This is a preprint of: “Dynamics of two Einstein-Friedmann cosmological models”, Inna Basak, Jaume Llibre, Eur. Phys. J. Plus, vol. 136, 8:1–7, 2021. DOI: [10.1140/epjp/s13360-020-01000-8] DYNAMICS OF TWO EINSTEIN-FRIEDMANN COSMOLOGICAL MODELS INNA BASAK AND JAUME LLIBRE Abstract. We describe completely the dynamics of the two Einstein-Friedmann cosmological models, which can be characterized by the Hamiltonians 1 H = (p2y − p2x ) + e2x V (y), 2 whith the cosmological potentials V (y) = eλy , or V (y) = (a + by)ey with λab 6= 0. 1. Introduction and statement of the main results The present work is devoted to the Einstein-Friedmann cosmological models, which can be characterized by the Hamiltonian 1 2 (1) H= (p − p2x ) + e2x V (y), 2 y where V (y) = eλy or V (y) = (a + by)ey with λab 6= 0 are cosmological potentials. For more details on these two special models see subsections 2.2 and 3.1 [10], and for more details on the general Einstein-Friedmann cosmological models see [4, 7]. The Hamiltonian system with two degrees of freedom associated to the Hamil- tonian 1 (2) H = (p2y − p2x ) + e2x+λy 2 is ∂H ẋ = = −px , ∂px ∂H ẏ = = py , ∂py (3) ∂H p˙x = − = −2e2x+yλ , ∂x ∂H p˙y = − = −λe2x+yλ . ∂y The above Hamiltonian H has the additional first integral (4) F = 2py − λpx , 2010 Mathematics Subject Classification. 37J05, 37J15, 37J35. Key words and phrases. Hamiltonian systems, Einstein-Friedmann cosmological models. 1 2 I. BASASK AND J. LLIBRE as it is easy to check. The rank of the 2 × 4 matrix  2x+yλ  2e λe2x+yλ −px py 0 0 −λ 2 is two except at a zero Lebesgue measure set in the phase space E = R4 with coordinates (x, y, px , py ). Therefore the first integrals H and F are independent. One can verify that the Poisson bracket {H, F } = 0, so the first integrals H and F are in involution. Therefore, the Hamiltonian system (3) is completely integrable in the sense of the Liouville-Arnold Theorem, see for instance [1, 2]. Clearly, the above Hamiltonian system has no equilibrium points. Therefore, all values (h, f ) for the map H × F are regular, in particular the level (H, F )−1 (0, f ) is regular, so if (H, F )−1 (0, f ) is not empty, it is a 2-dimensional manifold, for more details see for instance [5]. Since H and F are first integrals, the sets Ih = {(x, y, px , py ) ∈ E : H(x, y, px , py ) = h}, (5) If = {(x,Ty, px , py ) ∈ E : F (x, y, px , py ) = f }, Ihf = Ih If . are invariant by the Hamiltonian flow, i.e. if an orbit solution of the Hamiltonian system has a point in one of the previous three sets, the whole orbit is contained in that set. From physical reasons we are only interested in the dynamics of the Hamiltonian system (3) on the energy level H = 0, see [10]. Following the Liouville-Arnold The- orem, since the values (0, f ) are regular for all f ∈ R, every connected component of the invariant 2-manifold I0f is diffeomorphic either to a torus, to a cylinder or to a plane, see Theorem 3, and the dynamics on them are conjugated to a linear flow when the flow is complete, i.e. when the orbits are defined for all time t ∈ R. Theorem 1. The following statements hold. (a) If f = 0 and |λ| > 2, then I0f is empty. (b) If f 6= 0 and |λ| 6= 2, then I0f is an invariant 2-dimensional manifold diffeomorphic to two copies of R2 . (c) Assume λ = 2. If f = 0, then I0f is empty. If f 6= 0, then I0f is an invariant 2-dimensional manifold diffeomorphic to R2 . (d) Assume λ = −2. If f = 0, then I0f is empty. If f 6= 0, then I0f is an invariant 2-dimensional manifold diffeomorphic to R2 . (d) All the orbits of the Hamiltonian system (3) restricted to I0f come from the infinity and go to infinity. Theorem 1 is proved in section 2. Now we