stoilow compactification S2 is also different. As a consequence, a smooth real-valued function defined on S2 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 H, this leads to qualitatively different dynamics on D, D∗, and A(R). By considering some simple but natural choices of extended scalar potentials on S2, we use numerical methods to compute various trajectories on D∗ and A(R), thus illustrating the rich dynamics of such models, which is not entirely visible in the gradient flow approximation.The paper is organized as follows. Section2 briefly recalls the definition of generalized α-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 S2. Section 6 discusses the hyperbolic punctured disk. After giving the explicit form of the hyperbolic metric on D∗, 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 S2. 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 S2. In particular, this illustrates how the dynamics of our models differs on A(R) and D∗. 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 i denotes the imaginary unit. The Poincaré half plane is the upper half plane with complex coordinate τ: (1)H=τ∈C∣Im⁡τ>0,endowed with its unique complete metric of Gaussian curvature -1, which is given by (2)dsH2=1Im⁡τ2dτ2.The real coordinates on H are denoted by x=def.Reτ and y=def.Im⁡τ. The complex coordinate on the hyperbolic disk, the hyperbolic punctured disk, or an annulus is denoted by u. We define therescaled Planck mass through (3)M0=def.M23,where M is the (reduced) Planck mass.


## 2. Generalizedα-Attractor Models

### 2.1. Definition of the Models

Let(Σ,G) be a noncompact oriented, connected, and complete two-dimensional Riemannian manifold without boundary (called thescalar manifold) and Φ:Σ→R be a smooth function (called thescalar potential). We say that (Σ,G) ishyperbolic if the metric G has constant Gaussian curvature equal to -1. We say that Σ istopologically finite and that (Σ,G) isgeometrically finite if the fundamental group π1(Σ) is finitely generated. Let G=3αG, where α>0 is a fixed positive real number.LetX be any four-dimensional manifold which can support Lorentzian metrics. The Einstein-Scalar theory defined by the triplet (Σ,G,Φ) includes four-dimensional gravity (described by a Lorentzian metric g defined on X) and a smooth map φ:X→Σ, with action [1]: (4)Sg,φ=∫XM22Rg-12Trgφ∗G-Φ∘φvolg. Here volg is the volume form of (X,g), R(g) is the scalar curvature of g, and M is the (reduced) Planck mass.WhenX is diffeomorphic with R4 and g is a FLRW metric with flat spatial section, solutions of the equations of motion for action (4) for which φ depends only on the cosmological time t define the class of so-calledgeneralizedα-attractor models [1]. In this case, the map I∋t→φ(t)∈Σ (where I is a real interval) defines a curve in Σ 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 Σ to the Poincaré half plane H by using the covering map πH:H→Σ which uniformizes (Σ,G) to H. This allows one to determine the cosmological trajectories φ(t) by projecting to Σ the trajectories φ~(t) of a “lifted” model defined on H. The lifted model is governed by the following system of second-order nonlinear ordinary differential equations [1, eq. (7.4)]: (5)x¨-2yx˙y˙+1M323αx˙2+y˙2y2+2Φ~x,y1/2x˙+13αy2∂xΦ~x,y=0,y¨+1yx˙2-y˙2+1M323αx˙2+y˙2y2+2Φ~x,y1/2y˙+13αy2∂yΦ~x,y=0,where ˙=def.d/dt and t is the cosmological time while x=Reτ, y=Imτ are the Cartesian coordinates on the Poincaré half plane with complex coordinate τ and Φ~=def.Φ∘πH:H→R is thelifted potential. Let u0 be any point of Σ and let τ0∈H be chosen such that πH(τ0)=u0. An initial velocity vector v0=u˙0∈Tu0Σ defined at u0 on D∗ and its unique lift v~0=τ˙0∈Tτ0H through the differential of πH at τ0 are related through (6)v0=dτ0πHv~0.Notice that the differential of πH at τ0 is a bijective linear map because πH is a covering map and hence a local diffeomorphism. Writing τ=x+iy and τ0=x0+iy0, we have v~0=v~0x+iv~0y with real v~0x, v~0y. As shown in [1], a cosmological trajectory φ(t) on Σ with initial condition (u0,τ0) can be written as φ(t)=πH(φ~(t)), where φ~(t)=x(t)+iy(t) is the solution of lifted system (5) with initial conditions:(7)x0=x0,y0=y0,x˙0=v~0x,y˙0=v~0y.Eliminating the Planck Mass. Let Φ~=(1/M0)Φ~ and α=(1/M0)α, where M0=M2/3 is the rescaled Planck mass (3). Then (5) becomes (8)x¨-2yx˙y˙+3αx˙2+y˙2y2+2Φ~x,y1/2x˙+13αy2∂xΦ~x,y=0,y¨+1yx˙2-y˙2+3αx˙2+y˙2y2+2Φ~x,y1/2y˙+13αy2∂yΦ~x,y=0,showing that we can eliminate the Planck mass from the equations provided that we measure both α and Φ~ (and hence also Φ) in units of M0. The numerical solutions extracted in latter sections of this paper were obtained using system (8), after performing such a rescaling of α and Φ.


## 2.1. Definition of the Models

Let(Σ,G) be a noncompact oriented, connected, and complete two-dimensional Riemannian manifold without boundary (called thescalar manifold) and Φ:Σ→R be a smooth function (called thescalar potential). We say that (Σ,G) ishyperbolic if the metric G has constant Gaussian curvature equal to -1. We say that Σ istopologically finite and that (Σ,G) isgeometrically finite if the fundamental group π1(Σ) is finitely generated. Let G=3αG, where α>0 is a fixed positive real number.LetX be any four-dimensional manifold which can support Lorentzian metrics. The Einstein-Scalar theory defined by the triplet (Σ,G,Φ) includes four-dimensional gravity (described by a Lorentzian metric g defined on X) and a smooth map φ:X→Σ, with action [1]: (4)Sg,φ=∫XM22Rg-12Trgφ∗G-Φ∘φvolg. Here volg is the volume form of (X,g), R(g) is the scalar curvature of g, and M is the (reduced) Planck mass.WhenX is diffeomorphic with R4 and g is a FLRW metric with flat spatial section, solutions of the equations of motion for action (4) for which φ depends only on the cosmological time t define the class of so-calledgeneralizedα-attractor models [1]. In this case, the map I∋t→φ(t)∈Σ (where I is a real interval) defines a curve in Σ 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 Σ to the Poincaré half plane H by using the covering map πH:H→Σ which uniformizes (Σ,G) to H. This allows one to determine the cosmological trajectories φ(t) by projecting to Σ the trajectories φ~(t) of a “lifted” model defined on H. The lifted model is governed by the following system of second-order nonlinear ordinary differential equations [1, eq. (7.4)]: (5)x¨-2yx˙y˙+1M323αx˙2+y˙2y2+2Φ~x,y1/2x˙+13αy2∂xΦ~x,y=0,y¨+1yx˙2-y˙2+1M323αx˙2+y˙2y2+2Φ~x,y1/2y˙+13αy2∂yΦ~x,y=0,where ˙=def.d/dt and t is the cosmological time while x=Reτ, y=Imτ are the Cartesian coordinates on the Poincaré half plane with complex coordinate τ and Φ~=def.Φ∘πH:H→R is thelifted potential. Let u0 be any point of Σ and let τ0∈H be chosen such that πH(τ0)=u0. An initial velocity vector v0=u˙0∈Tu0Σ defined at u0 on D∗ and its unique lift v~0=τ˙0∈Tτ0H through the differential of πH at τ0 are related through (6)v0=dτ0πHv~0.Notice that the differential of πH at τ0 is a bijective linear map because πH is a covering map and hence a local diffeomorphism. Writing τ=x+iy and τ0=x0+iy0, we have v~0=v~0x+iv~0y with real v~0x, v~0y. As shown in [1], a cosmological trajectory φ(t) on Σ with initial condition (u0,τ0) can be written as φ(t)=πH(φ~(t)), where φ~(t)=x(t)+iy(t) is the solution of lifted system (5) with initial conditions:(7)x0=x0,y0=y0,x˙0=v~0x,y˙0=v~0y.Eliminating the Planck Mass. Let Φ~=(1/M0)Φ~ and α=(1/M0)α, where M0=M2/3 is the rescaled Planck mass (3). Then (5) becomes (8)x¨-2yx˙y˙+3αx˙2+y˙2y2+2Φ~x,y1/2x˙+13αy2∂xΦ~x,y=0,y¨+1yx˙2-y˙2+3αx˙2+y˙2y2+2Φ~x,y1/2y˙+13αy2∂yΦ~x,y=0,showing that we can eliminate the Planck mass from the equations provided that we measure both α and Φ~ (and hence also Φ) in units of M0. The numerical solutions extracted in latter sections of this paper were obtained using system (8), after performing such a rescaling of α and Φ.


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

LetΣ^ denote the Kerekjarto-Stoilow (a.k.a. end) compactification of Σ (see [10, 11]) and identify Σ with its image in Σ^ through the embedding map j:Σ→Σ^. A smooth scalar potential Φ:Σ→R is calledglobally well-behaved [1] if there exists a smooth function Φ^:Σ^→R whose restriction to Σ equals Φ, in which case Φ^ is uniquely determined by Φ through continuity. Since all elementary hyperbolic surfaces (Σ,G) are planar, their end compactification Σ^ is diffeomorphic with the unit sphere S2={(x1,x2,x3)∈R3|x12+x22+x32=1}, so in this case globally well-defined scalar potentials on Σ correspond bijectively to smooth functions Φ^:S2→R.Letψ and θ be spherical coordinates on S2; thus (9)x1=sin⁡ψcos⁡θ,x2=sin⁡ψsin⁡θ,x3=cos⁡ψ,where ψ∈[0,π] and θ∈[0,2π). Any smooth map Φ^:S2→R is square-integrable with respect to the round Lebesgue measure sin⁡ψdψdθ on S2 and admits the Laplace-Fourier series expansion: (10)Φ^ψ,θ=∑l=0∞∑m=-llClmYlmψ,θ,where Ylm are the complex spherical harmonics and (11)Clm=∫S2dψdθsin⁡ψYlmψ,θ¯Φ^ψ,θ∈C.The series in (10) convergesuniformly to Φ^ on S2 since Φ^ is smooth (see [12]). Recall that Ylm(ψ,θ)=Plm(cos⁡ψ)eimθ, where Plm(x) are the associated Legendre functions. Since Φ^ is real-valued, expansion (10) reduces to (12)Φ^ψ,θ=∑l=0∞∑m=-llPlmcos⁡ψAlmcos⁡mθ+Blmsin⁡mθ, where Alm and Blm are real constants. Equivalently, we have (13)Φ^ψ,θ=∑l=0∞∑m=-llDlmYlmψ,θ,where Dlm∈R and Ylm 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 Φ^ by truncating away the contributions with l greater than some cutoff value.Some Particular Choices for Φ ^. If only the modes with l=0 and l=1 (s and p orbitals) are present, then we have (14)Φ^ψ,θ=a+bcos⁡ψ+csin⁡ψcos⁡θ+dsin⁡ψsin⁡θ, where a,b,c,d are real constants and we used the expressions (15)P00x=1,P10x=x,P11x=-1-x2,P1-1x=121-x2,Y00=12π,Y10=34πcos⁡ψ,Y11=34πsin⁡ψcos⁡θ,Y1,-1=34πsin⁡ψsin⁡θ. The constant term in (14) corresponds to the s orbital (Y00) while the terms with prefactors b, c, and d correspond to the orbitals pz(Y10), px(Y11), and py(Y1,-1).Forc=d=0, two simple choices are a=b=(1/2)M0 and a=-b=(1/2)M0, where M0 is the rescaled Planck mass (3). These give the following θ-independent potentials, which involve only the orbitals s and pz and are shown in Figure 1: (16)Φ^+ψ=M0cos2ψ2,Φ^-ψ=M0sin2ψ2.Notice that Φ^+ has a maximum at ψ=0 (north pole) and a minimum at ψ=π (south pole) while Φ^- has a minimum at ψ=0 (north pole) and a maximum at ψ=π (south pole).Figure 1
Plot ofΦ^+/M0 (blue) and Φ^-/M0 (orange) as functions of ψ∈[0,π]. The value ψ=0 corresponds to the north pole of S2, while ψ=π corresponds to the south pole.Another simple choice isb=d=0 and a=c=M0, which corresponds to a linear combination of the s and px orbitals and gives the extended potential: (17)Φ^0ψ,θ=M01+sin⁡ψcos⁡θ=M01+x1. Unlike Φ^±, this function does not have extrema at the north or south pole. Instead, it has two extrema along the equator of S2, having a maximum (equal to 2M0) at the point (ψ,θ)=(π/2,0) and a minimum (equal to 0) at (ψ,θ)=(π/2,π). Notice that Φ^± and Φ^0 are Morse functions on S2, so the potentials derived from them on a planar surface (whose end compactification is S2) will be compactly Morse in the sense of [1].A Universal Approach to Globally Well-Behaved Potentials. The techniques of passing to the end compactification and lifting to the Poincaré half plane introduced in [1] allow for a uniform treatment of globally well-behaved scalar potentials in generalized α-attractor models for which (Σ,G) is geometrically finite. This is summarized in the commutative diagram (18), where j is the smooth embedding of Σ into its end compactification Σ^ and πH is the uniformizing map from H:(18)Any smooth real-valued function Φ^ defined on Σ^ induces a globally well-behaved scalar potential Φ on Σ through the formula Φ=Φ^∘j, while any globally well-behaved scalar potential Φ on Σ lifts to a smooth function Φ~=Φ∘πH defined on H. When the maps j and πH are known, Φ~ can be recovered from Φ^ as the composition Φ~=Φ^∘χ, where χ:H→Σ^ is the composite map χ=def.j∘πH. Notice that j and χ are smooth maps, while πH is holomorphic when Σ is endowed with the complex structure which corresponds to the conformal class of the metric G. The maps j,πH and χ differ for distinct geometrically finite hyperbolic surfaces (Σ^,G) having the same end compactification Σ^, which means that the same “universal” extended potential Φ^ defined on Σ^ can induce markedly different globally well-behaved potentials Φ and lifted potentials Φ~ for different hyperbolic surfaces of the same genus.WhenΣ is a planar surface, the end compactification Σ^ coincides with the unit sphere and Φ^ can be expanded into real spherical harmonics as explained above. This induces uniformly convergent expansions: (19)Φ=∑l=0∞∑m=-llDlmYlm∘j,Φ~=∑l=0∞∑m=-llDlmYlm∘χ of Φ and Φ~. In the next sections, we determine explicitly the maps j, πH, and χ for the planar elementary surfaces D, D∗, and A(R) and the maps Φ and Φ~ induced by the choices Φ^=Φ^± and Φ^=Φ^0 given above. This illustrates how the same function Φ^ leads to different dynamics of the generalized α-attractor models associated with distinct planar surfaces. For the three elementary surfaces, we will construct the map j:Σ→Σ^=S2 by first diffeomorphically (but not biholomorphically!) identifying Σ with the complex plane C of complex coordinate ζ (or with C with a point removed) and then identifying the latter with the once- or twice-punctured sphere by using stereographic projection from the north pole of S2: (20)ζ=cot⁡ψ2eiθ∈C.


