Two-field Cosmological $\alpha$-attractors with Noether Symmetry

We study Noether symmetries in two-field cosmological $\alpha$-attractors, investigating the case when the scalar manifold is an elementary hyperbolic surface. This encompasses and generalizes the case of the Poincare 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 $\alpha$-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.


Introduction
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 n s (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. 4 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 4 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 [7,8,9,10,11,12,13], 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 [14,15,16].
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 [22,23,24,25,26] 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. 5 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 compactifications. 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.
Most of the time, the equations of motion of two-field cosmological models are solved numerically in the literature. See, in particular, [15,16,17] 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 [27,28,29,30]. 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.
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 5 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. manifold. Here we will apply the Noether symmetry method to the two-field generalized α-attractors of [14,15,16]. 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 α-parameter 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 6 . 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.
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 cosmo-logically 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.
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 where R is the scalar curvature of the 4d space-time metric g µν , the fields φ I with I = 1, 2 are two real scalars and the non-linear sigma-model metric G IJ (φ) is a complete hyperbolic metric, i.e. a complete metric of constant negative Gaussian curvature K 7 . For brevity, The simplest example is obtained by taking the scalar manifold to be the Poincaré 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 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 7 It is convenient to write the Gaussian curvature as K = − const α in terms of an arbitrary positive parameter α. (There are differing conventions in the literature, namely: either K = − 1 3α , K = − 2 3α or K = − 1 2α .) It was shown in [14] that such models have universality properties similar to those of [1,2], hence the name 'α-attractors'. 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 [15,16,17]. 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: 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 G IJ , which admits a U(1) isometry parameterized by θ and so, in particular, for any of the elementary hyperbolic surfaces. Namely: • Poincaré disk: When G IJ is the metric on the hyperbolic disk D, the action (2.1) can be written as: in terms of a complex scalar Z = φ 1 + iφ 2 . Writing the latter as: and performing the field redefinition: we find that (2.3) acquires the form (2.2) with the following function f (ϕ): • Hyperbolic punctured disk: For G IJ the metric on the hyperbolic punctured disk D * , the action (2.1) can be written as: (2.7) Hence, the field redefinition: transforms it into (2.2), where now the function f (ϕ) is: • Hyperbolic annulus: When G IJ is the metric on a hyperbolic annulus A, (2.1) acquires the form: where C R ≡ π 2 lnR . This can be transformed to the expression in (2.2) by the redefinition: which leads to the following function f (ϕ): 3 Noether symmetries in two-field α-attractors We now investigate under what conditions the action (2.2), namely: has a Noether symmetry. As usual, we will consider the following Ansatz for the fourdimensional inflationary metric: as well as spatially-homogeneous scalar fields ϕ(x µ ) = ϕ(t) and θ(x µ ) = θ(t). Substituting these in (3.1), we obtain: 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 ,q i ) in terms of some generalized configuration space coordinates q i and the corresponding generalized velocitiesq i . To achieve this, we use integration by parts in theä term in (3.3). This allows us to write the action per unit spatial volume in (3.3) as dt L , with the following Lagrangian density:

The Noether system
Recall that a symmetry generator is a vector field X defined on T M, which preserves the Lagrangian: where L X 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: where the coefficients λ a,ϕ,θ are functions of the configuration space coordinates {a, ϕ, θ}.
Hence, the condition (3.5) becomes: 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 velocitiesȧ, ϕ andθ 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 λ a,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): 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: 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.

