Generalized -Attractor Models from Elementary Hyperbolic Surfaces
We consider generalized -attractor models whose scalar potentials are globally well-behaved and whose scalar manifolds are elementary hyperbolic surfaces. Beyond the Poincaré disk , such surfaces include the hyperbolic punctured disk and the hyperbolic annuli of modulus . For each elementary surface, we discuss its decomposition into canonical end regions and give an explicit construction of the embedding into the Kerekjarto-Stoilow compactification (which in all three cases is the unit sphere), showing how this embedding allows for a universal treatment of globally well-behaved scalar potentials upon expanding their extension in real spherical harmonics. For certain simple but natural choices of extended potentials, we compute scalar field trajectories by projecting numerical solutions of the lifted equations of motion from the Poincaré half plane through the uniformization map, thus illustrating the rich cosmological dynamics of such models.
In , we introduced a class of cosmological models which form a wide generalization of -attractors [2–7]. Such models are defined by cosmological solutions (with flat spatial section) of the Einstein theory of gravity coupled to two real scalar fields, the dynamics of the latter being described by a nonlinear sigma model whose scalar manifold is a noncompact, oriented, and topologically finite borderless surface endowed with a complete metric of constant Gaussian curvature . The scalar potential is given by a smooth real-valued function defined on . The rescaled scalar field metric has Gaussian curvature equal to ; hence is a geometrically finite hyperbolic surface in the sense of . Using the uniformization theorem of Koebe and Poincaré  and the theory of surface groups (Fuchsian groups without elliptic elements),  discussed the basic features of cosmological dynamics and inflation in such models.
In the present paper, we consider the nongeneric situation when is elementary, focusing on special features which arise in this case. As explained in , an elementary hyperbolic surface is isometric with the hyperbolic disk , with the hyperbolic punctured disk or with the hyperbolic annulus of modulus , where is a real number. Each of these surfaces is planar, having the unit sphere as its end compactification. This allows one to parameterize globally well-behaved  scalar potentials through the coefficients of the Laplace expansion of their global extension to the end compactification, which in this case reduces to an expansion in real spherical harmonics. Both the hyperbolic metric and a fundamental polygon are known explicitly for and . In particular, one can describe explicitly the hyperbolic geometry of such surfaces and one can compute scalar field trajectories on by determining trajectories of an appropriate lift of the model to the Poincaré half plane and projecting them to or to through the uniformization map. When the scalar potential is constant, this illustrates how the different projections from to and can take the same trajectory on the Poincaré half plane into qualitatively different trajectories on the corresponding elementary hyperbolic surface. The explicit embedding of , , and into their common Kerekjarto-Stoilow compactification is also different. As a consequence, a smooth real-valued function defined on will generally induce rather different globally well-behaved scalar potentials on the disk, the punctured disk, and the annulus. When combined with the different projections from , this leads to qualitatively different dynamics on , , and . By considering some simple but natural choices of extended scalar potentials on , we use numerical methods to compute various trajectories on and , thus illustrating the rich dynamics of such models, which is not entirely visible in the gradient flow approximation.
The paper is organized as follows. Section 2 briefly recalls the definition of generalized -attractor models and the lift of their cosmological evolution equation to the Poincaré half plane. In Section 3, we consider globally well-behaved scalar potentials on topologically finite planar oriented surfaces (of which the elementary surfaces are particular cases), showing how they can be parameterized through the coefficients of the Laplace expansion of their extension to the end compactification. In Section 4, we review the classification of elementary hyperbolic surfaces and some of their basic properties. Section 5 recalls the case of the hyperbolic disk, showing how it fits into the general approach developed in  and how well-behaved scalar potentials can be described through the Laplace expansion of their extension to . Section 6 discusses the hyperbolic punctured disk. After giving the explicit form of the hyperbolic metric on , of a partial isometric embedding into three-dimensional Euclidean space, and of the decomposition into horn and cusp regions, we compute certain scalar field trajectories for a few globally well-behaved scalar potentials which are induced naturally from . This illustrates the rich dynamics of our models on the punctured hyperbolic disk. We also discuss inflationary regions and the number of e-folds for various trajectories and provide an explicit example of an inflationary trajectory which produces between 50 and 60 e-folds, thus showing that such models can match observational constraints. Section 7 performs a similar analysis for hyperbolic annuli, using globally well-behaved scalar potentials induced by the same choices of functions on . In particular, this illustrates how the dynamics of our models differs on and . Section 8 comments briefly on the relation of such models with observational cosmology, while Section 9 contains our conclusions and sketches a few directions for further research.