## 4. Elementary Hyperbolic Surfaces

A (complete) hyperbolic surface(Σ,G) is calledelementary if it is conformally equivalent with a simply connected or doubly connected regular domain (a regular domain D⊂C is calleddoubly connected if its complement in the Riemann sphere has two connected components, which happens iff π1(D)≃Z) contained in the complex plane. This amounts to the condition that the uniformizing surface group Γ of (Σ,G) is the trivial group or a cyclic subgroup of PSL(2,R) of parabolic or hyperbolic type.Any simply connected regular domain is conformally equivalent with the unit diskD={u∈C∣u<1} (and hence with the upper half plane H). Such a domain admits a unique complete hyperbolic metric, known as the Poincaré metric. Any doubly connected regular domain is conformally equivalent to one of the following, when the latter is endowed with the complex structure inherited from the complex plane:(i)
The punctured planeC∗=C∖{0}(ii)
The punctured unit diskD∗=def.D∖{0}(iii)
The annulusA(R)=def.{u∈C∣1/R<|u|<R} of modulus μ=2log⁡R>0, where R>1.When endowed with its usual complex structure, the punctured plane does not support a complete hyperbolic metric. On the other hand, the punctured diskD∗ and annulus A(R) admit uniquely determined complete hyperbolic metrics. Notice that both D∗ and A(R) are homeomorphic with (open) cylinders. Due to this fact, the hyperbolic punctured disk D∗ is also called theparabolic cylinder while the hyperbolic annuli A(R) are also calledhyperbolic cylinders [8]. Summarizing, the list of elementary hyperbolic surfaces is as follows:(1)
The hyperbolic diskD (which is isometric with the Poincaré half plane H)(2)
The hyperbolic punctured diskD∗ (uniformized to H by a parabolic cyclic subgroup of PSL(2,R))(3)
The hyperbolic annuliA(R) for R>1 (uniformized to H by a hyperbolic cyclic subgroup of PSL(2,R)).The explicit form of the hyperbolic metric is known in all cases, as is a fundamental polygon [13–15] for D∗ and A(R). This allows one to study the cosmological dynamics of α-attractor models defined by such surfaces either directly on (Σ,G) or (as explained in [1]) by lifting to the hyperbolic disk or to the Poincaré half plane.For elementary hyperbolic surfaces, the isometry classification of the ends is as follows [1, 8, 16]:(i)
The hyperbolic diskD has a single end, known as aplane end.(ii)
The hyperbolic punctured diskD∗ has two ends. One of these is acusp end, the other being ahorn end.(iii)
The hyperbolic annulusA(R) has two ends, both of which arefunnel ends.All elementary hyperbolic surfaces are planar (i.e., of genus zero). As explained in [1], this implies that their end compactification [10, 11] is the unit sphere S2. On the other hand, the conformal boundary ∂∞GΣ [1, 17, 18] differs in the three cases:(i)
For the hyperbolic disk, we have∂∞D=S1, where the circle corresponds to the plane end.(ii)
For the hyperbolic punctured disk, we have∂∞D∗={0}⊔S1, where the origin corresponds to the cusp end and S1 corresponds to the horn end.(iii)
For the hyperbolic annulus, we have∂∞A(R)=S1⊔S1, each of the circles corresponding to a funnel end.Remark 1.
By a theorem of Hilbert, a complete hyperbolic surface cannot be embedded isometrically into Euclidean 3-dimensional space. However, incomplete regions of such a surface can be embedded isometrically (and we shall see examples of such partial embedding in latter sections). Notice that one can sometimes find isometric embedding of complete hyperbolic surfaces intonon-Euclidean 3-dimensional space, such as the well-known embedding of the Poincaré disk as a sheet of a hyperboloid defined inside three-dimensional Minkowski space.


## 5. The Hyperbolic Disk

The cosmological model defined by the hyperbolic diskD coincides with the two-field α-attractor model of [7], which was discussed extensively in the literature. The purpose of this section is to show how this fits into the general theory developed in [1].


### 5.1. Semigeodesic Coordinates

The unit diskD={u∈C∣u<1} admits a unique complete hyperbolic metric, which is given by (21)dsD2=λD2u,u¯du2,whereλDu,u¯=21-u2. In polar coordinates given by ρ=u∈(0,1) and θ=arg⁡(u)∈(0,2π), the metric becomes (22)dsD2=41-ρ22dρ2+ρ2dθ2. Semigeodesic coordinates (r,θ) for D are obtained by the change of variables: (23)ρ=def.tanh⁡r2∈0,1i.e.r=2arctanh⁡ρ=log⁡1+ρ1-ρ∈0,+∞.This maps the unit disk (diffeomorphically, but not conformally) to the complex plane with polar coordinates (r,θ) and complex coordinate (24)ζ=reiθ=2arctanh⁡ρeiθ and brings the metric to the form (25)dsD2=dr2+sinh2rdθ2. The single end of D (which is called aplane end) corresponds to r→+∞, while the center of D corresponds to r→0.


### 5.2. The End Compactification ofD

The end compactificationD^ of the hyperbolic disk coincides with the Alexandroff compactification of the ζ-plane, which by the stereographic projection (20) is identified with the unit sphere S2. The north pole ψ=0 corresponds to the plane end at r=|ζ|→∞, while the south pole ψ=π corresponds to r=0, that is, to the center of D. In spherical stereographic coordinates (ψ,θ), the Poincaré metric (25) becomes (26)dsD2=14dψ2sin2ψ/2+sinh2cotψ2dθ2.Remark 2.
The compact Riemann surface into whichD={u∈C∣u<1} is embedded holomorphically [17, 18] is the Riemann sphere CP1 associated with the u-plane. The coordinate u is given by (27)u=def.u1u2foru2≠0, where (u1,u2)∈C2∖{0,0} are the homogeneous coordinates of CP1.


### 5.3. Globally Well-Behaved Scalar Potentials onD

A potentialΦ:D→R is globally well-behaved on D iff there exists a smooth function Φ^:S2→R such that (28)Φr,θ=Φ^2arccot⁡r,θ; that is, (29)Φρ,θ=Φ^2arccot⁡2arctanh⁡ρ,θ. The condition that Φ^ is smooth on S2 implies, in particular, that Φ^(ψ,θ) has a finite limit for ψ→0 and hence Φ(r,θ) has a θ-independent limit for r→+∞, that is, for u→1. Expansion (13) gives the uniformly convergent series: (30)Φρ,θ=∑l=0∞∑m=-llDlmYlm2arccot⁡2arctanh⁡ρ,θ. To obtain inflationary behavior with the scalar field rolling from the plane end toward the interior of D, one can require that Φ^ has a local maximum at the north pole of S2. In the simplest models, one can take Φ^ to have only two critical points, namely, a global maximum at the north pole and a global minimum at the south pole. In that case, Φ has a global minimum at u=0 (the center of D) and increases monotonically to θ-independent finite value as u grows from zero to 1 (toward the conformal boundary ∂∞D=S1 of D).In polar coordinates(ρ,θ), the extended potentials (16) of Section 3 correspond to the following globally well-behaved scalar potentials on D: (31)Φ+=M0r21+r2=M0log⁡1+ρ/1-ρ21+log⁡1+ρ/1-ρ2,Φ-=M01+r2=M01+log⁡1+ρ/1-ρ2,where ρ=u. These potentials are shown in Figures 2(a) and 2(b), which illustrate the characteristic stretching toward the end when the potential is expressed in semigeodesic coordinates (r,θ) (with respect to the fact that the two locally defined real scalar fields of the sigma model have canonical kinetic terms). The supremum of Φ+ corresponds to ρ→1 (being equal to M0), while the infimum is attained at ρ=0 (where Φ+ vanishes). On the other hand, Φ- tends to its vanishing infimum for ρ→1 and has a maximum at ρ=0 (where it equals M0). Notice that only Φ+ leads to standard α-attractor behavior.Figure 2
The potentialsΦ± on the Poincaré disk.
(a)
Plot ofΦ+/M0 (blue) and Φ-/M0 (orange) as functions of ρ∈[0,1). The value ρ=0 corresponds to the center of D, while ρ=1 corresponds to the conformal boundary ∂∞D. Only the potential Φ+ leads to α-attractor behavior when inflation takes place near the plane end (b)
Plot ofΦ+/M0 (blue) and Φ-/M0 (orange) as functions of r∈[0,+∞). The value r=0 corresponds to the center of D, while r=+∞ corresponds to the plane end. Only the potential Φ+ leads to α-attractor behavior when inflation takes place near the plane endUsing the relationsin⁡ψ=2sin⁡ψ/2cos⁡(ψ/2)=2r/(1+r2), choice (17) gives the following potential on D: (32)Φ0=M01+2r1+r2cos⁡θ=M01+2log⁡1+ρ/1-ρ1+log⁡1+ρ/1-ρ2cos⁡θ.


## 5.1. Semigeodesic Coordinates

The unit diskD={u∈C∣u<1} admits a unique complete hyperbolic metric, which is given by (21)dsD2=λD2u,u¯du2,whereλDu,u¯=21-u2. In polar coordinates given by ρ=u∈(0,1) and θ=arg⁡(u)∈(0,2π), the metric becomes (22)dsD2=41-ρ22dρ2+ρ2dθ2. Semigeodesic coordinates (r,θ) for D are obtained by the change of variables: (23)ρ=def.tanh⁡r2∈0,1i.e.r=2arctanh⁡ρ=log⁡1+ρ1-ρ∈0,+∞.This maps the unit disk (diffeomorphically, but not conformally) to the complex plane with polar coordinates (r,θ) and complex coordinate (24)ζ=reiθ=2arctanh⁡ρeiθ and brings the metric to the form (25)dsD2=dr2+sinh2rdθ2. The single end of D (which is called aplane end) corresponds to r→+∞, while the center of D corresponds to r→0.


## 5.2. The End Compactification ofD

The end compactificationD^ of the hyperbolic disk coincides with the Alexandroff compactification of the ζ-plane, which by the stereographic projection (20) is identified with the unit sphere S2. The north pole ψ=0 corresponds to the plane end at r=|ζ|→∞, while the south pole ψ=π corresponds to r=0, that is, to the center of D. In spherical stereographic coordinates (ψ,θ), the Poincaré metric (25) becomes (26)dsD2=14dψ2sin2ψ/2+sinh2cotψ2dθ2.Remark 2.
The compact Riemann surface into whichD={u∈C∣u<1} is embedded holomorphically [17, 18] is the Riemann sphere CP1 associated with the u-plane. The coordinate u is given by (27)u=def.u1u2foru2≠0, where (u1,u2)∈C2∖{0,0} are the homogeneous coordinates of CP1.