Solving equations (E1), (E2) and (E4)
Substituting (3.9) in equation (E1), we obtain the following first order ODE: Its general solution is: where A is an arbitrary integration constant.
Equating the expressions for λ a obtained from (E1) and (E2) in (3.8), we have: Substituting (3.9) in (3.12), we find the following set of equations: 8 where k = const. Using (3.11) in the first equation of (3.13) gives: Let us now consider equation (E4). Substituting (3.9), (3.11) and (3.14) in this equation gives: This, together with the last relation in (3.13), implies that: Using the second equation of (3.13) in (3.16), we end up with the following ODE: whose general solution is: where b 1,2 = const. Using this in (3.13), we find: Note that our results above for A 1 (a), A 2 (a), Φ 1 (ϕ) and Φ 2 (ϕ) are consistent with those of [31], except that b 1 was set to zero in that work.
However, recall that we are only interested in functions f , such that the sigma-model metric in (3.1), namely the metric ds 2 = dϕ 2 + f (ϕ)dθ 2 , 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 as well as an equation for Φ 3 (ϕ) which we will write down shortly. Using (3.11) allows us to solve (3.21) as: where we have set an additive integration constant to zero in order to ensure that dΦ 3 dϕ = 0 and dΘ 2 dθ = 0. 9 Upon using (3.20) and (3.22), equation (E5) reduces to the following algebraic relation: 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: Comparing the last relation with (3.23), we conclude that which implies: where Φ 0 = const. Substituting (3.26) in (3.23), we obtain: (3.27) and thus Clearly, for arbitrary f (ϕ), the expression in (3.27) is not compatible with the Φ 1 (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 ds 2 = dϕ 2 + f (ϕ)dθ 2 is constant and negative. This restricts the form of the function f . To see how, let us compute the Gaussian curvature in question: 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: 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 Φ 1 (ϕ), we have to take 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. 10 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: 10 In [14], the normalization K = − 1 3α was imposed for any hyperbolic surface. In the present work, however, the coefficients of proportionality between K and 1 α 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.
Substituting these three cases for f (ϕ) in relation (3.27) and comparing with (3.19) gives the following conditions for the existence of a solution: while Φ 3 (ϕ) is given by (3.26) in all three cases. Also, for any function f , the solutions for A 1,2,3 (a) are: as can be seen in (3.11), (3.14) and (3.22).

Solving equation (E3)
Next, we consider equation (E3) of the system (3.8). Substituting the solutions for A 1,2,3 (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: Substituting (3.27) in (3.36) gives: It is easy to check that the expression 3f 2 −2f ′2 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: In fact, one can show directly that 3f 2 −2f ′2

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| = 3 8 , we obtain: Let us now study equation in (3.38) for each of the three values of q given in (3.39).

Solving equation (E7): the scalar potential
So far, we have found functions λ a,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: We begin by substituting (3.9) and (3.51) into (E7). Then, using the solutions for A 1,2,3 (a) given in (3.35) as well as the last relation in (3.13) (namely Θ 2 = kΘ 1 ), we find that (E7) reduces to: Note that here we have not used any particular form of the function f . Hence, for any f (ϕ), and thus for any Φ 1,2,3 (ϕ) and Θ 1,2,3 (θ), we have the pair of equations where p = const. Clearly, then, one has two separate equations for the two functionsṼ (ϕ) andV (θ). Let us now study these two equations for each of the three types of elementary hyperbolic surface.
Using (3.42) and (3.44) inside (3.53), we obtain: Therefore, for the case of the hyperbolic disk, the form of the scalar potential, that is compatible with Noether's symmetry, is: where V 0 = 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 C 2 = 0 and p c = −2, we haveV (θ) = const × cos 2 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.
• Hyperbolic punctured disk: Using the D * expressions from (3.34) and (3.33) in (3.53), we have: Now, substituting the solutions for Θ 1,3 from (3.46) and (3.47) inside (3.53), we end The solution of the last equation is: 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).

New variables: cyclic coordinate
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 L X L = 0 will become: 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: which shows that the generalized momentum p w ≡ ∂L ∂ẇ is conserved. To find such coordinates, we must solve the conditions i X du = 0 , i X dv = 0 and i X dw = 1, which amount to the system: 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.

