Two-field cosmological α-attractors with Noether symmetry

Published for SISSA by Springer Received: October Revised: March Accepted: April Published: April 24, 20, 18, 24, 2018 2019 2019 2019 Lilia Anguelova,a Elena Mirela Babalicb and Calin Iuliu Lazaroiuc a Institute for Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, Tsarigradsko Chaussee 72, Sofia, Bulgaria b Department of Theoretical Physics, Horia Hulubei National Institute for Physics and Nuclear Engineering (IFIN-HH), Str. Reactorului no. 30, Bucharest-Magurele, Romania c Center for Geometry and Physics, Institute for Basic Science (IBS), Pohang 37673, Republic of Korea E-mail: [email protected], [email protected], [email protected] Abstract: We study Noether symmetries in two-field cosmological α-attractors, investigating the case when the scalar manifold is an elementary hyperbolic surface. This encompasses and generalizes the case of the Poincaré disk. We solve the conditions for the existence of a ‘separated’ Noether symmetry and find the form of the scalar potential compatible with such, for any elementary hyperbolic surface. For this class of symmetries, we find that the α-parameter must have a fixed value. Using those Noether symmetries, we also obtain many exact solutions of the equations of motion of these models, which were studied previously with numerical methods. Keywords: Cosmology of Theories beyond the SM, Effective Field Theories, Sigma Models ArXiv ePrint: 1809.10563 Open Access, c The Authors. Article funded by SCOAP3 . https://doi.org/10.1007/JHEP04(2019)148 JHEP04(2019)148 Two-field cosmological α-attractors with Noether symmetry Contents 1 Introduction 2 2 Two-field cosmological α-attractor models 4 6 7 8 9 11 13 4 New variables: cyclic coordinate 4.1 Finding the coordinates u and v 4.2 Finding the cyclic coordinate w 15 15 18 5 Equations of motion for the Poincaré disk 5.1 Lagrangian in the new variables 5.2 Solutions 5.2.1 Special cases: m = 0, −1, −2 5.2.2 Generic case: m 6= 0, −1, −2 20 21 23 24 26 6 Equations of motion for the hyperbolic punctured disk 6.1 Lagrangian in the new variables 6.2 Solutions 28 29 30 7 Equations of motion for the hyperbolic annuli 7.1 Lagrangian in the new variables 7.2 Solutions 7.2.1 Special cases: m = 0, 1, 2 7.2.2 Generic case: m 6= 0, 1, 2 32 33 34 35 38 8 Summary and discussion 39 A Elementary hyperbolic surfaces 43 B Nontrivial trajectories for m = 0 B.1 Poincaré disk B.2 Hyperbolic punctured disk B.3 Hyperbolic annuli 44 44 46 50 –1– JHEP04(2019)148 3 Noether symmetries in two-field α-attractors 3.1 The Noether system 3.2 Solving equations (E1), (E2) and (E4) 3.3 Solving equations (E5) and (E6) 3.4 Solving equation (E3) 3.5 Solving equation (E7): the scalar potential 1 Introduction 1 By ‘effectively’ single-field models we mean two-field models on the Poincaré disk, in which however one studies only radial trajectories. The importance of the hyperbolic geometry of the scalar manifold is much more manifest in the recent works [7–13], which investigated novel behavior due to trajectories with nontrivial angular motion on the Poincaré disk. Note that this kind of trajectories had already been considered in a much wider context in the earlier references [14–16]. 2 The main conceptual objection can be summarized as follows. Effective field theory considerations clearly indicate the necessity to include quantum (in particular, non-perturbative) effects in order to obtain dS minima, while those string theory dS-related considerations which are sufficiently rigorous at present are essentially classical (relying on nontrivial background fluxes). So there should be no surprise at the difficulty, which can likely be overcome only upon developing a better non-perturbative understanding of string theory. –2– JHEP04(2019)148 A period of an accelerated expansion in the Early Universe is thought to be necessary for explaining the large-scale properties of the present day Universe. The standard description of such an inflationary stage is given by coupling the space-time metric to one or more fundamental scalars, which have a nontrivial potential that temporarily dominates the energy density of the Universe. There is, in fact, a wide variety of such inflationary models. A particular class, called α-attractors [1, 2] (see also the earlier related works [3, 4]), stands out as being in an especially good agreement with the current observational data. This class of models has certain universal predictions for the important cosmological observables ns (scalar spectral index) and r (tensor-to-scalar ratio). It has been understood that the key reason for this is a specific property of the kinetic terms of the scalars. More precisely, they are characterized by hyperbolic geometry [5, 6]. In fact, the original works on α-attractors focused mostly on effectively single field models.1 The widest generalization in the context of two-field models, which brings into sharp focus the essential role of the hyperbolic geometry of the scalar kinetic terms and of uniformization theory, was introduced in [14] and further explored in [15, 16] by considering models whose scalar manifolds are arbitrary hyperbolic surfaces, which can be much more complicated than the Poincaré disk. Although single-field inflationary models are the most studied, it is quite natural to consider models with more than one scalar field. The reason is that the underlying particle physics descriptions, including string compactifications, usually contain many scalars. So it makes sense to expect, in the context of a fundamental theory of matter and gravity, that more than one field would play an important role during an inflationary stage. In view of very recent developments in the literature, there may also be another motivation to be interested in multi-field cosmological models. Namely, it was conjectured in [18] that quantum gravity requires the scalar potential to satisfy a certain condition, which excludes dS minima and seems to be in severe tension with single-field slow-roll inflationary models [19, 20]. It was argued in [21] that one can reconcile slow-roll inflation with the conjecture of [18] by considering multi-field models. One should note, however, that there are already serious objections [22–26] to that conjecture, whose only motivation is that it is rather difficult to find well-under-control stringy constructions that have (meta-)stable dS minima.2 It could be helpful, in sorting out arguments for or against the conjecture, to better understand multi-field inflationary models and their embeddings in string compacti- fications. Regardless of whether one is motivated by the conjecture of [18] or by the general expectation that more than one scalar field could play an important role for inflation, it is natural to be interested in two-field models as the simplest case of multi-field ones. The Noether symmetry method was already applied to one-field α-attractor models of inflation in [31]. However, due to the limitation to a single scalar field, that analysis could not illustrate the essential role played by the hyperbolic geometry of the scalar manifold. Here we will apply the Noether symmetry method to the two-field generalized α-attractors of [14–16]. A key feature of this class of models is that the scalar manifold is a hyperbolic surface. For a Riemannian 2-manifold, hyperbolicity amounts to the condition that the Gaussian curvature is constant and negative. In fact, it is inversely proportional to the αparameter of these models. We will focus on the simplest class of hyperbolic surfaces, called elementary, of which there are three types: the Poincaré disk, the hyperbolic punctured disk and the hyperbolic annuli (see, for example, [15]). Using a separation-of-variables Ansatz, we show that two-field α-attractor models, with scalar manifold given by any elementary hyperbolic surface, have a ‘separated’ Noether symmetry for a certain form of the scalar potential. The existence of such a symmetry requires a different form of the scalar potential for each of the three types of elementary hyperbolic surface. The hyperbolic geometry of the scalar kinetic terms will play an essential role in this derivation. It turns out that the special kind of Noether symmetry, which we find using the separation of variables Ansatz, not only selects a particular form of the scalar potential, but also fixes the value of the otherwise arbitrary α-parameter.3 That a specific value of the α-parameter is required for a separated Noether symmetry may seem unexpected. However, it is also very intriguing. Recall that it is not uncommon, especially in the context of string theory, to have particular points in a certain parameter space, where an (enhanced) symmetry occurs, although there is no such symmetry at generic points of that parameter space. It would be very interesting to understand whether this peculiar feature can help find specific embeddings of two-field α-attractor models with a separated Noether symmetry in a more fundamental particle physics framework. 3 This condition may be relaxed for more general Noether symmetries, which are not of the separationof-variables type. We hope to say more on this in a future publication. –3– JHEP04(2019)148 Most of the time, the equations of motion of two-field cosmological models are solved numerically in the literature. See, in particular, [15–17] for such numerical investigations in two-field α-attractor models. Our goal here will be to find exact solutions by imposing the requirement that the model possesses a Noether symmetry. This method is well-known in the context of extended theories of gravity, where it has long been used to find classes of exact solutions [27–30]. The basic idea is that the presence of a Noether symmetry constrains the form of an otherwise arbitrary function in the action (in our context, the scalar potential) and allows one to simplify the equations of motion. In general, this method does not give all solutions of the field equations, but only a certain subset. However, having exact solutions to analyze is often more informative conceptually than performing numerical analysis. Furthermore, the relevant Noether symmetry may have a deeper meaning, if the two-field models under consideration could be embedded in some fundamental particle physics setup, like a class of string theory compactifications. 2 Two-field cosmological α-attractor models Generalized two-field α-attractors are a class of inflationary models obtained from Einstein gravity coupled to a non-linear sigma-model with two real scalar fields, whose target space (known as the scalar manifold ) is a hyperbolic surface. This system is described by the action   Z R 1 I J 4 √ − GIJ (φ) ∂φ ∂φ − V (φ) , (2.1) S = d x −g 2 2 where R is the scalar curvature of the 4d space-time metric gµν , the fields φI with I = 1, 2 are two real scalars and the non-linear sigma-model metric GIJ (φ) is a complete hyperbolic metric, i.e. a complete metric of constant negative Gaussian curvature K.4 For brevity, we use the notation ∂φI ∂φJ ≡ g µν ∂µ φI ∂ν φJ . The simplest example is obtained by taking the scalar manifold to be the Poinaré disk D. In this case, using polar coordinates on D and considering only radial trajectories, one recovers the original one-field α-attractors of [1, 2]. It was understood in [5, 6] that the in terms of an arbitrary positive It is convenient to write the Gaussian curvature as K = − const α 1 2 parameter α. (There are differing conventions in the literature, namely: either K = − 3α , K = − 3α or 1 K = − 2α .) It was shown in [14] that such models have universality properties similar to those of [1, 2], hence the name ‘α-attractors’. 