## 5.3. Globally Well-Behaved Scalar Potentials onD

A potentialΦ:D→R is globally well-behaved on D iff there exists a smooth function Φ^:S2→R such that (28)Φr,θ=Φ^2arccot⁡r,θ; that is, (29)Φρ,θ=Φ^2arccot⁡2arctanh⁡ρ,θ. The condition that Φ^ is smooth on S2 implies, in particular, that Φ^(ψ,θ) has a finite limit for ψ→0 and hence Φ(r,θ) has a θ-independent limit for r→+∞, that is, for u→1. Expansion (13) gives the uniformly convergent series: (30)Φρ,θ=∑l=0∞∑m=-llDlmYlm2arccot⁡2arctanh⁡ρ,θ. To obtain inflationary behavior with the scalar field rolling from the plane end toward the interior of D, one can require that Φ^ has a local maximum at the north pole of S2. In the simplest models, one can take Φ^ to have only two critical points, namely, a global maximum at the north pole and a global minimum at the south pole. In that case, Φ has a global minimum at u=0 (the center of D) and increases monotonically to θ-independent finite value as u grows from zero to 1 (toward the conformal boundary ∂∞D=S1 of D).In polar coordinates(ρ,θ), the extended potentials (16) of Section 3 correspond to the following globally well-behaved scalar potentials on D: (31)Φ+=M0r21+r2=M0log⁡1+ρ/1-ρ21+log⁡1+ρ/1-ρ2,Φ-=M01+r2=M01+log⁡1+ρ/1-ρ2,where ρ=u. These potentials are shown in Figures 2(a) and 2(b), which illustrate the characteristic stretching toward the end when the potential is expressed in semigeodesic coordinates (r,θ) (with respect to the fact that the two locally defined real scalar fields of the sigma model have canonical kinetic terms). The supremum of Φ+ corresponds to ρ→1 (being equal to M0), while the infimum is attained at ρ=0 (where Φ+ vanishes). On the other hand, Φ- tends to its vanishing infimum for ρ→1 and has a maximum at ρ=0 (where it equals M0). Notice that only Φ+ leads to standard α-attractor behavior.Figure 2
The potentialsΦ± on the Poincaré disk.
(a)
Plot ofΦ+/M0 (blue) and Φ-/M0 (orange) as functions of ρ∈[0,1). The value ρ=0 corresponds to the center of D, while ρ=1 corresponds to the conformal boundary ∂∞D. Only the potential Φ+ leads to α-attractor behavior when inflation takes place near the plane end (b)
Plot ofΦ+/M0 (blue) and Φ-/M0 (orange) as functions of r∈[0,+∞). The value r=0 corresponds to the center of D, while r=+∞ corresponds to the plane end. Only the potential Φ+ leads to α-attractor behavior when inflation takes place near the plane endUsing the relationsin⁡ψ=2sin⁡ψ/2cos⁡(ψ/2)=2r/(1+r2), choice (17) gives the following potential on D: (32)Φ0=M01+2r1+r2cos⁡θ=M01+2log⁡1+ρ/1-ρ1+log⁡1+ρ/1-ρ2cos⁡θ.


## 6. The Hyperbolic Punctured Disk

The hyperbolic punctured diskD∗ (also known as the “parabolic cylinder” [8]) is the simplest example of a hyperbolic surface with a cusp end. It also has a horn end.


### 6.1. The Hyperbolic Metric

The punctured unit diskD∗={u∈C∣0<u<1} admits a unique complete hyperbolic metric given by [19] (33)dsD∗2=λD∗2u,u¯du2,whereλD∗u,u¯=1ulog⁡1/u.In particular, Reu and Imu are isothermal coordinates. In polar coordinates (ρ,θ) defined through (34)u=ρeiθρ=u∈0,1, the metric takes the form (35)dsD∗2=1ρlog⁡ρ2dρ2+ρ2dθ2. The center of D∗ corresponds to the cusp end, while the bounding circle of D∗ corresponds to the horn end (see below). The Euclidean circle at ρ=e-2π is a horocycle of hyperbolic length 1. Notice that D∗ has infinite hyperbolic area.


### 6.2. Diffeomorphism to the Punctured Plane

One can introduce an orthogonal coordinate system(r,θ)∈(0,+∞)×(0,2π) on D∗ through the coordinate transformation: (36)r=def.1log⁡1/ρ=1log⁡ρ∈0,+∞i.e.ρ=e-1/r∈0,1. This gives a diffeomorphism between D∗ and the punctured complex plane C∗=def.C∖{0} with complex coordinate: (37)ζ=reiθ. In this coordinate system, metric (33) takes the form (38)dsD∗2=1r2dr2+r2dθ2.The center of D∗ corresponds to r→0, while the bounding circle of D∗ corresponds to r→+∞.


### 6.3. Partial Isometric Embedding into Euclidean 3-Dimensional Space

One can isometrically embed the portion0<u<1/e of the hyperbolic punctured disk into Euclidean R3 as the open half tractricoid (the surface obtained by revolving an open half of a tractrix along its asymptote [20])  E defined in cylindrical coordinates (r,θ,x3) (where x1=rcos⁡θ and x2=rsin⁡θ) by the parametric equations: (39)r=1cosh⁡t∈0,1,x3=t-tanh⁡tt∈0,+∞. Indeed, it is easy to see that the Euclidean metric of R3 induces a metric on E which coincides with (38). This is the classical pseudosphere model of Beltrami (see Figure 3).Figure 3
Upper half of a tractricoid (pseudosphere). The horocycle of length one is drawn in light blue. It separates the half tractricoid into a cusp (upper part) and a portion of a horn (lower part). The latter is bounded by a horocycle of length2π.


### 6.4. The End Compactification ofD∗

The stereographic projection (20) identifies S2 with the one-point compactification of the ζ-plane. This shows explicitly that S2 is the end compactification of D∗, where the north pole ψ=0 corresponds to the horn end and the south pole ψ=π corresponds to the cusp end. The embedding j:D∗→S2 is given by (40)ψ=2arctan⁡log⁡ρ,θ=θ.


### 6.5. Semigeodesic Coordinates

The further change of variables(41)r=12πe-ri.e.r=-log⁡2πr=log⁡log⁡ρ2π∈-∞,+∞ brings metric (38) to the form (42)dsD∗2=dr2+e-2r2π2dθ2, where r∈R. In particular, (r,θ) are semigeodesic coordinates. The center u=0 of D∗ corresponds to r→+∞ while the bounding circle |u|=1 of D∗ corresponds to r→-∞. The horocycle at r=0 (i.e., ρ=e-2π) has length 1.


### 6.6. The Hyperbolic Cusp

Let(43)κ=def.e-2π. As mentioned above, the Euclidean circle u=κ has hyperbolic length 1. Thehyperbolic cusp (cf. [1]) corresponds to the portion of D∗ lying inside this circle (see Figure 3), which is the open punctured disk: (44)C=def.u∈C∣0<u<κ⊂D∗, endowed with the restriction of the metric (33); notice that the restricted metric is not complete. In coordinates (r,θ), the metric on C is obtained by restricting (38) to the range r∈(0,r0), where (45)r0=def.12π. In semigeodesic coordinates (r,θ) the cusp metric is given by (42) with the restriction r∈(0,+∞). Notice that C has hyperbolic area equal to 1.


### 6.7. The Hyperbolic Horn

By definition, thehyperbolic horn is the annulus: (46)H=u∈C∣κ<u<1⊂D∗,endowed with the (incomplete) restriction of metric (33). In coordinates (r,θ), the horn metric is obtained from (38) by restricting the range of r to (r0,+∞). In coordinates (r,θ), the metric takes form (42), with the restriction r∈(-∞,0). Defining r′=def.-r, this can be brought to the form (47)dsH2=dr′2+e2r′2π2dθ2,withr′∈0,+∞, where the bounding circle u=1 of D∗ corresponds to r′→+∞.


### 6.8. Canonical Uniformization toH

The punctured disk is uniformized to the Poincaré half plane (1) with complex coordinate τ by the parabolic cyclic group ΓP=def.P⊂PSL(2,R) generated by the translation: (48)τ⟶τ+1, which corresponds to the parabolic element: (49)P=def.1101∈PSL2,R. This fixes the point τ=∞∈∂∞H. The uniformization map is (50)u=πHτ=e2πiτ. The hyperbolic cusp C is the projection through πH of the cusp domain: (51)CH=τ∈H∣Imτ>1, which is bounded by the horocycle: (52)cH=τ∈H∣Imτ=1. This horocycle is tangent to the conformal boundary of H at the point τ=∞, which projects through πH to the cusp ideal point of the end compactification of D∗.A fundamental polygon for the action ofΓP on H is given by the semi-infinite vertical strip (53)DH=τ∈H∣0<Reτ<1 and has vertices at the points (see Figure 4(a)): (54)A:τ=∞,B:τ=0,C:τ=1. The Poincaré side pairing maps (AB) into (AC) through transformation (48), which generates ΓP. The relative cusp neighborhood [1] with respect to DH is the intersection CH=def.CH∩DH. A lifted scalar potential Φ~=Φ∘πH is given by (55)Φ~τ=Φe2πiτ, being invariant under translation (48). In particular, the restrictions of Φ~ to the sides (AB) and (AC) agree through the Poincaré pairing.Figure 4
Fundamental polygons for the uniformization ofD∗ to the Poincaré half plane and to the hyperbolic disk.
(a)
A fundamental polygonDH for the punctured disk on H. The relative cusp neighborhood CH of the vertex A corresponds to the shaded region (b)
A fundamental polygonDD for the punctured disk on D. The shaded region is the relative cusp neighborhood CD of the vertex A′


### 6.9. Canonical Uniformization toD

For completeness and comparison with [1], we also give the canonical uniformization of D∗ to the hyperbolic disk. The Cayley transformation (56)u=i-τiτ-1 is an isometry from H to D. When uniformizing D∗ to D, the fundamental polygon DH becomes a hyperbolic triangle DD with vertices at the following points, which correspond, respectively, to the points A, B, and C of (54) through the Cayley transformation (57)A′:u=i,B′:u=-i,C′:u=1 and a free side connecting the points B′ and C′ (see Figure 4(b)). The sides of this triangle are a segment connecting A′ to B′ (which passes through the origin of D), the portion of ∂∞D connecting B′ to C′, and Euclidean circular arc orthogonal to ∂∞D which connects A′ to C′. The hyperbolic cusp neighborhood CD of the vertex A′ is bounded by the horocycle: (58)cD=t2-it∈DD∣t∈R, which is tangent to ∂∞D at the point A′. The intersection of CD with DD is the relative cusp neighborhood CD with respect to DD (see [1]), which is the image of CH through the Cayley transformation.


### 6.10. Globally Well-Behaved Scalar Potentials onD∗

A scalar potentialΦ:Σ→R is globally well-behaved on D∗ iff there exists a smooth map Φ^:S2→R such that(59)Φr,θ=Φ^2arccot⁡r,θ;that is, (60)Φρ,θ=Φ^2arctan⁡log⁡ρ,θ,where ρ=u. Expansion (13) gives the uniformly convergent series: (61)Φρ,θ=∑l=0∞∑m=-llDlmYlm2arctan⁡log⁡ρ,θ.For choices (16) and (17), we find (62)Φ+=M0r21+r2=M011+log⁡ρ2,Φ-=M011+r2=M0log⁡ρ21+log⁡ρ2,Φ0=M01+2rcos⁡θ1+r2=M01+2log⁡ρ1+log⁡ρ2cos⁡θ.These potentials are shown in Figure 5. Notice that Φ+ leads to α-attractor behavior if inflation takes place near the horn end, while Φ- leads to α-attractor behavior if inflation takes place near the cusp end [1]. The extended potentials Φ^± have maxima and minima at the two ideal points of the end compactification of D∗ (which correspond to the north and south poles of S2). On the other hand, Φ^0 doesnot have extrema at the ideal points; its extrema coincide with those of Φ0, being located inside D∗. The minimum (equal to zero) is at the point u=-1/e≈-0.36 while the maximum (equal to 2M0) is at u=+1/e≈+0.36.Figure 5
The potentialsΦ± and Φ0 on the hyperbolic punctured disk.
(a)
Plot ofΦ+/M0 (blue) and Φ-/M0 (orange) as functions of ρ∈(0,1). The value ρ=0 corresponds to the cusp end, while ρ=1 corresponds to the horn end (b)
Level plot ofΦ0/M0 on the punctured disk. Darker tones indicate lower values of Φ0/M0Remark 3.
Using (41), we find the following expressions in semigeodesic coordinates: (63)Φ+r=M01+2π2e2r,Φ-r=M01+e-2r/2π2,Φ0r,θ=M01+4πe-rcos⁡θ2π2+e-2r.Lift of the Potentials Φ ± and Φ 0 to H. Consider the well-behaved scalar potentials Φ± and Φ0 on D∗ given in (62). Let x=Reτ and y=def.Imτ. Then the covering map (50) reads (64)u=πHτ=e2πiτ=e-2πycos⁡2πx+isin⁡2πx, which gives (65)ρ=u=e-2πy,θ=arg⁡u=2πxmod2π.Hence the potentials Φ± and Φ0 have the following lifts to H (see Figure 6): (66)Φ~+=M011+2π2y2,Φ~-=M0y22π-2+y2,Φ~0=M01+4πycos⁡2πx1+4π2y2. The function Φ~0(τ) (which is periodic under τ→τ+1) attains its minimum min⁡Φ~0=0 at the points τ=n+1/2+i/2π≈n+0.5+0.16i (with n∈Z), while the maximum max⁡Φ~0=2M0 is attained for τ=n+i/2π≈n+0.16i with n∈Z.Figure 6
The lifted potentialsΦ~± and Φ~0.
(a)
Plot ofΦ~+/M0 (blue) and Φ~-/M0 (orange) as functions of y=Imτ∈(0,+∞). The values y=0 and y=+∞ correspond, respectively, to the horn and cusp ends (b)
Level plot ofΦ~0/M0. Higher values of Φ~0 correspond to lighter tones. The fundamental domain DH is bounded by the red lines