consider the Hamiltonians of the form (6) H = p2y − p2x + ex V (y), that can be obtain from Hamiltonian (1) by applying canonical transformation {x, y, px , py } → {1/2x, 1/2y, 2px , 2py } EINSTEIN-FRIEDMANN COSMOLOGICAL MODELS 3 and substituting {H, V (y)} → {−2H, −2V (y)}. Following [10] we consider the potential (7) V (y) = (a + by)ey with ab 6= 0. Then the Hamiltonian system with Hamiltonian (6) has the first integral (8) F (x, y, px , py )p2y − = px py + ex+y (a + b + by)/2. The condition {H, F } = 0 proves that F is a first integral of the following Hamil- tonian system ∂H ẋ = = −2px , ∂px ∂H ẏ = = 2py , ∂py (9) ∂H p˙x = − = −ex+y (a + by), ∂x ∂H p˙y = − = −ex+y (a + b + by). ∂y Therefore this Hamiltonian system is completely integrable in the sense of Liouville- Arnold Theorem, because the rank of the 2 × 4 matrix   ex+y (a + by) ex+y (a + b + by) −2px 2py ex+y (a + b + by)/2 ex+y (a + 2b + by)/2 −py 2py − px is two except in a zero Lebesgue measure set of the phase space E = R4 again in the coordinates (x, y, px , py ). Hence the first integrals H and F are independent. As in the previous section the above Hamiltonian system has no equilibrium points and the levels (H, F )−1 (0, f ) are regular. Theorem 2. The following statements hold for the Hamiltonian system (9). (a) The set I0f is empty if b < 0 and f ≤ 0, otherwise I0f is an invariant 2-dimensional manifold diffeomorphic to two copies of R2 . (b) All the orbits of the Hamiltonian system (3) restricted to I0f come from the infinity and go to infinity. Theorem 2 is proved in section 3. 2. Proof of Theorem 1 In order to describe the dynamics of a complete integrable Hamiltonian system of two degrees of freedom we shall use the Liouville-Arnold Theorem, which can be stated as follows, for more details see [1, 2, 5]. Theorem 3. Assume that a Hamiltonian system with two degrees of freedom de- fined on the phase space E has two independent first integrals H and F in involution. Then the following statements hold. 4 I. BASASK AND J. LLIBRE (a) Suppose that (h, f ) is a regular value of the function (H, F ), so I0f is a 2-dimensional submanifold of E invariant under the flow of the system. ∗ (b) Under the hypothesis (a) if the flow on a connected component Ihf of Ihf is complete, then Ihf is diffeomorphic either to the torus S × S , or to the ∗ 1 1 cylinder S1 × R, or to the plane R × R. ∗ (c) Under the hypothesis (b) the flow on Ihf is conjugated to a linear flow on either S1 × S1 , or S1 × R, or R × R. Unfortunately we cannot apply to our two Hamiltonian systems (3) and (9) the results of Theorem 3 because we do not know if the flows of these Hamiltonian systems are complete, because we cannot obtain explicitly the solutions of these systems in function of the time. But as we shall see later on we can apply to them the Markus–Neumann–Peixoto theorem. From F (x, y, px , py ) = f we obtain that 1 (10) py = (f + λpx ). 2 Assume that |λ| = 6 2. Then, from   1 H x, y, px , (f + λpx ) = 0, 2 we get that p f λ − 2 f 2 − 2 (λ2 − 4) e2x+λy (11) px = ± . 4 − λ2 Substituting px from (11) in (10) we obtain p 2f − λ f 2 − 2 (λ2 − 4) e2x+λy (12) py = ± . 4 − λ2 In order to prove Theorem 1 we must characterize the topology of the 2-dimensional manifolds I0f given by (13) ( ! ) p p f λ − 2 f 2 − 2 (λ2 − 4) e2x+λy 2f − λ f 2 − 2 (λ2 − 4) e2x+λy 2 x, y, ± ,± : (x, y) ∈ R . 