4 –4– JHEP04(2019)148 We also find many exact solutions of the equations of motion of two-field α-attractor models, which admit a separated Noether symmetry. To achieve this, we transform the relevant Lagrangian to a new system of generalized coordinates, which is adapted to the Noether symmetry. We investigate each of the elementary hyperbolic surfaces in detail and find a variety of exact solutions of the field equations in each case. The organization of the present paper is the following. In section 2, we briefly review the action for the class of cosmological models known as generalized two-field α-attractors. The two-dimensional scalar manifold of those models is a hyperbolic surface. We write down the action for each elementary hyperbolic surface, namely the Poincaré disk, the hyperbolic punctured disk and the hyperbolic annuli. In section 3, we write the cosmologically relevant point-particle Lagrangian (the so-called ‘minisuperspace Lagrangian’) and impose the condition that it has a Noether symmetry. This leads to a coupled system of seven PDEs. Using a separation-of-variables Ansatz, we find solutions of that system for each elementary hyperbolic surface, in particular determining the form of the scalar potential which is compatible with the separated Noether symmetry. In section 4, we find new generalized coordinates that are adapted to this Noether symmetry. In sections 5, 6 and 7, we investigate the equations of motion of the two-field α-attractor Lagrangian in the new coordinate system for the Poincaré disk, hyperbolic punctured disk and hyperbolic annuli respectively. We find many exact solutions in each of the three cases. Section 8 summarizes our results and briefly mentions some directions for further research. Appendix A recalls the basic definitions and properties of elementary hyperbolic surfaces (whose geometry is described in detail in reference [15]). Appendix B illustrates some of the new exact solutions. where now the two real scalars are ϕ and θ and all the information about the hyperbolic geometry of the sigma-model metric is contained in the function f (ϕ). Such a rewriting can be achieved for any metric GIJ , which admits a U(1) isometry parameterized by θ and so, in particular, for any of the elementary hyperbolic surfaces. Namely: • Poincaré disk. When GIJ is the metric on the hyperbolic disk D, the action (2.1) can be written as:   Z ∂Z∂ Z̄ R 4 √ SD = d x −g − 3α − V (Z) , (2.3) 2 (1 − Z Z̄)2 in terms of a complex scalar Z = φ1 + iφ2 . Writing the latter as: Z = ρeiθ (2.4) and performing the field redefinition:  ϕ ρ = tanh √ 6α  , we find that (2.3) acquires the form (2.2) with the following function f (ϕ): ! r 2 3α sinh2 ϕ . fD (ϕ) = 2 3α (2.5) (2.6) • Hyperbolic punctured disk. For GIJ the metric on the hyperbolic punctured disk D∗ , the action (2.1) can be written as:   Z  √ α R 2 2 2 4 (∂ρ) + ρ (∂θ) − V (ρ, θ) . (2.7) − SD∗ = d x −g 2 (ρ ln ρ)2 Hence, the field redefinition: ϕ= √ 2α ln(| lnρ|) –5– (2.8) JHEP04(2019)148 universal properties of the latter arise from the hyperbolic geometry of the Poincaré disk. Later, reference [14] considered a very wide generalization of the Poincaré disk models, obtained by taking the scalar manifold to be an arbitrary hyperbolic surface and showed that the universal properties of the original one-field α-attractors persist under certain conditions. Specific examples of generalized two-field α-attractors were explored in more detail in [15–17]. In particular, [15] studied α-attractors whose scalar manifold is an elementary hyperbolic surface, i.e. the Poincaré disk, the punctured hyperbolic disk or a hyperbolic annulus. We briefly review their definitions and properties in appendix A. Our goal here will be to show that, for each of the elementary hyperbolic surfaces, the cosmological model obtained from the action (2.1) possesses a Noether symmetry for a certain value of the parameter α and a particular form of the scalar potential V (φ). To achieve this goal, it will be useful to rewrite (2.1) in the form:   Z R 1 f (ϕ) 2 2 4 √ − (∂ϕ) − (∂θ) − V (ϕ, θ) , (2.2) S = d x −g 2 2 2 transforms it into (2.2), where now the function f (ϕ) is: ! r 2 ϕ . fD∗ (ϕ) = 2α exp − α (2.9) • Hyperbolic annulus. When GIJ is the metric on a hyperbolic annulus A, (2.1) acquires the form:   Z 2  αCR R 2 2 2 4 √ (∂ρ) + ρ (∂θ) − V (ρ, θ) , (2.10) − SA = d x −g 2 [ρ cos(CR lnρ)]2 3 Noether symmetries in two-field α-attractors We now investigate under what conditions the action (2.2), namely:   Z R 1 f (ϕ) 2 2 4 √ − (∂ϕ) − (∂θ) − V (ϕ, θ) , S = d x −g 2 2 2 (3.1) has a Noether symmetry. As usual, we will consider the following Ansatz for the fourdimensional inflationary metric: ds24 = −dt2 + a2 (t)d~x2 , (3.2) as well as spatially-homogeneous scalar fields ϕ(xµ ) = ϕ(t) and θ(xµ ) = θ(t). Substituting these in (3.1), we obtain:   Z 2 ϕ̇2 f (ϕ) 2 4 3 3(ȧ + aä) + + θ̇ − V (ϕ, θ) . (3.3) S = d xa a2 2 2 Note that, since here a, ϕ and θ depend only on time, the action per unit spatial volume in (3.3) can be viewed as the classical action of a mechanical system with three degrees of freedom. To use the Noether method, we have to rewrite the Lagrangian in (3.3) in canonical form, namely as L(q i , q̇ i ) in terms of some generalized configuration space coordinates q i and the corresponding generalized velocities q̇ i . To achieve this, we use integration by parts in the ä term in (3.3). This allows us to write the action per unit spatial volume in (3.3) R as dt L , with the following Lagrangian density: a3 ϕ̇2 a3 f (ϕ) θ̇2 + − a3 V (ϕ, θ) . (3.4) 2 2 In this point-like Lagrangian, we can view {a, ϕ, θ} as generalized coordinates on the configuration space M = R2 × S1 . Then {a, ȧ, ϕ, ϕ̇, θ, θ̇} provide coordinates on the corresponding tangent bundle T M. Let us now write down the conditions for (3.4) to have a Noether symmetry. L = −3aȧ2 + –6– JHEP04(2019)148 π . 2 lnR̂ This can be transformed to the expression in (2.2) by the   √ 1 + sin(CR lnρ ) , (2.11) ϕ = 2α ln cos(CR lnρ ) which leads to the following function f (ϕ):   ϕ 2 2 fA (ϕ) = 2αCR cosh √ . (2.12) 2α where CR ≡ redefinition: 3.1 The Noether system Recall that a symmetry generator is a vector field X defined on T M, which preserves the Lagrangian: LX L = 0 , (3.5) where LX is the Lie derivative along X. In fact, to generate a Noether symmetry of L, the vector field X has to be of the specific form: ∂ ∂ ∂ ∂ ∂ ∂ + λ̇a + λϕ + λ̇ϕ + λθ + λ̇θ , (3.6) X = λa ∂a ∂ ȧ ∂ϕ ∂ ϕ̇ ∂θ ∂ θ̇ λa ∂L ∂L ∂L ∂L ∂L ∂L =0 . + λ̇a + λϕ + λ̇ϕ + λθ + λ̇θ ∂a ∂ ȧ ∂ϕ ∂ ϕ̇ ∂θ ∂ θ̇ (3.7) Let us now investigate the implications of this condition for the Lagrangian (3.4). First, note that all terms in (3.7) are either quadratic in the generalized velocities ȧ, ϕ̇ and θ̇ or contain no velocity at all. So we can view the left-hand side of (3.7) as a second degree polynomial in the generalized velocities. Since we want to find functions λa,ϕ,θ (a, ϕ, θ) , for which the symmetry condition (3.7) is satisfied identically, we have to require that each coefficient of this polynomial vanishes separately. Therefore, computing the various terms in (3.7) for the Lagrangian (3.4), we find the following coupled system (where in brackets we indicate the corresponding coefficient of the velocity polynomial): (E1) (coeff. of ȧ2 ) : (E2) (coeff. of ϕ̇2 ) : (E3) (coeff. of θ̇2 ) : (E4) (coeff. of ȧϕ̇) : (E5) (coeff. of ȧθ̇) : (E6) (coeff. of ϕ̇θ̇) : (E7) (ind. of velocity) : ∂λa =0, ∂a ∂λϕ 3 λa + a =0, 2 ∂ϕ a ∂λθ 3 f (ϕ)λa + (∂ϕ f )λϕ + a f (ϕ) =0, 2 2 ∂θ ∂λϕ ∂λa + a2 =0, −6 ∂ϕ ∂a ∂λa ∂λθ −6 + a2 f (ϕ) =0, ∂θ ∂a ∂λϕ ∂λθ + f (ϕ) =0, ∂θ ∂ϕ λa + 2a 3V λa + aVϕ λϕ + aVθ λθ = 0 . (3.8) In the next subsections, we will show that equations (E1)-(E6) can be solved for any function f (ϕ), such that the scalar manifold metric in (3.1) is hyperbolic, i.e. with a constant negative Gaussian curvature. Then, equation (E7) determines a particular form of the scalar potential. As in [31], we will look for solutions with the following separationof-variables Ansatze: λa (a, ϕ, θ) = A1 (a)Φ1 (ϕ)Θ1 (θ) , λϕ (a, ϕ, θ) = A2 (a)Φ2 (ϕ)Θ2 (θ) , λθ (a, ϕ, θ) = A3 (a)Φ3 (ϕ)Θ3 (θ) . –7– (3.9) JHEP04(2019)148 where the coefficients λa,ϕ,θ are functions of the configuration space coordinates {a, ϕ, θ}. Hence, the condition (3.5) becomes: Let us begin by considering equations (E1), (E2) and (E4), which do not depend on f (ϕ) and hence have the same form for any elementary hyperbolic surface. 3.2 Solving equations (E1), (E2) and (E4) Substituting (3.9) in equation (E1), we obtain the following first order ODE: A1 (a) + 2a dA1 =0. da (3.10) Its general solution is: ∂λa 1 ∂λϕ = . ∂a 3 ∂ϕ (3.12) Substituting (3.9) in (3.12), we find the following set of equations:5 A2 (a) dA1 = da 3 , Φ1 (ϕ) = k dΦ2 dϕ , Θ1 (θ) = 1 Θ2 (θ) , k (3.13) where k = const.. Using (3.11) in the first equation of (3.13) gives: 3 A . (3.14) 2 a3/2 Let us now consider equation (E4). Substituting (3.9), (3.11) and (3.14) in this equation gives: dΦ1 −8 Θ1 (θ) + 3Φ2 (ϕ)Θ2 (θ) = 0 . (3.15) dϕ This, together with the last relation in (3.13), implies that: A2 (a) = − Φ2 = 8 dΦ1 3k dϕ (3.16) Using the second equation of (3.13) in (3.16), we end up with the following ODE: d2 Φ2 (ϕ) 3 − Φ2 (ϕ) = 0 , dϕ2 8 (3.17) whose general solution is: Φ2 (ϕ) = b1 sinh r 3 ϕ 8 ! + b2 cosh r 3 ϕ 8 ! , (3.18) where b1,2 = const.. Using this in (3.13), we find: ! !# r r r " 3 3 3 Φ1 (ϕ) = k b1 cosh ϕ + b2 sinh ϕ . 8 8 8 (3.19) Note that our results above for A1 (a), A2 (a), Φ1 (ϕ) and Φ2 (ϕ) are consistent with those of [31], except that b1 was set to zero in that work. 5 For convenience, as well as for easier comparison with [31], we have assigned the to the first equation in (3.13). –8– 1 3 coefficient in (3.12) JHEP04(2019)148 A (3.11) A1 (a) = √ , a where A is an arbitrary integration constant. Equating the expressions for λa obtained from (E1) and (E2) in (3.8), we have: 3.3 Solving equations (E5) and (E6) Now we turn to equations (E5) and (E6) of (3.8). We will see below that, for an arbitrary function f (ϕ), the (E5)-(E6) system does not have a solution compatible with (3.19). However, recall that we are only interested in functions f , such that the sigma-model metric in (3.1), namely the metric ds2 = dϕ2 + f (ϕ)dθ2 , is hyperbolic. We will show now that, for any such f (ϕ), equations (E5) and (E6) can be solved in a manner compatible with (3.19). Let us begin by substituting (3.9) in (E5). This gives and A1 (a) − βa2 dΘ1 dθ with dA3 =0 da c = const. with β = const. , (3.20) (3.21) as well as an equation for Φ3 (ϕ) which we will write down shortly. Using (3.11) allows us to solve (3.21) as: 2 A A3 (a) = − , (3.22) 3 βa3/2 3 where we have set an additive integration constant to zero in order to ensure that dΦ dϕ 6= 0 dΘ2 6 and dθ 6= 0. Upon using (3.20) and (3.22), equation (E5) reduces to the following algebraic relation: 6β 1 Φ1 (ϕ) . (3.23) Φ3 (ϕ) = c f (ϕ) Let us now consider equation (E6) of (3.8). Substituting (3.14) and (3.22), one finds that the a-dependence factors out of this equation. Then, using the third relation of (3.13) together with (3.16), equation (E6) reduces to: dΦ3 6β 1 dΦ1 =− . dϕ c f (ϕ) dϕ (3.24) Comparing the last relation with (3.23), we conclude that 1 dΦ1 1 dΦ3 =− , Φ3 dϕ Φ1 dϕ (3.25) which implies: Φ3 (ϕ) = Φ0 , Φ1 (ϕ) (3.26) where Φ0 = const.. Substituting (3.26) in (3.23), we obtain: s c Φ0 p f (ϕ) Φ1 (ϕ) = 6β 6 The sixth equation in (3.8) implies that constant in (3.22). dΦ3 dϕ = 0 and –9– dΘ2 dθ (3.27) = 0 if there is a non-vanishing additive JHEP04(2019)148 Θ3 (θ) = c and thus Φ3 (ϕ) = r 1 6βΦ0 p . c f (ϕ) (3.28) K=− 1 (2f f ′′ − f ′2 ) , 4 f2 (3.29) where f ′ ≡ ∂ϕ f . Imposing the condition that K = const. < 0 , we can view (3.29) as an ODE for f (ϕ). Solving it, we obtain:  p i2 h p |K| ϕ + C2ϕ sinh |K| ϕ f (ϕ) = C1ϕ cosh with ϕ C1,2 = const. . (3.30) Substituting (3.30) in (3.27), we find that the result has the same form as (3.19). To completely match the two expressions for Φ1 (ϕ), we have to take |K| = 3 . 8 (3.31) Note that this will restrict the value of the α-parameter in each of the three cases with f given by (2.6), (2.9) and (2.12), as we will see shortly.7 Let us now compare in more detail the solution (3.27), with f given by (3.30), to the expression in (3.19), for each elementary hyperbolic surface. The general form of f in (3.30) reduces to the specific form, in each of the three cases listed in equations (2.6), (2.9) and (2.12), for the following respective choices of the integration constants: C1ϕ = 0 f = fD : f = f D∗ : (C1ϕ ) = 2α and : 2 (C1ϕ ) and f = fA 2 2 = 2αCR and 3α , 2 C2ϕ = −C1ϕ , 2 (C2ϕ ) = C2ϕ = 0 . (3.32) Substituting these three cases for f (ϕ) in relation (3.27) and comparing with (3.19) gives 7 1 was imposed for any hyperbolic surface. In the present work, In [14], the normalization K = − 3α however, the coefficients of proportionality between K and α1 are different for each of the elementary hyperbolic surfaces. This follows from writing the relevant kinetic terms with the normalizations given in (2.3), (2.7) and (2.10), which is convenient for easier comparison with most of the literature. – 10 – JHEP04(2019)148 Clearly, for arbitrary f (ϕ), the expression in (3.27) is not compatible with the Φ1 (ϕ) solution found in (3.19). However, we are interested only in functions f (ϕ), for which the scalar manifold metric in (3.1) is hyperbolic. In other words, we are only considering f (ϕ) such that the Gaussian curvature K of the metric ds2 = dϕ2 + f (ϕ)dθ2 is constant and negative. This restricts the form of the function f . To see how, let us compute the Gaussian curvature in question: the following conditions for the existence of a solution: s 1 2αc Φ0 D : b1 = 0 , b2 = k 3β s 2 2αcΦ0 D∗ : b1 = , b2 = −b1 3k β s 2CR 2αcΦ0 , b2 = 0 A : b1 = 3k β , α= 16 9 , α= 4 3 , , α= 4 3 . , (3.33) while Φ3 (ϕ) is given by (3.26) in all three cases. Also, for any function f , the solutions for A1,2,3 (a) are: A1 (a) = A a1/2 , A2 (a) = − 3 A 2 a3/2 , A3 (a) = − 2 A 3 βa3/2 , (3.35) as can be seen in (3.11), (3.14) and (3.22). 3.4 Solving equation (E3) Next, we consider equation (E3) of the system (3.8). Substituting the solutions for A1,2,3 (a) given in (3.35), we find that the a-dependence drops out from (E3). Then, using the third relation in (3.13), as well as (3.16) and (3.20), we find that (E3) reduces to:   4 ′ ′ (3.36) 3 f (ϕ)Φ1 (ϕ) − f Φ1 Θ1 (θ) − 8Φ1 (ϕ)Θ′′1 (θ) = 0 . 3 Substituting (3.27) in (3.36) gives:
3f 2 − 2f ′2 Θ1 (θ) − 8f (ϕ)Θ′′1 (θ) = 0 . (3.37) Since the form of f (ϕ) is fixed for each elementary hyperbolic surface, equation (3.37) is an ODE for the function Θ1 (θ). This ODE admits solutions if and only if the following condition is satisfied: Θ′′1 (θ) 3f 2 − 2f ′2 = = const. ≡ q . (3.38) Θ1 (θ) 8f – 11 – JHEP04(2019)148 To recapitulate, we have shown that, upon choosing integration constants satisfying the constraints (3.33), the solutions of equations (E5) and (E6) are compatible with those of (E1), (E2) and (E4). More explicitly, the solutions for the functions Φ 1,2 (ϕ) in the three cases of interest have the form: ! ! r r r 3 3 3 D: Φ1 (ϕ) = kb2 sinh ϕ , Φ2 (ϕ) = b2 cosh ϕ , 8 8 8 ! ! r r r 3 3 3 ∗ D : Φ1 (ϕ) = kb1 , Φ2 (ϕ) = −b1 exp − exp − ϕ ϕ , 8 8 8 ! ! r r r 3 3 3 , Φ2 (ϕ) = b1 sinh (3.34) cosh ϕ ϕ , A: Φ1 (ϕ) = kb1 8 8 8 2 ′2 −2f It is easy to check that the expression 3f 8f is indeed constant in each of the three cases of interest, namely the Poincaré disk, the punctured hyperbolic disk and the hyperbolic annuli. More precisely, substituting f respectively from (2.6), (2.9) and (2.12) gives: qD = −1 , q D∗ = 0 2 2 qA = CR , . (3.39) ′2 Let us now study equation in (3.38) for each of the three values of q given in (3.39). • Poincaré disk. For q = −1, relation (3.38) gives: Θ′′1 (θ) + Θ1 (θ) = 0 , (3.41) Θ1 (θ) = C1 sin θ + C2 cos θ . (3.42) Θ2 (θ) = k (C1 sin θ + C2 cos θ) , (3.43) Θ3 (θ) = c (C1 cos θ − C2 sin θ) . (3.44) with the obvious solution Then (3.13) implies: whereas (3.20) gives: • Hyperbolic punctured disk. For q = 0, equation (3.38) becomes: Θ′′1 (θ) = 0 , (3.45) whose solution can be written as: Θ1 (θ) = C3 θ + θ0 with C3 , θ0 = const. . (3.46) Using (3.46) as well as (3.13) and (3.20), we find: Θ2 (θ) = k(C3 θ + θ0 ) and Θ3 (θ) = c C3 . (3.47) 2 , equation (3.38) takes the form: • Hyperbolic annulus. For q = CR 2 Θ1 (θ) = 0 , Θ′′1 (θ) − CR (3.48) which has the general solution Θ1 (θ) = C4 cosh(CR θ) + C5 sinh(CR θ) . (3.49) Hence, (3.13) and (3.20) give: Θ2 (θ) = k [C4 cosh(CR θ) + C5 sinh(CR θ)] , Θ3 (θ) = c CR [C4 sinh(CR θ) + C5 cosh(CR θ)] . – 12 – (3.50) JHEP04(2019)148 −2f In fact, one can show directly that 3f 8f = const. for any f (ϕ) , such that the scalar manifold metric in (3.1) is hyperbolic. Namely, using the form of f (ϕ) given in (3.30) with |K| = 83 , we obtain:  3 ϕ 2 3f 2 − 2f ′2 = (C1 ) − (C2ϕ )2 . (3.40) 8f 8 3.5 Solving equation (E7): the scalar potential So far, we have found functions λa,ϕ,θ (a, ϕ, θ) , which solve equations (E1)-(E6) of the Noether system (3.8). Now we will show that the last equation of that system, namely (E7), determines the scalar potential V (ϕ, θ), if the latter is assumed to have the separation of variables form: V (ϕ, θ) = Ṽ (ϕ)V̂ (θ) . (3.51) Note that here we have not used any particular form of the function f . Hence, for any f (ϕ), and thus for any Φ1,2,3 (ϕ) and Θ1,2,3 (θ), we have the pair of equations i h 3 Ṽ (ϕ)Φ1 (ϕ) − k2 Ṽ ′ (ϕ)Φ2 (ϕ) V̂ ′ (θ) Θ3 (θ) , (3.53) = p = 2 V̂ (θ) Θ1 (θ) 3β Ṽ (ϕ)Φ3 (ϕ) where p = const.. Clearly, then, one has two separate equations for the two functions Ṽ (ϕ) and V̂ (θ). Let us now study these two equations for each of the three types of elementary hyperbolic surface. • Poincaré disk. Substituting the D expressions from (3.34) and (3.33) into (3.53), we find the following equation for Ṽ (ϕ): q h q i 2 3 p 3 − 2 sinh 8 c 8ϕ dṼ (ϕ) (3.54) + q  Ṽ (ϕ) = 0 . q  dϕ 3 3 ϕ cosh ϕ sinh 8 8 Its general solution has the form: Ṽ (ϕ) = Ṽ0 cosh 2 r ! ! r p 3 3 ϕ coth c ϕ , 8 8 (3.55) where Ṽ0 is an integration constant. Using (3.42) and (3.44) inside (3.53), we obtain: dV̂ (θ) p (C1 sin θ + C2 cos θ) − V̂ (θ) = 0 , dθ c (C1 cos θ − C2 sin θ) whose solution is: p V̂ (θ) = V̂0 [C1 cos θ − C2 sin θ]− c with V̂0 = const.. – 13 – (3.56) (3.57) JHEP04(2019)148 We begin by substituting (3.9) and (3.51) into (E7). Then, using the solutions for A1,2,3 (a) given in (3.35) as well as the last relation in (3.13) (namely Θ2 = kΘ1 ), we find that (E7) reduces to:   k 2 3 Ṽ (ϕ)Φ1 (ϕ) − Ṽ ′ (ϕ)Φ2 (ϕ) V̂ (θ)Θ1 (θ) − Ṽ (ϕ)Φ3 (ϕ)V̂ (θ)Θ3 (θ) = 0 . (3.52) 2 3β Therefore, for the case of the hyperbolic disk, the form of the scalar potential, that is compatible with Noether’s symmetry, is: ! ! r r p p 3 3 2 (3.58) ϕ coth c ϕ [C1 cos θ − C2 sin θ]− c , V (ϕ, θ) = V0 cosh 8 8 • Hyperbolic punctured disk. Using the D∗ expressions from (3.34) and (3.33) in (3.53), we have: r   q 3 3 dṼ (ϕ) p ϕ 2 Ṽ (ϕ) = 0 , (3.59) + e 1− dϕ 2 2c whose solution is: Ṽ (ϕ) = Ṽ0 exp − Now, substituting the solutions for up with: dV̂ (θ) − dθ r 3 p ϕ+ e 2 2c q 3 ϕ 2 ! . (3.60) Θ1,3 from (3.46) and (3.47) inside (3.53), we end   p θ0 V̂ (θ) = 0 . θ+ c C3 (3.61) The solution of the last equation is:    pθ θ θ0 V̂ (θ) = V̂0 exp . + c 2 C3 (3.62) Note that p = 0 again gives a result independent of θ and thus leads to an effectively single-field system. It may be interesting to investigate this special case further and to see whether or how it differs from the single-field system studied in [31] (which arises from taking p = 0 for the Poincaré disk). • Hyperbolic annulus. Finally, from the A expressions in (3.34) and (3.33), substituted in (3.53), we obtain: q h q i p 2 3 3 − 2 cosh 2 8 cCR 8ϕ dṼ (ϕ) (3.63) + q  q  Ṽ (ϕ) = 0 . dϕ 3 3 sinh ϕ cosh ϕ 8 8 Hence, in this case the solution for Ṽ is: ! ! r r p 3 3 2 ϕ coth cCR ϕ . Ṽ (ϕ) = Ṽ0 sinh2 8 8 – 14 – (3.64) JHEP04(2019)148 where V0 = const.. Note that this expression reduces to the single-field result of [31] for p = 0. It is also worth pointing out that the θ-dependence in (3.57) allows as a special case the particular form needed for natural inflation. Indeed, by taking C2 = 0 and pc = −2, we have V̂ (θ) = const. × cos2 θ. In that regard, it may be interesting to make a connection to the recent considerations of [8] on realizing natural inflation in two-field attractor models. Now substituting (3.49) and (3.50) in (3.53), one finds the following equation for V̂ : dV̂ (θ) p [C4 cosh(CR θ) + C5 sinh(CR θ)] − V̂ (θ) = 0 , dθ cCR [C4 sinh(CR θ) + C5 cosh(CR θ)] (3.65) whose solution is given by: V̂ (θ) = V̂0 [C4 sinh(CR θ) + C5 cosh(CR θ)] 4 p cC 2 R . (3.66) New variables: cyclic coordinate ∂L =0 . ∂w (4.1) This will simplify the relevant equations of motion significantly, as we will see below. Note that, due to (4.1), the Euler-Lagrange equation for w becomes: d ∂L =0 , dt ∂ ẇ (4.2) which shows that the generalized momentum pw ≡ ∂∂L ẇ is conserved. To find such coordinates, we must solve the conditions iX du = 0 , iX dv = 0 and iX dw = 1, which amount to the system: ∂u ∂u ∂u + λϕ + λθ =0 , ∂a ∂ϕ ∂θ ∂v ∂v ∂v + λϕ + λθ =0 , λa ∂a ∂ϕ ∂θ ∂w ∂w ∂w λa + λϕ + λθ =1 , ∂a ∂ϕ ∂θ λa (4.3) Since the first two equations in (4.3) are formally identical, the general solutions for u(a, ϕ, θ) and v(a, ϕ, θ) will have the same form. Ensuring different functions for u and v will be due to choosing different values for (some of) the constants that characterize this general form, as will become clear below. 4.1 Finding the coordinates u and v In this subsection we consider the first equation in (4.3), namely λa ∂u ∂u ∂u + λϕ + λθ =0 . ∂a ∂ϕ ∂θ As already pointed out, this will enable us to find not only u, but v as well. – 15 – (4.4) JHEP04(2019)148 In this section, we will look for a suitable coordinate transformation (a, ϕ, θ) → (u, v, w), such that w is the cyclic coordinate corresponding to the symmetry with generator X that we found above. This will be very useful for finding analytical solutions of the α-attractor equations of motion for the following reason. In the new variables the symmetry generator ∂ will have the form X = ∂w and thus the condition LX L = 0 will become: We will look for solutions with the separation of variables Ansatz: u(a, ϕ, θ) = Au (a)Φu (ϕ)Θu (θ) . (4.5) Using (3.9), (4.5) and the last relation in (3.13) (i.e. Θ2 = kΘ1 ), equation (4.4) reduces to:   A1 Φ1 A′u Φu + kA2 Φ2 Au Φ′u Θ1 Θu = −A3 Φ3 Au Φu Θ3 Θ′u . (4.6) Separating out the θ-dependence gives: (4.7) A1 Φ1 A′u Φu + kA2 Φ2 Au Φ′u + cθ A3 Φ3 Au Φu = 0 (4.8) and for some cθ = const.. Now, substituting A1,2,3 (a) from (3.35) in (4.8), we find that the a-dependence factors out provided that: dAu Au (a) = ca da a (4.9) for some ca = const.. The last equation is solved by: Au (a) = aca , (4.10) where for convenience we have set the overall multiplicative integration constant to one. 8 Substituting (4.10) and (3.35) in (4.8) gives: 3 dΦu (ϕ) kΦ2 + 2 dϕ   2 cθ Φ3 − ca Φ1 Φu (ϕ) = 0 . 3β (4.11) This equation has different coefficients for each elementary hyperbolic surface, since the functions Φ1,2,3 (ϕ) differ in each case (see equation (3.34)). Before specializing to the various cases, we can further simplify (4.11) by using the expressions (3.27)–(3.28) and (3.16) for Φ1,2,3 in terms of the function f (ϕ). This allow us to bring (4.11) to the form: dΦu (ϕ) + dϕ  2cθ c  − c2a f (ϕ) Φu (ϕ) = 0 . f ′ (ϕ) (4.12) To recapitulate, the solution for Au (a) is independent of f and is given by (4.10). On the other hand, the solutions for Φu (ϕ) and Θu (θ) do depend on the form of the function f and are determined by equations (4.7) and (4.12), respectively. Let us now find Φu and Θu for each type of elementary hyperbolic surface. 8 Note that, to preform a coordinate transformation (a, ϕ, θ) → (u, v, w), we only need a particular solution of the system (4.3). – 16 – JHEP04(2019)148 Θ3 (θ) dΘu (θ) = cθ Θ1 (θ)Θu (θ) dθ • Poincaré disk. For f (ϕ) given in (2.6) with α = 16 9 (see the corresponding row in (3.33)), we find that (4.12) acquires the form: q h q i 2 2ca 3 cθ 3 − sinh 8 c 3 8ϕ dΦu (ϕ) (4.13) + q  Φu (ϕ) = 0 . q  dϕ 3 3 ϕ cosh ϕ sinh 8 8 (4.14) where we have again set the overall integration constant to one for convenience. Note that the single-field result for Φu (ϕ) in (4.14) is obtained by taking cθ = 0. Then, setting ca = 3, we find from (4.14) and (4.10) the same particular solution for u(a, ϕ) = Au (a)Φu (ϕ), as that in [31]. Now, substituting (3.42) and (3.44) in (4.7), we find: dΘu (θ) cθ (C1 sin θ + C2 cos θ) − Θu (θ) = 0 , dθ c (C1 cos θ − C2 sin θ) whose solution is: Θu (θ) = (C1 cos θ − C2 sin θ)− cθ c (4.15) (4.16) with the overall integration constant once again set to one. • Hyperbolic punctured disk. Taking f (ϕ) as in (2.9) with α = 43 (in accordance with the D∗ row of (3.33)), equation (4.12) becomes: !# r r " 3 2ca cθ 3 dΦu (ϕ) + − exp ϕ Φu (ϕ) = 0 . (4.17) dϕ 8 3 c 2 This ODE is solved by " cθ ca exp Φu (ϕ) = exp − √ ϕ + 2c 6 r 3 ϕ 2 !# , (4.18) where again the overall integration constant has been set to one. The solution of (4.7), after substituting (3.46) and (3.47), is given by:    cθ θ θ θ0 Θu (θ) = exp , + c 2 C3 (4.19) where the overall integration constant was set to one. • Hyperbolic annulus. For f (ϕ) given by (2.12) with α = 34 (as in the A line of (3.33)), equation (4.12) becomes: q h q i cθ 2 2ca 3 3 cosh − 2 8 cCR 3 8ϕ dΦu (ϕ) (4.20) + q  Φu (ϕ) = 0 . q  dϕ 3 3 ϕ cosh ϕ sinh 8 – 17 – 8 JHEP04(2019)148 This equation has the general solution: " !# 2ca !# cθ " r r c 3 3 3 Φu (ϕ) = coth cosh ϕ , ϕ 8 8 Hence, the solution in this case is " Φu (ϕ) = coth r 3 ϕ 8 !# cθ cC 2 R " sinh r 3 ϕ 8 !# 2ca 3 . (4.21) Finally, the solution of (4.7), after substituting (3.49) and (3.50), has the form: cθ 2 Θu (θ) = [C4 sinh(CR θ) + C5 cosh(CR θ)] cCR . (4.22) 4.2 Finding the cyclic coordinate w Now we will consider the last equation in (4.3), namely: λa ∂w ∂w ∂w + λϕ + λθ =1 . ∂a ∂ϕ ∂θ (4.23) As usual, we will make the separation of variables Ansatz: w(a, ϕ, θ) = Aw (a)Φw (ϕ)Θw (θ) . (4.24) Substituting (4.24), (3.9) and (3.35) in (4.23), it is easy to realize that the a-dependence can be canceled within each term by taking Aw (a) = 1 3/2 a . A (4.25) Using (4.25) and the relation Θ2 = kΘ1 (see (3.13)), we find that (4.23) acquires the form:  2 3 Φ1 Φw − kΦ2 Φ′w Θ1 Θw − Φ3 Φw Θ3 Θ′w = 1 . 2 3β (4.26) Now we will show that one can remove the ϕ-dependence in (4.26) by a suitable choice of the function Φw (ϕ). The result will be an equation for Θw (θ). Indeed, let us take: Φw (ϕ) = φ0 Φ3 (ϕ) with φ0 = const. . (4.27) Then, obviously, the second term in (4.26) becomes independent of ϕ. In addition, one can show that the combination [Φ1 Φw − kΦ2 Φ′w ] in the first term, with Φw given by (4.27), is a constant for each of the three cases in (3.34). In fact, one can see directly that this – 18 – JHEP04(2019)148 Remark on the coordinate v: so far, we have found a function u(a, ϕ, θ) = Au (a)Φu (ϕ)Θu (θ), for each of the three cases under consideration, that solves the first equation in (4.3). As mentioned above, the second equation in (4.3) is then solved by a function v(a, ϕ, θ) = Av (a)Φv (ϕ)Θv (θ), such that Av , Φv and Θv have the same general form as their u-indexed counterparts. To ensure that v is a different function, one has to choose different values of the constants ca and cθ than those taken for the function u. combination is constant for any function f (ϕ) compatible with the hyperbolic geometry of the scalar manifold. Indeed, using (3.16), (4.27), (3.26) and (3.27), we obtain:
  8 ′2 φ0 c f 2 − 23 f ′2 φ0 2 ′ Φ1 − Φ1 = . (4.28) Φ1 Φw − kΦ2 Φw = Φ0 3 6β f where for convenience we also wrote the result in terms of the constant q defined in (3.38). We are finally ready to extract an ODE for Θw (θ). Substituting (4.27) and (4.29) into (4.26) gives:  2 φ0  c q Θ1 (θ) Θw (θ) − Θ3 (θ) Θ′w (θ) = 1 . (4.30) 3 β Then, using (3.20) and setting 3β 2c in order to simplify the equation, we obtain from (4.30): (4.31) φ0 = − Θ′1 (θ) Θ′w (θ) − q Θ1 (θ) Θw (θ) − 1 = 0 . (4.32) Let us now solve the last equation for each type of elementary hyperbolic surface. • Poincaré disk. In this case q = −1 (see (3.39)) and Θ1 (θ) is given by (3.42). Therefore, (4.32) becomes: [C1 cos θ − C2 sin θ] dΘw + [C1 sin θ + C2 cos θ] Θw (θ) − 1 = 0 . dθ (4.33) The general solution of the last equation can be written as: Θw (θ) = sin θ + Ĉθ [C1 cos θ − C2 sin θ] C1 with Ĉθ = const. . (4.34) Note that, upon redefinition of the integration constant Ĉθ , the solution can also be written as: cos θ + Ĉθ [C1 cos θ − C2 sin θ] (4.35) Θw (θ) = C2 or as:   1 sin θ cos θ + Ĉθ [C1 cos θ − C2 sin θ] . (4.36) + Θw (θ) = 2 C1 C2 The last form might seem preferable, since it is symmetric with respect to interchange of the trigonometric functions sin and cos. However, this form requires both C1 6= 0 and C2 6= 0. On the other hand, the forms (4.34) and (4.35) allow one to take respectively the limits C2 = 0 and C1 = 0. Since we will be particularly interested in the limit C2 = 0, we will use the form (4.34) in what follows (although we will comment more on using (4.36) below). – 19 – JHEP04(2019)148 Now recall relation (3.40), which holds for any f (ϕ) of the form (3.30) with |K| = 83 . Using this relation, we find that (4.28) implies: i 4φ c φ0 c h ϕ 2 0 2 Φ1 Φw − kΦ2 Φ′w = q , (4.29) (C1 ) − (C2ϕ ) = 6β 9 β • Hyperbolic punctured disk. In this case q = 0 (see (3.39)). Also, Θ1 (θ) has the form (3.46). Substituting these in (4.32), we obtain the ODE: C3 Θ′w (θ) − 1 = 0 , which has the solution: Θw (θ) = θ + const. . C3 (4.37) (4.38) Similarly to the D case above, the general solution of (4.39) can be written in three equivalent ways, namely: Θw (θ) = or   sinh(CR θ) C sinh(C θ) + C cosh(C θ) + C̃ 4 5 R R θ 2C CR 5 Θw (θ) = − or Θw (θ) = 5 1 2 2CR    cosh(CR θ) + C̃θ C4 sinh(CR θ) + C5 cosh(CR θ) 2 CR C4 sinh(CR θ) cosh(CR θ) − C5 C4  (4.40) (4.41)   + C̃θ C4 sinh(CR θ) + C5 cosh(CR θ) . (4.42) Equations of motion for the Poincaré disk In this section our goal will be to find solutions to the equations of motion of the Lagrangian (3.4) for the case of the Poincaré disk. For that purpose, we will first rewrite the Lagrangian in terms of the new coordinates (u, v, w) with the cyclic variable w. As already pointed out, this will lead to a significant simplification of the equations that will enable us to find analytical solutions. Let us begin by summarizing the relevant results, which we have obtained so far for the two-field cosmological model based on the Poincaré disk. For f (ϕ) given by (2.6) with α = 16 9 as in (3.33), the Lagrangian (3.4) has the form: ! r 3 ϕ̇2 3 a 4 L = −3aȧ2 + + a3 sinh2 ϕ θ̇2 − a3 V (ϕ, θ) . (5.1) 2 3 8 We found that (5.1) has a certain Noether symmetry, when the scalar potential is of the form (3.58), namely: ! ! r r p p 3 3 V (ϕ, θ) = V0 cosh2 (5.2) ϕ coth c ϕ [C1 cos θ − C2 sin θ]− c . 8 8 – 20 – JHEP04(2019)148 2 . Using this and the • Hyperbolic annulus. In this case, relation (3.39) gives q = CR relevant Θ1 (θ) expression (3.49), we find that (4.32) acquires the form:   CR C4 sinh(CR θ) + C5 cosh(CR θ) Θ′w (θ)   2 C4 cosh(CR θ) + C5 sinh(CR θ) Θw (θ) = 1 . (4.39) − CR Also, according to (4.10), (4.14), (4.16), (4.25), (4.27) and (4.34), the general form of the new variables u, v and w, with the latter being the cyclic coordinate corresponding to the Noether symmetry of section 3, is the following: u r " r u(a, ϕ, θ) = aca cosh v 3 coth !# 2cva " 3 3 coth ϕ 8 ! r 3 sinh ϕ sin θ , 8 v(a, ϕ, θ) = aca cosh w(a, ϕ, θ) = Cw a3/2 3 ϕ 8 !# 2cua " r r !# cuθ (C1 cos θ − C2 sin θ)− !# cvθ (C1 cos θ − C2 sin θ)− 3 ϕ 8 3 ϕ 8 c c cu θ c cv θ c (5.3) q β 1 where for convenience we have denoted Cw ≡ − AC c Φ0 and have taken Ĉθ = 0 in (4.34). 1 Note that, to obtain this expression for the coefficient Cw , one has to take into account (4.31), as well as the relevant coefficient for Φ1 (ϕ) according to (3.33)–(3.34). Finally, we have labeled the ca and cθ constants, characterizing the functions u(a, ϕ, θ) and v(a, ϕ, θ) , with upper u and v indices, respectively, to underline the fact that their values in the two cases are independent of each other. Now we are ready to rewrite the Lagrangian in terms of the variables (u, v, w) and to study the resulting equations of motion. An important remark is in order, though, before we embark on that investigation. Namely, the Lagrangian (5.1) is subject to the Hamiltonian constraint EL = 0, where EL = ∂L i q̇ − L ∂ q̇ i (5.4) is the energy function corresponding to any point particle Lagrangian L(q i , q̇ i ) with generalized coordinates q i . It is well-known that the Hamiltonian EL is conserved on any solutions of the Euler-Lagrange equations, i.e. that for such solutions one has EL = const.. So imposing the constraint EL = 0 (which is equivalent to the first order Einstein equation, often also called Friedman constraint) only results in a relation between the integration constants of the Euler-Lagrange equations; see for example [27]. Instead of just using the Hamiltonian constraint at the end of the computation, in order to eliminate one of the integration constants, it is tempting to try to utilize it from the start, in order to facilitate the search for solutions. However, since this constraint is generally (highly) non-linear, there is no guarantee that it will make a crucial difference for that purpose. In particular, for the cases that we will investigate below, it will turn out not to be useful in our search for analytical solutions. 5.1 Lagrangian in the new variables To obtain the Lagrangian in terms of the new variables u, v and w, we only need a particular coordinate transformation (a, ϕ, θ) → (u, v, w). Hence, we can choose convenient values u,v for the arbitrary constants cu,v in (5.3). Particularly simple (and convenient for a and cθ – 21 – JHEP04(2019)148 " comparison with [31]) expressions are obtained for the following choices: cuθ = 0 , cua = 3 cvθ = −c and , cva = 3 2 . (5.5) Substituting (5.5) in (5.3) gives:9 ! 3 u = a3 cosh2 ϕ , 8 ! r 3 ϕ (C1 cos θ − C2 sin θ) , v = a3/2 sinh 8 ! r 3 w = Cw a3/2 sinh ϕ sin θ . 8 r Note that here we need C1 6= 0, to ensure that v and w are independent variables. However, this was already tacitly assumed when using the Θw (θ) solution (4.34) in (5.3); to allow for C1 = 0, one would have to use the form (4.35) instead. From now on, we will work with the coordinate transformation (5.6), whose inverse transformation is: " #     1/3 1 v C2 2 1 2 a = u− , w + + Cw2 C12 w Cw  #−1/2  " r r 2  2 u 1 C2 1 v  , + 2 arccoth + ϕ=2 3 w2 C12 w Cw Cw    Cw v C2 θ = arccot . (5.7) + C1 w Cw Note that, when θ = const. , the variables v and w coincide up to a constant and the resulting expressions in (5.6) and (5.7) are consistent with the single-field ones obtained in [31]. Now, substituting (5.7) in (5.1)–(5.2), we find that in the new variables the Lagrangian is: 1 u̇2 4 1 2 4 1 L=− + v̇ + 3 u 3 C12 3 Cw2  C2 1 + 22 C1  p 8 C2 u 2c +1 ẇ + v̇ ẇ − V . p 0 3 C12 Cw vc 2 (5.8) As already mentioned above, the single-field case is obtained for w = const. × v and p = 0. In that case, the Lagrangian (5.8) is consistent with that in [31]. Also, note that the mixed term drops out for C2 = 0, which is exactly the special case relevant for natural inflation as mentioned below (3.58).10 9 As mentioned earlier, here we use (4.34), since we are interested in encompassing the special case with C2 = 0. For a discussion of the coordinate transformation and resulting Lagrangian, when using the form of the Θw solution in (4.36), see footnote 10 below. 10 Note that, if we had used the Θw solution in (4.36) q of (5.6)  (still with Ĉθ = 0), then the third line q would have been modified to w = Cw a3/2 sinh 3 8 ϕ (C1 cos θ + C2 sin θ) with Cw = − 2AC11 C2 – 22 – β cΦ0 and JHEP04(2019)148 (5.6) Before we begin looking for solutions, let us underline again that (5.8) is subject to the Hamiltonian constraint EL = 0, where EL = ∂L ∂L ∂L u̇ + v̇ + ẇ − L , ∂ u̇ ∂ v̇ ∂ ẇ (5.11) as discussed above. 5.2 Solutions It is convenient to introduce the notation: ẅ = − C2 Cw v̈ , C12 + C22 (5.14) and introduce the function ũ(t) ≡ p u(t) . (5.15) Substituting (5.14) and (5.15) in the second and third equations of (5.13) gives: 3 ũm+2 v̈ − V0 C0 m m+1 = 0 , 8 v m+1 3 ũ ¨ − V0 (m + 2) ũ =0 , 8 vm (5.16) where for convenience we have denoted C0 ≡ C12 + C22 . C1,2 6= 0. Then, the inverse transformation would be: (  2  )1/3 C1 (w − vCw )2 + C22 (w + vCw )2 a = u− , 2 4C1 C22 Cw   r √ 2C C C 2 u 1 2 w  , ϕ = 2 arccoth q 3 C12 (w − vCw )2 + C22 (w + vCw )2   C2 (w + vCw ) θ = arccot . C1 (w − vCw ) That would lead to the following Lagrangian:   p 2 (C22 − C12 ) 1 2 1 (C12 + C22 ) u 2c +1 1 u̇2 2 + v̇ + + ẇ v̇ ẇ − V . L=− 0 2 3 u 3 C12 C22 Cw 3Cw C12 C22 v p/c Clearly, in this case, the mixed v̇ ẇ term would vanish for C1 = C2 . – 23 – (5.9) (5.10) JHEP04(2019)148 p . (5.12) c Then the Euler-Lagrange equations of (5.8) are:   C22 C2 Cw 1 + 2 ẅ + v̈ = 0 , C1 C12   m C2 u 2 +1 8 1 v̈ + 2 ẅ = 0 , V0 m m+1 − v 3 C12 C1 Cw m  u m2 +2 2uü − u̇2 − 3V0 +1 =0 . (5.13) 2 vm Note that in the single-field limit, which for us is given by p = 0 (equivalently, m = 0) and v = const. × w, this system is in complete agreement with [31]. To simplify the system (5.13), let us express ẅ from the first equation, namely m≡ Before we begin solving (5.16), let us make an important remark. Equation (5.14) can be solved immediately for w in terms of v. One of the integration constants in this solution is determined by the constant of motion Σ0 , that is due to the Noether symmetry. Indeed, in general, Σ0 is given by:   C22 8 1 8 C2 ∂L 1 + = ẇ + v̇ . (5.17) Σ0 ≡ ∂ ẇ 3 Cw2 3 C12 Cw C12 The first equation in (5.13) (equivalently, equation (5.14)) is precisely the time derivative of (5.17), due to the fact that w is a cyclic coordinate. So the general solution for w is: where Σ̂0 = we have: 2 2 3 C 1 Cw 8 (C12 +C22 ) Σ0 C2 Cw v(t) + Σ̂0 t + C0w , (C12 + C22 ) (5.18) and C0w = const. . Hence, using (5.18) and (5.15) in (5.11), 4 v̇ 2 3 C12 Cw2 ũm+2 4 EL = − ũ˙ 2 + + + V Σ2 . 0 3 3 (C12 + C22 ) vm 16 (C12 + C22 ) 0 (5.19) As alluded to earlier, the constraint EL = 0 is highly nonlinear and we have not found it helpful in looking for exact solutions. So we will utilize it only at the end, in order to fix one of the integration constants of the solutions of (5.16) that we will manage to find. Now let us turn to solving the system (5.16). It simplifies significantly for three special choices of m, namely m = 0, −1, −2. We will begin by investigating these special cases in order of increasing complexity. Finally, we will address the generic case with m 6= 0, −1, −2. 5.2.1 Special cases: m = 0, −1, −2 The simplest special cases are m = 0 and m = −2. So we will consider them first, before turning to the m = −1 case. • m = 0 case. In this case, the system (5.16) reduces to: v̈ = 0 , ¨ − 3 V0 ũ = 0 , ũ 4 (5.20) v(t) = C1v t + C2v ,  p   p  1 1 u u 3V0 t + C2 sinh 3V0 t , ũ(t) = C1 cosh 2 2 (5.21) with the general solution: u = const. and C v = const.. where C1,2 1,2 Substituting (5.21) in (5.19) with m = 0 gives:   4 (C1v )2 3 C12 Cw2 EL = (C1u )2 − (C2u )2 V0 + + Σ2 . 2 2 3 (C1 + C2 ) 16 (C12 + C22 ) 0 – 24 – (5.22) JHEP04(2019)148 w(t) = − Hence, we can enforce the Hamiltonian constraint EL = 0, for example, by taking: (C1v )2 =   9 3 2 (C1 + C22 ) (C2u )2 − (C1u )2 V0 − C12 Cw2 Σ20 . 4 64 (5.23) Note that, depending on the choice of integration constants, these m = 0 solutions can have either w = const. × v (which is the single-field limit) or w 6= const. × v. In appendix B.1 we illustrate genuine two-field trajectories obtained in the latter case for certain values of the integration constants. 3 v̈ + V0 C0 v = 0 , 4 ¨=0 , ũ (5.24)   p   p 1 1 v 3V0 C0 t + C2 cos 3V0 C0 t , v(t) = sin 2 2 ũ(t) = C1u t + C2u . (5.25) whose general solution is: C1v Substituting (5.25) in (5.19) with m = −2, we find:   4 C12 Cw2 3 EL = (C1v )2 + (C2v )2 V0 − (C1u )2 + Σ2 . 3 16 (C12 + C22 ) 0 (5.26) So, to ensure that EL = 0, we can take for instance: (C1u )2 =  C12 Cw2 9 3 v 2 (C1 ) + (C2v )2 V0 + Σ2 . 4 64 (C12 + C22 ) 0 (5.27) Note that, for m = −2 and C2 = 0 , our scalar potential is of the kind relevant for natural inflation, namely V ∼ cos2 θ . It would be interesting to compare the solution with Noether symmetry obtained here to the considerations of [8]. • m = −1 case. In this case, the system (5.16) becomes: 3 v̈ + V0 C0 ũ = 0 , 8 ¨ − 3 V0 v = 0 . ũ 8 Denoting for convenience 3 Q ≡ V0 , 8 (5.28) (5.29) we find from the first equation: ũ = − v̈ . QC0 – 25 – (5.30) JHEP04(2019)148 • m = −2 case. In this case, (5.16) acquires the form: Differentiating (5.30) twice gives: (4) ¨=−v ũ , QC0 where v (4) ≡ d4 v . dt4 (5.31) Substituting (5.31) in the second equation of (5.28), we end up with: v (4) + Q2 C0 v = 0 . (5.32) Recall that C0 > 0 by definition. So the general solution of (5.32) has the form: +C3v cosh(ωt) sin(ωt) + C4v sinh(ωt) sin(ωt) with ω ≡ q 1/4 1 2 Q C0 = 1 4 √ 1/4 3V0 C0 (5.33) . Hence, using (5.30), the solution for ũ(t) is: ũ = C̃1v sinh(ωt) sin(ωt) + C̃2v cosh(ωt) sin(ωt) −C̃3v sinh(ωt) cos(ωt) − C̃4v cosh(ωt) cos(ωt), (5.34) √ where C̃iv = Civ / C0 for i = 1, . . . , 4. Using (5.33) and (5.34) in (5.19) with m = −1, we obtain: EL = 3 C12 Cw2 (C2v C3v − C1v C4v ) 2 p V + 0 2 + C 2 ) Σ0 . 2 2 16 (C C1 + C2 1 2 (5.35) Clearly, we can ensure that EL = 0 by choosing appropriately any one of the intev gration constants C1,...,4 in terms of the remaining constants in (5.35). 