### 6.11. Cosmological Trajectories on the Hyperbolic Punctured Disk

In this subsection, we present examples of numerically computed trajectories onD∗ for the vanishing scalar potential and for the globally well-behaved scalar potentials Φ± and Φ0. These were obtained as explained in Section 2.2, by numerically computing solutions of system (8) on the Poincaré half plane for the corresponding lifted potentials and then projecting these trajectories to the hyperbolic punctured disk using explicitly known uniformization map (50) (which is equivalent to (65)).Trajectories for Vanishing Scalar Potential. To understand the effect of the hyperbolic metric on the dynamics, we start with the case of a vanishing scalar potential Φ=0. Then Φ~=0 and one immediately checks that straight lines given by constant functions x(t)=x0, y(t)=y0 are solutions of (5) for any initial point (x0,y0)∈H, with initial velocity zero. This means that a scalar field starting “at rest” remains at rest for all times. On the other hand, numerical computation shows that any solution of (5) tends to the real axis for t→+∞, irrespective of its initial conditions. As a consequence, any solution defined on D∗ (which is obtained by projecting a solution defined on H through map (50)) will tend toward the horn end as t→+∞. This shows that the hyperbolic metric acts as an effective force which repulses φ away from the cusp end. Notice that a global trajectory defined on D∗ can have cusp and self-intersection points and hence that it need not correspond to an embedded curve in D∗.Figure7 shows five trajectories on H and their projections to D∗ for Φ=0 and α=M0/3, with the initial conditions shown in Table 1.Table 1
Initial conditions for five trajectories on the Poincaré half plane.
Trajectory τ 0 v~0 Orange 0.3+0.159i 0 Yellow 0.01+0.009i 0 Red 0.1+0.2i 2+3i Blue i 1+i Magenta 0.1 i 1.3 + 7 iFigure 7
Numerical solutions forΦ=0 and α=M0/3.
(a)
Trajectories forΦ~=0 on the Poincaré half plane. The solutions shown in orange and yellow are stationary (b)
Projection of the trajectories shown at the left to the hyperbolic punctured diskTrajectories for Φ -. Five lifted trajectories (and their projections to D∗) for α=M0/3 and Φ=Φ- with the initial conditions given in Table 1 are shown in Figure 8. Since Φ^- has a maximum at the cusp end (center of the disk) and a minimum at the horn end, it reinforces the effect of the hyperbolic metric, together with which it produces an effective repulsion away from the cusp end. In particular, the two trajectories which start with vanishing initial velocity are no longer stationary but evolve for t→+∞ to the funnel end (see the orange and yellow trajectories in the figures). Any other trajectory (irrespective of its initial velocity) also evolves for t→+∞ to the funnel end.Figure 8
Numerical solutions forΦ=Φ- and α=M0/3.
(a)
Trajectories forΦ~=Φ~- on the Poincaré half plane, drawn over a level plot of Φ~- on H (b)
Projection of the trajectories shown at the left to the hyperbolic punctured disk. We also show a level plot ofΦ- on D∗Trajectories for Φ +. Five lifted trajectories for α=M0/3 and Φ=Φ+ (and their projections to D∗) with the initial conditions given in Table 1 are shown in Figure 9. Since Φ^+ has a minimum at the cusp end (which corresponds to the center of the disk), it produces an attractive force toward the cusp end, which acts as a counterbalance to the repulsive effect of the hyperbolic metric.Figure 9
Numerical solutions forΦ=Φ+ and α=M0/3.
(a)
Trajectories forΦ~=Φ~+ on the Poincaré half plane and a level plot of Φ+ on  H (b)
Projection of the trajectories shown at the left to the hyperbolic punctured disk and a level plot ofΦ~+ on D∗Trajectories for Φ 0. Five lifted trajectories (and their projections to D∗) for α=M0/3 and Φ=Φ0 with the initial conditions given in Table 1 are shown in Figure 10.Figure 10
Numerical solutions forΦ=Φ0 and α=M0/3.
(a)
Trajectories forΦ~=Φ~0 on the Poincaré half plane (b)
Projection of the trajectories shown at the left to the hyperbolic punctured diskFigure11 shows in more detail the behavior of the magenta trajectory of Figure 10(a) near the local minimum of Φ~0 located at τ=3/2+i/2π (which projects to u=-1/e) and of its projection to D∗. For this solution, φ evolves in a spiral around the minimum of Φ0, until it settles at the minimum.Figure 11
Detail of the magenta trajectory.
(a)
Detail of the magenta trajectory shown in Figure10(a) near one minimum of Φ~0 (b)
Projection of the trajectory shown at the left to the hyperbolic punctured disk


### 6.12. Inflationary Regions and Number of e-Folds

Recall from [21] the expressions for the Hubble parameter and the critical Hubble parameter: (67)Ht=13M03αx˙t2+y˙t2yt2+2Φ~xt,yt,Hct=1M0Φ~t3.For the inflationary regions the following inequality should be satisfied: (68)Ht<Hct;meanwhile the number of e-folds N is given by integrating H(t) over the first inflationary time interval tI:(69)N=∫0tIHtdt.Analyzing the five trajectories with the initial conditions given in Table 1, we find that only the orange and yellow trajectories start in inflationary regime for each of the three potentials Φ+,Φ-,Φ0, while the other three trajectories do not start in inflationary regime for any of these potentials. We also notice that the yellow trajectory in Φ0 is in inflationary regime for all times. Calculating the number of e-folds, we find for the orange trajectory the values 0.44 for the potential Φ0 and 0.75 for Φ±, while the yellow trajectory has 55 e-folds for the potential Φ+ (so it satisfies the observational requirement of 50–60 e-folds) and 0.3 for Φ-. Varying the initial conditions of the yellow trajectory in the potential Φ+ (namely, varying Imτ0 in the range [0.0094,0.0096]) produces many other trajectories with N lying in the observationally expected range of [50,60] e-folds.


## 6.1. The Hyperbolic Metric

The punctured unit diskD∗={u∈C∣0<u<1} admits a unique complete hyperbolic metric given by [19] (33)dsD∗2=λD∗2u,u¯du2,whereλD∗u,u¯=1ulog⁡1/u.In particular, Reu and Imu are isothermal coordinates. In polar coordinates (ρ,θ) defined through (34)u=ρeiθρ=u∈0,1, the metric takes the form (35)dsD∗2=1ρlog⁡ρ2dρ2+ρ2dθ2. The center of D∗ corresponds to the cusp end, while the bounding circle of D∗ corresponds to the horn end (see below). The Euclidean circle at ρ=e-2π is a horocycle of hyperbolic length 1. Notice that D∗ has infinite hyperbolic area.


## 6.2. Diffeomorphism to the Punctured Plane

One can introduce an orthogonal coordinate system(r,θ)∈(0,+∞)×(0,2π) on D∗ through the coordinate transformation: (36)r=def.1log⁡1/ρ=1log⁡ρ∈0,+∞i.e.ρ=e-1/r∈0,1. This gives a diffeomorphism between D∗ and the punctured complex plane C∗=def.C∖{0} with complex coordinate: (37)ζ=reiθ. In this coordinate system, metric (33) takes the form (38)dsD∗2=1r2dr2+r2dθ2.The center of D∗ corresponds to r→0, while the bounding circle of D∗ corresponds to r→+∞.


## 6.3. Partial Isometric Embedding into Euclidean 3-Dimensional Space

One can isometrically embed the portion0<u<1/e of the hyperbolic punctured disk into Euclidean R3 as the open half tractricoid (the surface obtained by revolving an open half of a tractrix along its asymptote [20])  E defined in cylindrical coordinates (r,θ,x3) (where x1=rcos⁡θ and x2=rsin⁡θ) by the parametric equations: (39)r=1cosh⁡t∈0,1,x3=t-tanh⁡tt∈0,+∞. Indeed, it is easy to see that the Euclidean metric of R3 induces a metric on E which coincides with (38). This is the classical pseudosphere model of Beltrami (see Figure 3).Figure 3
Upper half of a tractricoid (pseudosphere). The horocycle of length one is drawn in light blue. It separates the half tractricoid into a cusp (upper part) and a portion of a horn (lower part). The latter is bounded by a horocycle of length2π.


## 6.4. The End Compactification ofD∗

The stereographic projection (20) identifies S2 with the one-point compactification of the ζ-plane. This shows explicitly that S2 is the end compactification of D∗, where the north pole ψ=0 corresponds to the horn end and the south pole ψ=π corresponds to the cusp end. The embedding j:D∗→S2 is given by (40)ψ=2arctan⁡log⁡ρ,θ=θ.


## 6.5. Semigeodesic Coordinates

The further change of variables(41)r=12πe-ri.e.r=-log⁡2πr=log⁡log⁡ρ2π∈-∞,+∞ brings metric (38) to the form (42)dsD∗2=dr2+e-2r2π2dθ2, where r∈R. In particular, (r,θ) are semigeodesic coordinates. The center u=0 of D∗ corresponds to r→+∞ while the bounding circle |u|=1 of D∗ corresponds to r→-∞. The horocycle at r=0 (i.e., ρ=e-2π) has length 1.


## 6.6. The Hyperbolic Cusp

Let(43)κ=def.e-2π. As mentioned above, the Euclidean circle u=κ has hyperbolic length 1. Thehyperbolic cusp (cf. [1]) corresponds to the portion of D∗ lying inside this circle (see Figure 3), which is the open punctured disk: (44)C=def.u∈C∣0<u<κ⊂D∗, endowed with the restriction of the metric (33); notice that the restricted metric is not complete. In coordinates (r,θ), the metric on C is obtained by restricting (38) to the range r∈(0,r0), where (45)r0=def.12π. In semigeodesic coordinates (r,θ) the cusp metric is given by (42) with the restriction r∈(0,+∞). Notice that C has hyperbolic area equal to 1.


## 6.7. The Hyperbolic Horn

By definition, thehyperbolic horn is the annulus: (46)H=u∈C∣κ<u<1⊂D∗,endowed with the (incomplete) restriction of metric (33). In coordinates (r,θ), the horn metric is obtained from (38) by restricting the range of r to (r0,+∞). In coordinates (r,θ), the metric takes form (42), with the restriction r∈(-∞,0). Defining r′=def.-r, this can be brought to the form (47)dsH2=dr′2+e2r′2π2dθ2,withr′∈0,+∞, where the bounding circle u=1 of D∗ corresponds to r′→+∞.


## 6.8. Canonical Uniformization toH

The punctured disk is uniformized to the Poincaré half plane (1) with complex coordinate τ by the parabolic cyclic group ΓP=def.P⊂PSL(2,R) generated by the translation: (48)τ⟶τ+1, which corresponds to the parabolic element: (49)P=def.1101∈PSL2,R. This fixes the point τ=∞∈∂∞H. The uniformization map is (50)u=πHτ=e2πiτ. The hyperbolic cusp C is the projection through πH of the cusp domain: (51)CH=τ∈H∣Imτ>1, which is bounded by the horocycle: (52)cH=τ∈H∣Imτ=1. This horocycle is tangent to the conformal boundary of H at the point τ=∞, which projects through πH to the cusp ideal point of the end compactification of D∗.A fundamental polygon for the action ofΓP on H is given by the semi-infinite vertical strip (53)DH=τ∈H∣0<Reτ<1 and has vertices at the points (see Figure 4(a)): (54)A:τ=∞,B:τ=0,C:τ=1. The Poincaré side pairing maps (AB) into (AC) through transformation (48), which generates ΓP. The relative cusp neighborhood [1] with respect to DH is the intersection CH=def.CH∩DH. A lifted scalar potential Φ~=Φ∘πH is given by (55)Φ~τ=Φe2πiτ, being invariant under translation (48). In particular, the restrictions of Φ~ to the sides (AB) and (AC) agree through the Poincaré pairing.Figure 4
Fundamental polygons for the uniformization ofD∗ to the Poincaré half plane and to the hyperbolic disk.
(a)
A fundamental polygonDH for the punctured disk on H. The relative cusp neighborhood CH of the vertex A corresponds to the shaded region (b)
A fundamental polygonDD for the punctured disk on D. The shaded region is the relative cusp neighborhood CD of the vertex A′


## 6.9. Canonical Uniformization toD

For completeness and comparison with [1], we also give the canonical uniformization of D∗ to the hyperbolic disk. The Cayley transformation (56)u=i-τiτ-1 is an isometry from H to D. When uniformizing D∗ to D, the fundamental polygon DH becomes a hyperbolic triangle DD with vertices at the following points, which correspond, respectively, to the points A, B, and C of (54) through the Cayley transformation (57)A′:u=i,B′:u=-i,C′:u=1 and a free side connecting the points B′ and C′ (see Figure 4(b)). The sides of this triangle are a segment connecting A′ to B′ (which passes through the origin of D), the portion of ∂∞D connecting B′ to C′, and Euclidean circular arc orthogonal to ∂∞D which connects A′ to C′. The hyperbolic cusp neighborhood CD of the vertex A′ is bounded by the horocycle: (58)cD=t2-it∈DD∣t∈R, which is tangent to ∂∞D at the point A′. The intersection of CD with DD is the relative cusp neighborhood CD with respect to DD (see [1]), which is the image of CH through the Cayley transformation.


