N -field cosmology in hyperbolic field space: stability and general solutions
NN -field cosmology in hyperbolic field space: stability and general solutions Perseas Christodoulidis ∗ and Andronikos Paliathanasis † Van Swinderen Institute for Particle Physics and Gravity,University of Groningen, Nijenborgh 4, 9747 AG Groningen, The Netherlands Institute of Systems Science, Durban University of Technology, Durban 4000, South Africa (Dated: January 26, 2021)We study the dynamics of N fields on a hyperbolic field space and a symmetric potential in thepresence of a fluid. We list all late-time solutions and investigate their stability. Furthermore, forthe case of two scalar fields with exponential potential we prove that the field equations are Liouvilleintegrable and we provide for the first time the general solution for a region of the parameter space. PACS numbers: 98.80.-k, 95.35.+d, 95.36.+xKeywords: Cosmology; Scalar field; Chiral Cosmology; Exact solution;
1. INTRODUCTION
Over the last years two-field models with a symmetric potential and hyperbolic field space have been extensivelystudied in the literature in the context of multi-field inflation [1, 2] or late-time universe [3, 4]. These models havedisplayed interesting phenomenology while remaining observationally viable [2, 5–8]. Though most works have focusedon the two-field regime, certain many-field constructions have also been proposed e.g. [9, 10]. Similarly, some progresshas been made in the derivation of general solutions in scalar-field cosmology. Multi-field generalizations, on the otherhand, have been proved more challenging and up to date only a few solution are known for arbitrary number of fields[11–15].The existence of exact and analytic solutions is an essential property for the mathematical description of a physicaltheory. Although a dynamical system can be solved by using numerical techniques we do not know that the numericaltrajectories correspond always to real solution of the problem, thus we should investigate if the dynamical systemposses the integrability property. There are various techniques for the study of the integrability in the literature. Incosmological studies dues to the fact that the gravitational field equations for scalar field theories admit a minisu-perspace description techniques from analytic mechanics can be applied. The theory of similarity transformations forthe derivation of conservation laws has been applied in [26–30] while some other approaches can be found in [31–33].Another important approach for the study of a cosmological model is the determination of the stationary points.The later points can be used for the determination of the asymptotic behaviour of a specific theory and to extractimportant information and criteria for the cosmological evolution of the specific model [34–37].In this work we will investigate the N -field generalization of the two-field hyperbolic problem in the presence ofa perfect fluid. We will first list all critical-point solutions and investigate their stability. Next, we will apply theNoether method to derive two-field general solutions in some cases and then extend them to N -fields.The paper is organized as follows: in Sec. 3 we revisit the stability analysis for the two-field hyperbolic problem inthe presence of a fluid. Next, we generalize the discussion for N fields in Sec. 4 and list all new solutions as well astheir stability properties. In Sec. 5 we calculate the general solution for a subset of the parameter space using theNoether method. Finally, in Sec. 6 we offer our conclusions.
2. CHIRAL (MULTI-FIELD) COSMOLOGY