5.2.2 Generic case: m 6= 0, −1, −2 For m 6= 0, −2 the two equations in (5.16) are always coupled. In principle, one could use a procedure similar to that used for the m = −1 case above, in order to reduce the system to a single fourth order ODE. Namely, we can express ũ from the first equation ¨ in in terms of v and v̈. Upon differentiating this expression twice, we would obtain ũ (4) terms of v and its derivatives up to and including v . Finally, substituting the results ¨ . . . , v (4) ) in the second equation of (5.16), we would end up with a for ũ(v, v̈) and ũ(v, single 4th order ODE for v(t). However, this equation is generally nonlinear and rather messy. Alternatively, one could substitute the expression for ũ(v, v̈), resulting from the first equation of (5.16), into (5.19) in order to obtain a 3rd order ODE for v(t) from the constraint EL = 0. This equation, however, is also highly nonlinear and quite messy. Despite not being able to solve (5.16) analytically in full generality, we will nevertheless manage to find particular classes of solutions for any m < −2 or m > 0. For that purpose, let us first note that the two equations in (5.16), together, imply the relation: ¨ ũ = (m + 2) v̈ v . C0 m ũ (5.36) – 26 – JHEP04(2019)148 v(t) = C1v cosh(ωt) cos(ωt) + C2v sinh(ωt) cos(ωt) So we can view (5.36) and one of the equations in (5.16) as the two independent equations to solve. An obvious Ansatz solving (5.36) is v= 1/2 ± C0 r m ũ . m+2 (5.37) Depending on the sign of the ũ term,11 the solutions of (5.38) are: ũ(t) = C1u sinh(ω̃t) + C2u cosh(ω̃t) (±1)m (m + 2) > 0 (5.39) (±1)m (m + 2) < 0 , (5.40) m (5.41) for and ũ(t) = C1u sin(ω̃t) + C2u cos(ω̃t) for where 1 ω̃ = 2 s 3 −m/2 V0 C 0 | (±1)m (m + 2)| 2  m+2 m 2 . Substituting (5.37) and (5.39) in (5.19), we have: EL = −m/2 V0 C 0 (±1)m  m+2 m m 2   3 C12 Cw2 Σ2 , (C2u )2 − (C1u )2 + 16 (C12 + C22 ) 0 (5.42) while using (5.37) and (5.40) inside (5.19) gives: EL = −m/2 V0 C 0 (±1)m  m+2 m m 2   3 C12 Cw2 Σ2 . (C2u )2 + (C1u )2 + 16 (C12 + C22 ) 0 (5.43) Clearly, one can always satisfy the Hamiltonian constraint EL = 0 for the expression (5.42), upon fixing suitably an integration constant. On the other hand, (5.43) can vanish only for the minus sign in (±1)m together with m being odd. In that case, the condition (±1)m (m + 2) < 0, together with the earlier requirement m ∈ (−∞, −2) ∪ (0, ∞), implies that m > 0. Hence, the system (5.16) has particular solutions with ũ(t) of the form (5.40), only for the “−” sign of the v(t) expression in (5.37), as well as m odd and positive. To illustrate the above considerations, let us write down, for example, the particular solutions for m = 1: r C0 v(t) = ũ(t) , 3 ũ(t) = C1u sinh(ω̃t) + C2u cosh(ω̃t) 11 Note that this sign is correlated with the choice of sign in (5.37). – 27 – (5.44) JHEP04(2019)148 However, notice that, in order to have real solutions with this Ansatz, we need to assume that m < −2 or m > 0. Substituting (5.37) in any of the two equations of (5.16), we end up with:  m 3 m+2 2 −m/2 m ¨ ũ − V0 C0 (±1) (m + 2) ũ = 0 . (5.38) 8 m and v(t) = − r C0 ũ(t) , 3 ũ(t) = C1u sin(ω̃t) + C2u cos(ω̃t) with ω̃ = 3 2 q 31/2 2 −1/2 V0 C 0 and (C1u )2 = √ 2 2 3 C 1 Cw 2 u 2 16 V C 1/2 Σ0 ± (C2 ) 0 0 (5.45) , where the “+” corresponds 6 Equations of motion for the hyperbolic punctured disk In this section, we will look for solutions of the equations of motion for the case of the hyperbolic punctured disk. Let us begin with a summary of the necessary results from the previous sections. For the hyperbolic punctured disk, we have α = 43 according to (3.33). Hence, using (2.9), the Lagrangian (3.4) acquires the form: ! r 3 a3 ϕ̇2 4 3 2 + a exp − ϕ θ̇2 − a3 V (ϕ, θ) , L = −3aȧ + (6.1) 2 3 2 where the potential is V (ϕ, θ) = V0 exp − r p 3 ϕ+ e 2 2c q 3 ϕ 2 ! exp  p  θ2 2c (6.2) in accord with (3.60) and (3.62); note that, for technical simplicity, here and in the following we will only consider the θ0 = 0 case. In addition, from (4.10), (4.18), (4.19), (4.25), (4.27) and (4.38), we have: " !# r  u  u u c c c 3 a cu θ u(a, ϕ, θ) = a a exp − √ ϕ + exp ϕ exp θ θ2 2c 2 2c 6 !# " r  v  cvθ c cva 3 cva √ ϕ+ exp ϕ exp θ θ2 v(a, ϕ, θ) = a exp − 2c 2 2c 6 ! r 3 ϕ , (6.3) w(a, ϕ, θ) = Cw a3/2 θ exp − 8 – 28 – JHEP04(2019)148 to (5.44) and the “−” to (5.45). In view of the m = −1 case considered above, relation (5.36) also seems to suggest looking for solutions with an Ansatz of the form ũ = const. × v̈. Unlike (5.37), however, such an Ansatz would lead to two independent equations for v since it does not solve identically (5.36), but instead brings it in the form v (4) −const1 ×v = 0. Indeed, substituting the same Ansatz ũ = const. × v̈ in any of the two equations in (5.16), would lead to an equation of the form v̈ + const2 × v = 0 . Since in general const1 6= const22 , the two equations for v(t) would be incompatible. One can ensure const1 = const22 by viewing it as a constraint relating V0 , C0 and m and then solving it for one of those constants in terms of the other two. In that case, one would still end up with a solution of the same kind as (5.39) or (5.40), but with at least one of the previously arbitrary integration constants now fixed. q β 1 where we have denoted Cw = − AC cΦ0 and have taken const. = 0 in (4.38) for conve3 nience. Notice that we used (4.31), (3.26) and the D∗ lines for Φ1 (ϕ) in (3.33)–(3.34), in order to obtain the expression for Cw given above. 6.1 Lagrangian in the new variables To rewrite the Lagrangian (6.1) in terms of (u, v, w), let us first choose suitably the conu,v stants cu,v a and cθ in (6.3). It is convenient to take: , cua = 3 2 cvθ = c and , cva = 0 . Using (6.4), the coordinate transformation (6.3) becomes: ! r 3 3/2 ϕ , u = a exp − 8   q3 ϕ 2 2 +θ  e , v = exp 2 ! r 3 w = Cw a3/2 θ exp − ϕ , 8 whose inverse transformation has the form: 1/3  w2 2/3 , 2 ln v − 2 2 a=u u Cw   w2 ϕ = (2/3)1/2 ln 2 ln v − 2 2 , u Cw w θ= . uCw (6.4) (6.5) (6.6) Substituting (6.6) in (6.1)–(6.2) gives: 8 8 u 4 1 2 L = − u̇2 ln v − u̇ v̇ + ẇ − V0 u2 v m , 3 3 v 3 Cw2 (6.7) where for convenience we introduced the notation m≡ p , c (6.8) as in the previous section. Notice that the Lagrangian (6.7) can be simplified upon exchanging v for a new variable v̂, defined through: v̂ = u ln v . (6.9) Indeed, equation (6.9) implies that v = ev̂/u . Substituting this in (6.7), we find: 8 4 1 2 L = − u̇ v̂˙ + ẇ − V0 u2 emv̂/u . 3 3 Cw2 – 29 – (6.10) JHEP04(2019)148 cuθ = 0 6.2 Solutions Let us now turn to investigating the equations of motion of the Lagrangian (6.10). Clearly, the w-equation is: ẅ = 0 , (6.11) w(t) = Σ∗ t + C0w , (6.12) Hence, we immediately have: where C0w = const. and Σ∗ is the Noether symmetry constant of motion, up to a numerical factor. Substituting (6.10) and (6.12) in the general expression (5.4), we find the Hamiltonian: 8 4 Σ2∗ EL = − u̇ v̂˙ + V0 u2 emv̂/u + . (6.13) 3 3 Cw2 As in the previous section, the constraint EL = 0 will not turn out to be helpful in finding new analytical solutions, due to its non-linearity. So we will use it only at the end, to fix one of the integration constants of the solutions of the Euler-Lagrange equations that we find. The u and v̂ equations of motion, following from (6.10), are: 3 ü − V0 m u emv̂/u = 0 , 8 3 v̂¨ + V0 (mv̂ − 2u)emv̂/u = 0 . 8 (6.14) Note that, due to the unusual kinetic term, the first equation in (6.14) arises from the d ∂L v̂-variation, i.e. from ∂L ∂ v̂ − dt ∂ v̂˙ = 0, while the second one comes from the u-variation. Clearly, taking m = 0 simplifies greatly the system (6.14). So let us consider this case first. – 30 – JHEP04(2019)148 Recall also that (6.10) is subject to the Hamiltonian constraint EL = 0, as discussed in the previous section. Before we begin looking for solutions of the equations of motion, it is worth making a couple of remarks. First, one can easily see from (6.9) that the expression for v̂ is not of the form (4.5), and consequently not of the same form as the u and v solutions in (6.3). Note, however, that the separation of variables Ansatz (4.5) only enables us to find a particular class of solutions of (4.4). Furthermore, for any u and v satisfying the latter equation, the expression uf (v), where f (v) is an arbitrary function of v, is clearly a solution of (4.4) too. Finally, let us comment on the single-field case, which is again obtained for m = 0. At first sight, it might seem that there is a problem, as (6.5) implies w = const. × u for θ = const., whereas the Lagrangian (6.10) depends explicitly on u, and not on v̂, after setting m = 0. However, this is exactly the correct dependence, since the Lagrangian does not contain the usual kinetic terms for u and v̂, but only the mixed u̇v̂˙ term. As a result, the u-variation gives the v-equation of motion and vice-versa. This will become apparent shortly. • Special case: m = 0. In this case, (6.14) acquires the form: ü = 0 , 3 v̂¨ − V0 u = 0 . 4 (6.15) Recall that, as pointed out above, m = 0 corresponds to the single-field limit. Obviously, in view of (6.11), the first equation in (6.15) is consistent with the single-field identification w = const. × u when θ = const., that we discussed in subsection 6.1. The solutions of (6.15) are: (6.16) where Ci∗ with i = 1, . . . , 4 are integration constants. Note that this is quite different from the analogous solutions in the Poincaré disk case, given in (5.21). It may be worth exploring further what distinguishing features that may lead to for the punctured disk case, even with just one scalar field. Now, substituting (6.16) in (6.13) with m = 0, we obtain: EL = (C2∗ )2 V0 − 8 ∗ ∗ 4 Σ2∗ . C C + 3 1 3 3 Cw2 (6.17) To ensure that EL = 0 , we can take for example:   1 8 ∗ ∗ 4 Σ2∗ ∗ 2 . (C2 ) = C C − V0 3 1 3 3 Cw2 (6.18) Note that the solutions above can have w 6= const.×u, even though m = 0, depending on the choice of integration constants. In appendix B.2 we illustrate such two-field solutions for particular values of the constants. • Generic case: m 6= 0. Now let us consider the generic case with m 6= 0 . Then, one  couldsolve the first equation in (6.14) algebraically for v̂, obtaining u v̂ = m ln 3V80 m uü . Substituting this expression in the second equation of (6.14), one would find a fourth order ODE for u(t). However, the resulting equation is highly non-linear and thus cannot besolved analytically in full generality. Alternau ln 3V80 m üu in the constraint EL = 0, in order to tively, one could substitute v̂ = m obtain a third order ODE for u(t). This equation, though, is also highly nonlinear and unwieldy. So we will pursue a different route instead. Namely, we will use a certain Ansatz that will enable us to find particular analytical solutions for any m > 0 . Notice that, from the first equation in (6.14), we have emv̂/u = this in the second equation of (6.14), we end up with: m(uv̂¨ + üv̂) − 2uü = 0 . – 31 – 8 ü 3V0 m u . Substituting (6.19) JHEP04(2019)148 u = C1∗ t + C2∗ , 3 1 v̂ = V0 C1∗ t3 + V0 C2∗ t2 + C3∗ t + C4∗ , 8 8 Clearly, one can view (6.19) and one of (6.14) as the two independent equations to solve. Therefore, an obvious Ansatz, that solves (6.19) identically, is: v̂ = u . m (6.20) Substituting (6.20) in any of the two equations in (6.14), we obtain: ü(t) − 3 V0 me u(t) = 0 . 8 (6.21) u(t) = C1∗ sinh(ω∗ t) + C2∗ cosh(ω∗ t) for m>0 (6.22) m<0 , (6.23) and u(t) = C1∗ sin(ω∗ t) + C2∗ cos(ω∗ t) where ω∗ = r 3 V0 e|m| . 8 Let us now impose the Hamiltonian constraint. in (6.13), we obtain: m>0 : for (6.24) Substituting (6.20) and (6.22)   4 Σ2∗ , EL = eV0 (C2∗ )2 − (C1∗ )2 + 3 Cw2 (6.25) while substituting (6.20) and (6.23) in (6.13) gives: m<0 :   4 Σ2∗ . EL = eV0 (C2∗ )2 + (C1∗ )2 + 3 Cw2 (6.26) Clearly, one can ensure that (6.25) satisfies the constraint EL = 0 by fixing suitably C1∗ or C2∗ . On the other hand, the expression (6.26), following from (6.23), is incompatible with the Hamiltonian constraint. Hence, this constraint allows only particular solutions of the form (6.22). Note that (6.20), together with (6.9), implies that v = const.. Nevertheless, this is not a degenerate case, since from (6.6) we can see that all of a(t), ϕ(t) and θ(t) are nontrivial functions. For v = const., however, it is evident that ϕ and θ become functionally dependent. So this particular solution corresponds to yet another effectively single-field system, although it has m 6= 0. It would be very interesting to understand whether there is a deeper reason for this outcome. 7 Equations of motion for the hyperbolic annuli In this section, we turn to finding analytical solutions of the equations of motion for the hyperbolic annuli case. As before, we begin by summarizing the relevant results from sections 3 and 4. – 32 – JHEP04(2019)148 The solutions of this equation are: 4 3 In the A case, we have from (3.33) that α = the Lagrangian (3.4) becomes: a3 ϕ̇2 4 3 2 + a CR cosh2 L = −3aȧ2 + 2 3 with potential given by: r V (ϕ, θ) = V0 sinh2 . Using this and (2.12), we find that r ! 3 ϕ θ̇2 − a3 V (ϕ, θ) 8 (7.1) ! ! r p p 3 3 2 2 ϕ coth cCR ϕ [C4 sinh(CR θ) + C5 cosh(CR θ)] cCR , (7.2) 8 8 u(a, ϕ, θ)=a v(a, ϕ, θ)=a cu a " cva " sinh r sinh r w(a, ϕ, θ)=Cw a3/2 cosh !# 2cua " 3 ϕ 8 !# 2cva " 3 ϕ 8 r coth r coth r 3 3 !# 3 ϕ 8 !# 3 ϕ 8 cu θ cC 2 R cv θ cC 2 R [C4 sinh(CR θ) + C5 cosh(CR θ)] [C4 sinh(CR θ) + C5 cosh(CR θ)] ! 3 ϕ sinh(CR θ) , 8 cu θ cC 2 R cv θ cC 2 R (7.3) where we have used (4.10), (4.21), q (4.22), (4.25), (4.27) and (4.40). Also, for convenience 1 we have denoted Cw ≡ − ACR C5 cΦβ 0 and have taken C̃θ = 0 in (4.40). Finally, note that, similarly to sections 5 and 6, we have obtained the expression for Cw here by using (3.26), the A lines for Φ1 (ϕ) in (3.33)–(3.34), (4.27), (4.31) and (4.40). 