Non-integrability of the axisymmetric Bianchi IX cosmological model via Differential Galois Theory
Primitivo Acosta-Humánez, Juan J. Morales-Ruiz, Teresinha J. Stuchi
NN ON - INTEGRABILITY OF THE AXISYMMETRIC B IANCHI IX COSMOLOGICAL MODEL VIA D IFFERENTIAL G ALOIS T HEORY
Primitivo ACOSTA-HUMÁNEZ
Universidad Simón Bolívar, Barranquilla - ColombiaInstituto Superior de Formación Docente Salomé UreñaSantiago de los Caballeros - Dominican Republic [email protected]
Juan J. MORALES-RUIZ
Depto. de Matemática AplicadaUniversidad Politécnica de MadridMadrid, Spain [email protected]
Teresinha J. STUCHI
Instituto de FísicaUniversidade Federale de Rio de JaneiroRio de Janeiro, Brazil [email protected] A BSTRACT
We investigate the integrability of an anisotropic universe with matter and cosmological constantformulated as Bianchi IX models. The presence of the cosmological constant causes the existenceof a critical point in the finite part of the phase space. The separatrix associated to this Einstein’sstatic universe is entirely contained in an invariant isotropic plane forgetting the singularity at theorigin. This invariant plane of isotropy is an integrable sub-space of the Taub type. In this paper weanalyse the differential Galois group of the second order variational equations to this plane in order toapply the integrability theorem of the second author with Ramis and Simó. The main result is that themodel is non-integrable by meromorphic functions.
Keywords
Bianchi IX models · cosmological constant · differential Galois theory · integrability · Poincaré sections.
MSC 2010.
Introduction
Belinskii, Khalatnikov and Lifishitz [8] started the question of chaotic behaviour of general Bianchi IX models inRelativistic Cosmology. The interest in the chaoticity (or not) of Bianchi IX models has been mainly focused on theMixmaster case. Vacuum Bianchi IX models with three scale factors was taken by Misner [17]). Also [9, 10] studied thethree dimensional Bianchi IX model from the real dynamics point of view. The absence of an invariant (or topological)characterisation of chaos in the model, i.e., standard chaotic indicators such as Lyapunov exponents being coordinatedependent and therefore questionable [14] ,[12]. For discussions of the issue of chaotic dynamics on these models werefer to the works of Barrow [7]. Cornish and Levin [13] proposed to quantify chaos in the Mixmaster universe bycalculating the dimensions of fractal basin boundaries in initial-conditions sets for the full dynamics. Calzeta [12] andEl Hasi proved integrability under adiabatic approximation, but the tori breaked up without the adiabatic terms.We recall that the conjunction of a cosmological constant and anisotropy implies in the existence of a critical point, theEinstein universe, in the finite region of phase space. This point is linearly a saddle center that implies the existenceof a centre manifold tangent to the linear periodic orbits around E. From this centre manifold the stable and unstablemanifolds which are topologically cylinders carry the dynamics away and to the inflationary region. This was the objectof the work of H.P. Oliveira et al. (see [22]) for the axisymmetric Bianchi IX model. This is the model we study in thispaper. This problem was numerically shown non integrable; these cylinders, when the full dynamics is considered, a r X i v : . [ m a t h . D S ] F e b ON - INTEGRABILITY OF A XISYMMETRIC B IANCHI IX MODEL VIA D IFFERENTIAL G ALOIS T HEORY intercept each other in homoclinic and heteroclinic connections causing a chaotic dynamics. This has as consequence achaotic escape in the inflationary regime. However, as the iterates cross are q the scale factor the model breaks downand so there is no recurrence. In [20], see also [18], is showed that the three degrees of freedom Bianchi IX models,without cosmological constant is no-integrable by rational functions. The axisymmetric version has so far no analyticproof of non integrability.In this paper we show that the two degrees of freedom axisymmetric Bianchi IX model with cosmological constant isnon integrable by meromorphic functions. We apply the following theorem, see A and [21], to prove the meromorphicnon integrability: Theorem 3.
Assume that a complex analytical Hamiltonian system is integrable by meromorphic first integrals in aneighborhood of the integral curve z = φ ( t ) . Then the identity components ( G k ) , k ≥ , of the Galois groups of thevariational equations along the Riemman surface Γ are commutative. Moreover, the singularities of VE k are of regulartype. We note that k is the order of the variational equations VE k which in our case is k = 2 because the first order variation, k = 1 does not decide the matter of integrability.In section 1 we present the axisymmetric model. In section 2 the calculation of Galois groups for first and secondvariational equations are made. In section 3 we illustrate the behaviour of the system with Poincaré sections. Anoverview about Differential Galois Theory and non-integrability of Hamiltonian systems are presented in A. Finally,application of Kovacic algorithm to compute the solutions and Galois of the first variational equation is presented in B. The axisymmetric Bianchi IX models, see [25], derived from the full three scales model, H ( A, B, C, P A , P B , P C ) considering B = C , ˙ B = ˙ C , with cosmological constant λ is: H ( A, B, P A , P B ) = P A P B B − AP A B + 2 A − A B − λAB (1)It obeys the Einstein equation : G µν − λg µν = T µν . and the line element is obviously ds = dt − A ( t ) ( ω ) − B ( t ) (( ω ) + ( ω ) ) .The introduction of the cosmological constant is one of the cornerstones in the paradigm of inflation. The inflationaryconstant plays a fundamental role in the gravitational dynamics model, by inducing an expansion of the scales towardsthe De Sitter configuration.The equations of motion corresponding to Eq. (1) are given by the Hamiltonian system ˙ A = P B B − AP A B ˙ B = P A B ˙ P A = P A B − A B + 2 λB ˙ P B = P A P B B − AP A B − A B + 4 λAB (2)This system, Eq. (2), has a critical point E in the finite region of the phase space with coordinates E : A = B = 1 √ λ , P A = P B = 0 , with associated energy E = 1 / √ λ . This point corresponds to the static Einstein universe. From the point E emanatethe stable and unstable manifolds which goes to infinity to the right of E and in the left the unstable and stable branchmeet at the origin. The origin is a singular point: The Friedmann universe2 ON - INTEGRABILITY OF A XISYMMETRIC B IANCHI IX MODEL VIA D IFFERENTIAL G ALOIS T HEORY
Furthermore system (2) has an invariant plane Γ : A = B , P B = 2 P A that gives us a family of integral curves, the Taubfamily, see [27], defined by the one degree of freedom Hamiltonian E = y x + 32 x − λx , (3)where we denote x := A = B and y := P A = P B = 4 x ˙ x . As we want to compute the variational equations along the separatrix of the Taub family as a second order variationalequation, it is convenient to write de hamiltonian equations as second order differential equations ¨ A = A B − A B + A ˙ B B + λA − ˙ A ˙ BB ¨ B = A B − B + λB − ˙ B B (4)The invariant plane Γ is given in these coordinates by A = B , ˙ A = ˙ B , and the Taub family along this plane isparametrized by coordinates x := A = B and ˙ x = ˙ A = ˙ B and given by ¨ x = − x + λx − ˙ x x (5)The first integral of the Hamiltonian restricted to the invariant plane Γ gives the integral curves of the above equationwhich in the coordinates ( x, ˙ x ) , through Eq. (3) because y = 4 x ˙ x , are expressed as the family of cubics Γ E : 6 x ˙ x + 3 x − λx − E = 0 , (6)for generic values of λ and E . The variational equations of the Hamiltonian system (4) along a particular solution of Γ is given by the equations ¨ ψ A = (cid:16) B − A B + ˙ B B + λ (cid:17) ψ A − ˙ BB ˙ ψ A + (cid:16) λAB − A ˙ BB − AB + A ˙ B B + AB (cid:17) ψ B + (cid:16) A ˙ BB − ˙ AB (cid:17) ˙ ψ B ¨ ψ B = A B ψ A + ( − A B + B + λ + ˙ B B ) ψ B − ˙ B B ˙ ψ B (7)where the coefficients are restricted to the integral curve Γ : A = A ( t ) , ˙ A = ˙ A ( t ) , B = B ( t ) and ˙ B = ˙ B ( t ) .For Γ = Γ E , being ψ = ψ A , ζ = ψ B , x = A = B , ˙ x = ˙ A = ˙ B , we obtain the VE (variational equations) along theTaub solution, ¨ ψ = − ˙ xx ψ (cid:48) + (cid:16) (cid:0) ˙ xx (cid:1) − x + λ (cid:17) ψ − (cid:16) xx + 2 (cid:0) ˙ xx (cid:1) − x − λ (cid:17) ζ ¨ ζ = − ˙ xx ζ (cid:48) − (cid:16) xx + (cid:0) ˙ xx (cid:1) + x − λ (cid:17) ζ + x ψ (8)We set the following notations that will be useful along the rest of the paper: C ( x ) := 4 λx − x + 2 E, C ( x ) := 35 λx + 210 x + 4 E (9)Now in order to obtain a differential equation over the Riemann sphere P (i.e. with rational coefficients), it isconvenient to apply the so-called Hamiltonian algebrization procedure , see [2, 4]. This procedure starts with a rationalHamiltonian change of variable , that is, making a change of variables t (cid:55)→ x such that ( ˙ x ) = α ( x ) ∈ C ( x ) , forinstance, ˙ x = ± (cid:112) α ( x ) . This change of variable is obtained by the using of equation of Γ E , equation (6), and then wehave α = C ( x )12 x , ψ (cid:48) = √ α (cid:98) ψ (cid:48) , ζ (cid:48) = √ α (cid:98) ζ (cid:48) , ON - INTEGRABILITY OF A XISYMMETRIC B IANCHI IX MODEL VIA D IFFERENTIAL G ALOIS T HEORY where C ( x ) is given in Eq. (9). Applying Hamiltonian algebrization procedure, where Eq. (5) provides ¨ x , we obtainthe algebraic variational equations (AVE) (cid:98) ψ (cid:48)(cid:48) = − λx − x + ExC ( x ) (cid:98) ψ (cid:48) + λx − x + Ex C ( x ) (cid:98) ψ + xC ( x ) (cid:98) ζ (cid:98) ζ (cid:48)(cid:48) = − λx − x + ExC ( x ) (cid:98) ζ (cid:48) + λx − x + Ex C ( x ) (cid:98) ζ + xC ( x ) (cid:98) ψ , (10)for generic values of λ and E , which correspond to the algebraic form of variational equations given in equation (8).After a hard work combining computations by hand with Maple, we solve the algebraic form of the variational equation(10). Therefore one basis of solutions of the variational equation (10) is given by B fve = (cid:26)(cid:18) (cid:98) ψ ( x ) (cid:98) ζ ( x ) (cid:19) , (cid:18) (cid:98) ψ ( x ) (cid:98) ζ ( x ) (cid:19) , (cid:18) (cid:98) ψ ( x ) (cid:98) ζ ( x ) (cid:19) , (cid:18) (cid:98) ψ ( x ) (cid:98) ζ ( x ) (cid:19)(cid:27) , where (cid:98) ψ i ( x ) , (cid:98) ζ i ( x ) are given by (cid:98) ψ ( x ) = (cid:98) ζ ( x ) = (cid:113) C ( x ) x = √ α (cid:98) ψ ( x ) = (cid:98) ζ ( x ) = (cid:113) C ( x ) x (cid:82) (cid:113) xC ( x ) dx = (cid:98) ψ ( x ) (cid:82) dxx (cid:98) ψ ( x ) (cid:98) ψ ( x ) = − (cid:98) ζ ( x ) = (cid:113) C ( x ) x exp (cid:18) − i (cid:82) √ E λ +2450) dxC ( x ) (cid:98) ψ ( x ) (cid:19)(cid:98) ψ ( x ) = − (cid:98) ζ ( x ) = (cid:113) C ( x ) x exp (cid:18) i (cid:82) √ E λ +2450) dxC ( x ) (cid:98) ψ ( x ) (cid:19) (11) Remark 1.