Finding the coordinates u and v
In this subsection we consider the first equation in (4.3), namely As already pointed out, this will enable us to find not only u, but v as well.
We will look for solutions with the separation of variables Ansatz: Using (3.9), (4.5) and the last relation in (3.13) (i.e. Θ 2 = kΘ 1 ), equation (4.4) reduces to: Separating out the θ-dependence gives: and for some c θ = const. Now, substituting A 1,2,3 (a) from (3.35) in (4.8), we find that the a-dependence factors out provided that: for some c a = const. The last equation is solved by: where for convenience we have set the overall multiplicative integration constant to one. 11 Substituting (4.10) and (3.35) in (4.8) gives: This equation has different coefficients for each elementary hyperbolic surface, since the functions Φ 1,2,3 (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 Φ 1,2,3 in terms of the function f (ϕ). This allow us to bring (4.11) to the form: To recapitulate, the solution for A u (a) is independent of f and is given by (4.10). On the other hand, the solutions for Φ u (ϕ) and Θ u (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 Φ u and Θ u for each type of elementary hyperbolic surface.
• 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: This equation has the general solution: where we have again set the overall integration constant to one for convenience. Note that the single-field result for Φ u (ϕ) in (4.14) is obtained by taking c θ = 0. Then, setting c a = 3, we find from (4.14) and (4.10) the same particular solution for u(a, ϕ) = A u (a)Φ u (ϕ), as that in [31]. Now, substituting (3.42) and (3.44) in (4.7), we find: whose solution is: with the overall integration constant once again set to one.
• Hyperbolic punctured disk: Taking f (ϕ) as in (2.9) with α = 4 3 (in accordance with the D * row of (3.33)), equation (4.12) becomes: This ODE is solved by 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: where the overall integration constant was set to one.
• Hyperbolic annulus: For f (ϕ) given by (2.12) with α = 4 3 (as in the A line of (3.33)), equation (4.12) becomes: Hence, the solution in this case is Finally, the solution of (4.7), after substituting (3.49) and (3.50), has the form: Remark on the coordinate v: such that A v , Φ v and Θ v 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 c a and c θ than those taken for the function u.

Finding the cyclic coordinate w
Now we will consider the last equation in (4.3), namely: As usual, we will make the separation of variables Ansatz: 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 Using (4.25) and the relation Θ 2 = kΘ 1 (see (3.13)), we find that (4.23) acquires the form: Now we will show that one can remove the ϕ-dependence in (4.26) by a suitable choice of the function Φ w (ϕ). The result will be an equation for Θ w (θ). Indeed, let us take: Then, obviously, the second term in (4.26) becomes independent of ϕ. In addition, one in the first term, with Φ w given by (4.27), is a constant for each of the three cases in (3.34). In fact, one can see directly that this 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: Now recall relation (3.40), which holds for any f (ϕ) of the form (3.30) with |K| = 3 8 . Using this relation, we find that (4.28) implies: 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 Θ w (θ). Substituting (4.27) and (4.29) into (4.26) gives: Then, using (3.20) and setting in order to simplify the equation, we obtain from (4.30): 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 Θ 1 (θ) is given by (3.42). Therefore, (4.32) becomes: The general solution of the last equation can be written as: Note that, upon redefinition of the integration constantĈ θ , the solution can also be written as: or as: 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 C 1 = 0 and C 2 = 0. On the other hand, the forms (4.34) and (4.35) allow one to take respectively the limits C 2 = 0 and C 1 = 0. Since we will be particularly interested in the limit C 2 = 0, we will use the form (4.34) in what follows (although we will comment more on using (4.36) below).
• Hyperbolic punctured disk: In this case q = 0 (see (3.39)). Also, Θ 1 (θ) has the form (3.46). Substituting these in (4.32), we obtain the ODE: which has the solution: • Hyperbolic annulus: In this case, relation (3.39) gives q = C 2 R . Using this and the relevant Θ 1 (θ) expression (3.49), we find that (4.32) acquires the form: Similarly to the D case above, the general solution of (4.39) can be written in three equivalent ways, namely:

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: We found that (5.1) has a certain Noether symmetry, when the scalar potential is of the form (3.58), namely: 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: where for convenience we have denoted C w ≡ − 1 AC 1 β c Φ 0 and have takenĈ θ = 0 in (4.34). Note that, to obtain this expression for the coefficient C w , one has to take into account (4.31), as well as the relevant coefficient for Φ 1 (ϕ) according to (3.33)- (3.34). Finally, we have labeled the c a and c θ 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 is the energy function corresponding to any point particle Lagrangian L(q i ,q i ) with generalized coordinates q i . It is well-known that the Hamiltonian E L is conserved on any solutions of the Euler-Lagrange equations, i.e. that for such solutions one has E L = const.
So imposing the constraint E L = 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.

Lagrangian in the new variables
Note that here we need C 1 = 0, to ensure that v and w are independent variables.
However, this was already tacitly assumed when using the Θ w (θ) solution (4.34) in (5.3); to allow for C 1 = 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: 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: 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 C 2 = 0, which is exactly the special case relevant for natural inflation as mentioned below (3.58). 13 12 As mentioned earlier, here we use (4.34), since we are interested in encompassing the special case with C 2 = 0. For a discussion of the coordinate transformation and resulting Lagrangian, when using the form of the Θ w solution in (4.36), see footnote 13 below. 13 Note that, if we had used the Θ w solution in (4.36) (still withĈ θ = 0), then the third line of (5.6) Before we begin looking for solutions, let us underline again that (5.8) is subject to the Hamiltonian constraint E L = 0, where as discussed above.

Solutions
It is convenient to introduce the notation: Then the Euler-Lagrange equations of (5.8) are: 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 expressẅ from the first equation, namelÿ , (5.14) would have been modified to w = C w a 3/2 sinh and C 1,2 = 0. Then, the inverse transformation would be: That would lead to the following Lagrangian: Clearly, in this case, the mixedvẇ term would vanish for C 1 = C 2 . and introduce the functionũ (t) ≡ u(t) .
(5.15) Substituting (5.14) and (5.15) in the second and third equations of (5.13) gives: where for convenience we have denoted C 0 ≡ C 2 1 + C 2 2 . 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 Σ 0 , that is due to the Noether symmetry.
Indeed, in general, Σ 0 is given by: 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: 2 ) Σ 0 and C w 0 = const . Hence, using (5.18) and (5.15) in (5.11), we have: As alluded to earlier, the constraint E L = 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 = 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:v with the general solution: where C u 1,2 = const and C v 1,2 = const. Substituting (5.21) in (5.19) with m = 0 gives: Hence, we can enforce the Hamiltonian constraint E L = 0, for example, by taking: 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 = const × v. In Appendix B.1 we illustrate genuine two-field trajectories obtained in the latter case for certain values of the integration constants.
• m = −2 case: In this case, (5.16) acquires the form: v whose general solution is: Substituting (5.25) in (5.19) with m = −2, we find: So, to ensure that E L = 0, we can take for instance: Note that, for m = −2 and C 2 = 0 , our scalar potential is of the kind relevant for natural inflation, namely V ∼ cos 2 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: v Denoting for convenience Recall that C 0 > 0 by definition. So the general solution of (5.32) has the form: 0 . Hence, using (5.30), the solution forũ(t) is: whereC Using (5.33) and (5.34) in (5.19) with m = −1, we obtain: 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: 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:ü Depending on the sign of theũ term 14 , the solutions of (5.38) are: while using (5.37) and (5.40) inside (5.19) gives: To illustrate the above considerations, let us write down, for example, the particular solutions for m = 1: where the "+" corresponds 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 formũ = const ×v. 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) − const 1 × v = 0. Indeed, substituting the same Ansatzũ = const ×v in any of the two equations in (5.16), would lead to an equation of the formv + const 2 × v = 0 . Since in general const 1 = const 2 2 , the two equations for v(t) would be incompatible. One can ensure const 1 = const 2 2 by viewing it as a constraint relating V 0 , C 0 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.