7.1 Lagrangian in the new variables In the hyperbolic annuli case, it is convenient to choose the constants, defining the coordinate transformation (a, ϕ, θ) → (u, v, w), as follows: cuθ = 0 , cua = 3 2 and 2 cvθ = c CR , cva = 3 2 Substituting (7.4) in (7.3), we find: ! r 3 3/2 ϕ , u = a sinh 8 ! r 3 3/2 v = a cosh ϕ [C4 sinh(CR θ) + C5 cosh(CR θ)] , 8 ! r 3 3/2 w = Cw a cosh ϕ sinh(CR θ) . 8 . (7.4) (7.5) Note that, to have independent functions for v and w, we need C5 6= 0 in (7.5). However, we have already assumed that by choosing to use inside (7.3) the form of the Θw (θ) solution, given by (4.40). – 33 – JHEP04(2019)148 according to (3.64) and (3.66). In addition, the new variables u, v and w, with w being the cyclic coordinate, have the form: Using (7.6) in (7.1) and (7.2), we obtain the following action:   8 C4 4 2 4 1 2 4 1 vm C42 2 L = u̇ − v̇ + ẇ + v̇ ẇ − V 1 − , 0 3 3 C52 3 Cw2 3 C52 Cw um−2 C52 (7.6) (7.7) where for convenience we have denoted m= p 2 . cCR (7.8) Note that for C4 = ±C5 the ẇ2 term in (7.7) drops out, whereas for C4 = 0 the mixed v̇ ẇ term vanishes. Finally, recall also that the Lagrangian (7.7) is subject to the Hamiltonian constraint EL = 0. Due to its non-linearity, this constraint again will be of practical use only for fixing an integration constant of the solutions of the Euler-Lagrange equations. 7.2 Solutions Let us now look for solutions of the equations of motion of (7.7). In order to keep the ẇ2 term in the Lagrangian, we will assume that C42 6= C52 .12 Then the w-equation immediately gives: C4 Cw ẅ = − 2 v̈ . (7.9) C5 − C42 The solution of the latter is w(t) = − C4 Cw v(t) + Σ̂0 t + C0w , (C52 − C42 ) (7.10) C2C2 where C0w = const. and Σ̂0 = 83 (C 25−Cw2 ) Σ0 with Σ0 being the Noether symmetry con5 4 stant of motion. Substituting (7.7) and (7.10) in the general expression (5.4), we obtain the Hamiltonian: EL = v̇ 2 3 C52 Cw2 4 2 4 vm u̇ − + + V Σ2 . 0 m−2 3 3 (C52 − C42 ) u 16 (C52 − C42 ) 0 (7.11) Using (7.9), we find that the u and v Euler-Lagrange equations acquire the form: v m−1 3 v̈ − V0 Ĉ0 m m−2 = 0 , 8 u vm 3 ü − V0 (m − 2) m−1 = 0 , 8 u 12 We will comment on the degenerate C4 = ±C5 case in an appropriate place below. – 34 – (7.12) JHEP04(2019)148 The inverse of the transformation (7.5) is: )1/3 # ("   1 C4 2 v 1 , − 2 w 2 − u2 − a= Cw C52 w Cw  " #1/2  r  2 v w 1 1 C4 2  , ϕ=2 − 2 arccoth − 2 3 u C5 w Cw Cw    C4 Cw v 1 . arccoth − θ= CR C5 w Cw where we have denoted Ĉ0 ≡ C52 − C42 . One can easily notice that the system (7.12) becomes exactly the same as (5.16) under the simultaneous formal substitutions m → −m and V0 → −V0 . However, we would like to keep V0 > 0, in order to have a positive-definite scalar potential. So we will view (7.12) as a different system, albeit quite similar to (5.16). Clearly, the special choices of m, that simplify significantly (7.12), are m = 0, 1, 2. Let us consider them first, before turning to the generic case with m 6= 0, 1, 2 . 7.2.1 Special cases: m = 0, 1, 2 As in section 5, we begin with the simplest cases, namely m = 0 and m = 2. v̈ = 0 , 3 ü + V0 u = 0 . 4 (7.13) v(t) = C1v t + C2v ,   p   p 1 1 3V0 t + C2u cos 3V0 t . u(t) = C1u sin 2 2 (7.14) Hence, the solutions are: Substituting (7.14) in (7.11) gives:   3 C52 Cw2 4 (C1v )2 + Σ2 . EL = (C1u )2 + (C2u )2 V0 − 3 (C52 − C42 ) 16 (C52 − C42 ) 0 Hence, to impose the constraint EL = 0, we can take for instance:   3 9 (C1v )2 = (C52 − C42 ) (C1u )2 + (C2u )2 V0 + C52 Cw2 Σ20 . 4 64 (7.15) (7.16) Note that these m = 0 solutions can have either w = const. × v (single field limit) or w 6= const. × v (genuine two-field case), depending on how the integration constants are chosen. In appendix B.3 we illustrate such genuine two-field solutions for certain values of the constants. • m = 2 case. In this case, (7.12) gives: 3 v̈ − V0 Ĉ0 v = 0 , 4 ü = 0 . (7.17) Obviously, then, the solution for u(t) is: u(t) = C1u t + C2u . (7.18) However, unlike C0 in section 5, Ĉ0 here can have either sign. So we have the following two cases for v(t):   q   q 1 1 3V0 Ĉ0 t + C2v cosh 3V0 Ĉ0 t for Ĉ0 > 0 , (7.19) v(t) = C1v sinh 2 2   q   q 1 1 v v 3V0 |Ĉ0 | t + C2 cos 3V0 |Ĉ0 | t for Ĉ0 < 0 . (7.20) v(t) = C1 sin 2 2 – 35 – JHEP04(2019)148 • m = 0 case. From (7.12), we now have: Using (7.18), together with either (7.19) or (7.20), inside (7.11) gives:   3 C52 Cw2 4 Σ2 . EL = (C2v )2 − (C1v )2 V0 + (C1u )2 + 3 16 (C52 − C42 ) 0 Clearly, we can ensure that EL = 0 by fixing suitably one of the integration constants in the last expression. • m = 1 case. Now we obtain from (7.12): (7.21) The system (7.21) can be reduced to a single ODE by expressing u from the first equation, namely: 3 v̈ with Q ≡ V0 , (7.22) u= 8 QĈ0 and substituting this result in the second equation. One then finds: v (4) + Q2 Ĉ0 v = 0 . (7.23) Depending on the sign of Ĉ0 , equation (7.23) has the following solutions:  For Ĉ0 > 0 we have: v(t) = C1v cosh(ωt) cos(ωt) + C2v sinh(ωt) cos(ωt) +C3v cosh(ωt) sin(ωt) + C4v sinh(ωt) sin(ωt) , (7.24) q 1/2 1 where ω = Substituting (7.24), as well as the resulting u(t) 2 QĈ0 . from (7.22), into (7.11) gives: EL =  (C1v C4v − C2v C3v ) 3 C52 Cw2 p V + Σ2 . 0 16 (C52 − C42 ) 0 C52 − C42 (7.25) For Ĉ0 < 0, the solution of (7.23) is: v(t) = C1v sin(ω̂t) + C2v cos(ω̂t) + C3v sinh(ω̂t) + C4v cosh(ω̂t) , q where ω̂ = Q |Ĉ0 |1/2 . Now (7.11) becomes:  (C1v )2 + (C2v )2 + (C3v )2 − (C4v )2 C52 Cw2 3 2 p EL = V + 0 2 − C 2 ) Σ0 . 2 2 16 (C C4 − C5 5 4  (7.26) (7.27) Clearly, one can ensure that both (7.25) and (7.27) satisfy the Hamiltonian constraint EL = 0 by fixing appropriately an integration constant. – 36 – JHEP04(2019)148 3 v̈ − V0 Ĉ0 u = 0 , 8 3 ü + V0 v = 0 . 8 Remark on Ĉ0 = 0. As noted in the beginning of section 7.2, when Ĉ0 ≡ C52 − C42 = 0, one cannot use equation (7.9). Instead, the w equation of motion gives v̈ = 0, with the solution v(t) = Ĉ1 t + Ĉ2 (7.28) for any m. Then, the remaining two equations of motion acquire the form: • m = 0: in this case, the solutions of (7.29) are:  p   p  1 1 u = Ĉ3 sin 3V0 t + Ĉ4 cos 3V0 t 2 2 w = Ĉ5 t + Ĉ6 . Evaluating the Hamiltonian on these solutions gives:   3 EL = Ĉ32 + Ĉ42 V0 − Cw2 Σ20 + Ĉ5 Σ0 , 16 where we have used that Ĉ1 = stant of motion. 2 3 C5 C w 8 C 4 Σ0 (7.29) (7.30) (7.31) with Σ0 being the Noether symmetry con- • m = 2: now (7.29) has the following solutions: u(t) = Ĉ3 t + Ĉ4   1 3 Cw C52 1 3 2 Ĉ1 t + Ĉ2 t + Ĉ5 t + Ĉ6 . w(t) = − V0 4 C4 6 2 (7.32) 3 4 EL = Ĉ22 V0 + Ĉ32 − Cw2 Σ20 + Ĉ5 Σ0 , 3 16 (7.33) Thus, the Hamiltonian becomes: where again we have used Ĉ1 = 2 3 C 5 Cw 8 C4 Σ0 . • m = 1: in this case, the solutions of (7.29) are given by: 1 3 V0 Ĉ1 t3 − V0 Ĉ2 t2 + Ĉ3 t + Ĉ4 (7.34) 16 16   1 8 3 V0 Cw C52 1 V0 Ĉ1 t5 + V0 Ĉ2 t4 − Ĉ3 t3 − 8Ĉ4 t2 + Ĉ5 t + Ĉ6 . w= 128 C4 20 4 3 u=− Hence the Hamiltonian gives: 4 3 EL = Ĉ2 Ĉ4 V0 + Ĉ32 − Cw2 Σ20 + Ĉ5 Σ0 , 3 16 where we have substituted Ĉ1 = (7.35) 2 3 C5 Cw 8 C4 Σ0 . Obviously, in all three special cases with Ĉ0 = 0 one can satisfy the Hamiltonian constraint EL = 0 by fixing suitably one of the integration constants. – 37 – JHEP04(2019)148 Cw C52 v m−1 3 =0 . ẅ + V0 m 8 C4 um−2 3 vm ü − V0 (m − 2) m−1 = 0 . 8 u So we have the following u(t) and w(t) solutions: 7.2.2 Generic case: m 6= 0, 1, 2 This case is similar to that considered in subsection 5.2.2. More precisely, for arbitrary m it is not possible to find the exact solutions of equations (7.12) in full generality. However, just as in section 5.2.2, we will be able to find particular classes of exact solutions. Unlike there though, here we will find solutions for any m 6= 0, 2. To begin, let us observe that, together, the two equations in (7.12) imply the relation: Ĉ0 m ü u = (m − 2) v̈ v . (7.36) To obtain real and nontrivial solutions from this Ansatz, we need that other words: Ĉ0 > 0 and m ∈ (−∞, 0) ∪ (2, ∞) Ĉ0 m (m−2) > 0, or in (7.38) or Ĉ0 < 0 and m ∈ (0, 2) . (7.39) Substituting (7.37) in either equation of (7.12), one finds: !m 3 Ĉ0 m 2 m ü − V0 (±1) (m − 2) u = 0 . 8 m−2 (7.40) The solutions of (7.40) depend on the sign of the combination (±1)m (m − 2), namely: u(t) = C1u sinh(ω̂t) + C2u cosh(ω̂t) for (±1)m (m − 2) > 0 (7.41) (±1)m (m − 2) < 0 , (7.42) and u(t) = C1u sin(ω̂t) + C2u cos(ω̂t) where for v u 1u 3 ω̂ = t V0 | (±1)m (m − 2)| 2 2 Ĉ0 m m−2 !m 2 . (7.43) Note that the (±1) in (7.40) is correlated with the ± sign in (7.37). Let us now impose the Hamiltonian constraint. Substituting (7.37) and (7.41) into (7.11) gives: !m 2  u 2  3 C52 Cw2 Ĉ m 0 Σ2 , (7.44) (C2 ) − (C1u )2 + EL = V0 (±1)m m−2 16 (C52 − C42 ) 0 whereas using (7.37) and (7.42) in (7.11) leads to: !m 2  u 2  Ĉ m 3 C52 Cw2 0 Σ2 . EL = V0 (±1)m (C2 ) + (C1u )2 + m−2 16 (C52 − C42 ) 0 – 38 – (7.45) JHEP04(2019)148 One can take (7.36) and one of (7.12) as the two independent equations. An Ansatz that solves (7.36) is given by: s Ĉ0 m v = ± u . (7.37) (m − 2) 1) Ĉ0 > 0 : m odd and m > 2, together with the minus sign in (7.37). 2) Ĉ0 < 0 : 0 < m < 2 and a plus sign in (7.37). As an example of the above considerations, let us write down the particular solutions for m = 3 and Ĉ0 > 0. In that case, the + sign in (7.37) leads to the solution q v(t) = 3Ĉ0 u(t) u(t) = C1u sinh(ω̂t) + C2u cosh(ω̂t) , (7.46) while the − sign gives: q v(t) = − 3Ĉ0 u(t) u(t) = C1u sin(ω̂t) + C2u cos(ω̂t) , (7.47) q 1/2 3/2 where ω̂ = 32 3 2 V0 Ĉ0 . As a final remark note that, among the particular solutions above, there are solutions with m = 1, obtained for Ĉ0 < 0 as can be seen from (7.39). However, we already found the most general solution with m = 1 and Ĉ0 < 0 in subsection 7.2.1, namely (7.26) together with (7.22). Hence, the latter must contain as special cases the particular m = 1 solutions coming from (7.41) and (7.42), together with (7.37). One can verify that this is indeed v or the pair C v of integration constants the case, upon setting to zero either the pair C1,2 3,4 in (7.26). 8 Summary and discussion We studied two-field cosmological α-attractors whose scalar manifold is any elementary hyperbolic surface. We imposed the requirement that these models have a Noether symmetry and found those solutions of the symmetry conditions which follow from a separation-ofvariables Ansatz. In particular, we showed that such separated Noether symmetries exist only for a certain value of the parameter α. To prove these results, we rewrote the cosmologically relevant Lagrangian in canonical form, i.e. as L(q i , q̇ i ) in terms of generalized – 39 – JHEP04(2019)148 Clearly, there is no problem to satisfy EL = 0 for the expression in (7.44), by choosing suitably C1u or C2u . On the other hand, for (7.45) a more careful discussion is needed. Unlike in sections 5 and 6, now the Σ20 term can have either sign. Let us consider first Ĉ0 ≡ C52 − C42 > 0. In that case, either m < 0 or m > 2; see (7.38). Only m > 2, however, can ensure EL = 0. The reason is that, since the Σ20 term in (7.45) is positive, we need the V0 term to be negative, which can only be achieved for (±1)m = −1. Then, the condition (±1)m (m − 2) < 0 in (7.42) implies that m > 2. Now let us consider Ĉ0 < 0, in which case 0 < m < 2 according to (7.39). Hence the condition (±1)m (m − 2) < 0 implies that (±1)m = +1, which is exactly what is needed to have a positive V0 term in (7.45), when the Σ20 term is negative. To summarize, the Hamiltonian constraint allows solutions with u(t) as in (7.42) only in the following two parts of the parameter space: coordinates {q i } = (a, ϕ, θ), where a(t) is the metric scale factor and ϕ(t), θ(t) are the two scalar fields. A generic Noether symmetry generator has the form (3.6), where λa,ϕ,θ (a, ϕ, θ) are functions on configuration space such that (3.5) is satisfied. With the separation of variables Ansatz, we found that the functions λa,ϕ,θ have the following form for the elementary hyperbolic surfaces: • Poincaré disk (D): λa = 3 Akb2 2 (C1 sin θ + C2 cos θ) sinh a1/2 (C1 sin θ + C2 cos θ) cosh 3 λϕ = − Akb2 2r a3/2 3 3 (C1 cos θ − C2 sin θ) λθ = − Akb2 q  4 2 3 a3/2 sinh 8ϕ q q 3 8 3 8 ϕ ϕ   , (8.1) where A, k, b2 , C1 , C2 are constants. Notice that this is effectively a two-parameter family of Noether symmetries, since three of the five parameters occur only in the combination Akb2 and the latter appears only as an overall multiplier, which thus can be factored out of the symmetry condition (3.7). • Hyperbolic punctured disk (D∗ ): λa = 1 2 r 3 Akb1 2  q  (C3 θ + θ0 ) exp − 38 ϕ a1/2  q  (C3 θ + θ0 ) exp − 38 ϕ 3 Akb1 2 a3/2q   r 3 ϕ exp 8 3 3 λθ = − Akb1 C3 3/2 4 2 a λϕ = , (8.2) where A, k, b1 , C3 , θ0 are constants. The same comment as below equations (8.1) applies, namely (8.2) effectively gives a two-parameter family of symmetries. • Hyperbolic Annulus (A): λa = 1 2 r 3 Akb1 2 [C4 cosh(CR θ) + C5 sinh(CR θ)] cosh a1/2 [C4 cosh(CR θ) + C5 sinh(CR θ)] sinh 3 λϕ = − Akb1 2r a3/2 3 3 Akb1 [C4 sinh(CR θ) + C5 cosh(CR θ)] λθ = − q  4 2 CR 3 a3/2 cosh ϕ q q , 3 8 3 8 ϕ ϕ   (8.3) 8 where A, k, b1 , C4 , C5 are constants. Again (8.3) is a two-parameter family of Noether symmetries. – 40 – JHEP04(2019)148 1 2 r In (8.1)–(8.3), we have collected the results of (3.9), (3.35), (3.34), (3.33), (3.26), (3.42)– (3.44), (3.46), (3.47), (3.49) and (3.50). Note