We know that if the first variational equation is fuchsian, then the next variational equations are fuchsiantoo. Thus, the solutions given by Eq. (11) have no exponential behaviour around the singular points. In fact, it ispossible to verify the existence of enough logarithms in the integrand in the exponentials of (cid:98) ψ i , i = 3 , , in Eq. (11) . We recall that cubic polynomials C ( x ) and C ( x ) are given in Eq. (9). Thus, the basis field for Eq. (10) is K = C ( x ) and the Picard-Vessiot extension is L = K ( (cid:98) ψ i , (cid:98) ψ (cid:48) i ) , i = 1 , , , . For instance, the differential Galois group of Eq. (11)is a subgroup of GL (4 , C ) . Thus, we shall obtain the normal variational equation due to it is well known classificationof subgroups of SL (2 , C ) , see [16] and references therein. We observe that in Γ E , the equation (8) is satisfied by the change of variable χ := ψ − ζ , leading to the normalvariational equation: ¨ χ + (cid:18) ˙ xx (cid:19) ˙ χ + (cid:18) x − ˙ x x − λ (cid:19) χ = 0 (12)being x = x ( t ) , ˙ x = ˙ x ( t ) the Taub solution in Γ E . Recalling that α = C ( x )12 x , we obtain the algebraic form of VE corresponding to equation (12), through the change of variables ( x ( t ) , ˙ x ( t ) , χ, ˙ χ, ¨ χ ) (cid:55)→ ( x, √ α, η, √ αη, α (cid:48) η (cid:48) + αη (cid:48)(cid:48) ) , which is the AVE given by η (cid:48)(cid:48) + p ( x ) η (cid:48) + q ( x ) η = 0 ,p ( x ) = λx − x + ExC ( x ) , q ( x ) = − λx +27 x − Ex C ( x ) . (13)Now, the invariant normal form of the equation (13) can be obtained as d ξdx − g ( x ) ξ = 0 , g ( x ) = p ( x )4 + 12 dp ( x ) dx − q, ξ = η · exp (cid:18) (cid:90) xx p ( t ) dt (cid:19) Thus, the invariant normal form of equation (13) corresponds to4 ON - INTEGRABILITY OF A XISYMMETRIC B IANCHI IX MODEL VIA D IFFERENTIAL G ALOIS T HEORY d ξdx = g ( x ) ξ, ξ = η (cid:112) xC ( x ) g ( x ) = λ x − λx +128 Eλx +315 x − Ex +5 E x C ( x ) . (14)We remark that the discriminant for the cubic equation C ( x ) = 0 , i.e., for λx − x + 2 E = 0 , is ∆ = − E λ − λ and it is an important element for the galoisian analysis of the above differential equation, that is, equation (14).Furthermore, the differential equations (7), (10), (12), (13) and (14) over the invariant plane Γ E have the sameconnected identity component of the Galois group because they are linked through change of variables (algebraic) thatallows to preserve the connected identity component of the Galois group, see [18].We recall that a Fuchsian differential equation is a linear homogeneous ordinary differential equation with analyticcoefficients in the complex domain whose singular points are all regular singular points, see [1]. Therefore, thedifferential equations (10), (12), (13) and (14) over the invariant plane have the same set of singularities whichis, using the Cardano’s formula for the cubic equation, given by the set of poles of g including ∞ as follows: Γ = { , ∞ , ρ , ρ , ρ } , where ρ j = ν j (cid:115) − E λ + i √ ν j (cid:115) − E λ − i √ , ν = 1 , j = 0 , , . For suitability, we write the cube roots as follows: ρ j = ν j κ λ + ν j κ , κ = (cid:118)(cid:117)(cid:117)(cid:116)(cid:32) − E + (cid:114) E λ − λ (cid:33) λ , ν = 1 , j = 0 , , . These five singularities are of the regular type, which implies that these differential equations are fuchsian differentialequations with five regular singularities.To obtain the solutions of Eq. (13) and Eq. (14), we consider ∆ (cid:54) = 0 , E (cid:54) = 0 and λ (cid:54) = 0 . Due to χ = ψ − ζ we have η = (cid:98) ψ − (cid:98) ζ . Thus, setting µ = i (cid:112) E λ + 2450) (cid:54) = 0 we have the following solutions for η in Eq. (13), according to Eq. (11): η ( x ) = (cid:98) ψ ( x ) − (cid:98) ζ ( x ) = (cid:113) C ( x ) x exp (cid:18) µ (cid:82) √ xdxC ( x ) √ C ( x ) (cid:19) η ( x ) = (cid:98) ψ ( x ) − (cid:98) ζ ( x ) = (cid:113) C ( x ) x exp (cid:18) − µ (cid:82) √ xdxC ( x ) √ C ( x ) (cid:19) η ( x ) = (cid:98) ψ ( x ) − (cid:98) ζ ( x ) = 0 η ( x ) = (cid:98) ψ ( x ) − (cid:98) ζ ( x ) = 0 (15)For suitability we reordered the basis of solutions of Eq. (11) to obtain as basis of solutions of Eq. (13) the setconformed by η ( x ) = (cid:98) ψ ( x ) − (cid:98) ζ ( x ) = (cid:98) ψ ( x ) and η ( x ) = (cid:98) ψ ( x ) − (cid:98) ζ ( x ) = (cid:98) ψ ( x ) in Eq. (15). Now, thesolutions of Eq. (14) are given by the expression ξ k ( x ) = (cid:112) xC ( x ) η k ( x ) = 32 (cid:112) xC ( x ) (cid:98) ψ − k ( x ) , k = 1 , , which is given as follows: ξ ( x ) = (cid:113) C ( x ) x · (cid:112) C ( x ) · exp (cid:18) µ (cid:82) √ xdxC ( x ) √ C ( x ) (cid:19) ξ ( x ) = (cid:113) C ( x ) x · (cid:112) C ( x ) · exp (cid:18) − µ (cid:82) √ xdxC ( x ) √ C ( x ) (cid:19) (16)Now we are interested in the obtaining of the identity connected component of the Galois group for Eq. (14) to applyTheorem 2, see A for details and theoretical background. We recall that the differential field for Eq. (14) is K = C ( x ) and the corresponding Picard Vessiot extension is L = K ( ξ ( x ) , ξ (cid:48) ( x )) . We observe that the logarithmic derivatives5 ON - INTEGRABILITY OF A XISYMMETRIC B IANCHI IX MODEL VIA D IFFERENTIAL G ALOIS T HEORY of the solutions are algebraic functions, therefore the differential Galois Group G := G ( L /K ) of V E is not aconnected group. We denote by K the algebraic functions over K , contained in the Picard-Vessiot L . Therefore, theconnected identity component of the differential Galois group, i.e., ( G ) = G ( L /K ) . Now, to compute the Galoisgroup G ( L /K ) we start Writing Eq. (14) as a linear differential system, thus we obtain (cid:18) ξξ (cid:48) (cid:19) (cid:48) = (cid:18) g ( x ) 0 (cid:19) (cid:18) ξξ (cid:48) (cid:19) (17)The fundamental matrix of Eq. (17) is U = (cid:18) ξ ξ ξ (cid:48) ξ (cid:48) (cid:19) (18)We compute the action of the Galois Group G ( L /K ) over the fundamental matrix (18), therefore σ , element of G ( L /K ) , satisfies σU = σ (cid:18) ξ ξ ξ (cid:48) ξ (cid:48) (cid:19) = (cid:18) ξ ξ ξ (cid:48) ξ (cid:48) (cid:19) A σ , A σ = (cid:18) δ δ − (cid:19) ∈ G m . Since ( G m , · ) is isomorphic to the multiplicative group ( C ∗ , · ) , we conclude that G ( L /K ) is an abelian group. Werecall that a Picard-Vessiot extension is purely transcendental if and only if its Galois group is connected, see A. For theinterested reader, in B we present in detailed way the solutions and Galois group G = G ( L /K ) , with K = C ( x ) , bythe direct application of Kovacic Algorithm.As we are looking for obstructions to the integrability of the axisymmetric Bianchi IX cosmological model, taking intoaccount Theorem 3, we must to consider the second variational equation over K , because unfortunately the identitycomponent of the Galois group of the first variational equation is abelian. In this way, we should compute the secondvariational equation V E . To obtain it V E , see A and references [21, 5], we make the change of variables ( A, B ) (cid:55)→ ( A s + (cid:15)C A + (cid:15) D A , B s + (cid:15)C B + (cid:15) D B ) , (19)where ( As, Bs ) is a particular solution of Eq. (4). Next step is to apply the change of variable given in Equation (19)into Equation (4). Using the invariant plane Γ = Γ E , provided by A = B, ˙ A = ˙ B and collecting the coefficients of (cid:15) we obtain the second variational equation related with Eq. (7). The second variational equation related to Eq. (8) comesfrom the collecting of coefficient of (cid:15) after the restriction of Γ = Γ E in Equation (7), written in terms of ˆ ψ and ˆ ζ , i.e.we get the second variational equation along the Taub solution. Recall that Γ = Γ E is provided by Equation (6) (somecomputations were made with Maple).Now, in a similar way as for Equation (12), we write one equation coming from the difference ε = ˆ ψ − ˆ ζ , which nowreads ¨ ε + (cid:0) ˙ xx (cid:1) ˙ ε + (cid:16) x − ˙ x x − λ (cid:17) ε = f ( χ , ˙ χ , ¨ χ , ψ , ˙ ψ , x, ˙ x ) ,f ( χ , ˙ χ , ¨ χ , ψ , ˙ ψ , x, ˙ x ) = (cid:16)(cid:16) λx + ˙ x x − x (cid:17) ψ + ˙ xx ˙ ψ (cid:17) χ − x χ ¨ χ + (cid:16) λx + ˙ x x − x (cid:17) χ − x ˙ χ − (3 ˙ xx ψ + x ˙ ψ ) ˙ χ − x ψ ¨ χ − ˙ xx χ ˙ χ , (20)where x = x ( t ) , ˙ x = ˙ x ( t ) is the Taub solution in Γ E , ψ is one solution of equation (8) and χ is one solution ofequation (12). Applying again the algebrization process through the Hamiltonian change of variable x = x ( t ) , such asfor Equation (10), we obtain the algebraic form for equation (20) as follows. ε (cid:48)(cid:48) + p ( x ) ε (cid:48) + q ( x ) ε = (cid:101) fx C ( x ) , (cid:101) f = (28 λx − x + 2 E ) η − x (4 λx − η η (cid:48) − x C ( x ) η η (cid:48)(cid:48) + xC ( x ) (cid:99) ψ (cid:48) η + (28 λx − x + 2 E ) (cid:98) ψ η − x (28 λx − x + 2 E ) (cid:98) ψ η (cid:48) − x C ( x ) (cid:98) ψ (cid:48) η (cid:48) − x C ( x ) (cid:98) ψ (cid:48) η (cid:48)(cid:48) , (21)where p ( x ) and q ( x ) are given in Eq. (13), (cid:98) ψ is solution of Eq. (10) (given explicitly in Eq. (11)) and η is onesolution of Eq. (13). Moreover, due to η (cid:48)(cid:48) = − pη (cid:48) − qη and (cid:98) ψ = η we can simplify the expression for (cid:101) f as follows (cid:101) f = − (52 λx − x + 14 E ) η − x (4 λx − λx − x + (2 E + 12) x − E )( η (cid:48) ) − (28 λx − λx − x + (14 E + 99) x − E ) η · η (cid:48) ON - INTEGRABILITY OF A XISYMMETRIC B IANCHI IX MODEL VIA D IFFERENTIAL G ALOIS T HEORY