The Chiral cosmological model belongs to the family of the Einstein-nonlinear σ -model where in the Einstein-HilbertAction two scalar fields minimally coupled to gravity are introduced such that the gravitational Action Integral to bewritten as [17–20] S = (cid:90) √− g d X (cid:18) R − g µν ∇ µ φ ∇ ν φ − g µν F ( φ ) ∇ µ ψ ∇ ν ψ − V ( φ ) (cid:19) + S m , (1) ∗ Electronic address: [email protected] † Electronic address: [email protected] a r X i v : . [ g r- q c ] J a n where there is a coupling in the kinetic term between the two scalar fields. When the coupling function F ( φ ) isconstant the Action Integral (1) describes two quintessence; however, in Chiral cosmology the two dynamics of thetwo fields evolve in a space of constant nonzero curvature, that is, F ( φ ) = e κφ . At this point it is important tomention that we refer to a two-dimensional space defined by the kinetic terms of the scalar fields and not in thebackground space with metric g µν and Ricciscalar R .According to the cosmological principle for the background space we assume that of spatially flat Friedmann–Lemaˆıtre–Robertson–Walker (FLRW) described by the line elementd s = − N ( t ) d t + a ( t ) (cid:0) d X + d Y + d Z (cid:1) , (2)where a ( t ) is the scale factor and N ( t ) the lapse function. Moreover we assume that the scalar fields inherit thesymmetries of the background space, that is, φ ( x µ ) = φ ( t ), ψ ( x µ ) = ψ ( t ), while the Action Integral S m describes anideal gas with energy density ρ , pressure p and constant equation of state parameter p = wρ .Hence, the gravitational field equations are [20] 3 H = ρ f + ρ, (3) − (cid:16) H + 3 H (cid:17) = p f + P, (4)in which ρ f , p f are the energy density and pressure components of the two scalar fields, that is, ρ f = 12 ˙ φ + 12 e κφ ˙ ψ + V ( φ ) , (5) p f = 12 ˙ φ + 12 e κφ ˙ ψ − V ( φ ) . (6)Finally, the equation of motions for the two fields are¨ φ + 3 H ˙ φ − κ e κφ ˙ ψ + V ,φ ( φ ) = 0 , (7)¨ ψ + 3 H ˙ ψ + κ ˙ φ ˙ ψ = 0 , (8)while the ideal gas satisfies the continuity equation˙ ρ + 3 H (1 + w ) ρ = 0 , (9)from where it follows ρ = ρ m a − w ) .At this point it is important to mention that the gravitational field equations can be derived by the variation ofthe point-like Lagrangian L (cid:16) a, ˙ a, φ, ˙ φ, ψ, ˙ ψ (cid:17) = 12 N ( t ) (cid:16) − a ˙ a + a (cid:16) ˙ φ + e κφ ˙ ψ (cid:17)(cid:17) − N ( t ) a V ( φ ) + N ( t ) ρ m a − w m . (10)That it is an important observation, because we can apply techniques from analytic mechanics for the determinationof exact solutions. As far as, the scalar field potential V ( φ ) is concerned, for most part of this paper we will assume theexponential function V ( φ ) = V e λφ , which as it has been shown before provides interesting physical results [23, 24].In order to study the asymptotic behaviour of the previous model it is better to switch from cosmic time to thee-folding number defined from d N = H d t . In this way the set of evolution equations for the scalar fields and thefluid can be written as an autonomous dynamical system. Introducing a new variable z = ρ/H , describing the fluidenergy density, the Friedman constraint becomes3 = 12 v i v i + z + V , (11)which implies that the allowed values for the field velocities and z should satisfy12 v i v i + z ≤ . (12)The slow-roll parameter becomes (cid:15) = 12 v i v i + 12 (1 + w ) z , (13)while the potential satisfies VH = 3 − (cid:15) + 12 ( w − z . (14)The evolution equations for the fields and the fluid are( v i ) (cid:48) + Γ ijk v j v k + (3 − (cid:15) )( v i + λ i ) + 12 ( w − λ i z = 0 , (15) z (cid:48) + (3 + 3 w − (cid:15) ) z = 0 . (16)However, critical points for the velocities are not expected to be found when a generic field metric is consideredbecause Christoffel symbols are field dependent. Instead, it is better to study equations for the normalized velocities √ G ii v i (no sum is assumed in i ) that enter the definition of (cid:15) . For the field metric with an isometry these velocitiesare defined as y ≡ φ (cid:48) , x ≡ (cid:112) F ( φ ) ψ (cid:48) . (17)The variables of the two-field dynamical system in first order form are φ, ψ, y, x, z .