## 6.10. Globally Well-Behaved Scalar Potentials onD∗

A scalar potentialΦ:Σ→R is globally well-behaved on D∗ iff there exists a smooth map Φ^:S2→R such that(59)Φr,θ=Φ^2arccot⁡r,θ;that is, (60)Φρ,θ=Φ^2arctan⁡log⁡ρ,θ,where ρ=u. Expansion (13) gives the uniformly convergent series: (61)Φρ,θ=∑l=0∞∑m=-llDlmYlm2arctan⁡log⁡ρ,θ.For choices (16) and (17), we find (62)Φ+=M0r21+r2=M011+log⁡ρ2,Φ-=M011+r2=M0log⁡ρ21+log⁡ρ2,Φ0=M01+2rcos⁡θ1+r2=M01+2log⁡ρ1+log⁡ρ2cos⁡θ.These potentials are shown in Figure 5. Notice that Φ+ leads to α-attractor behavior if inflation takes place near the horn end, while Φ- leads to α-attractor behavior if inflation takes place near the cusp end [1]. The extended potentials Φ^± have maxima and minima at the two ideal points of the end compactification of D∗ (which correspond to the north and south poles of S2). On the other hand, Φ^0 doesnot have extrema at the ideal points; its extrema coincide with those of Φ0, being located inside D∗. The minimum (equal to zero) is at the point u=-1/e≈-0.36 while the maximum (equal to 2M0) is at u=+1/e≈+0.36.Figure 5
The potentialsΦ± and Φ0 on the hyperbolic punctured disk.
(a)
Plot ofΦ+/M0 (blue) and Φ-/M0 (orange) as functions of ρ∈(0,1). The value ρ=0 corresponds to the cusp end, while ρ=1 corresponds to the horn end (b)
Level plot ofΦ0/M0 on the punctured disk. Darker tones indicate lower values of Φ0/M0Remark 3.
Using (41), we find the following expressions in semigeodesic coordinates: (63)Φ+r=M01+2π2e2r,Φ-r=M01+e-2r/2π2,Φ0r,θ=M01+4πe-rcos⁡θ2π2+e-2r.Lift of the Potentials Φ ± and Φ 0 to H. Consider the well-behaved scalar potentials Φ± and Φ0 on D∗ given in (62). Let x=Reτ and y=def.Imτ. Then the covering map (50) reads (64)u=πHτ=e2πiτ=e-2πycos⁡2πx+isin⁡2πx, which gives (65)ρ=u=e-2πy,θ=arg⁡u=2πxmod2π.Hence the potentials Φ± and Φ0 have the following lifts to H (see Figure 6): (66)Φ~+=M011+2π2y2,Φ~-=M0y22π-2+y2,Φ~0=M01+4πycos⁡2πx1+4π2y2. The function Φ~0(τ) (which is periodic under τ→τ+1) attains its minimum min⁡Φ~0=0 at the points τ=n+1/2+i/2π≈n+0.5+0.16i (with n∈Z), while the maximum max⁡Φ~0=2M0 is attained for τ=n+i/2π≈n+0.16i with n∈Z.Figure 6
The lifted potentialsΦ~± and Φ~0.
(a)
Plot ofΦ~+/M0 (blue) and Φ~-/M0 (orange) as functions of y=Imτ∈(0,+∞). The values y=0 and y=+∞ correspond, respectively, to the horn and cusp ends (b)
Level plot ofΦ~0/M0. Higher values of Φ~0 correspond to lighter tones. The fundamental domain DH is bounded by the red lines


## 6.11. Cosmological Trajectories on the Hyperbolic Punctured Disk

In this subsection, we present examples of numerically computed trajectories onD∗ for the vanishing scalar potential and for the globally well-behaved scalar potentials Φ± and Φ0. These were obtained as explained in Section 2.2, by numerically computing solutions of system (8) on the Poincaré half plane for the corresponding lifted potentials and then projecting these trajectories to the hyperbolic punctured disk using explicitly known uniformization map (50) (which is equivalent to (65)).Trajectories for Vanishing Scalar Potential. To understand the effect of the hyperbolic metric on the dynamics, we start with the case of a vanishing scalar potential Φ=0. Then Φ~=0 and one immediately checks that straight lines given by constant functions x(t)=x0, y(t)=y0 are solutions of (5) for any initial point (x0,y0)∈H, with initial velocity zero. This means that a scalar field starting “at rest” remains at rest for all times. On the other hand, numerical computation shows that any solution of (5) tends to the real axis for t→+∞, irrespective of its initial conditions. As a consequence, any solution defined on D∗ (which is obtained by projecting a solution defined on H through map (50)) will tend toward the horn end as t→+∞. This shows that the hyperbolic metric acts as an effective force which repulses φ away from the cusp end. Notice that a global trajectory defined on D∗ can have cusp and self-intersection points and hence that it need not correspond to an embedded curve in D∗.Figure7 shows five trajectories on H and their projections to D∗ for Φ=0 and α=M0/3, with the initial conditions shown in Table 1.Table 1
Initial conditions for five trajectories on the Poincaré half plane.
Trajectory τ 0 v~0 Orange 0.3+0.159i 0 Yellow 0.01+0.009i 0 Red 0.1+0.2i 2+3i Blue i 1+i Magenta 0.1 i 1.3 + 7 iFigure 7
Numerical solutions forΦ=0 and α=M0/3.
(a)
Trajectories forΦ~=0 on the Poincaré half plane. The solutions shown in orange and yellow are stationary (b)
Projection of the trajectories shown at the left to the hyperbolic punctured diskTrajectories for Φ -. Five lifted trajectories (and their projections to D∗) for α=M0/3 and Φ=Φ- with the initial conditions given in Table 1 are shown in Figure 8. Since Φ^- has a maximum at the cusp end (center of the disk) and a minimum at the horn end, it reinforces the effect of the hyperbolic metric, together with which it produces an effective repulsion away from the cusp end. In particular, the two trajectories which start with vanishing initial velocity are no longer stationary but evolve for t→+∞ to the funnel end (see the orange and yellow trajectories in the figures). Any other trajectory (irrespective of its initial velocity) also evolves for t→+∞ to the funnel end.Figure 8
Numerical solutions forΦ=Φ- and α=M0/3.
(a)
Trajectories forΦ~=Φ~- on the Poincaré half plane, drawn over a level plot of Φ~- on H (b)
Projection of the trajectories shown at the left to the hyperbolic punctured disk. We also show a level plot ofΦ- on D∗Trajectories for Φ +. Five lifted trajectories for α=M0/3 and Φ=Φ+ (and their projections to D∗) with the initial conditions given in Table 1 are shown in Figure 9. Since Φ^+ has a minimum at the cusp end (which corresponds to the center of the disk), it produces an attractive force toward the cusp end, which acts as a counterbalance to the repulsive effect of the hyperbolic metric.Figure 9
Numerical solutions forΦ=Φ+ and α=M0/3.
(a)
Trajectories forΦ~=Φ~+ on the Poincaré half plane and a level plot of Φ+ on  H (b)
Projection of the trajectories shown at the left to the hyperbolic punctured disk and a level plot ofΦ~+ on D∗Trajectories for Φ 0. Five lifted trajectories (and their projections to D∗) for α=M0/3 and Φ=Φ0 with the initial conditions given in Table 1 are shown in Figure 10.Figure 10
Numerical solutions forΦ=Φ0 and α=M0/3.
(a)
Trajectories forΦ~=Φ~0 on the Poincaré half plane (b)
Projection of the trajectories shown at the left to the hyperbolic punctured diskFigure11 shows in more detail the behavior of the magenta trajectory of Figure 10(a) near the local minimum of Φ~0 located at τ=3/2+i/2π (which projects to u=-1/e) and of its projection to D∗. For this solution, φ evolves in a spiral around the minimum of Φ0, until it settles at the minimum.Figure 11
Detail of the magenta trajectory.
(a)
Detail of the magenta trajectory shown in Figure10(a) near one minimum of Φ~0 (b)
Projection of the trajectory shown at the left to the hyperbolic punctured disk


## 6.12. Inflationary Regions and Number of e-Folds

Recall from [21] the expressions for the Hubble parameter and the critical Hubble parameter: (67)Ht=13M03αx˙t2+y˙t2yt2+2Φ~xt,yt,Hct=1M0Φ~t3.For the inflationary regions the following inequality should be satisfied: (68)Ht<Hct;meanwhile the number of e-folds N is given by integrating H(t) over the first inflationary time interval tI:(69)N=∫0tIHtdt.Analyzing the five trajectories with the initial conditions given in Table 1, we find that only the orange and yellow trajectories start in inflationary regime for each of the three potentials Φ+,Φ-,Φ0, while the other three trajectories do not start in inflationary regime for any of these potentials. We also notice that the yellow trajectory in Φ0 is in inflationary regime for all times. Calculating the number of e-folds, we find for the orange trajectory the values 0.44 for the potential Φ0 and 0.75 for Φ±, while the yellow trajectory has 55 e-folds for the potential Φ+ (so it satisfies the observational requirement of 50–60 e-folds) and 0.3 for Φ-. Varying the initial conditions of the yellow trajectory in the potential Φ+ (namely, varying Imτ0 in the range [0.0094,0.0096]) produces many other trajectories with N lying in the observationally expected range of [50,60] e-folds.


## 7. Hyperbolic Annuli

Hyperbolic annuli (also known as “hyperbolic cylinders” [8]) have a single modulus and two funnel ends.


### 7.1. The Hyperbolic Metric

LetR>1 be a real number. The annulus A(R)={u∈C∣1/R<u<R} of modulus μ=2log⁡R>0 admits a unique complete hyperbolic metric, which is given by [19] (70)dsR2=λRu,u¯2du2,whereλRu,u¯=π2log⁡R1ucos⁡πlog⁡u/2log⁡R.In particular, Reu and Imu are isothermal coordinates. Notice that the transformation u→1/u¯ is an isometry. In polar coordinates (ρ,θ) defined through (71)u=ρeiθ,withρ=u∈1R,R,the metric takes the form(72)dsR2=π2log⁡R2dρ2+ρ2dθ2ρcos⁡πlog⁡ρ/2log⁡R2.For any ρ∈(1/R,R), the Euclidean circle cρ=def.{u∈C∣u=ρ} has hyperbolic circumference given by (73)lRcρ=π2log⁡Rcos⁡πlog⁡ρ/2log⁡R.We have(74)lRcρ=lRc1/ρ.Notice that lR(cρ) increases from π2/log⁡R to infinity as ρ increases from 1 to R and as ρ decreases from 1 to 1/R. In particular, the minimum hyperbolic length is attained for the circle c1 of Euclidean radius ρ=1, which is the only closed hyperbolic geodesic of A(R) and has hyperbolic length: (75)l=def.lRc1=π2log⁡R=2π2μ,known as thehyperbolic circumference of A(R). The lines (76)teiθ∣t∈1R,Rare geodesics of infinite length. Notice that the hyperbolic area of A(R) is infinite. Relation (75) gives (77)R=Rl=eπ2/l.


### 7.2. Diffeomorphism to the Punctured Disk and to the Punctured Plane

The annulusA(R) is diffeomorphic (but not biholomorphic!) with the punctured unit disk with complex coordinate u′ through the map: (78)u′=fu=u-1/RR-1/Ruu∈D∗u∈AR,whose inverse is given by(79)u=f-1u′=R-1/Ru′+1/Ru′u′∈ARu′∈D∗. Notice that f maps the Euclidean circle u=1/R to u′=0 and the Euclidean circle u=R to the Euclidean circle u′=1. Composing f with the map (36) gives a diffeomorphism from A(R) to the punctured complex plane with coordinate: (80)ζ=reiθ,where(81)r=-1logu′=1log⁡R-1/R/u-1/R=1log⁡R-1/R/ρ-1/R.The funnel end at u=1/R corresponds to ζ=0 while the funnel end at u=R corresponds to ζ=∞.


### 7.3. The End Compactification ofA(R)

The end compactification ofA(R) is the unit sphere S2 with polar coordinates (ψ,θ)∈(0,π)×(0,2π), which is mapped to the ζ-plane through stereographic projection (20). The funnel end at u=1/R corresponds to the south pole, while the funnel end at |u|=R corresponds to the north pole. The explicit embedding of A(R) into S2 is given by (82)ψ=2arctan⁡logR-1/Rρ-1/R,θ=θ.


### 7.4. The Hyperbolic Funnel

Letl>0 be a positive real number and Rl be defined as in (77). By definition, ahyperbolic funnel (cf. [1]) of circumference l>0 is the annulus: (83)Fl=u∈C∣1Rl<u<1⊂ARl, endowed with the restriction of the metric (70). Since u→1/u¯ is an isometry of A(R), the funnel is isometric with the annulus 1<u<Rl (endowed with the restriction of (70)). Hence A(R) decomposes as the disjoint union of two funnels and the closed geodesic c1. Notice that a funnel has infinite hyperbolic area. The funnel is diffeomorphic (but not conformally equivalent!) with the punctured unit disk through the map: (84)u′=u-1/Rl1-1/Rluu∈D∗u∈Fl,which takes the funnel end u→1/Rl to the center u′→0 of the unit disk and the circle c1 into the bounding circle of the unit disk.


### 7.5. Semigeodesic Coordinates onFl

Consider orthogonal coordinates(q,θ) on Fl given by (85)q=πμ1cosπ/μlog⁡ρ∈πμ,+∞, where ρ=u and θ∈(0,2π) is the polar angle in the u-plane. The limit ρ→1/R corresponds to q→∞, while ρ→1 corresponds to q→π/μ=l/2π. In these coordinates, the metric on Fl becomes (86)dsFl2=1q2-π/μ2dq2+q2dθ2=1q2-l/2π2dq2+q2dθ2.The coordinates θ and(87)r=def.arccosh⁡2πlq=arccosh⁡1cosπ/μlog⁡ρ∈0,+∞are semigeodesic on Fl. In these coordinates, the metric takes the form (88)dsFl2=dr2+l22π2cosh2rdθ2.The limit ρ→1/R corresponds to r→+∞ while ρ→1 corresponds to r→0. The parameter l>0 is the hyperbolic length of the geodesic at r=0.


### 7.6. Partial Isometric Embedding of the Funnel into Euclidean 3-Dimensional Space

One can isometrically embed the annulusFl0⊂Fl corresponding to the range (89)l2π=πμ<q≤1+πμ2=1+l2π2into Euclidean R3 as the surface of revolution defined by the parametric equations (see [20, Chap. 3C] or [22, Chap. 15]): (90)x1=qrcos⁡θ,x2=qrsin⁡θ,x3=ξr,withr∈0,arcsinh⁡2πl,where(91)qr=l2πcosh⁡r∈l2π,1+l2π2see87ξr=def.∫0rdr′1-l2π2sinh2r′=-iEir,-l2π2,and E(τ,m) denotes the elliptic integral of the second kind. This is one of the three types of (incomplete) classical surfaces of revolution in R3 of constant Gaussian curvature equal to -1, namely, a surface of “hyperboloid type” (see Figure 14). The other two are the pseudosphere/tractricoid (which corresponds to a portion of the hyperbolic cusp) and the surface of “conical type.”