Now, writing η (cid:48) in terms of η and x we have η (cid:48) = (cid:18) xC ( x ) C (cid:48) ( x ) − C ( x ) C ( x ) + 2 µx / C ( x )2 xC ( x ) C ( x ) (cid:19) η and therefore (cid:101) f becomes (cid:101) f = P ( √ x ) C ( x ) η = P ( √ x ) C ( x ) (cid:112) xC ( x ) ξ , P ( √ x ) = (cid:88) k =0 a k ( √ x ) k , where η = ξ √ xC ( x ) and ξ is given in Eq. (16). The coefficients of the polynomial P are a = − λ , a = 0 , a = 4900 λ , a = − λ µ, a = λ ,a = − λ µ, a = λ (( − E + 96775) λ − µ ) , a = − λµ,a = 1120 λ E + µ λ + 1771350 λ, a = − λµ ( E + ) ,a = (21560 E + 632100) λ + 2 µ , a = − λEµ + 2520 µ,a = − λE − µ + 15400 λ − µ + 6460650 , a = ( − E − µ,a = 64 λE + Eµ + 231420 E + 1367100 , a = − Eµ ( E + 16) ,a = 1784 E + 52080 E, a = − E µ, a = − E ( E − . In a similar way as for equations (13) and (14), we obtain the invariant form of equation (21) through the change ofdependent variable ε = ω (cid:112) xC ( x ) as follows. ω (cid:48)(cid:48) = g ( x ) ω + P ( √ x ) x / C ( x ) / C ( x ) ξ ( x ) , (22)where g ( x ) is given in equation (14).We must solve an inhomogeneous second order differential equation, Eq. (22). Thus, computing the wronskian W := W ( ξ ( x ) , ξ ( x )) = 18 i (cid:112) E λ + 14700 (23)and using the method of variation of parameters we obtain as solutions of (22) the following ω ( x ) = c ξ ( x ) + c ξ ( x ) + ω p ( x ) ω p ( x ) = − ξ ( x ) W (cid:82) P ( √ x ) x C ( x ) / C ( x ) ξ ( x ) dx + ξ ( x ) W (cid:82) P ( √ x ) x / C ( x ) / C ( x ) ξ ( x ) dx (24)It is convenient to write the inhomogeneous equation, Eq. (22), as an homogeneous one, using as a new variable ϕ = ξ .Thus, ω (cid:48)(cid:48) = g ( x ) ω + s ( x ) ϕ ( x ) , s ( x ) = P ( √ x ) x / C ( x ) / C ( x ) ϕ (cid:48) = 2 h ( x ) ϕ, h ( x ) = (log( ξ ( x ))) (cid:48) (25)The key point is h ( x ) = (log( ξ ( x ))) (cid:48) belongs to K = C ( x ) , i.e., the field of coefficients of Eq. (25) becomes K .Rewritten it in matrix form (cid:32) ωω (cid:48) ϕ (cid:33) (cid:48) = (cid:32) g ( x ) 0 s ( x )0 0 2 h ( x ) (cid:33) (cid:32) ωω (cid:48) ϕ (cid:33) . (26)The fundamental matrix of Eq. (26) is Φ = ξ ξ ω p ξ (cid:48) ξ (cid:48) ω (cid:48) p ϕ . (27)The Picard-Vessiot Extension of Eq. (26) is K ( ξ , ξ (cid:48) , ω p , ω (cid:48) p ) ⊃ K . We compute the action of the Galois Group G over the fundamental matrix (27), recall that σ , element of G , satisfies σ ( ξ ) = δξ and σ ( ξ ) = δ − ξ . Therefore σ ( ϕ ) = σ ( ξ ) = δ ϕ . Finally, setting I := (cid:90) P ( √ x ) x C ( x ) / C ( x ) ξ ( x ) dx := (cid:90) y ( x ) ξ ( x ) dx, ON - INTEGRABILITY OF A XISYMMETRIC B IANCHI IX MODEL VIA D IFFERENTIAL G ALOIS T HEORY I := (cid:90) P ( √ x ) x / C ( x ) / C ( x ) ξ ( x ) dx := (cid:90) y ( x ) ξ ( x ) dx, we observe σ ( I (cid:48) ) = y σ ( ξ ) and σ ( I (cid:48) ) = y σ ( ξ ) , because y i ∈ K , i = 1 , . Hence, σ ( I ) = δI + γ and σ ( I ) = δ I + γ . Taking into account the expression for ω p in Eq. (24), as W ∈ K ⊂ K , we obtain σ ( ω p ) = σ (cid:18) − ξ W I + ξ W I (cid:19) = δγ ξ + δ − γ ξ + δ ω p , where δ ∈ C ∗ , γ , γ ∈ C . Therefore, σ satisfies σ ξ ξ ω p ξ (cid:48) ξ (cid:48) ω (cid:48) p ϕ = ξ ξ ω p ξ (cid:48) ξ (cid:48) ω (cid:48) p ϕ B σ , B σ = δ δγ δ − δ − γ δ . The matrix B σ ∈ G can be written as follows B σ = δ δ −
00 0 δ · (cid:32) γ γ (cid:33) . According to [15, §8.4], G = D (cid:110) N , is the semidirect product between the Diagonal subgroup of G (left factor in thedecomposition of B σ ) and the Normal Unipotent subgroup of G (right factor in the decomposition of B σ ).Then we state the following conjecture. Conjecture 1.
We will assume that, generically for λ and E , γ and γ are not simultaneously zero By the differential Galois theory the above conjecture means that either I or I do not belong to L = C ( z, ξ , ξ (cid:48) ) , i.e.,these integrals are not expressed as algebraic functions of z , ξ and ξ (cid:48) . This conjecture is natural because, as usuallyhappens, when you integrate functions in a differential field you do not obtain functions in the same field. Moreover,the proof of this conjecture seems to us that it would be quite involved: as far as we know there are not algorithms, inthe literature, to solve this problem. More elementary situations were studied in the papers [3, 6].As the Galois group G is connected and non abelian, using Theorem 3, we obtain the result that we are looking for: Theorem 1.