3. RECAP OF THE TWO-FIELD PROBLEM WITH A PERFECT FLUID3.1. Hyperbolic field metric
Specializing to the hyperbolic field metric d s = d φ + e κφ d ψ , (18)the system in first order form is φ (cid:48) = y , (19) ψ (cid:48) = xe − κ/ φ , (20) y (cid:48) + (3 − (cid:15) )( y + λ ) − κ x + 12 ( w − λz = 0 , (21) x (cid:48) + (cid:16) − (cid:15) + κ y (cid:17) x = 0 , (22) z (cid:48) + (3 + 3 w − (cid:15) ) z = 0 . (23)The second equation can be discarded because ψ is a cyclic variable and does not affect dynamics and, moreover, weobserve that the last three equations do not depend on φ and so the first equation can also be omitted; the reduced3 × w that were missing for some solutions of theaformentioned work.For our set of variables the eigenvalues of the Jacobian matrix follow straightforwardly from our analysis becausethat matrix can always be written in block-diagonal form. We have the following critical points:1. First, we have scalar field domination solutions, which generalize the solutions presented in [24, 25] with theaddition of z = 0. Their stability properties remain unchanged with the additinal requirement (cid:15) < / w ).This happens because the stability matrix acquires an upper triagonal form with zeros below the main diagonaland so the presence of the fluid does not affect eigenvalues (for this type of solutions). There are three types ofscalar-dominated solutions:(a) The scalar-field gradient solution, ( y, x, z ) gr = ( − λ, , . (24)Motion is aligned with the potential gradient flow. It is stable provided − (cid:114) κ − κ < λ < (cid:114) κ − κ , | λ | < √ , | λ | < (cid:112) w ) . (25)The solution is depicted at the left panel of Fig. 1. FIG. 1:
The numerical solutions for a wide range of initial conditions (drawn uniformly from the surface defined from (cid:15) ≈ ).Left: For λ = κ = 1 and w = 0 the solution asymptotes to the gradient critical point. Right: For a fluid with w = 2 , λ = 3 and κ = − the solution asymptotes to kinetic domination. Blue dots correspond to the respective critical points and thesemi-transparent blue surface denotes the region of definition for x, y, z . (b) The scalar-field kinetic domination solution given as( y, x, z ) kin = (cid:16) ±√ , , (cid:17) , (26)and is stable for | λ | > √ , λ · κ < , w > , (27)(see left panel of Fig. 1).(c) The hyperbolic solution ( y, x, z ) hyper = (cid:32) − κ + λ , ± √ √ λ + κλ − κ + λ , (cid:33) . (28)This solution exists provided k (cid:54) = − λ , λ + κλ − > λ > (cid:114) κ − κ , λ < − (cid:114) κ − κ , (29)and it is stable whenever the previous two solutions are unstable, namely when the following conditionsare satisfied κ > , λ > (cid:114) κ − κ , λκ + λ <
12 (1 + w ) (30) κ < , λ < − (cid:114) κ − κ λκ + λ <
12 (1 + w ) . (31)The solution is depicted at the left panel of Fig. 2.2. The second type of solutions describe fluid domination z = 3, where fields have zero values. In order for this tobe a solution of the dynamical system the parenthesis of Eq. (23) should be zero and this gives the value of (cid:15)(cid:15) = 32 (1 + w ) , (32)(this value of (cid:15) is compatible with Eq. (21)). These solutions are stable for w < −
1, which yields (cid:15) <
0, andthus describe contracting universes.
FIG. 2:
The numerical solutions assuming a fluid with w = 0 for a wide range of initial conditions (drawn uniformly fromthe surface defined from (cid:15) ≈ ). Left: For λ = 1 , κ = 15 the solution asymptotes to one of the two hyperbolic critical points(depending on the initial x ). Right: For λ = 2 . , κ = 1 the solution asymptotes to the scaling critical point. Blue dots correspondto the respective critical points and the semi-transparent blue surface denotes the region of definition for x, y, z .