### 7.7. Canonical Uniformization toH

The hyperbolic annulusA(Rl) is uniformized to the upper half plane by the hyperbolic cyclic subgroup generated by the transformation: (92)τ⟶elτ,which corresponds to the hyperbolic element (93)Hl=def.el/200e-l/2∈PSL2,Rand fixes the points τ=0 and τ=∞ lying on ∂∞H. The uniformization map is (94)u=πHτ=Rle2πi/llog⁡τ=eπ2/l+2πi/llog⁡τ,which gives(95)ρ=eπ2/l-2π/larg⁡τ,θ=2πllog⁡τmod2π.A fundamental polygon is given by (see Figure 15(a)) (96)DH=τ∈H∣el<τ<e2l,l>0.This is a hyperbolic quadrilateral with two free sides and vertices located at the points (97)A:τ=-e2l,B:τ=-el,C:τ=el,D:τ=e2l.The funnel Fl is the projection of the relative funnel neighborhood [1]: (98)FHl=τ∈DH∣Reτ<0=τ∈DH∣arg⁡τ∈π2,π.


### 7.8. Canonical Uniformization toD

When passing to the disk model, the fundamental domainDH is mapped by the Cayley transformation (56) to a hyperbolic quadrilateral DD with vertices located at the following points, which are obtained from the points A,B,C,D defined in (97) by applying (56): (99)A′:u=-i+e2lie2l+1,B′:u=-i+eliel+1,C′:u=i-eliel-1,D′:u=i-e2lie2l-1.The free sides (A′B′) and (C′D′) are portions of ∂∞D (see Figure 15(b)), while the sides (B′C′) and (A′D′) are arc segments of Euclidean circles which are orthogonal to ∂∞D.


### 7.9. Globally Well-Behaved Scalar Potentials onA(R)

A scalar potentialΦ on A(R) is globally well-behaved iff there exists a smooth function Φ^:S2→R such that(100)Φr,θ=Φ^2arccot⁡r,θ;that is,(101)Φρ,θ=Φ^2arctanlogR-1/Rρ-1/R,θ.Expansion (13) gives the uniformly convergent series: (102)Φρ,θ=∑l=0∞∑m=-llDlmYlm2arctan⁡logR-1/Rρ-1/R,θ.For choices (16), we find (103)Φ+=M0r21+r2=M011+log⁡R-1/R/ρ-1/R2,Φ-=M011+r2=M0log⁡R-1/R/ρ-1/R21+log⁡R-1/R/ρ-1/R2,where ρ=u∈(1/R,R) and we used (81). The potentials Φ± are plotted in Figure 16(a) for the case l=π2 (R=e). For choice (17), we find (104)Φ0=M01+2r1+r2cos⁡θ=M01+2log⁡R-1/R/ρ-1/R1+log⁡R-1/R/ρ-1/R2cos⁡θ.Recall that Φ^0 has two extrema on S2, which are located at (ψ,θ)=(π/2,0) (maximum) and (ψ,θ)=(π/2,π) (minimum). At each of these points, relation (20) gives r=cot⁡(π/4)=1, so (81) gives ρ=ρ0, where (105)ρ0=def.1R+1eR-1R.It follows that the two critical points of Φ0 on A(R) are located on the real axis at(i)
uM=+ρ0, where Φ0 attains its maximum (which equals 2M),(ii)
um=-ρ0, where Φ0 attains its minimum (which equals zero).The level curves ofΦ0 are shown in Figure 16(b) for the case l=π2 (R=e), which gives ρ0=(e2+e-1)/e2=1+1/e-1/e2≈1.23.Lift of Φ ± and Φ 0 to H. The globally well-behaved scalar potentials (103) and (104) lift to the following potentials on H (see Figures 17(a) and 17(b)):(106)Φ~+τ=M011+log⁡R-1/R/ρτ-1/R2,Φ~-τ=M0log⁡R-1/R/ρτ-1/R21+log⁡R-1/R/ρτ-1/R2,Φ~0τ=M01+2log⁡R-1/R/ρτ-1/R1+log⁡R-1/R/ρτ-1/R2cos2πllog⁡τ,where ρ(τ)=eπ2/l-(2π/l)arg⁡(τ) (see (95)). The lifted potential Φ~0 has local maxima at the inverse image points of uM=+ρ0 and local minima at the inverse image points of um=-ρ0 (see (105)). Using (95), we find that these are located at(i)
(maxima)τ=τM(n)=def.enl+i(l/2π)logR/ρ0 (where Φ~0 equals 2M0),(ii)
(minima)τ=τm(n)=def.e(n+1/2)l+i(l/2π)logR/ρ0 (where Φ~0 vanishes),withn∈Z being an arbitrary integer. Hence all extrema of Φ~0 lie on the half-line L0 through the origin which makes an angle θ0=(l/2π)logR/ρ0 with the x-axis of the τ-plane. The maxima and minima alternate along this half-line and the ratio between the absolute values of two consecutive maxima or two consecutive minima equals el; in particular, the extrema accumulate toward the origin along L0. The fundamental domain DH contains exactly one of the minima, namely, that located at τ=e(3/2)l+iθ0. On the other hand, the two nonfree sides of DH contain the two consecutive maxima located at τ=el+iθ0 and τ=e2l+iθ0; these two maxima of Φ~0 are identified by the projection πH.In Figure17(b), we show the level curves of Φ~0 for the case l=π2 (R=e), which gives ρ0=1+1/e-1/e2≈1.23 and θ0=(π/2)1-log⁡1+1/e-1/e2≈0.39π. In this case, the absolute value of the ratio of successive minima or maxima of Φ~0 equals eπ2≈1.93×104. Due to the large size of this ratio, we chose for clarity to display the level curves of Φ~ on a region of the “semilogarithmic upper half plane.” The latter has complex coordinate T=X+iY (where X=ReT and Y=ImT>0), being related to the region (107)R=def.τ∈H∣τ>1of the Poincaré half plane through the coordinate transformation: (108)T=logτeiarg⁡τ,τ=eTeiarg⁡T. Notice that R contains only those extrema of Φ~0 which have absolute value larger than one. In the semilogarithmic half plane, these extrema are located at T=nleiθ0 with n∈Z>0 (maxima) and T=(n+1/2)leiθ0 with n∈Z≥0 (minima), lying equally spaced on the half-line in the T-plane which passes through the origin at angle θ0 with the x-axis. The Cartesian coordinates X,Y and x=Reτ, y=Imτ are related through (109)x=XeX2+Y2X2+Y2,y=YeX2+Y2X2+Y2.Figure 17(b) shows the level curves of the function Φ~0(X,Y) in the region defined by X<2l and Y<2l (where 2l=2π2≈19.7), which contains the image of the following annular region of the Poincaré half plane: (110)A=def.τ∈H∣1<τ<e2l.Notice that A contains a copy of the fundamental domain DH.


### 7.10. Lift of the Cosmological Model toH

We now present examples of trajectories onA(R) for l=π2 (R=e, modulus μ=2) for the vanishing scalar potential and for the globally well-behaved scalar potentials Φ± and Φ0. These were obtained as explained in Section 2.2, by numerically computing solutions of the system (8) on the Poincaré half plane for the corresponding lifted potentials and then projecting these trajectories to the hyperbolic punctured disk using the explicitly known uniformization map (94), which is equivalent to (95).Trajectories for Vanishing Scalar Potential. Figure 18 shows five trajectories (orange, yellow, red, blue, and magenta) for α=M0/3, l=π2 (R=e), and Φ=0, with the initial conditions given in Table 1. In this case, vanishing initial velocity leads to two stationary trajectories (the orange and yellow dots), while the hyperbolic geometry produces an effective attraction toward the outer funnel end for |u|>1 and toward the inner funnel end for u<1. The red, blue, and magenta trajectories start in the funnel region given by u>1 (with velocities pointing toward the outer funnel end) and hence evolve toward that end.Trajectories for Φ -. Figure 19 shows five trajectories (orange, yellow red, blue, and magenta) for α=M0/3, l=π2 (R=e), and Φ=Φ-, with the initial conditions given in Table 1. In this case, the potential produces a repulsive force away from the inner funnel end and an attraction force toward the outer end, thus accentuating the effect of the hyperbolic metric on the trajectories shown.Trajectories for Φ +. Figure 20 shows five trajectories (orange, yellow red, blue, and magenta) for α=M0/3, l=π2 (R=e) and Φ=Φ+, with the initial conditions given in Table 1. In this case, the potential induces an attractive force toward the inner funnel end. As a consequence, each of the red, blue, and magenta trajectories turns at some point in A(R) and evolves back toward the inner funnel end.Trajectories for Φ 0 . Figure 21 shows five trajectories (orange, yellow, red, blue, and magenta) for α=M0/3, l=π2 (R=e), and Φ=Φ0, with the initial conditions given in Table 1. Figure 22 shows a detail of the yellow trajectory on both H and A(R). It spirals in a complicated manner around a minimum point of Φ~0 and projects to the single minimum of Φ0 on A(R) located at u=-ρ0≈-1.23. For t→∞, the trajectory reaches the minimum point.


### 7.11. Inflationary Regions and the Number of e-Folds

Using relations (67)–(69), we find that, among the five trajectories with initial conditions given in Table 1, the orange and yellow trajectories start in inflationary regime for all three potentials Φ+,Φ-, and Φ0 (see Figures 12, 13, and 23), while the blue trajectory is always inflationary in potential Φ0 but is never inflationary for the other potentials. The orange and yellow trajectories give, respectively, 76 and 74 e-folds in the potential Φ+ after the first inflationary regime, which lasts for a cosmological time of t=2223 s, respectively, t=2184 s. The other trajectories do not start in the inflationary regime for any of the three potentials. It is easy to see that one can rescale the potential Φ+ by a positive constant in order to reach a phenomenologically appropriate value N∈[50,60] for each of the orange and yellow trajectories in this potential.Figure 12
Plot ofH(t)/M0 (black) and Hc(t)/M0 (green) for the yellow trajectory in the potential Φ+. For better clarity, we truncated the plot at t=200 s.Figure 13
Plots ofH(t)/M0 (black) and Hc(t)/M0 (green) for the yellow trajectory in the potential Φ0 and for the orange trajectory in the potentials Φ0 and Φ-, respectively. For better clarity, we truncated the plots at convenient values of t.Figure 14
Hyperbolic surface of revolution of hyperboloid type. The upper half corresponds to the regionFl0 of Fl. The circle drawn in light blue is the closed geodesic c1 of length l.Figure 15
Fundamental domains for the uniformization of the hyperbolic annulus to the Poincaré half plane and to the hyperbolic disk.
(a)
Fundamental domainDH for A(R) on the Poincaré half plane. The shaded region is the relative funnel neighborhood FH corresponding to the free side on the left (b)
Fundamental domainDD for A(R) on the Poincaré disk. The shaded region is the relative funnel neighborhood of the free side on the leftFigure 16
The potentialsΦ± and Φ0 on the annulus of modulus μ=2 (l=π2, R=e). For Φ+ and Φ-, we indicate only the dependence of ρ=|u|, since these two potentials do not depend on the polar angle θ. Notice that Φ+ tends to its infimum at the inner funnel end, while Φ- tends to its infimum at the outer funnel end. Similarly, Φ+ and Φ- tend to their suprema at opposite funnel ends.
(a)
Plot ofΦ+/M0 (blue) and Φ-/M0 (orange) as functions of ρ∈(1/R,R)=(e-1,e) for l=π2. The values ρ=1/R and ρ=R correspond to the two funnel ends of A(R) (b)
Level plot ofΦ0/M0 on A(R) for l=π2. Darker tones indicate lower values of Φ0. In this case, the maximum of Φ0 is located at uM=(e2+e-1)/e2, while the minimum is located at um=-uM=-(e2+e-1)/e2Figure 17
The lifted potentialsΦ~± and Φ~0 for the annulus of modulus μ=2 (l=π2, R=e).
(a)
Plot ofΦ~+/M0 (blue) and Φ~-/M0 (orange) as functions of arg⁡(τ)∈(0,π) for l=π2. The values arg⁡(τ)=0 and arg⁡(τ)=π correspond to the two funnel ends of A(R) (b)
Level plot ofΦ~0/M0 in the coordinates X,Y for l=π2. Darker tones indicate lower values of Φ~0. The maxima of Φ~0 are indicated as blue dots, while the minima are shown as small red squaresFigure 18
Five trajectories forΦ=0, α=M0/3, and l=π2, with initial conditions as in Table 1.
(a)
ForΦ=0, the five trajectories on H with initial conditions given in Table 1 coincide with those shown in Figure 7(a) (b)
Projection toA(R) of the trajectories shown at the left. The projection differs from that for D∗ (cf. Figure 7(b)) since the uniformization maps are differentFigure 19
Five trajectories forΦ=Φ-, α=M0/3, and l=π2 with initial conditions of Table 1.
(a)
Trajectories onH for Φ~=Φ~-, superimposed on a level plot of Φ~- on H (b)
Projection toA(R) of the trajectories shown at the left, superimposed on a level plot of Φ- on A(R)Figure 20
Five trajectories forΦ=Φ+, α=M0/3, and l=π2, with initial conditions of Table 1.
(a)
Trajectories onH for Φ~=Φ~+, superimposed on a level plot of Φ~- on H (b)
Projection toA(R) of the trajectories shown at the left, superimposed on a level plot of Φ+ on A(R)Figure 21
Five trajectories forΦ=Φ0 when α=M0/3 and l=π2.
(a)
Trajectories onH for Φ~=Φ~0. The yellow trajectory is not stationary, although at the scale shown it looks like a dot (b)
Projection toA(R) of the five trajectories shown at the leftFigure 22
Detail of the yellow trajectory shown in Figure21.
(a)
Detail of the lifted yellow trajectory shown in Figure21(a) near one minimum of Φ~0 (b)
Projection of the yellow trajectory to the hyperbolic annulus. The trajectory spirals in a complicated manner around the minimum ofΦ0, until it settles at the minimum pointFigure 23
Plots ofH(t)/M0 (black) and Hc(t)/M0 (green) for the orange and yellow trajectories in the potential Φ+. For better clarity, we truncated the plot at t=60 s. For both trajectories, inflation stops after a finite time (indicated in the text), though this is not visible at the scale of the figures.