Assuming Conjecture 1, the Axisymmetric Bianchi IX cosmological model is not meromorphicallyintegrable for generic values of ( E, λ ) . To illustrate the conjecture about the non integrability of the axisymmetric Bianchi IX models with cosmologicalconstant and reduced to two degrees of freedom, we calculate Poincaré sections through the surface P B = 0 and ˙ P B > . It is not our intention to make a through investigation of the section phase space, so we have chosen to viewselected values of the ( λ, energy ) pair.First of all let us look at the behaviour of the dynamics taking a fixed value of λ , λ = 0 . , and varying the energy: . , . , . and . . We have left in all figures some island regions empty to make the viewing better. Around A = 1 the iterates scape to infinity to the De Sitter attractor. When the value of the energy increases the islands becomebigger. This feature appears more clearly in the bottom panels of fig 1 which have higher energy valuesOne could want to know how the sections vary with a fixed value of the energy and different values of λ . In fig 2 top wehave chosen E = 0 . and varied λ from . to . . Note that the quantity of islands increase as λ increases. As in fig1 the border of the Poincaré section are escaping points to larger regions of the phase space.In fig 3 we show the sections when most of the iterates have escaped and return to the still ordered region; in fig 3 top λ = 0 . and the energy varies from . to . ; in the bottom case λ = 1 . and the energy goes from . to . . Note that the iterates which are not in the islands have the appearance of stable or unstable manifold sembracing the island region. This is the point to recall that there is a saddle centre equilibrium point in the dynamics in A = B = √ λ = E . This point is a scatterer for the dynamics taking away and bringing back iterates of the Poincarésection.An interesting feature is found for the system with energy equals to zero and λ = 0 . The fixed point in this case A = B = √ λ = E → ∞ . In Fig 5 we show 30000 initial conditions integrated each 300. The behaviour is continuosand no trajectory scape to infinity. 8 ON - INTEGRABILITY OF A XISYMMETRIC B IANCHI IX MODEL VIA D IFFERENTIAL G ALOIS T HEORY
Figure 1: ( A, P A ) projection of the section P B = 0 , ˙ P B > . λ is fixed at . and the energy has the following values . , . (top) and . , . (bottom). Note the increase of the island region as the energy increases.Figure 2: ( A, P A ) projection. The energy is fixed at . . Top, from left to right : λ = 0 . and . ; bottom: λ = 1 . and . from left to right. The region with more islands is the one with larger value of λ .9 ON - INTEGRABILITY OF A XISYMMETRIC B IANCHI IX MODEL VIA D IFFERENTIAL G ALOIS T HEORY
Figure 3: Top: λ = 1 . and energy = 0 . (left) and . (right). Bottom λ = 1 . with energy = 0 . (left)and . (right). The more organized region is destroyed and there tongs of iterates embracing the reducedisland region.Figure 4: Top: left, ABP A projection of some trajectories with zero energy and λ = 3 . ; right, the same apparentperiodic trajectories projected in the plane ( A, P A ) . Bottom: left, the periodic orbits are interrupted by escapingtrajectories around . . The saddle centre is located at A f = − . ; right, the ( A, B ) projection showing theexit in the scale B . 10 ON - INTEGRABILITY OF A XISYMMETRIC B IANCHI IX MODEL VIA D IFFERENTIAL G ALOIS T HEORY
Figure 5: Right: ( A, P A ) and left ( A, B ) projection. Compare the scales with these of the previous figure. Since thecritical point was moved to infinity by taking λ = 0 the family of periodic orbits remains all over the phase space.Obviously there is no escaping orbits. A Differential Galois Theory and Hamiltonians
Differential Galois Theory, see [11, 23] among others, also known as Picard-Vessiot theory, is the Galois theory of lineardifferential equations. We recall that rational functions over C are denoted by C ( x ) . Consider differential equations inthe form L := y (cid:48)(cid:48) + ay (cid:48) + by = 0 , being a, b ∈ C ( x ) , being K = C ( x ) the differential field of L . Suppose that y , y is a basis of solutions of L . Let F = K ( y , y , y (cid:48) , y (cid:48) ) be the differential extension of K such that C is the field ofconstants for K and F . In these terms, we say that F , the smallest differential field containing K and { y , y , y (cid:48) , y (cid:48) } ,is the Picard-Vessiot extension of K for L . Therefore, the group of all K -differential automorphisms of F over K iscalled the differential Galois group , or simply the Galois group , of F over K and is denoted by G ( F/K ) . In terms ofdifferential Galois theory the notion of integrability is the following:We say that the differential equation L is integrable if and only if the Picard-Vessiot extension F ⊃ K is obtained as atower of differential fields K = F ⊂ F ⊂ · · · ⊂ F m = F such that F i = F i − ( η ) for i = 1 , . . . , m , where either1. η is algebraic over F i − , that is η satisfies a polynomial equation with coefficients in F i − .2. η is primitive over F i − , that is η (cid:48) ∈ F i − .3. η is exponential over F i − , that is η (cid:48) /η ∈ F i − .Kolchin Theorem states the relation between the integrability of of L and the solvability of the connected identitycomponent of its differential Galois group as follows: Kolchin Theorem.