3. Finally, we find the scaling solution with the fluid and at least one of the fields non-zero. Again we require theparenthesis of Eq. (23) to vanish and so (cid:15) has the same value as in fluid domination. The solution is( y, x, z ) scal = (cid:18) − w + 1) λ , , − w ) λ (cid:19) , (33)which exists provided z scal ≥ λ ≥ w ) . (34)The eigevalues of the Jacobian matrix are m = 32 λ [( w + 1) κ + ( w − λ ] , (35) m ± = 34 (cid:16) w − ± (cid:112) ( w − + 8( w − − w − w + 1) λ − (cid:17) , (36)and they are non-positive for w > − w, λ, κ − < w < , κλ < − w w . (37)For w = 0 we recover the relations mentioned in Ref. [3]. The solution is illustrated at the right panel of Fig. 2.Note that there are no real solutions with y, x, z (cid:54) = 0. We will briefly comment on the case of a general field metric with isometry in ψ d s = d φ + F ( φ ) d ψ . (38)In this case κ ≡ F ,φ /F is field dependent and Eq. (19) can not be omitted. Choosing an exponential potential, or apotential that asymptotes to an exponential at e.g. −∞ , all previous solutions (except for the hyperbolic one) mayexist for appropriate choices of the metric function (see Fig. 3 for a model with F = e φ ), while the hyperbolic solutionis replaced by a de Sitter asymptotic state y, x →
0. This can be shown as follows: if the gradient or kinetic solutions
FIG. 3:
The numerical solutions assuming a fluid with w = 0 and φ = 2 for a wide range of initial conditions (drawnuniformly from the surface defined from (cid:15) ≈ ). Left: For λ = 1 the solution asymptotes to the gradient critical point. Right:For λ = 2 . the solution asymptotes to the scaling critical point. Blue dots correspond to the respective critical points and thesemi-transparent blue surface denotes the region of definition for x, y, z . FIG. 4:
Numerical solutions that asymptote to the de Sitter critical point with φ = 2 , λ = 1 . are unstable then a solution, that resembles the hyperbolic one, can be obtained if κ diverges to plus/minus infinityat the boundary of the space. In this case the combination κy is required to be constant and so the parenthesis ofEq. (22) can vanish. Plugging back into Eq. (21) gives the asymptotic solution for y and x which has exactly the sameform as in the hyperbolic case, albeit κ is field dependent and growing in norm [4, 24]. Even though a proper stabilityanalysis for a field-dependent κ requires study of the dynamical system at infinity, we can use a simpler argumentto understand the behaviour of these solutions. For the 4 × φ . Since φ will eventually roll towards decreasing values ofthe potential (otherwise the Friedman constraint would be violated) no instability related to this marginal directionwill be present. Therefore, one can apply the previous formulae for solutions and stability criteria after taking thelimit φ → ∞ . Note though that convergenece towards the de Sitter critical point is much slower compared to othercritical points, while the system may pass through other critical points first (see Fig. 4 for a model with F = e φ aswell as the discussion in Ref. [4]).
4. THE N - FIELD HYPERBOLIC SOLUTION The generalization of the hyperbolic problem will depend on the parametrization of the hyperbolic space in morethan two dimensions. We choose the following form of the field metricd s = d φ + (cid:88) e κ i φ d ψ i , (39)with Ricci scalar R = − (cid:16)(cid:88) κ i (cid:17) − (cid:88) κ i . (40)Equations of motion are φ (cid:48) = y , (41) ψ (cid:48) i = x i e − κ i φ , (42) y (cid:48) + (3 − (cid:15) )( y + λ ) − (cid:88) κ i x i + 12 ( w − λz = 0 , (43) x (cid:48) i + (cid:16) − (cid:15) + κ i y (cid:17) x i = 0 , (44) z (cid:48) + (3 + 3 w − (cid:15) ) z = 0 . (45)To analyze the problem we will distinguish between two cases. κ i are all equal In the symmetric