Notations and Conventions. All manifolds considered are smooth, connected, oriented, and paracompact (hence also second-countable). All homeomorphisms and diffeomorphisms considered are orientation-preserving. By definition, a Lorentzian four-dimensional manifold has “mostly plus” signature. The symbol denotes the imaginary unit. The Poincaré half plane is the upper half plane with complex coordinate : endowed with its unique complete metric of Gaussian curvature , which is given by The real coordinates on are denoted by and . The complex coordinate on the hyperbolic disk, the hyperbolic punctured disk, or an annulus is denoted by . We define the rescaled Planck mass through where is the (reduced) Planck mass.
2. Generalized -Attractor Models
2.1. Definition of the Models
Let be a noncompact oriented, connected, and complete two-dimensional Riemannian manifold without boundary (called the scalar manifold) and be a smooth function (called the scalar potential). We say that is hyperbolic if the metric has constant Gaussian curvature equal to . We say that is topologically finite and that is geometrically finite if the fundamental group is finitely generated. Let , where is a fixed positive real number.
Let be any four-dimensional manifold which can support Lorentzian metrics. The Einstein-Scalar theory defined by the triplet includes four-dimensional gravity (described by a Lorentzian metric defined on ) and a smooth map , with action : Here is the volume form of , is the scalar curvature of , and is the (reduced) Planck mass.
When is diffeomorphic with and is a FLRW metric with flat spatial section, solutions of the equations of motion for action (4) for which depends only on the cosmological time define the class of so-called generalized -attractor models . In this case, the map (where is a real interval) defines a curve in which obeys an invariantly defined nonlinear second-order ordinary differential equation (which is locally equivalent with a system of two second-order equations). We refer the reader to  for a general discussion of such models.
2.2. Lift to the Poincaré Half Plane
As explained in , the cosmological equations of motion can be lifted from to the Poincaré half plane by using the covering map which uniformizes to . This allows one to determine the cosmological trajectories by projecting to the trajectories of a “lifted” model defined on . The lifted model is governed by the following system of second-order nonlinear ordinary differential equations [1, eq. ]: where and is the cosmological time while , are the Cartesian coordinates on the Poincaré half plane with complex coordinate and is the lifted potential. Let be any point of and let be chosen such that . An initial velocity vector defined at on and its unique lift through the differential of at are related through Notice that the differential of at is a bijective linear map because is a covering map and hence a local diffeomorphism. Writing and , we have with real , . As shown in , a cosmological trajectory on with initial condition can be written as , where is the solution of lifted system (5) with initial conditions:
Eliminating the Planck Mass. Let and , where is the rescaled Planck mass (3). Then (5) becomes showing that we can eliminate the Planck mass from the equations provided that we measure both and (and hence also ) in units of . The numerical solutions extracted in latter sections of this paper were obtained using system (8), after performing such a rescaling of and .
3. Laplace Expansion of Globally Well-Behaved Scalar Potentials
Let denote the Kerekjarto-Stoilow (a.k.a. end) compactification of (see [10, 11]) and identify with its image in through the embedding map . A smooth scalar potential is called globally well-behaved  if there exists a smooth function whose restriction to equals , in which case is uniquely determined by through continuity. Since all elementary hyperbolic surfaces are planar, their end compactification is diffeomorphic with the unit sphere , so in this case globally well-defined scalar potentials on correspond bijectively to smooth functions .
Let and be spherical coordinates on ; thus where and . Any smooth map is square-integrable with respect to the round Lebesgue measure on and admits the Laplace-Fourier series expansion: where are the complex spherical harmonics and The series in (10) converges uniformly to on since is smooth (see ). Recall that , where are the associated Legendre functions. Since is real-valued, expansion (10) reduces to where and are real constants. Equivalently, we have where and are the real (a.k.a. tesseral) spherical harmonics, which correspond to orbitals. This expansion is again uniformly convergent and gives a systematic way to approximate by truncating away the contributions with greater than some cutoff value.
Some Particular Choices for . If only the modes with and ( and orbitals) are present, then we have where are real constants and we used the expressions The constant term in (14) corresponds to the orbital ) while the terms with prefactors , , and correspond to the orbitals , , and .
For , two simple choices are and , where is the rescaled Planck mass (3). These give the following -independent potentials, which involve only the orbitals and and are shown in Figure 1: Notice that has a maximum at (north pole) and a minimum at (south pole) while has a minimum at (north pole) and a maximum at (south pole).