The equation L is integrable if and only if the connected identity component of the differentialGalois group is a solvable group. We denote by ( G ( F/K )) the connected identity component of the differential Galois group of the extension F overthe differential field K . Picard-Vessiot extension is purely transcendental if and only if its Galois group is connected,that is, ( G ( F/K )) = G ( F/K ) , see [11] and references therein for further details.The reduced form, also known as invariant normal form, of the equation L is as follows: R := ζ (cid:48)(cid:48) = rζ, r ∈ K. We recall that equation R can be obtained from equation L through the change of variable y = e − (cid:82) a ζ, r = a a (cid:48) − b. Through the change of variable v = ζ (cid:48) /ζ we get the associated Riccati equation to equation R as follows: v (cid:48) = r − v , v = ζ (cid:48) ζ , ON - INTEGRABILITY OF A XISYMMETRIC B IANCHI IX MODEL VIA D IFFERENTIAL G ALOIS T HEORY where r is given by the above equation. Moreover, the above Riccatti equation has one algebraic solution over thedifferential field K if and only if the differential equation R is integrable.The concepts related with integrability of Hamiltonian Systems are very well known. The Hamilton Equations for theHamiltonian H are ˙ x i = ∂H/∂y i , ˙ y i = − ∂H/∂x i , i = 1 , ..., n One says that X H = ( ∂H/∂y i , − ∂H/∂x i ) , i = 1 , ..., n , is completely integrable or Liouville integrable if there are n functions f = H , f ,..., f n , such that(1) they are functionally independent i.e., the 1-forms df i i = 1 , , ..., n , are linearly independent over a dense open set U ⊂ M , ¯ U = M ;(2) they form an involutive set, { f i , f j } = 0 , i, j = 1 , , ..., n .We remark that in virtue of item (2) above the functions f i , i = 1 , ..., n are first integrals of X H . It is very importantto be precise regarding the degree of regularity of these first integrals. Here we assume that the first integrals aremeromorphic.Given a dynamical system of first order, ˙ z = X H ( z ) , we can write the variational equations along a particular integral curve z = φ ( t ) of the Hamiltonian vector field X H ˙ ξ = X (cid:48) H ( φ ( t )) ξ. We denote by Γ the Riemann surface representing the integral curve z = z ( t ) which is not an equilibrium point of thevector field X H . The following theorem is included in a new Theory of Integrability of Dynamical Systems usingDifferential Galois Theory, see [19, 20] and see also [18]: Theorem 2 (Meromorphically non-integrability theorem, [19]) . Assume a complex analytic Hamiltonian system ismeromorphically completely integrable in a neighborhood of the integral curve z = φ ( t ) . Then the identity componentsof the Galois groups of the variational equations are commutative groups. The previous theorem has been generalized to higher order variational equations VE k along Γ , with k > , V E beingthe first variational equation VE.The “fundamental” solution of VE k of our dynamical system is given by ( φ (1) ( t ) , φ (2) ( t ) , . . . , φ ( k ) ( t )) , where φ ( z, t ) = φ ( z , t ) + φ (1) ( t )( z − z ) + . . . + 1 k ! φ ( k ) ( t )( z − z ) k + . . . the Taylor series up to order k of the flow φ ( z, t ) with respect to the variable z at the point ( z , t ) . That is, φ ( k ) ( t ) = ∂ k ∂z k φ ( z , t ) . The initial conditions are φ (1) (0) = Id m and φ ( j ) (0) = 0 for all j > . Since through a linearization process we can consider the equations VE k as linear differential equations, we can talkabout their Picard-Vessiot extensions and about their Galois groups G k . Thus, we arrive to the following theorem, see[21]. Theorem 3 (Higher order variational equation theorem, [21]) . Assume that a complex analytical Hamiltonian systemis integrable by meromorphic first integrals in a neighborhood of the integral curve z = φ ( t ) . Then the identitycomponents ( G k ) , k ≥ , of the Galois groups of the variational equations along Γ are commutative. Recently, in [3, 5], the higher order variational equation theorem where applied to obtain non-integrability of planarvector fields and Painlevé Equations. The explicit formulas presented in [3, 5] are also useful here. In particular, toobtain the first and second variational equation of a Hamiltonian system ˙ z = X H , with particular solution γ ( t ) , wemake the changes of variables x i = x is ( t ) + (cid:15)w i + (cid:15) r i , y i = y is ( t ) + (cid:15)χ i + (cid:15) φ i , γ ( t ) = ( x s , . . . , x ns , y s , . . . , y ns ) , ON - INTEGRABILITY OF A XISYMMETRIC B IANCHI IX MODEL VIA D IFFERENTIAL G ALOIS T HEORY into the Hamilton equations ˙ z = X H ( z ) . The first variational equation is obtained collecting the coefficients of (cid:15) in theprevious Hamilton equations, while the second variational equation is obtained by the collecting of the coefficient of (cid:15) in the previous Hamilton equations. B Application of Kovacic Algorithm