that, clearly, one can absorb the overall Akb1,2 factors in (8.1)–(8.3) inside the arbitrary constants θ0 and Ci , i = 1, . . . , 5; we have kept them explicit to facilitate tracing how the above results are obtained throughout section 3. The requirement for the existence of a Noether symmetry restricts the form of the scalar potential. We showed that, to be compatible with the symmetries (8.1)–(8.3), the Lagrangian (3.4) has to have the following form: a3 ϕ̇2 4 3 ˜2 + a f (ϕ) θ̇2 − a3 V (ϕ, θ) , 2 3 (8.4) where ! (8.5) ! ! r 3 3 m VD = V0 cosh ϕ coth ϕ [C1 cos θ − C2 sin θ]−m , 8 8 ! r    q 3 3 m θ0 θ ϕ 2 ∗ VD = V0 exp − , ϕ+ e + exp m θ 2 2 2 C3 ! ! r r 3 3 VA = V0 sinh2 ϕ cothm ϕ [C4 sinh(CR θ) + C5 cosh(CR θ)]m 8 8 (8.6) f˜D = sinh r 3 ϕ 8 ! , f˜D∗ = exp − r 3 ϕ 8 ! , f˜A = CR cosh r 3 ϕ 8 and 2 r with V0 and m being arbitrary constants. Note that the scalar potentials (8.6) depend, in each of the three cases, only on those two parameters of the corresponding Noether symmetries (8.1)–(8.3), which are essential, as should be the case. Furthermore, we simplified the Euler-Lagrange equations of (8.4), for each of the three elementary hyperbolic surface cases, by transforming to generalized coordinates adapted to the corresponding Noether symmetry. This enabled us to find many exact solutions. For some values of the parameter m (the special cases in sections 5, 6 and 7), we found the most general solutions of the equations of motion. For the rest of the m-parameter space (the generic m cases), we found classes of particular solutions.13 An obvious open direction to pursue further is to investigate the physical consequences of the solutions which we have found. More precisely, one should explore what kinds of Hubble parameter, as a function of time, these solutions give. Furthermore, what parts of the parameter space lead to inflationary expansion and/or to actual attractor behavior of the solutions. It would also be interesting to understand how the results of [8] on obtaining natural inflation in two-field α-attractors relate to our considerations. We already pointed out above that the relevant θ-dependent part of the scalar potential can be obtained as a special case of VD in (8.6). Hence, it is worth exploring whether one can find a realization of hypernatural inflation which is compatible with the Noether symmetry investigated 13 For the punctured disk case, we considered only θ0 = 0 for simplicity. – 41 – JHEP04(2019)148 L = −3aȧ2 + Acknowledgments L.A. would like to thank the Stony Brook Simons Workshop in Mathematics and Physics and the Mainz Institute for Theoretical Physics for hospitality during the completion of 14 Often a useful first step in that direction is the successful embedding in supergravity. In that regard, the recent work [34], on dS constructions in multi-field no-scale supergravity models, may be of great relevance. – 42 – JHEP04(2019)148 here. In a similar vein, it is very interesting to understand whether considerations on primordial non-Gaussianity (along the lines of [32]) or dark energy (along the lines of [33]) in α-attractor models can be compatible with our Noether symmetry. It is also worth investigating possible embeddings into suitable classes of string compactifications. 14 In this context, the special value of the α-parameter required by our separated Noether symmetry might play an important role. It could be related to a point of enhanced symmetry in some larger parameter space. Or it could be a manifestation of a moduli stabilization mechanism, if the α-parameter becomes a modulus in the underlying compactification. Another important problem (on which we plan to report in the near future) is to find more general solutions to the Noether symmetry conditions that do not rely on the separation of variables Ansatz. Indications are that such solutions have an elegant mathematical theory, though only a subclass of them restricts the value of the α-parameter (equivalently, the Gaussian curvature) of the scalar manifold. It would be interesting to compare the solutions of the equations of motion in the presence of such more general symmetries to the solutions of the field equations that we obtained here. This might uncover some characteristic features of cosmological behavior which arise in the presence of separated Noether symmetries when compared to more general symmetries. A different line of investigation is to extend the study of Noether symmetries to twofield models defined on arbitrary hyperbolic surfaces and to general multifield models and to explore their description in the Hamiltonian approach. A proper formulation of this problem requires the geometric approach to Noether symmetries provided by the jet bundle formalism. While the Noether approach requires the Lagrangian formulation discussed in the present paper, we should mention that classical cosmological dynamics can also be studied using the formulation used in [14–17], which is obtained by solving the Friedmann equation in order to eliminate the cosmological scale factor a(t). As explained in those references, this leads to a geometric system of non-linear second order ODEs which involves only the scalar fields φI . In fact, the Friedmann equation provides an energy shell constraint which must be imposed on the Lagrangian system described by (3.4) in order to isolate those solutions of the Euler-Lagrange equations which are of actual cosmological relevance. Due to the non-holonomic character of this equation, the resulting geometric system of ODEs for φI does not generally admit a non-constrained Lagrangian formulation. This system of ODEs defines a dissipative geometric dynamical system on the tangent bundle of the scalar manifold, which can be studied with the methods of dynamical systems theory [36]. In particular, symmetries of the cosmological model could be studied directly at this level using Lie’s theory of symmetries of systems of ODEs, which in this setting has an elegant geometric formulation. We hope to address this topic in the future. this work. L.A. has received partial support from the Bulgarian NSF grant DN 08/3 and the bilateral grant STC/Bulgaria-France 01/6. E.M.B. has been supported mainly by the Romanian Ministry of Research and Innovation, grant PN 18 09 01 01/2018. The work of L.A. and E.M.B has also been partially supported by the ICTP - SEENET-MTP project NT-03 Cosmology - Classical and Quantum Challenges. The work of C. I. L. was supported by grant IBS-R003-D1. A Elementary hyperbolic surfaces • Poincaré disk. The Poincaré disk D is the open subset of the complex plane C defined by the condition |z| < 1 , (A.1) endowed with the complete hyperbolic metric: ds2 = 4 dzdz̄ . (1 − z z̄)2 (A.2) For various reasons, some going as far back as [35], in the literature on cosmological α-attractors this metric appears in the scalar kinetic terms with a different overall constant factor. One can transform (A.2) to polar coordinates ρ and θ, determined via z ≡ ρeiθ with ρ ∈ [0, 1), and then, by changing suitably the radial variable, to semigeodesic coordinates (see [15]). This is what is achieved with the redefinition (2.5) that maps the action (2.3) into the form (2.2). • Hyperbolic punctured disk. The hyperbolic punctured disk D∗ is the open subset of C defined by 0 < |z| < 1 , (A.3) 15 The hyperbolic plane does not have a boundary in the sense of manifold theory. However, one can define a conformal boundary for H (‘a boundary at infinity’, which is ∂ H = R ∪ {∞}) by using the conformal structure of the hyperbolic metric, in the same vein as for the Penrose conformal boundary in general relativity. See [14] and references therein for details and generalization. – 43 – JHEP04(2019)148 Any smooth and complete hyperbolic surface is isometric to a quotient of the hyperbolic plane H (the open upper half plane of the complex plane endowed with the Poincaré metric) by a discrete subgroup of its group of isometries PSL(2, R). Elements of PSL(2, R) are classified according to their fixed points. Elliptic elements have a single fixed point located in H, parabolic elements have a fixed point on the conformal boundary15 ∂ H of H and hyperbolic elements have two distinct fixed points on ∂ H. A complete hyperbolic surface is called elementary if it is isometric with H or with a quotient of H by a cyclic subgroup of PSL(2, R) (i.e. a group generated by a single element), which is of parabolic or of hyperbolic type. There are three types of elementary hyperbolic surfaces: the hyperbolic disk D (also called the Poincaré disk, since it is isometric with H), the hyperbolic punctured disk D∗ and the hyperbolic annuli A(R). The hyperbolic disk and hyperbolic punctured disk are unique up to isometry, while the isometry class of a hyperbolic annulus depends on a real modulus R > 1. We briefly discuss these hyperbolic surfaces in turn, referring the reader to [15] for more detail: endowed with the complete hyperbolic metric: ds2 =
1 2 2 2 dρ + ρ dθ , (ρ ln ρ)2 (A.4) where ρ = |z| and θ = arg(z) are polar coordinates on the complex plane. As explained in [15], one can transform this metric to semi-geodesic coordinates, i.e. to the form ds2 = dϕ2 + f (ϕ)dθ2 using a certain change of variables ϕ = ϕ(ρ). This is what the transformation (2.8) amounts to. 1 R̂ < |z| < R̂ where R̂ > 1 , (A.5) endowed with the complete hyperbolic metric (in polar coordinates): ds2 = 2 CR [ρ cos (CR lnρ)] 2 dρ2 + ρ2 dθ2
where CR ≡ π 2 ln R̂ . (A.6) The transformation (2.11), modulo an overall numerical factor, maps the metric (A.6) to the form ds2 = dϕ2 + f (ϕ)dθ2 , where ϕ ∈ (−∞, ∞). Note that ϕ < 0 corresponds to 1 < ρ < 1, while ϕ > 0 corresponds to 1 < ρ < R̂. R̂ We refer the reader to [15] for more detail on the geometry of elementary hyperbolic surfaces. B Nontrivial trajectories for m = 0 In this appendix we illustrate some of the exact solutions we have obtained in sections 5, 6 and 7. A comprehensive investigation of the phenomenological implications of all new solutions, in their entire parameter spaces, is a rather laborious effort that we leave for the future. Nevertheless, here we will illustrate, in a certain corner of parameter space, the existence of nontrivial two-field trajectories among our solutions for m = 0, in each of the three elementary hyperbolic surface cases.16 We will also consider the behavior of the Hubble parameters in the three cases, for the relevant parts of parameter space. B.1 Poincaré disk In section 5 we pointed out that, for the Poincare disk case, the single field limit is obtained when m = 0 and w = const. × v. Indeed, for m = 0 the scalar potential becomes: ! r 3 ϕ , (B.1) V (ϕ, θ) = V0 cosh2 8 16 The possibility of having nontrivial multi-field trajectories, even for a potential without angular dependence, was already shown in [10, 11]. – 44 – JHEP04(2019)148 • Hyperbolic annuli. The hyperbolic annulus A(R̂) is the open domain in the complex plane defined through: as can be seen from (8.6), while w = const. × v implies θ = const., as is evident from (5.7). However, by choosing suitably the integration constants in (5.18) and (5.21), one can have w 6= conts × v even for m = 0. Thus, one can obtain nontrivial (ϕ, θ) trajectories, even though the potential has no angular dependence. We will illustrate these trajectories in a certain part of parameter space. To underline their dependence on the parameters, we will explore how the trajectories change as we vary two of the integration constants, namely C1u and C2u , while keeping the rest fixed. Let us make the following convenient choices: 1 C1 = √ , C2 = 0 , Cw = 1 , V0 = 3 , Σ0 = 2 , C0w = 1 , C2v = 0 . 3 (B.2) Recall that the constant C1v is determined from the Hamiltonian constraint (5.23). To be able to solve the letter, one needs C2u 6= 0 and even |C2u | > |C1u |. We also have to take |C1u | > 1 for the choices in (B.2), in order to ensure a real and positive scale factor a(t) for  1 any t ≥ 0 . This can be understood by noting that a(t)|t=0 = (C1u )2 − (C0w )2 − 3(C2v )2 3 . Thus, if (C1u )2 − (C0w )2 − 3(C2v )2 < 0 , then a(t) becomes complex in a neighborhood of t = 0. So, to recapitulate, we need to take: 1 < |C1u | < |C2u | . (B.3) Now we are ready to investigate numerically the m = 0 solutions, obtained from substituting (5.18) and (5.21), together with (5.15) and (B.2), into (5.7). On figure 1 we u . On the left C u = const., while C u have plotted the scalar ϕ(t) for different choices of C1,2 1 2 varies. In this case, the initial value of ϕ at t = 0 stays the same, although the shape of the function ϕ(t) changes. In particular, increasing C2u increases ϕ. On the right of figure 1, C2u = const. and C1u varies. Clearly, now the initial value of ϕ also changes. However, – 45 – JHEP04(2019)148 u Figure 1. Plots of ϕ(t) for different values of the constants C1,2 . On the left , C1u = 2 and C2u u u takes the following values: C2 = 5 (solid line), C2 = 4 (dashed line) and C2u = 3 (dotted line). On the right , C2u = 5 and C1u takes the following values: C1u = 2 (solid line), C1u = 3 (dash-dotted line) and C1u = 4 (space-dotted line). Note that the solid lines on the left and right sides are the same curve. increasing C1u decreases ϕ. In all of the cases on figure 1, ϕ starts at a finite value at t = 0 and ϕ → 0 as t → ∞. Note that ϕ = 0 is precisely the minimum of the potential (B.1).