Another simple choice is and , which corresponds to a linear combination of the and orbitals and gives the extended potential: Unlike , this function does not have extrema at the north or south pole. Instead, it has two extrema along the equator of , having a maximum (equal to ) at the point and a minimum (equal to ) at . Notice that and are Morse functions on , so the potentials derived from them on a planar surface (whose end compactification is ) will be compactly Morse in the sense of .
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  allow for a uniform treatment of globally well-behaved scalar potentials in generalized -attractor models for which is geometrically finite. This is summarized in the commutative diagram (18), where is the smooth embedding of into its end compactification and is the uniformizing map from :Any smooth real-valued function defined on induces a globally well-behaved scalar potential on through the formula , while any globally well-behaved scalar potential on lifts to a smooth function defined on . When the maps and are known, can be recovered from as the composition , where is the composite map . Notice that and are smooth maps, while is holomorphic when is endowed with the complex structure which corresponds to the conformal class of the metric . The maps and differ for distinct geometrically finite hyperbolic surfaces having the same end compactification , which means that the same “universal” extended potential defined on can induce markedly different globally well-behaved potentials and lifted potentials for different hyperbolic surfaces of the same genus.
When is a planar surface, the end compactification coincides with the unit sphere and can be expanded into real spherical harmonics as explained above. This induces uniformly convergent expansions: of and . In the next sections, we determine explicitly the maps , , and for the planar elementary surfaces , , and and the maps and induced by the choices and given above. This illustrates how the same function leads to different dynamics of the generalized -attractor models associated with distinct planar surfaces. For the three elementary surfaces, we will construct the map by first diffeomorphically (but not biholomorphically!) identifying with the complex plane of complex coordinate (or with 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 :
4. Elementary Hyperbolic Surfaces
A (complete) hyperbolic surface is called elementary if it is conformally equivalent with a simply connected or doubly connected regular domain (a regular domain is called doubly connected if its complement in the Riemann sphere has two connected components, which happens iff ) contained in the complex plane. This amounts to the condition that the uniformizing surface group of is the trivial group or a cyclic subgroup of of parabolic or hyperbolic type.
Any simply connected regular domain is conformally equivalent with the unit disk (and hence with the upper half plane ). 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 plane (ii)The punctured unit disk (iii)The annulus of modulus , where .
When endowed with its usual complex structure, the punctured plane does not support a complete hyperbolic metric. On the other hand, the punctured disk and annulus admit uniquely determined complete hyperbolic metrics. Notice that both and are homeomorphic with (open) cylinders. Due to this fact, the hyperbolic punctured disk is also called the parabolic cylinder while the hyperbolic annuli are also called hyperbolic cylinders . Summarizing, the list of elementary hyperbolic surfaces is as follows:(1)The hyperbolic disk (which is isometric with the Poincaré half plane )(2)The hyperbolic punctured disk (uniformized to by a parabolic cyclic subgroup of )(3)The hyperbolic annuli for (uniformized to by a hyperbolic cyclic subgroup of ).
The explicit form of the hyperbolic metric is known in all cases, as is a fundamental polygon [13–15] for and . This allows one to study the cosmological dynamics of -attractor models defined by such surfaces either directly on or (as explained in ) 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 disk has a single end, known as a plane end.(ii)The hyperbolic punctured disk has two ends. One of these is a cusp end, the other being a horn end.(iii)The hyperbolic annulus has two ends, both of which are funnel ends.
All elementary hyperbolic surfaces are planar (i.e., of genus zero). As explained in , this implies that their end compactification [10, 11] is the unit sphere . On the other hand, the conformal boundary [1, 17, 18] differs in the three cases:(i)For the hyperbolic disk, we have , where the circle corresponds to the plane end.(ii)For the hyperbolic punctured disk, we have , where the origin corresponds to the cusp end and corresponds to the horn end.(iii)For the hyperbolic annulus, we have , 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 into non-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 disk coincides with the two-field -attractor model of , which was discussed extensively in the literature. The purpose of this section is to show how this fits into the general theory developed in .
5.1. Semigeodesic Coordinates
The unit disk admits a unique complete hyperbolic metric, which is given by In polar coordinates given by and , the metric becomes Semigeodesic coordinates for are obtained by the change of variables: This maps the unit disk (diffeomorphically, but not conformally) to the complex plane with polar coordinates and complex coordinate and brings the metric to the form The single end of (which is called a plane end) corresponds to , while the center of corresponds to .