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.

Lagrangian in the new variables
whose inverse transformation has the form: Substituting (6.6) in (6.1)-(6.2) gives: 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 variablev, defined through: Indeed, equation (6.9) implies that v = ev /u . Substituting this in (6.7), we find: Recall also that (6.10) is subject to the Hamiltonian constraint E L = 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 forv 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 onv, after setting m = 0. However, this is exactly the correct dependence, since the Lagrangian does not contain the usual kinetic terms for u andv, but only the mixeduv term. As a result, the u-variation gives the v-equation of motion and vice-versa. This will become apparent shortly.

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) Hence, we immediately have: where C w 0 = 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: As in the previous Section, the constraint E L = 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 andv equations of motion, following from (6.10), are: Note that, due to the unusual kinetic term, the first equation in (6.14) arises from thê v-variation, i.e. from ∂L ∂v − d dt ∂L ∂v = 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.
• Special case: m = 0 In this case, (6.14) acquires the form:ü 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: where C * i 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: To ensure that E L = 0 , we can take for example: Note that the solutions above can have w = 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 = 0 Now let us consider the generic case with m = 0 . 15 Then, one could solve the first equation in (6.14) algebraically forv, obtainingv = u m ln 8 3V 0 mü u . 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 be solved analytically in full generality. Alternatively, one could substitutev = u m ln 8 3V 0 mü u in the constraint E L = 0, in order to 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 e mv/u = 8 3V 0 mü u . Substituting this in the second equation of (6.14), we end up with: Clearly, one can view (6.19) and one of (6.14) as the two independent equations to solve.
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 = 0. It would be very interesting to understand whether there is a deeper reason for this outcome.

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.
In the A case, we have from (3.33) that α = 4 3 . Using this and (2.12), we find that the Lagrangian (3.4) becomes: with potential given by: 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:

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: (7.4) Substituting (7.4) in (7.3), we find: Note that, to have independent functions for v and w, we need C 5 = 0 in (7.5). However, we have already assumed that by choosing to use inside (7.3) the form of the Θ w (θ) solution, given by (4.40).
The inverse of the transformation (7.5) is: (7.6) Using (7.6) in (7.1) and (7.2), we obtain the following action: where for convenience we have denoted (7.8) Note that for C 4 = ±C 5 theẇ 2 term in (7.7) drops out, whereas for C 4 = 0 the mixedvẇ term vanishes. Finally, recall also that the Lagrangian (7.7) is subject to the Hamiltonian constraint E L = 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.