case κ i = κ the solutions presented in Sec. 3 carry over with the substitution x → (cid:80) x i . Thisbecomes apparent if we write the differential equation of the slow-roll parameter (cid:15) which is found by contractingEq. (15) with v i : (cid:15) (cid:48) + (3 − (cid:15) )(2 (cid:15) + λy ) + 12 ( w − λyz = 0 , (46)and substituting (cid:80) x i = 2 (cid:15) − y in Eq. (43) y (cid:48) + (3 − (cid:15) )( y + λ ) − κ (cid:15) − y ) + 12 ( w − λz = 0 . (47)This shows that the set of Eqs. (44) can be replaced with Eq. (46) and x i are left undetermined (a similar argumentwas used in Ref. [16]). κ i are different The situation is drastically different when κ i are different. In the next we list all critical-point solutions and theirstability properties.1. The analogue of the hyperbolic solution with y and all x i different than zero is inconsistent as it requires(3 − (cid:15) ) + κ i y = 0 , (48)to hold for every κ i . Therefore, we conclude that only one x j can be non-zero and x i = 0 for i (cid:54) = j . To study thestability we calculate the Jacobian matrix evaluated on the hyperbolic solution and we observe that it alwaysacquires a block diagonal form (some permutations of rows and columns may be necessary) (cid:18) A × ×N − N − × B N − ×N − (cid:19) , (49) FIG. 5:
The numerical solution for three fields and a wide range of initial conditions (drawn uniformly from the surface ofshpere with (cid:15) ≈ ) with λ = 3 , κ = 1 , z = 0 and κ = − (left) and κ = 10 (right). The blue dots correspond to the twohyperbolic solutions. where A × = 1( κ j + λ ) (cid:32) − κ j − λ )( κ j + 2 λ ) √ (cid:112) λ + κ j λ − κ j − λ ) − − (cid:113) (12 + κ j + λκ j ) (cid:112) λ + κ j λ − κ j − λ ) − (cid:33) , (50)is exactly the stability matrix of the reduced two-field problem, while the other matrix is diagonal B N − ×N − = diag (cid:18) − κ j − κ i ) κ j + λ , · · · , w − (cid:15) (cid:19) , (51)for i = 1 , · · · , N − (cid:54) = j . The eigenvalues of A need to satisfy the inequalities (30)-(31) (with the substitution κ → κ j ) while the rest N − B and so a stable solution requires κ j > κ i for κ j > , κ j < κ i for κ j < , (52)for i (cid:54) = j .2. The fluid domination and the scaling solution with non-zero y, z and x i = 0( y, x i , z ) scal = (cid:18) − w + 1) λ , , · · · , − w ) λ (cid:19) , (53)both exist with the same stability properties as previously.3. Finally, in addition to the usual kinetic-domination solution( y, x i , z ) kin = (cid:16) ±√ , , · · · , (cid:17) , (54)which is stable for | λ | > √ λκ i <
0, new solutions ( (cid:15) = 3) exist with y = 0 and x i (cid:54) = 0 which follow from κ i x i = 0 , (cid:88) x i = 6 . (55)For three fields the solution is trivially found to be( y, x , x , z ) kin = (cid:18) , ±√ (cid:114) κ κ − κ , ±√ (cid:114) − κ κ − κ , (cid:19) , (56)which exists provided κ κ <
0. It can be shown that the stability matrix evaluated on the solution contains atleast one positive eigenvalue and thus this kinetic domination is unstable. For N > N −
5. THE GENERAL SOLUTION
For the case without matter source we find that there are general solutions for the case where κ = ± (cid:0) λ + √ (cid:1) . Weconsider the case where κ = − (cid:16) λ + √ (cid:17) , (57)and let us select the lapse function as N ( t ) = a − −√ λ . For the latter selection the field equations admit theNoetherian conservation laws I (cid:16) a, ˙ a, φ, ˙ φ, ψ, ˙ ψ (cid:17) = dd t (cid:16) a √ λ e − ( λ + √ φ (cid:17) , (58) I (cid:16) a, ˙ a, φ, ˙ φ, ψ, ˙ ψ (cid:17) = a √ λ e − ( λ + √ φ ˙ ψ . (59)Applying the coordinate transformation a ( χ ( t ) , ξ ( t ) , ζ ( t )) = a (cid:32) (cid:0) √ λ + 6 (cid:1) (cid:16) χ ( t ) ζ ( t ) − ξ ( t ) (cid:17)(cid:33) ( √ λ ) − , (60) φ ( χ ( t ) , ξ ( t ) , ζ ( t )) = 2 λ + √ (cid:115)(cid:18) ( √ λ +6 ) (cid:16) χ ( t ) ζ ( t ) − ξ ( t ) (cid:17)(cid:19) χ ( t ) , (61) ψ ( χ ( t ) , ξ ( t ) , ζ ( t )) = ξ ( t ) χ ( t ) . (62)the point-like Lagrangian is expressed in the new coordinates as L (cid:16) χ, ˙ χ, ξ, ˙ ξ, ζ, ˙ ζ (cid:17) = 2 (cid:16) ˙ ξ − ˙ χ ˙ ζ (cid:17) − ¯ V χ − ¯ λ , ¯ V = 4 − √ λ √ λ V , ¯ λ = − √ λ . (63)The field equations are written as follows¨ ξ = 0 , ¨ χ = 0 , ¨ ζ − ¯ λ ¯ V χ ¯ λ − = 0 , (64)along with the constraint 2 (cid:16) ˙ ξ − ˙ χ ˙ ζ (cid:17) + ¯ V χ − ¯ λ = 0 . (65)Consequently, the analytic solution of the field equations is χ ( t ) = χ ( t − t ) , ξ ( t ) = ξ ( t − t ) , - - - - ( t ) w e ff ( t ) FIG. 6: Qualitative evolution of the effective equation of state parameter w eff = − −
23 ˙ HH for the analytic solution of ourconsideration for (cid:0) λ, χ , ζ , t , t , t , ¯ V (cid:1) = ( − , . , . , , . , , and ζ ( t ) = ¯ V χ ¯ λ − (cid:0) ¯ λ + 1 (cid:1) ( t − t ) ¯ λ +1 + ζ ( t − t ) , for ¯ λ (cid:54) = − , , (66)or ζ ( t ) = ¯ V χ ln ( t − t ) + ζ ( t − t ) , for ¯ λ = − , (67)with the constraint ξ − χ ζ = 0. For the special case of ¯ λ = 0, that is the potential is a cosmological constant with λ = 0, the exact solution is χ ( t ) = χ ( t − t ) , ξ ( t ) = ξ ( t − t ) , ζ ( t ) = ζ ( t − t ) , (68)with the constraint equation ξ − χ ζ = 0.In Figs. 6 and 7 we present the qualitative evolution of the effective equation of state parameter w eff = w eff ( a ),which is defined as w eff ( a ( t )) = − −
23 ˙ HH , for two different values of the parameter λ . We observe that the w eff ( a )is that of the hyperbolic expansion.It was pointed out recently in Ref. [22] that when the second field is phantom we can reconstruct the analyticsolution for the two-field model in a same way as before by setting ˜ ψ = iψ . Therefore, the analytic solution for thesecond case for κ = (cid:0) λ + √ (cid:1) is determined in a similar way and we omit the presentation. Indeed by considering ξ ( t ) = iξ ( t ) we obtain the coordinate transformation a ( χ ( t ) , ξ ( t ) , ζ ( t )) = a (cid:32) (cid:0) √ λ + 6 (cid:1) (cid:16) χ ( t ) ζ ( t ) + ξ ( t ) (cid:17)(cid:33) ( √ λ ) − , (69) φ ( χ ( t ) , ξ ( t ) , ζ ( t )) = 2 λ + √ (cid:115)(cid:18) ( √ λ +6 ) (cid:16) χ ( t ) ζ ( t ) + ξ ( t ) (cid:17)(cid:19) χ ( t ) , (70) ψ ( χ ( t ) , ξ ( t ) , ζ ( t )) = ξ ( t ) χ ( t ) , (71)1 - - ( t ) w e ff ( t ) FIG. 7: Qualitative evolution of the effective equation of state parameter w eff = − −
23 ˙ HH for the analytic solution of ourconsideration for (cid:0) λ, χ , ζ , t , t , t , ¯ V (cid:1) = (cid:0) √ , , ζ , , , , (cid:1) and the second field is phantom which produce the same second-order differential equations as before. Consequently, the solution for the variables χ ( t ) , ξ ( t ) and ζ ( t ) is the same as before, while the constraint equation is ξ + χ ζ = 0. In Figs. 6 and 7 we presentthe qualitative evolution of the effective equation of state parameter w eff for the exact solutions we presented in thisSection. Consider now the existence of an additional matter field. We can conclude that when the equation of state parameter w ( λ ) = − √ λ then the solutions found in the previous section hold for this cosmological model as well, with onlydifference the constraint equations of the integration constants ξ , χ , ζ are now ρ m = 2 ξ − χ ζ , (72)for the Chiral model and ρ m = ξ + χ ζ , (73)for the Chiral-quintom model.We observe that for w m = − , point-like Lagrangian (10) describes the gravitational field equations for the case ofa FLRW spacetime with spatially curvature k = ρ m , that is, for the line elementd s = − N ( t ) d t + a ( t )1 − k ( X + Y + Z ) (cid:0) d X + d Y + d Z (cid:1) . (74)Hence, we also presented for the first time an analytic solution for a two-field model in a non-flat FLRW backgroundspace.