5.2. The End Compactification of
The end compactification of the hyperbolic disk coincides with the Alexandroff compactification of the -plane, which by the stereographic projection (20) is identified with the unit sphere . The north pole corresponds to the plane end at , while the south pole corresponds to , that is, to the center of . In spherical stereographic coordinates , the Poincaré metric (25) becomes
Remark 2. The compact Riemann surface into which is embedded holomorphically [17, 18] is the Riemann sphere associated with the -plane. The coordinate is given by where are the homogeneous coordinates of .
5.3. Globally Well-Behaved Scalar Potentials on
A potential is globally well-behaved on iff there exists a smooth function such that that is, The condition that is smooth on implies, in particular, that has a finite limit for and hence has a -independent limit for , that is, for . Expansion (13) gives the uniformly convergent series: To obtain inflationary behavior with the scalar field rolling from the plane end toward the interior of , one can require that has a local maximum at the north pole of . In the simplest models, one can take 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, has a global minimum at (the center of ) and increases monotonically to -independent finite value as grows from zero to (toward the conformal boundary of ).
In polar coordinates , the extended potentials (16) of Section 3 correspond to the following globally well-behaved scalar potentials on : where . 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 (with respect to the fact that the two locally defined real scalar fields of the sigma model have canonical kinetic terms). The supremum of corresponds to (being equal to ), while the infimum is attained at (where vanishes). On the other hand, tends to its vanishing infimum for and has a maximum at (where it equals ). Notice that only leads to standard -attractor behavior.
(a) Plot of (blue) and (orange) as functions of . The value corresponds to the center of , while corresponds to the conformal boundary . Only the potential leads to -attractor behavior when inflation takes place near the plane end
(b) Plot of (blue) and (orange) as functions of . The value corresponds to the center of , while corresponds to the plane end. Only the potential leads to -attractor behavior when inflation takes place near the plane end
Using the relation , choice (17) gives the following potential on :
6. The Hyperbolic Punctured Disk
The hyperbolic punctured disk (also known as the “parabolic cylinder” ) 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 disk admits a unique complete hyperbolic metric given by  In particular, and are isothermal coordinates. In polar coordinates defined through the metric takes the form The center of corresponds to the cusp end, while the bounding circle of corresponds to the horn end (see below). The Euclidean circle at is a horocycle of hyperbolic length . Notice that has infinite hyperbolic area.
6.2. Diffeomorphism to the Punctured Plane
One can introduce an orthogonal coordinate system on through the coordinate transformation: This gives a diffeomorphism between and the punctured complex plane with complex coordinate: In this coordinate system, metric (33) takes the form The center of corresponds to , while the bounding circle of corresponds to .
6.3. Partial Isometric Embedding into Euclidean 3-Dimensional Space
One can isometrically embed the portion of the hyperbolic punctured disk into Euclidean as the open half tractricoid (the surface obtained by revolving an open half of a tractrix along its asymptote ) defined in cylindrical coordinates (where and ) by the parametric equations: Indeed, it is easy to see that the Euclidean metric of induces a metric on which coincides with (38). This is the classical pseudosphere model of Beltrami (see Figure 3).
6.4. The End Compactification of
The stereographic projection (20) identifies with the one-point compactification of the -plane. This shows explicitly that is the end compactification of , where the north pole corresponds to the horn end and the south pole corresponds to the cusp end. The embedding is given by
6.5. Semigeodesic Coordinates
The further change of variables brings metric (38) to the form where . In particular, are semigeodesic coordinates. The center of corresponds to while the bounding circle of corresponds to . The horocycle at (i.e., ) has length .
6.6. The Hyperbolic Cusp
Let As mentioned above, the Euclidean circle has hyperbolic length . The hyperbolic cusp (cf. ) corresponds to the portion of lying inside this circle (see Figure 3), which is the open punctured disk: endowed with the restriction of the metric (33); notice that the restricted metric is not complete. In coordinates , the metric on is obtained by restricting (38) to the range , where In semigeodesic coordinates the cusp metric is given by (42) with the restriction . Notice that has hyperbolic area equal to .
6.7. The Hyperbolic Horn
By definition, the hyperbolic horn is the annulus: endowed with the (incomplete) restriction of metric (33). In coordinates , the horn metric is obtained from (38) by restricting the range of to . In coordinates , the metric takes form (42), with the restriction . Defining , this can be brought to the form where the bounding circle of corresponds to .