Solutions
Let us now look for solutions of the equations of motion of (7.7). In order to keep theẇ 2 term in the Lagrangian, we will assume that C 2 4 = C 2 5 . 16 Then the w-equation immediately gives:ẅ The solution of the latter is 4 ) Σ 0 with Σ 0 being the Noether symmetry constant of motion. Substituting (7.7) and (7.10) in the general expression (5.4), we obtain the Hamiltonian: (7.11) 16 We will comment on the degenerate C 4 = ±C 5 case in an appropriate place below. Using (7.9), we find that the u and v Euler-Lagrange equations acquire the form: where we have denotedĈ 0 ≡ C 2 5 − C 2 4 . One can easily notice that the system (7.12) becomes exactly the same as (5.16) under the simultaneous formal substitutions m → −m and V 0 → −V 0 . However, we would like to keep V 0 > 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 = 0, 1, 2 . Hence, the solutions are: Substituting (7.14) in (7.11) gives: Hence, to impose the constraint E L = 0, we can take for instance: Note that these m = 0 solutions can have either w = const × v (single field limit) or w = const × v (genuine two-field case), depending on how the integration constants are 17 Note that, for all m = 0 solutions below, the same kind of remark applies as in footnote 15. 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:v Obviously, then, the solution for u(t) is: However, unlike C 0 in Section 5,Ĉ 0 here can have either sign. So we have the following two cases for v(t): Using (7.18), together with either (7.19) or (7.20), inside (7.11) gives: Clearly, we can ensure that E L = 0 by fixing suitably one of the integration constants in the last expression.
Remark onĈ 0 = 0 : As noted in the beginning of Section 7.2, whenĈ 0 ≡ C 2 5 −C 2 4 = 0, one cannot use equation (7.9). Instead, the w equation of motion givesv = 0, with the solution v(t) =Ĉ 1 t +Ĉ 2 (7.28) for any m. Then, the remaining two equations of motion acquire the form: So we have the following u(t) and w(t) solutions: • m = 0: In this case, the solutions of (7.29) are: Evaluating the Hamiltonian on these solutions gives: where we have used thatĈ 1 = 3 8 C 2 5 Cw C 4 Σ 0 with Σ 0 being the Noether symmetry constant of motion.
• m = 2: Now (7.29) has the following solutions: Thus, the Hamiltonian becomes: where again we have usedĈ 1 = 3 • m = 1: In this case, the solutions of (7.29) are given by: Hence the Hamiltonian gives: where we have substitutedĈ 1 = 3 Obviously, in all three special cases withĈ 0 = 0 one can satisfy the Hamiltonian constraint E L = 0 by fixing suitably one of the integration constants. 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 = 0, 2.
Let us now impose the Hamiltonian constraint. Substituting (7.37) and (7.41) into (7.11) gives: whereas using (7.37) and (7.42) in (7.11) leads to: Clearly, there is no problem to satisfy E L = 0 for the expression in (7.44), by choosing suitably C u 1 or C u 2 . On the other hand, for (7.45) a more careful discussion is needed. Unlike in Sections 5 and 6, now the Σ 2 0 term can have either sign. Let us consider first C 0 ≡ C 2 5 − C 2 4 > 0. In that case, either m < 0 or m > 2; see (7.38). Only m > 2, however, can ensure E L = 0. The reason is that, since the Σ 2 0 term in (7.45) is positive, we need the V 0 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 considerĈ 0 < 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 V 0 term in (7.45), when the Σ 2 0 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: 1)Ĉ 0 > 0 : m odd and m > 2, together with the minus sign in (7.37).
As an example of the above considerations, let us write down the particular solutions for m = 3 andĈ 0 > 0. In that case, the + sign in (7.37) leads to the solution while the − sign gives: 0 . As a final remark note that, among the particular solutions above, there are solutions with m = 1, obtained forĈ 0 < 0 as can be seen from (7.39). However, we already found the most general solution with m = 1 andĈ 0 < 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 the case, upon setting to zero either the pair C v 1,2 or the pair C v 3,4 of integration constants in (7.26).