## 7.1. The Hyperbolic Metric

LetR>1 be a real number. The annulus A(R)={u∈C∣1/R<u<R} of modulus μ=2log⁡R>0 admits a unique complete hyperbolic metric, which is given by [19] (70)dsR2=λRu,u¯2du2,whereλRu,u¯=π2log⁡R1ucos⁡πlog⁡u/2log⁡R.In particular, Reu and Imu are isothermal coordinates. Notice that the transformation u→1/u¯ is an isometry. In polar coordinates (ρ,θ) defined through (71)u=ρeiθ,withρ=u∈1R,R,the metric takes the form(72)dsR2=π2log⁡R2dρ2+ρ2dθ2ρcos⁡πlog⁡ρ/2log⁡R2.For any ρ∈(1/R,R), the Euclidean circle cρ=def.{u∈C∣u=ρ} has hyperbolic circumference given by (73)lRcρ=π2log⁡Rcos⁡πlog⁡ρ/2log⁡R.We have(74)lRcρ=lRc1/ρ.Notice that lR(cρ) increases from π2/log⁡R to infinity as ρ increases from 1 to R and as ρ decreases from 1 to 1/R. In particular, the minimum hyperbolic length is attained for the circle c1 of Euclidean radius ρ=1, which is the only closed hyperbolic geodesic of A(R) and has hyperbolic length: (75)l=def.lRc1=π2log⁡R=2π2μ,known as thehyperbolic circumference of A(R). The lines (76)teiθ∣t∈1R,Rare geodesics of infinite length. Notice that the hyperbolic area of A(R) is infinite. Relation (75) gives (77)R=Rl=eπ2/l.


## 7.2. Diffeomorphism to the Punctured Disk and to the Punctured Plane

The annulusA(R) is diffeomorphic (but not biholomorphic!) with the punctured unit disk with complex coordinate u′ through the map: (78)u′=fu=u-1/RR-1/Ruu∈D∗u∈AR,whose inverse is given by(79)u=f-1u′=R-1/Ru′+1/Ru′u′∈ARu′∈D∗. Notice that f maps the Euclidean circle u=1/R to u′=0 and the Euclidean circle u=R to the Euclidean circle u′=1. Composing f with the map (36) gives a diffeomorphism from A(R) to the punctured complex plane with coordinate: (80)ζ=reiθ,where(81)r=-1logu′=1log⁡R-1/R/u-1/R=1log⁡R-1/R/ρ-1/R.The funnel end at u=1/R corresponds to ζ=0 while the funnel end at u=R corresponds to ζ=∞.


## 7.3. The End Compactification ofA(R)

The end compactification ofA(R) is the unit sphere S2 with polar coordinates (ψ,θ)∈(0,π)×(0,2π), which is mapped to the ζ-plane through stereographic projection (20). The funnel end at u=1/R corresponds to the south pole, while the funnel end at |u|=R corresponds to the north pole. The explicit embedding of A(R) into S2 is given by (82)ψ=2arctan⁡logR-1/Rρ-1/R,θ=θ.


## 7.4. The Hyperbolic Funnel

Letl>0 be a positive real number and Rl be defined as in (77). By definition, ahyperbolic funnel (cf. [1]) of circumference l>0 is the annulus: (83)Fl=u∈C∣1Rl<u<1⊂ARl, endowed with the restriction of the metric (70). Since u→1/u¯ is an isometry of A(R), the funnel is isometric with the annulus 1<u<Rl (endowed with the restriction of (70)). Hence A(R) decomposes as the disjoint union of two funnels and the closed geodesic c1. Notice that a funnel has infinite hyperbolic area. The funnel is diffeomorphic (but not conformally equivalent!) with the punctured unit disk through the map: (84)u′=u-1/Rl1-1/Rluu∈D∗u∈Fl,which takes the funnel end u→1/Rl to the center u′→0 of the unit disk and the circle c1 into the bounding circle of the unit disk.


## 7.5. Semigeodesic Coordinates onFl

Consider orthogonal coordinates(q,θ) on Fl given by (85)q=πμ1cosπ/μlog⁡ρ∈πμ,+∞, where ρ=u and θ∈(0,2π) is the polar angle in the u-plane. The limit ρ→1/R corresponds to q→∞, while ρ→1 corresponds to q→π/μ=l/2π. In these coordinates, the metric on Fl becomes (86)dsFl2=1q2-π/μ2dq2+q2dθ2=1q2-l/2π2dq2+q2dθ2.The coordinates θ and(87)r=def.arccosh⁡2πlq=arccosh⁡1cosπ/μlog⁡ρ∈0,+∞are semigeodesic on Fl. In these coordinates, the metric takes the form (88)dsFl2=dr2+l22π2cosh2rdθ2.The limit ρ→1/R corresponds to r→+∞ while ρ→1 corresponds to r→0. The parameter l>0 is the hyperbolic length of the geodesic at r=0.


## 7.6. Partial Isometric Embedding of the Funnel into Euclidean 3-Dimensional Space

One can isometrically embed the annulusFl0⊂Fl corresponding to the range (89)l2π=πμ<q≤1+πμ2=1+l2π2into Euclidean R3 as the surface of revolution defined by the parametric equations (see [20, Chap. 3C] or [22, Chap. 15]): (90)x1=qrcos⁡θ,x2=qrsin⁡θ,x3=ξr,withr∈0,arcsinh⁡2πl,where(91)qr=l2πcosh⁡r∈l2π,1+l2π2see87ξr=def.∫0rdr′1-l2π2sinh2r′=-iEir,-l2π2,and E(τ,m) denotes the elliptic integral of the second kind. This is one of the three types of (incomplete) classical surfaces of revolution in R3 of constant Gaussian curvature equal to -1, namely, a surface of “hyperboloid type” (see Figure 14). The other two are the pseudosphere/tractricoid (which corresponds to a portion of the hyperbolic cusp) and the surface of “conical type.”


## 7.7. Canonical Uniformization toH

The hyperbolic annulusA(Rl) is uniformized to the upper half plane by the hyperbolic cyclic subgroup generated by the transformation: (92)τ⟶elτ,which corresponds to the hyperbolic element (93)Hl=def.el/200e-l/2∈PSL2,Rand fixes the points τ=0 and τ=∞ lying on ∂∞H. The uniformization map is (94)u=πHτ=Rle2πi/llog⁡τ=eπ2/l+2πi/llog⁡τ,which gives(95)ρ=eπ2/l-2π/larg⁡τ,θ=2πllog⁡τmod2π.A fundamental polygon is given by (see Figure 15(a)) (96)DH=τ∈H∣el<τ<e2l,l>0.This is a hyperbolic quadrilateral with two free sides and vertices located at the points (97)A:τ=-e2l,B:τ=-el,C:τ=el,D:τ=e2l.The funnel Fl is the projection of the relative funnel neighborhood [1]: (98)FHl=τ∈DH∣Reτ<0=τ∈DH∣arg⁡τ∈π2,π.


## 7.8. Canonical Uniformization toD

When passing to the disk model, the fundamental domainDH is mapped by the Cayley transformation (56) to a hyperbolic quadrilateral DD with vertices located at the following points, which are obtained from the points A,B,C,D defined in (97) by applying (56): (99)A′:u=-i+e2lie2l+1,B′:u=-i+eliel+1,C′:u=i-eliel-1,D′:u=i-e2lie2l-1.The free sides (A′B′) and (C′D′) are portions of ∂∞D (see Figure 15(b)), while the sides (B′C′) and (A′D′) are arc segments of Euclidean circles which are orthogonal to ∂∞D.


## 7.9. Globally Well-Behaved Scalar Potentials onA(R)

A scalar potentialΦ on A(R) is globally well-behaved iff there exists a smooth function Φ^:S2→R such that(100)Φr,θ=Φ^2arccot⁡r,θ;that is,(101)Φρ,θ=Φ^2arctanlogR-1/Rρ-1/R,θ.Expansion (13) gives the uniformly convergent series: (102)Φρ,θ=∑l=0∞∑m=-llDlmYlm2arctan⁡logR-1/Rρ-1/R,θ.For choices (16), we find (103)Φ+=M0r21+r2=M011+log⁡R-1/R/ρ-1/R2,Φ-=M011+r2=M0log⁡R-1/R/ρ-1/R21+log⁡R-1/R/ρ-1/R2,where ρ=u∈(1/R,R) and we used (81). The potentials Φ± are plotted in Figure 16(a) for the case l=π2 (R=e). For choice (17), we find (104)Φ0=M01+2r1+r2cos⁡θ=M01+2log⁡R-1/R/ρ-1/R1+log⁡R-1/R/ρ-1/R2cos⁡θ.Recall that Φ^0 has two extrema on S2, which are located at (ψ,θ)=(π/2,0) (maximum) and (ψ,θ)=(π/2,π) (minimum). At each of these points, relation (20) gives r=cot⁡(π/4)=1, so (81) gives ρ=ρ0, where (105)ρ0=def.1R+1eR-1R.It follows that the two critical points of Φ0 on A(R) are located on the real axis at(i)
uM=+ρ0, where Φ0 attains its maximum (which equals 2M),(ii)
um=-ρ0, where Φ0 attains its minimum (which equals zero).The level curves ofΦ0 are shown in Figure 16(b) for the case l=π2 (R=e), which gives ρ0=(e2+e-1)/e2=1+1/e-1/e2≈1.23.Lift of Φ ± and Φ 0 to H. The globally well-behaved scalar potentials (103) and (104) lift to the following potentials on H (see Figures 17(a) and 17(b)):(106)Φ~+τ=M011+log⁡R-1/R/ρτ-1/R2,Φ~-τ=M0log⁡R-1/R/ρτ-1/R21+log⁡R-1/R/ρτ-1/R2,Φ~0τ=M01+2log⁡R-1/R/ρτ-1/R1+log⁡R-1/R/ρτ-1/R2cos2πllog⁡τ,where ρ(τ)=eπ2/l-(2π/l)arg⁡(τ) (see (95)). The lifted potential Φ~0 has local maxima at the inverse image points of uM=+ρ0 and local minima at the inverse image points of um=-ρ0 (see (105)). Using (95), we find that these are located at(i)
(maxima)τ=τM(n)=def.enl+i(l/2π)logR/ρ0 (where Φ~0 equals 2M0),(ii)
(minima)τ=τm(n)=def.e(n+1/2)l+i(l/2π)logR/ρ0 (where Φ~0 vanishes),withn∈Z being an arbitrary integer. Hence all extrema of Φ~0 lie on the half-line L0 through the origin which makes an angle θ0=(l/2π)logR/ρ0 with the x-axis of the τ-plane. The maxima and minima alternate along this half-line and the ratio between the absolute values of two consecutive maxima or two consecutive minima equals el; in particular, the extrema accumulate toward the origin along L0. The fundamental domain DH contains exactly one of the minima, namely, that located at τ=e(3/2)l+iθ0. On the other hand, the two nonfree sides of DH contain the two consecutive maxima located at τ=el+iθ0 and τ=e2l+iθ0; these two maxima of Φ~0 are identified by the projection πH.In Figure17(b), we show the level curves of Φ~0 for the case l=π2 (R=e), which gives ρ0=1+1/e-1/e2≈1.23 and θ0=(π/2)1-log⁡1+1/e-1/e2≈0.39π. In this case, the absolute value of the ratio of successive minima or maxima of Φ~0 equals eπ2≈1.93×104. Due to the large size of this ratio, we chose for clarity to display the level curves of Φ~ on a region of the “semilogarithmic upper half plane.” The latter has complex coordinate T=X+iY (where X=ReT and Y=ImT>0), being related to the region (107)R=def.τ∈H∣τ>1of the Poincaré half plane through the coordinate transformation: (108)T=logτeiarg⁡τ,τ=eTeiarg⁡T. Notice that R contains only those extrema of Φ~0 which have absolute value larger than one. In the semilogarithmic half plane, these extrema are located at T=nleiθ0 with n∈Z>0 (maxima) and T=(n+1/2)leiθ0 with n∈Z≥0 (minima), lying equally spaced on the half-line in the T-plane which passes through the origin at angle θ0 with the x-axis. The Cartesian coordinates X,Y and x=Reτ, y=Imτ are related through (109)x=XeX2+Y2X2+Y2,y=YeX2+Y2X2+Y2.Figure 17(b) shows the level curves of the function Φ~0(X,Y) in the region defined by X<2l and Y<2l (where 2l=2π2≈19.7), which contains the image of the following annular region of the Poincaré half plane: (110)A=def.τ∈H∣1<τ<e2l.Notice that A contains a copy of the fundamental domain DH.