On figure 2 we have plotted the trajectories ϕ(t), θ(t) obtained for the same values of u as in figure 1. At t = 0 these trajectories start at θ = π , while as t → ∞ the constants C1,2 2 they tend to ϕ = 0. In fact, it is more illuminating to plot them in polar coordinates. For easier comparison with the punctured disk and annuli cases, on figure 3 we plot these trajectories in terms of the canonical radial variable of the Poincaré disk ρ ∈ [0, 1) , which is related to ϕ via (2.5).17 Clearly, when C1u = const. and C2u varies, the starting point at t = 0 remains the same, although the shape of the trajectory changes. When C2u = const. and C1u varies, the starting point changes as well. In both cases, though, the trajectories start at t = 0 at a finite ρ and as t → ∞ they tend to ρ = 0 , or equivalently ϕ = 0 , which is the minimum of the potential (B.1). Finally, on figure 4 we plot the Hubble parameters H(t) = ȧ(t) a(t) for the same trajectories studied above. In all cases, H(t) → 1 as t → ∞. So the spacetimes, corresponding to these solutions, asymptote to dS space. Note that the horizontal axis starts at t = 0.4 only for better visibility of the distinctions between the graphs. In each case, H(0) is finite. For example, H(0) = 3.2 for the solid line, common for the left and right sides. B.2 Hyperbolic punctured disk The m = 0 potential for the hyperbolic punctured disk case is: ! r 3 V (ϕ, θ) = V0 exp − ϕ , 2 (B.4) as one can see from (8.6). To obtain the single-field limit, we also need w = const. × u (implying that θ = const.), as discussed in section 6. However, by appropriately choosing √ 17 Note that for the ranges of ϕ and ρ relevant here, relation (2.5) becomes ρ ≈ 86 ϕ . So in polar (ϕ,θ) coordinates, the trajectories are the same as on figure 3, up to a rescaling of the radial direction. – 46 – JHEP04(2019)148
Figure 2. The trajectories ϕ(t), θ(t) for the same values of the constants as in figure 1. The dot at one end of a trajectory denotes its starting point at t = 0. Figure 4. The Hubble parameters H(t) for the same values of the constants as in figure 1. the integration constants in (6.12) and (6.16), we can have w 6= const. × u although m = 0. So, in this case too, there are nontrivial two-field trajectories, even when the scalar potential does not depend on θ. Before turning to their numerical investigation, it will be useful to write down explicitly the inverse of (2.8). Substituting α = 34 , according to (3.33), gives:  √6  ρ = exp −e 4 ϕ , (B.5) where we have also used that by definition ρ < 1 (see appendix A). We will explore, again, the dependence of the (ϕ, θ) trajectories on the two integration constants characterizing u(t), namely C1∗ and C2∗ , while keeping all the other constants fixed. In the process, a certain complementarity between the two constants in u(t) will become even more apparent. It is convenient to take: Cw = 1 , V0 = 3 , Σ∗ = 2 , C0w = 0 , C4∗ = 1 , – 47 – (B.6) JHEP04(2019)148
Figure 3. The trajectories ρ(t), θ(t) , with ρ being the radial variable on the unit disk, for the same values of the constants as in figure 1. 
Figure 6. The trajectories ϕ(t), θ(t) for the same values of the constants as in figure 5. The dot at one end of a trajectory denotes its starting point at t = 0. while solving the constraint (6.18) for C3∗ . To ensure, with the choices (B.6), that the scale factor a(t) > 0 for every t ≥ 0 and that (6.18) can be solved, we need: C2∗ > 0 and C1∗ C3∗ > 2 . (B.7) Let us now turn to the numerical investigation of the solutions, obtained by substituting (6.12) and (6.16), together with (6.9) and (B.6), into (6.6). On figure 5 we plot ϕ(t); on the left C1∗ = const. and C2∗ varies, while on the right C2∗ = const. and C1∗ varies. In all cases ϕ → ∞ as t → ∞. This is in perfect agreement with the fact that the minimum of the potential (B.4) is achieved for ϕ → ∞. Note that, due to (B.5), ϕ → ∞ corresponds to
ρ → 0. On figure 6 we plot the trajectories ϕ(t), θ(t) for the same values of the constants as in figure 5. On the left, for different choices of C2∗ (with C1∗ fixed) the trajectories start – 48 – JHEP04(2019)148 ∗ Figure 5. Plots of ϕ(t) for different values of C1,2 . On the left , C1∗ = 1 and C2∗ takes the values: 1 ∗ ∗ ∗ C2 = 2 (solid line), C2 = 1 (dashed line) and C2 = 2 (dotted line). On the right , C2∗ = 1 and C1∗ takes the values: C1∗ = 1 (dashed line), C1∗ = 2 (space-dotted line) and C1∗ = 3 (dash-dotted line). Note that the dashed lines on the left and right sides are the same curve. 
Figure 8. The trajectories ρ(t), θ(t) for the same values of the constants as in figure 5. at t = 0 at different values of ϕ, while they all tend to ϕ → ∞ and θ = 5π 8 as t → ∞. On the right, for different values of C1∗ (with C2∗ fixed) all trajectories start at the same point, while for t → ∞ they tend to different values of θ. This is even more clear in polar (ϕ, θ) coordinates; see figure 7. For easier comparison with the disk and annuli cases, on figure 8 we also plot the same trajectories in polar (ρ, θ) coordinates, with ρ ∈ (0, 1) being the canonical radial variable of the hyperbolic punctured disk. Note that at t = 0, the different trajectories start at different ρ, but as t → ∞ they all tend to ρ = 0, which corresponds to the minimum of the scalar potential. Finally, on figure 9 we plot the Hubble parameters corresponding to the trajectories considered above. In all cases, H(t)|t=0 is finite and H(t) → 0 as t → ∞. This is in accordance with the fact that, for large t, the scalar ϕ → ∞ and thus the potential (B.4), i.e. the effective cosmological constant, tends to zero. So the spacetimes of these solutions tend to Minkowski space. This may represent a natural mechanism for relaxation of the – 49 – JHEP04(2019)148 Figure 7. The trajectories of figure 6 in polar (ϕ, θ) coordinates. cosmological constant. Or it may indicate that this class of models has to be considered only in a finite time-range, assuming that at later times a different effective description (for example, containing new fields) would become more appropriate. B.3 Hyperbolic annuli For the hyperbolic annuli case, the m = 0 potential is: ! r 3 2 V (ϕ, θ) = V0 sinh ϕ , 8 (B.8) according to (8.6). From section 7, it is clear that the single-field limit is obtained when, in addition, one has w = const. × v, which implies θ = const.. However, just like in appendices B.1 and B.2, one can have w 6= const. × v even when m = 0, for suitable choices of the integration constants in (7.10) and (7.14).18 So, again, one can have nontrivial (ϕ, θ) trajectories, even though the potential is independent of θ. To study numerically those trajectories, it will be convenient to use the canonical radial variable ρ of the hyperbolic annuli, which is related to ϕ via (2.11). Note that the inverse transformation (with α = 34 substituted) is: " √ !# 2 6 ln ρ = , (B.9) arctan tanh ϕ CR 8 where ϕ ∈ (−∞, ∞), with ϕ < 0 corresponding to ρ < 1 and ϕ > 0 corresponding to ρ > 1. As before, we will study numerically the dependence of the nontrivial two-field trajectories on the integration constants in u(t), i.e. on C1u and C2u , with all other constants fixed. For convenience, let us take the following values: 1 C4 = 0 , C5 = √ , Cw = 30 , V0 = 3 , Σ0 = 2 , C0w = 1 , C2v = 3 , R̂ = 2 , (B.10) 3 18 In this appendix, we will focus on the generic Ĉ0 6= 0 case in section 7. Note, however, that for m = 0, the solutions in the degenerate Ĉ0 = 0 case are of the same form as for Ĉ0 6= 0, as can be seen easily by comparing (7.10) and (7.14) to (7.28) and (7.30), although the m = 1 and m = 2 solutions in the two cases differ significantly. – 50 – JHEP04(2019)148 Figure 9. The Hubble parameters H(t) for the same values of the constants as in figure 5. 
Figure 11. The trajectories ϕ(t), θ(t) for the same values of the constants as in figure 10. with C1v determined from the Hamiltonian constraint (7.16). This, in particular, means that we are considering the annulus given by: 1 <ρ<2. 2 Note that, with the choices (B.10), we need to have: (C2u )2 < 23 , (B.11) (B.12) in order to ensure that a(t) > 0 for any t ≥ 0 . Finally, unlike in appendices B.1 and B.2, the Hamiltonian constraint in this case does not impose any restriction on the choices of C1u and C2u . Now we turn to studying numerically the m = 0 solutions, obtained from substituting (7.10) and (7.14), together with (B.10), into (7.6). On figure 10 we plot ϕ(t); on the – 51 – JHEP04(2019)148 u Figure 10. Plots of ϕ(t) for several values of C1,2 . On the left , C1u = 1 and C2u takes the values: 1 u u C2 = −4 (black line), C2 = − 2 (blue line) and C2u = 2 (magenta line). On the right , C2u = −4 and C1u takes the values: C1u = −3 (red line), C1u = 1 (black line) and C1u = 4 (green line). Note that the black lines on the left and right sides are the same curve. Figure 13. The Hubble parameters H(t) for the same values of the constants as in figure 10. left C1u = const. and C2u changes, while on the right C2u = const. and C1u changes. Note that in all cases ϕ(0) is finite; this is not obvious, because we have started the plots at t = 0.1 in order to make the overall features of the graphs better visible. Also, on the right side ϕ(0) = −1.67 for all three graphs. Notice that in all cases ϕ(t) oscillates around ϕ = 0 with an ever decreasing amplitude. Eventually, as t → ∞, the scalar ϕ(t) settle at ϕ = 0, which is the minimum of the potential (B.8). This is even more clear on figure 11, where
we plot the trajectories ϕ(t), θ(t) for the same values of the constants as in figure 10. The plots on figure 11 start at t = 0.2 , again for better visibility of the features of the graphs at large t. (They end at t = 140.) This obscures the fact that all trajectories on the right side start at the same point. To illustrate clearly the entirety of the trajectories, it is most useful to change variables from ϕ to the radial coordinate ρ ∈ ( 12 , 2) of the hy
perbolic annulus. On figure 12 we plot the trajectories ρ(t), θ(t) for the same values of the constants as in figure 10. We have restricted the plot to the segment with θ ∈ [0, π2 ] , – 52 – JHEP04(2019)148
Figure 12. The trajectories ρ(t), θ(t) for the same values of the constants as in figure 10. The dot at one end of a trajectory denotes its starting point at t = 0. Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. References [1] R. Kallosh, A. Linde and D. Roest, Superconformal inflationary α-attractors, JHEP 11 (2013) 198 [arXiv:1311.0472] [INSPIRE]. [2] R. Kallosh, A. Linde and D. Roest, Large field inflation and double α-attractors, JHEP 08 (2014) 052 [arXiv:1405.3646] [INSPIRE]. [3] R. Kallosh and A. Linde, Universality class in conformal inflation, JCAP 07 (2013) 002 [arXiv:1306.5220] [INSPIRE]. [4] R. Kallosh and A. Linde, Multi-field conformal cosmological attractors, JCAP 12 (2013) 006 [arXiv:1309.2015] [INSPIRE]. [5] R. Kallosh and A. 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It is also interesting to observe that trajectories, which start closer to ρ = 1, reach greater values of θ as t → ∞, although trajectories starting further from ρ = 1 have greater amplitudes early on. Finally, on figure 13 we plot the Hubble parameters for the same trajectories as in figures 10–12. We have restricted the range of t from below only to make the distinctions between the graphs, as well as their features, visible. For each curve, H(0) is finite and H(t) → 0 as t → ∞. This is in agreement with the fact that at late times ϕ settles at ϕ = 0 and so the potential (B.8) vanishes. This conclusion is similar to the one at the end of appendix B.2. However, the present case has the rather peculiar feature that H(t) exhibits a damped oscillations pattern. Thus, this class of models describes a kind of a cascading spacetime evolution. 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