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 separationof-variables 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 ,q i ) in terms of generalized 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 λ a,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 λ a,ϕ,θ have the following form for the elementary hyperbolic surfaces: • Poincaré disk (D): where A, k, b 2 , C 1 , C 2 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 Akb 2 and the latter appears only as an overall multiplier, which thus can be factored out of the symmetry condition (3.7).
• Hyperbolic Annulus (A): a 3/2 cosh and with V 0 and m being arbitrary constants. Note that the scalar potentials ( 18 For the punctured disk case, we considered only θ 0 = 0 for simplicity. 19 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.
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 [14,15,16,17], 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 φ I . In fact, the Friedmann equation provides an energy shell constraint which must be imposed on the Lagrangian system described by (3.4)  depends on a real modulus R > 1. We briefly discuss these hyperbolic surfaces in turn, referring the reader to [15] for more detail: • 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: 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 ≡ ρe iθ with ρ ∈ [0, 1), and then, by changing suitably the radial variable, to semi-geodesic coordinates (see [15]). This is what is achieved with the redefinition (2.5) that maps the action (2.3) into the form (2.2). 20 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.
• Hyperbolic punctured disk: The hyperbolic punctured disk D * is the open subset of C defined by endowed with the complete hyperbolic metric: 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 ds 2 = dϕ 2 + f (ϕ)dθ 2 using a certain change of variables ϕ = ϕ(ρ). This is what the transformation (2.8) amounts to.
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. 21 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: 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 = 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 C u 1 and C u 2 , while keeping the rest fixed. Let us make the following convenient choices: Recall that the constant C v 1 is determined from the Hamiltonian constraint (5.23). To be able to solve the letter, one needs C u 2 = 0 and even |C u 2 | > |C u 1 |. We also have to take |C u 1 | > 1 for the choices in (B.2), in order to ensure a real and positive scale factor a(t) for any t ≥ 0 . This can be understood by noting that a(t) , then a(t) becomes complex in a neighborhood of t = 0. So, to recapitulate, we need to take: 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 have plotted the scalar ϕ(t) for different choices of C u 1,2 . On the left C u 1 = const, while C u 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 C u 2 increases ϕ. On the right of 21 The possibility of having nontrivial multi-field trajectories, even for a potential without angular dependence, was already shown in [10,11].  Clearly, now the initial value of ϕ also changes. However, increasing C u 1 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 the constants C u 1,2 as in Figure 1. At t = 0 these trajectories start at θ = π 2 , while as t → ∞ 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) . 22 Clearly, when C u 1 = const and C u 2 varies, the starting point at t = 0 remains the same, although the shape of the trajectory changes.
When C u 2 = const and C u 1 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) =˙a (t) a(t) for the same trajectories studied above. In all cases, we have H(t) → 1 as t → ∞. So the spacetimes, corresponding to these solutions, asymptote to dS space. Note that the horizontal axis 22 Note that for the ranges of ϕ and ρ relevant here, relation (2.5) becomes ρ ≈

B.2 Hyperbolic punctured disk
The m = 0 potential for the hyperbolic punctured disk case is: 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 the integration constants in (6.12) and (6.16), we can have w = 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 = 4 3 , according to (3.33), gives: 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 C * 1 and C * 2 , 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: while solving the constraint (6.18) for C * 3 . 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:  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 C * 1 = const and C * 2 varies, while on the right C * 2 = const and C * 1 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 C * 2 (with C * 1 fixed) the trajectories start 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 C * 1 (with C * 2 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 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: 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 = const × v even when m = 0, for suitable choices of the integration constants in (7.10) and (7.14). 23 So, again, one can have nontrivial (ϕ, θ) 23 In this Appendix, we will focus on the genericĈ 0 = 0 case in Section 7. Note, however, that for m = 0, the solutions in the degenerateĈ 0 = 0 case are of the same form as forĈ 0 = 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.
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 = 4 3 substituted) is: 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 C u 1 and C u 2 , with all other constants fixed. For convenience, let us take the following values: with C v 1 determined from the Hamiltonian constraint (7.16). This, in particular, means that we are considering the annulus given by: Note that, with the choices (B.10), we need to have: (C u 2 ) 2 < 23 , (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 C u 1 and C u 2 . 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 left C u 1 = const and C u 2 changes, while on the right C u 2 = const and C u 1 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) settles 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   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 ∈ ( 1 2 , 2) of the hyperbolic 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, π 2 ] , in order to make the graphs better visible. Clearly, they all oscillate around ρ = 1 with decreasing amplitudes and, as t → ∞, they settle at ρ = 1 and different values of θ. Note that, due to (B.9), ρ = 1 corresponds precisely to ϕ = 0, 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. which is the minimum of the scalar potential (B.8). 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  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. It would be interesting to explore whether, considered in a finite-time range, a transient stage of this kind (at the time of horizon exit of the largest observable CMB scales) might be helpful for explaining low multipole-moment anomalies in the CMB.