6. DISCUSSION
In this work we studied the dynamics and the existence of analytic solutions for a multi-field cosmological modelin a spatially flat FLRW background space. In particular, we considered a cosmological model consisting of N scalarfields minimally coupled to gravity with a hyperbolic interaction in the kinetic terms in the presence of an ideal gas.For this gravitational model we studied the asymptotic behaviour of the field equations. We recovered previous results2for the two-field model, that is all critical points for the quintessence, i.e. of single-field theory, as well as a pair ofpoints which describe a hyperbolic solution where both fields contribute.In the multi-field scenario with N > Q i = E iA q A where the matrix E iA has as columns the components of the orthonormal vectors E iA = (cid:0) t i , n i , b i , · · · (cid:1) . (75)Here t i is the tangent unit vector, n i the normal vector, b i the binormal vector and so on. The projection along thetangent vector is related the curvature perturbation, while projections along the orthogonal directions are relatedto isocurvature perturbations. The covariant time derivatives of the orthonormal vectors satisfy the Frenet-Serretequations D t E iA = C BA E iB , (76)where the matrix C is antisymmetric with non-zero elements in the upper and lower diagonals.It is well known that the curvature perturbation ( Q σ ≡ Q i E i = q ) is sourced by the first isocurvature perturbation( Q s ≡ Q i E i = q ) [38, 39], while the rest orthogonal perturbations interact via a “mass matrix” as well as throughthe curvatures that appear in the Frenet-Serret equations (see e.g. Refs. [40–42] for examples including up to threefields and Refs. [43, 44] for a formal discussion including an arbitrary number of fields). More specifically, orthogonalfields are coupled through the following terms in the second order action2 ˙ q T · Ω · q + q T · Ω T · Ω · q − q T · M · q , (77)where Ω is the truncated matrix obtained from C after removing elements of the first row and column and M AB ≡ E iA E jB (cid:0) V ; ij + ˙ σ t k t l R kilj + 3 ω δ A δ B (cid:1) . (78)Extra orthogonal fields decouple from Q σ and Q s when the matrices Ω and M are block diagonal. A necessarycondition for this decoupling is the vanishing of the torsion of the N -dimensional field-space trajectory. Recall thatthe torsion of a curve is found by calculating the rate of change of the normal unit vector. Using the Frenet-Serretequations the torsion ( τ ) is defined from D t n i = − ωt i + τ b i , where t i , n i , b i are the first three unit vectors of theorthonormal frame at some point of the curve and ω is the turn rate. The case of zero torsion is reminiscent togeodesic motion where the curvature perturbation decouples from isocurvature perturbations.For the hyperbolic solution the vectors t i , n i have the first two components non-zero and the rest zero, which forcesthe next vectors in the series to have the first two components zero and the rest non-zero. This means that the matrix E iA is block diagonal. Using the Frenet-Serret equation for n i we find D t n i = − ωt i and, hence, the torsion is zero,which implies that Ω is block diagonal. Moreover, M turns out to be block diagonal and so we conclude that Q σ and Q s evolve independently from the rest fields. Therefore, the basic predictions for this model, namely the spectralindex and the tensor-to-scalar ratio, are identical to the two-field case. This is an important observation because itis clear that by adding additional scalar fields in this specific theory we do not get new physical results (regardinglate-time solutions), while no information can be extracted from current observations to support this N -field theorywith N >
2. It would be interesting to investigate whether this holds for non-Gaussianities and other higher-ordercorrelators as well.Finally, for the two-field model with an exponential potential we proved for the first time the Liouville integra-bility of the field equations while we derive the analytic solution of the model. We applied the theory of similaritytransformations to construct conservation laws. We found that for a specific combination of the exponents λ and κ (associated with the potential gradient and the curvature of the hyperbolic space respectively) new conservation lawsexist which facilitate the derivation of a closed-form solution of the dynamical system. We demonstrated that thesolution holds for the case of the presence of additional matter, while it can be used to construct the analytic solutionof the multi-field model in a non-spatially flat FLRW background space. Acknowledgments
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