To apply Kovacic Algorithm we follow the references [16] and [2]. Thus, in order to apply Kovacic Algorithm, for thegeneral case ∆ (cid:54) = 0 , E (cid:54) = 0 and λ (cid:54) = 0 , we start considering the set of poles of g that including the point at infinity isgiven by Υ = { , ρ , ρ , ρ , ∞} . Moreover, we observe that ◦ ( g c ) = 2 for all c ∈ Υ , i.e., the order of each pole of g is . Thus, we could fall in any case of Kovacic Algorithm and we should check conditions c and ∞ for each of them.Moreover, Laurent expansion of g around each c ∈ Υ . Therefore we have: g ( x ) = b x + . . . , around x = 0 , b = 516 g ( x ) = b ∞ x + . . . , around x = ∞ , b ∞ = 2 g ( x ) = b ρ ( x − ρ ) + . . . , around x = ρ , b ρ is given by λ κ (128 λ κ ρ − λρ + 128 Eλρ + 315 ρ − Eρ + 5 E )(3( κ / + λ )( κ / + λκ / + λ )) . In a similar way we obtain expressions for the coefficients of x − in the Laurent expansion of g around ρ and ρ respectively. Thus, b ρ and b ρ will include algebraic combinations of ρ , ρ and ρ with some terms of b ρ .For suitability, we start with case 1 and after we follow with case 3 to discard them, showing that solutions are providedby case 2. Thus, starting with case 1 of Kovacic we obtain [ √ g ] c = 0 and α ± c = ±√ b c . In particular, α +0 = 54 , α − = − , α + ∞ = 2 , α −∞ = − , α ± ρ j / ∈ Q . By step 2 we obtain that does not exists a polynomial of degree d satisfying at least one of the following options: d = α ±∞ − ( α ± + α ± ρ + α ± ρ + α ± ρ ) ∈ Z + . Therefore, case 1 of Kovacic Algorithm is discarded and now we consider the case 3. By step 1 we have the samevalues for b c as for the previous cases. Thus, for c ∈ Υ \ {∞} and n ∈ { , , } we have E c = { ± k (cid:112) b c : 0 ≤ k ≤ } ∩ Z , E ( n ) ∞ = (cid:26) ± kn (cid:112) b ∞ : 0 ≤ k ≤ , (cid:27) ∩ Z Therefore we obtain the following sets E = { k − ≤ k ≤ } , E ρ j = { } , E ( n ) ∞ = (cid:26) ± kn : 0 ≤ k ≤ (cid:27) , j = 1 , , . In particular we have E (4) ∞ = { ± k : 0 ≤ k ≤ } , E (6) ∞ = { ± k : 0 ≤ k ≤ } , E (12) ∞ = { ± k : 0 ≤ k ≤ } . By step two, the set D (cid:54) = ∅ is as follows: D = (cid:110) d ∈ Z + : d = n
12 ( e ( n ) ∞ − e − e ρ − e ρ − e ρ ) (cid:111) . That is, there exist some non-negative integer values of d satisfying d = 3 k − n k + 3) , ≤ k ≤ . Such values of d , depending of k as in E c and in E ( n ) ∞ , for n ∈ { , , } correspond to the following. n = 4 . In this case, d ∈ { k − ≤ k ≤ } , e ∈ { k − ≤ k ≤ } , e ∞ ∈ { k + 6 : 2 ≤ k ≤ } . n = 6 . In this case, d ∈ { k − ≤ k ≤ } , e ∈ { k − ≤ k ≤ } , e ∞ ∈ { k + 6 : 3 ≤ k ≤ } , n = 12 . In this case we cannot obtain there are not values for d ∈ Z + .13 ON - INTEGRABILITY OF A XISYMMETRIC B IANCHI IX MODEL VIA D IFFERENTIAL G ALOIS T HEORY . By step 2 we obtain the polynomial S = x ( x − ρ )( x − ρ )( x − ρ ) and the rational function θ is given by θ = n (cid:88) c ∈ Υ \{∞} e c x − c = n (cid:18) e x + e ρ x − ρ + e ρ x − ρ + e ρ x − ρ (cid:19) . Finally, by step 3, we set P n = − P d and P i − = − SP (cid:48) i + (( n − i ) S (cid:48) − Sθ ) P i − ( n − i )( i + 1) S gP i +1 . Due to this recurrence is not satisfied, because we arrive to P − (cid:54) = 0 for all d obtained previously with n ∈ { , } , wediscard case 3 of Kovacic Algorithm. Due to case 1 and case 3 were discarded, we proceed to analyse case 2 of KovacicAlgorithm.By step 1 of case 2 we obtain E c = { , − (cid:112) b c , (cid:112) b c , ∀ c ∈ Υ } ∩ Z . Now we should find the set D corresponding to the possible degrees of the polynomials P d . Then D = (cid:26) d ∈ Z + : d = 12 ( e ∞ − e − e ρ − e ρ − e ρ ) (cid:27) ,e ∈ {− , , } , e ∞ ∈ {− , , } , e ρ j = 2 . Therefore D = { } , because the only one option to have a non-negative integer for d is setting e ∞ = 8 and e c = 2 forall c ∈ Υ \ {∞} . For instance the polynomial P ( x ) = 1 and the rational function θ , corresponding to the step 2 isgiven by θ = (cid:88) c ∈ Υ \{∞} (cid:18) e c x − c (cid:19) = 1 x + 1 x − ρ + 1 x − ρ + 1 x − ρ . Finally, by step 3 we have that P and θ satisfy θ (cid:48)(cid:48) + 3 θθ (cid:48) + θ − gθ − g (cid:48) ≡ . Thus, the solution of the solutions ξ and ξ given by ξ , = e (cid:82) ω , where ω − θω + 12 ( ω (cid:48) + ω − g ) = 0 . For instance, the explicit solutions are given by ξ = (cid:113) C ( x ) x · (cid:112) C ( x ) · exp (cid:16) i (cid:82) (cid:113) x ( E λ +2450) C ( x ) C ( x ) dx (cid:17) ξ = (cid:113) C ( x ) x · (cid:112) C ( x ) · exp (cid:16) − i (cid:82) (cid:113) x ( E λ +2450) C ( x ) C ( x ) dx (cid:17) (28)We recall that the differential field for equation (14) is K = C ( x ) . Thus, the corresponding Picard Vessiot extension is L = C ( x ) (cid:104) ξ ( x ) , ξ (cid:48) ( x ) (cid:105) . Computing the differential Galois group, we observe that for σ ∈ Gal ( L/ C ( x )) , it satisfies σ ( ξ ( x ) · ξ ( x )) = ( ξ ( x ) · ξ ( x )) , σ (cid:0) (log ( ξ ( x ) ξ ( x ))) (cid:48) (cid:1) = (log ( ξ ( x ) ξ ( x ))) (cid:48) ,σ (cid:32)(cid:18) log (cid:18) ξ ( x ) ξ ( x ) (cid:19)(cid:19) (cid:48) (cid:33) = (cid:32)(cid:18) log (cid:18) ξ ( x ) ξ ( x ) (cid:19)(cid:19) (cid:48) (cid:33) ,ξ ( x ) · ξ ( x ) / ∈ C ( x ) , (cid:18) log (cid:18) ξ ( x ) ξ ( x ) (cid:19)(cid:19) (cid:48) / ∈ C ( x ) . That is, ( ξ ( x ) · ξ ( x )) = C ( x ) C ( x ) x . ON - INTEGRABILITY OF A XISYMMETRIC B IANCHI IX MODEL VIA D IFFERENTIAL G ALOIS T HEORY (log ( ξ ( x ) · ξ ( x ))) (cid:48) = 560 λ x + 1365 λx + 191 Eλx − x + 210 Ex − E xC ( x ) C ( x ) (cid:32) log (cid:18) ξ ( x ) ξ ( x ) (cid:19) (cid:48) (cid:33) = − x ( E λ + 2450) C ( x ) C ( x ) . Moreover, the wronskian of ξ ( x ) and ξ ( x ) , which was given in Eq. (23), is constant.Thus, we obtain that the differential Galois group is G ( L/K ) = D ∞ (the infinite dihedral group). Acknowledgements
PA-H acknowledges and thanks the support of COLCIENCIAS through grant numbers FP44842-013-2018 of the
Fondo Nacional de Financiamiento para la Ciencia, la Tecnología y la Innovación . JM is member of the UniversidadPolitécnica de Madrid research group
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