6.8. Canonical Uniformization to
The punctured disk is uniformized to the Poincaré half plane (1) with complex coordinate by the parabolic cyclic group generated by the translation: which corresponds to the parabolic element: This fixes the point . The uniformization map is The hyperbolic cusp is the projection through of the cusp domain: which is bounded by the horocycle: This horocycle is tangent to the conformal boundary of at the point , which projects through to the cusp ideal point of the end compactification of .
A fundamental polygon for the action of on is given by the semi-infinite vertical strip and has vertices at the points (see Figure 4(a)): The Poincaré side pairing maps into through transformation (48), which generates . The relative cusp neighborhood  with respect to is the intersection . A lifted scalar potential is given by being invariant under translation (48). In particular, the restrictions of to the sides and agree through the Poincaré pairing.
(a) A fundamental polygon for the punctured disk on . The relative cusp neighborhood of the vertex corresponds to the shaded region
(b) A fundamental polygon for the punctured disk on . The shaded region is the relative cusp neighborhood of the vertex
6.9. Canonical Uniformization to
For completeness and comparison with , we also give the canonical uniformization of to the hyperbolic disk. The Cayley transformation is an isometry from to . When uniformizing to , the fundamental polygon becomes a hyperbolic triangle with vertices at the following points, which correspond, respectively, to the points , , and of (54) through the Cayley transformation and a free side connecting the points and (see Figure 4(b)). The sides of this triangle are a segment connecting to (which passes through the origin of ), the portion of connecting to , and Euclidean circular arc orthogonal to which connects to . The hyperbolic cusp neighborhood of the vertex is bounded by the horocycle: which is tangent to at the point . The intersection of with is the relative cusp neighborhood with respect to (see ), which is the image of through the Cayley transformation.
6.10. Globally Well-Behaved Scalar Potentials on
A scalar potential is globally well-behaved on iff there exists a smooth map such thatthat is, where . Expansion (13) gives the uniformly convergent series: For choices (16) and (17), we find These potentials are shown in Figure 5. Notice that leads to -attractor behavior if inflation takes place near the horn end, while leads to -attractor behavior if inflation takes place near the cusp end . The extended potentials have maxima and minima at the two ideal points of the end compactification of (which correspond to the north and south poles of ). On the other hand, does not have extrema at the ideal points; its extrema coincide with those of , being located inside . The minimum (equal to zero) is at the point while the maximum (equal to ) is at .
(a) Plot of (blue) and (orange) as functions of . The value corresponds to the cusp end, while corresponds to the horn end
(b) Level plot of on the punctured disk. Darker tones indicate lower values of
Remark 3. Using (41), we find the following expressions in semigeodesic coordinates:
Lift of the Potentials and to . Consider the well-behaved scalar potentials and on given in (62). Let and . Then the covering map (50) reads which gives Hence the potentials and have the following lifts to (see Figure 6): The function (which is periodic under ) attains its minimum at the points (with ), while the maximum is attained for with .
(a) Plot of (blue) and (orange) as functions of . The values and correspond, respectively, to the horn and cusp ends
(b) Level plot of . Higher values of correspond to lighter tones. The fundamental domain 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 on for the vanishing scalar potential and for the globally well-behaved scalar potentials and . 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 . Then and one immediately checks that straight lines given by constant functions , are solutions of (5) for any initial point , 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 , irrespective of its initial conditions. As a consequence, any solution defined on (which is obtained by projecting a solution defined on through map (50)) will tend toward the horn end as . This shows that the hyperbolic metric acts as an effective force which repulses away from the cusp end. Notice that a global trajectory defined on can have cusp and self-intersection points and hence that it need not correspond to an embedded curve in .
(a) Trajectories for 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 disk
Trajectories for . Five lifted trajectories (and their projections to ) for and with the initial conditions given in Table 1 are shown in Figure 8. Since 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 to the funnel end (see the orange and yellow trajectories in the figures). Any other trajectory (irrespective of its initial velocity) also evolves for to the funnel end.
(a) Trajectories for on the Poincaré half plane, drawn over a level plot of on
(b) Projection of the trajectories shown at the left to the hyperbolic punctured disk. We also show a level plot of on
Trajectories for . Five lifted trajectories for and (and their projections to ) with the initial conditions given in Table 1 are shown in Figure 9. Since 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.
(a) Trajectories for on the Poincaré half plane and a level plot of on
(b) Projection of the trajectories shown at the left to the hyperbolic punctured disk and a level plot of on