## 7.10. Lift of the Cosmological Model toH

We now present examples of trajectories onA(R) for l=π2 (R=e, modulus μ=2) for the vanishing scalar potential and for the globally well-behaved scalar potentials Φ± and Φ0. These were obtained as explained in Section 2.2, by numerically computing solutions of the system (8) on the Poincaré half plane for the corresponding lifted potentials and then projecting these trajectories to the hyperbolic punctured disk using the explicitly known uniformization map (94), which is equivalent to (95).Trajectories for Vanishing Scalar Potential. Figure 18 shows five trajectories (orange, yellow, red, blue, and magenta) for α=M0/3, l=π2 (R=e), and Φ=0, with the initial conditions given in Table 1. In this case, vanishing initial velocity leads to two stationary trajectories (the orange and yellow dots), while the hyperbolic geometry produces an effective attraction toward the outer funnel end for |u|>1 and toward the inner funnel end for u<1. The red, blue, and magenta trajectories start in the funnel region given by u>1 (with velocities pointing toward the outer funnel end) and hence evolve toward that end.Trajectories for Φ -. Figure 19 shows five trajectories (orange, yellow red, blue, and magenta) for α=M0/3, l=π2 (R=e), and Φ=Φ-, with the initial conditions given in Table 1. In this case, the potential produces a repulsive force away from the inner funnel end and an attraction force toward the outer end, thus accentuating the effect of the hyperbolic metric on the trajectories shown.Trajectories for Φ +. Figure 20 shows five trajectories (orange, yellow red, blue, and magenta) for α=M0/3, l=π2 (R=e) and Φ=Φ+, with the initial conditions given in Table 1. In this case, the potential induces an attractive force toward the inner funnel end. As a consequence, each of the red, blue, and magenta trajectories turns at some point in A(R) and evolves back toward the inner funnel end.Trajectories for Φ 0 . Figure 21 shows five trajectories (orange, yellow, red, blue, and magenta) for α=M0/3, l=π2 (R=e), and Φ=Φ0, with the initial conditions given in Table 1. Figure 22 shows a detail of the yellow trajectory on both H and A(R). It spirals in a complicated manner around a minimum point of Φ~0 and projects to the single minimum of Φ0 on A(R) located at u=-ρ0≈-1.23. For t→∞, the trajectory reaches the minimum point.


## 7.11. Inflationary Regions and the Number of e-Folds

Using relations (67)–(69), we find that, among the five trajectories with initial conditions given in Table 1, the orange and yellow trajectories start in inflationary regime for all three potentials Φ+,Φ-, and Φ0 (see Figures 12, 13, and 23), while the blue trajectory is always inflationary in potential Φ0 but is never inflationary for the other potentials. The orange and yellow trajectories give, respectively, 76 and 74 e-folds in the potential Φ+ after the first inflationary regime, which lasts for a cosmological time of t=2223 s, respectively, t=2184 s. The other trajectories do not start in the inflationary regime for any of the three potentials. It is easy to see that one can rescale the potential Φ+ by a positive constant in order to reach a phenomenologically appropriate value N∈[50,60] for each of the orange and yellow trajectories in this potential.Figure 12
Plot ofH(t)/M0 (black) and Hc(t)/M0 (green) for the yellow trajectory in the potential Φ+. For better clarity, we truncated the plot at t=200 s.Figure 13
Plots ofH(t)/M0 (black) and Hc(t)/M0 (green) for the yellow trajectory in the potential Φ0 and for the orange trajectory in the potentials Φ0 and Φ-, respectively. For better clarity, we truncated the plots at convenient values of t.Figure 14
Hyperbolic surface of revolution of hyperboloid type. The upper half corresponds to the regionFl0 of Fl. The circle drawn in light blue is the closed geodesic c1 of length l.Figure 15
Fundamental domains for the uniformization of the hyperbolic annulus to the Poincaré half plane and to the hyperbolic disk.
(a)
Fundamental domainDH for A(R) on the Poincaré half plane. The shaded region is the relative funnel neighborhood FH corresponding to the free side on the left (b)
Fundamental domainDD for A(R) on the Poincaré disk. The shaded region is the relative funnel neighborhood of the free side on the leftFigure 16
The potentialsΦ± and Φ0 on the annulus of modulus μ=2 (l=π2, R=e). For Φ+ and Φ-, we indicate only the dependence of ρ=|u|, since these two potentials do not depend on the polar angle θ. Notice that Φ+ tends to its infimum at the inner funnel end, while Φ- tends to its infimum at the outer funnel end. Similarly, Φ+ and Φ- tend to their suprema at opposite funnel ends.
(a)
Plot ofΦ+/M0 (blue) and Φ-/M0 (orange) as functions of ρ∈(1/R,R)=(e-1,e) for l=π2. The values ρ=1/R and ρ=R correspond to the two funnel ends of A(R) (b)
Level plot ofΦ0/M0 on A(R) for l=π2. Darker tones indicate lower values of Φ0. In this case, the maximum of Φ0 is located at uM=(e2+e-1)/e2, while the minimum is located at um=-uM=-(e2+e-1)/e2Figure 17
The lifted potentialsΦ~± and Φ~0 for the annulus of modulus μ=2 (l=π2, R=e).
(a)
Plot ofΦ~+/M0 (blue) and Φ~-/M0 (orange) as functions of arg⁡(τ)∈(0,π) for l=π2. The values arg⁡(τ)=0 and arg⁡(τ)=π correspond to the two funnel ends of A(R) (b)
Level plot ofΦ~0/M0 in the coordinates X,Y for l=π2. Darker tones indicate lower values of Φ~0. The maxima of Φ~0 are indicated as blue dots, while the minima are shown as small red squaresFigure 18
Five trajectories forΦ=0, α=M0/3, and l=π2, with initial conditions as in Table 1.
(a)
ForΦ=0, the five trajectories on H with initial conditions given in Table 1 coincide with those shown in Figure 7(a) (b)
Projection toA(R) of the trajectories shown at the left. The projection differs from that for D∗ (cf. Figure 7(b)) since the uniformization maps are differentFigure 19
Five trajectories forΦ=Φ-, α=M0/3, and l=π2 with initial conditions of Table 1.
(a)
Trajectories onH for Φ~=Φ~-, superimposed on a level plot of Φ~- on H (b)
Projection toA(R) of the trajectories shown at the left, superimposed on a level plot of Φ- on A(R)Figure 20
Five trajectories forΦ=Φ+, α=M0/3, and l=π2, with initial conditions of Table 1.
(a)
Trajectories onH for Φ~=Φ~+, superimposed on a level plot of Φ~- on H (b)
Projection toA(R) of the trajectories shown at the left, superimposed on a level plot of Φ+ on A(R)Figure 21
Five trajectories forΦ=Φ0 when α=M0/3 and l=π2.
(a)
Trajectories onH for Φ~=Φ~0. The yellow trajectory is not stationary, although at the scale shown it looks like a dot (b)
Projection toA(R) of the five trajectories shown at the leftFigure 22
Detail of the yellow trajectory shown in Figure21.
(a)
Detail of the lifted yellow trajectory shown in Figure21(a) near one minimum of Φ~0 (b)
Projection of the yellow trajectory to the hyperbolic annulus. The trajectory spirals in a complicated manner around the minimum ofΦ0, until it settles at the minimum pointFigure 23
Plots ofH(t)/M0 (black) and Hc(t)/M0 (green) for the orange and yellow trajectories in the potential Φ+. For better clarity, we truncated the plot at t=60 s. For both trajectories, inflation stops after a finite time (indicated in the text), though this is not visible at the scale of the figures.


## 8. On the Relation to Observational Cosmology

As explained in [1], generalized α-attractor models based on any geometrically finite noncompact hyperbolic Riemann surface Σ and with a well-behaved scalar potential enjoy universal behavior near each end of Σ where the extended potential has a local maximum, in a certain one-field truncation near that end. Within this approximation, such models make the same predictions for the spectral index ns and the tensor to scalar ratio r as ordinary α-attractors, provided that inflation takes place sufficiently close to such an end and along a trajectory which proceeds radially from that end in canonical local semigeodesic coordinates. In the slow-roll approximation for motion along such a trajectory, one finds [1] (111)ns≈1-2N,r≈12αN2, where N is the number of e-folds during the inflationary period. For such trajectories, generalized α-attractors are therefore as promising for matching observational data as ordinary α-attractors, whose agreement with current observations is quite good [2–5]. For the models considered in this paper, such special trajectories are the radial trajectories on the punctured disk and on the annulus, when inflation takes place close to any of the components of the conformal boundary. We gave two explicit examples of such trajectories: (i)
The yellow trajectory for the hyperbolic punctured disk in potentialΦ+ (see Figure 9(b)): as discussed in Section 6.12, this trajectory produces 55 e-folds, a value which lies in the observationally expected range of 50–60 e-folds.(ii)
The orange and yellow trajectories for the hyperbolic annulus in potentialΦ+ (see Figure 20(b)): as discussed in Section 7.11, both of these trajectories produce around 75 e-folds, but a constant positive rescaling of the potential Φ+ allows one to bring the number of e-folds within the phenomenologically desired range N∈[50,60].Unlike the one-fieldα-attractors usually considered in the literature, generalized α-attractors are genuinetwo-field models and hence they can incorporate corrections to the traditional paradigm of inflationary cosmology, which assumes for simplicity that the inflaton is a single real scalar field. While current observational data can be successfully reproduced by various one-field models, they can also be reproduced by multifield models and it is generally deemed quite possible that, within the next decade, improved measurements could detect deviations from one-field model predictions. The recognition of this possibility has led to renewed interest in the study of multifield models [23–26] and in particular to the development of numerical methods for determining the effect of their cosmological perturbations [27–29] beyond the limitations of the SRST approximation [30, 31]. As a single example, see [32] for a recent investigation of constraints imposed on such models by Planck 2015 data [33].In the context of the elementary generalizedα-attractor models considered in this paper, deviations from the one-field paradigm could be visible, for example, for trajectories which depart slightly from the radial trajectories discussed above. This will affect the two-point correlators which determine cosmological observables, thus producing subleading corrections to relations (111). Generalized α-attractor models are also interesting for investigations of the postinflationary period of a given trajectory, for which generic two-field models have low predictive power since they depend on the choice of an arbitrary metric for the target manifold of the scalar fields. By contrast, generalized α-attractors provide a natural class of two-field models which have universal behavior in the truncated inflationary regime near the ends, while at the same time allowing for remarkable dynamical complexity beyond that regime. As shown in the previous sections, even the simplest instances of such models (namely, those based on elementary hyperbolic surfaces) already allow trajectories of considerable complexity, due to the interplay between the effective force induced by the hyperbolic geometry and that induced by the scalar potential. Such models could therefore play the role of a natural testing ground of two-field model technology, in a mathematically tractable framework which may allow one to develop insights deeper than those afforded by current approximations and by generic numerical methods.We end by mentioning that it is a nontrivial task to embed cosmological models with a single real scalar field within fundamental quantum theories of gravity such as string theory in a manner which is compatible with all phenomenological and self-consistency constraints. In particular, most scalar fields which arise naturally in closed string theory are complex-valued and one has to rely on special and quite finely tuned constructions when embedding single field models in string theory in a consistent and phenomenologically reasonable manner. Naturality arguments might therefore suggest that the inflaton could in fact be a complex-valued field in a fundamental theory of gravity, thus leading to a two-field cosmological model. In this context, we mention that generalizedα-attractors appear to have natural string-theoretic realizations which involve F-theory backgrounds with discrete fluxes, though a proper discussion of that construction (which involves the theory of modular curves and Shimura varieties) lies well outside the scope of the present paper.


## 9. Conclusions and Further Directions

We studied generalizedα-attractor models defined by elementary hyperbolic surfaces, showing how they fit into the framework developed in [1]. Following a “universal” approach to globally well-behaved scalar potentials, we showed how they can be approximated systematically using the Laplace expansion of their extension to the end compactification (which in such models is the unit sphere) and how a smooth real-valued map defined on the latter induces different potentials on each elementary hyperbolic surface. We also illustrated cosmological dynamics of generalized α-attractor models by numerically extracting various trajectories for the cases of D∗ and A(R), finding rather complex behavior even for relatively simple scalar potentials. From the universal perspective followed here and in [1], the difference between models with globally well-behaved scalar potential defined on various hyperbolic surfaces of the same genus is captured by two maps, namely, the uniformization map πH and the map j which embeds the given surface into its end compactification. For elementary surfaces, both of these maps can be constructed explicitly and hence their effects on the cosmological dynamics can be explored systematically.A similar approach could in principle be followed for any geometrically finite hyperbolic surface. In this regard, it would be natural to explore the large class of nonelementary planar surfaces, which form a classical subject in complex analysis and uniformization theory. For such surfaces, the uniformization mapπH is not usually known explicitly and, at least for general values of the moduli, it must be determined numerically. Despite this fact, a fundamental domain is known for any planar surface, as are certain other properties of the hyperbolic metric and of the uniformization map [34, 35]. This allows one to approach generalized α-attractor models whose scalar manifolds are given by such surfaces using the general algorithm proposed in [1, Sec. 7]. For example, it would be interesting to perform a detailed study of cosmological trajectories for some triply connected nonelementary planar surfaces such as the twice-punctured disk [36–39] and once-punctured annulus [40].We also commented briefly on the potential relevance of such models to observational cosmology. As pointed out in Section8, the models considered in this paper share the general features discussed in [1] and hence provide reasonable candidates for reproducing current observational constraints, similar to ordinary α-attractors. In particular, they easily support cosmological trajectories which produce the expected number of 50–60 e-folds. Of course, much deeper investigation of such models is needed before the question of their phenomenological relevance can be answered fully.



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