4 − λ2 4 − λ2 Note that when in the expression of py there is a plus (respectively minus) in the expression of px there is a plus (respectively minus). In short, when |λ| = 6 2 from (13) it follows that I0f is empty if f = 0 and |λ| > 2, otherwise I0f is an invariant 2-dimensional manifold diffeomorphic to two copies of R2 . This completes the statements (a) and (b) of Theorem 1. Assume λ = 2. Then from F (x, y, px , py ) = f we obtain again that py = (f + 2px )/2, and from H(x, y, px , (f +2px )/2) = 0, we have that px = −2e2(x+y)) /f −f /4. Substituting this expression of px into the previous expression of py we obtain that py = −2e2(x+y)) /f + f /4. In order to prove statement (c) of Theorem 1 we must characterize the topology of the 2-dimensional manifolds I0f given by    2e2(x+y)) 1 2e2(x+y)) 1 2 (14) x, y, − − f, − + f : (x, y) ∈ R . f 4 f 4 EINSTEIN-FRIEDMANN COSMOLOGICAL MODELS 5 Therefore, when λ = 2 from (14) it follows that I0f is empty if f = 0, otherwise I0f is an invariant 2-dimensional manifold diffeomorphic to R2 . This completes the statement (c) of Theorem 1. Assume now λ = −2. Then from F (x, y, px , py ) = f we obtain again that py = (f − 2px )/2, and from H(x, y, px , (f − 2px )/2) = 0, we have that px = 2e2(x−y)) /f + f /4. Substituting this expression of px into the previous expression of py we obtain that py = −2e2(x−y)) /f + f /4. Again in order to prove statement (d) of Theorem 1 we must characterize the topology of the 2-dimensional manifolds I0f given by    2e2(x−y)) 1 2e2(x−y)) 1 2 (15) x, y, + f, − + f : (x, y) ∈ R . f 4 f 4 Therefore, when λ = −2 from (15) it follows that I0f is empty if f = 0, otherwise I0f is an invariant 2-dimensional manifold diffeomorphic to R2 . This completes the statement (d) of Theorem 1. In order to prove statement (e) we need some preliminary results. A phase portrait of the Hamiltonian system (3) restricted to the 2-dimensional manifold I0f is the decomposition of I0f as union of the orbits of this differential system. Two phase portraits on I0f1 and on I0f2 are topologically equivalent if there is a homeomorphism h : I0f1 −→ I0f2 which send orbits in I0f1 into orbits of I0f2 , preserving or reversing the sense of all the orbits. A separatrix of the Hamiltonian system (3) restricted to the 2-dimensional man- ifold I0f is one of following orbits: the equilibrium points, the limit cycles, and the two orbits at the boundary of every hyperbolic sector of an equilibrium point, see for more details on the separatrices [6, 8]. Recall that a limit cycle of a differential system is a periodic orbit isolated in the set of all periodic orbits of the differential system. For a definition of a hyperbolic sector see page 18 of [3]. The set of all separatrices of the Hamiltonian system (3) restricted to the 2- dimensional manifold I0f , denoted by Σ0f , is a closed set (see [8]). Here A canonical region of I0f is an open connected component of I0f \ Σ0f . The union of the set Σ0f with an orbit of each canonical region form the separatrix con- figuration of the Hamiltonian system (3) restricted to the 2-dimensional manifold I0f and is denoted by Σ00f . We say that the flow of the Hamiltonian system (3) restricted to a 2-dimensional manifold I0f is parallel if it is topologically equivalent to one of the following flows: (i) The flow defined on R2 by the differential system ẋ = 1, ẏ = 0, which it is called the strip flow. (ii) The flow defined on R2 \ {0} by the differential system given in polar coor- dinates by r0 = 0, θ0 = 1, which it is called the annulus flow. (iii) The flow defined on R2 \ {0} by the differential system given in polar coor- dinates by r0 = r, θ0 = 0, which it is called the spiral or nodal flow. 6 I. BASASK AND J. LLIBRE A main result is the following: The flow at every canonical region of a flow on a 2-dimensional manifold is parallel, given by either a strip, an annular or a spiral flow, see [8]. Two separatrix configurations Σ00f1 and Σ00f2 are topologically equivalent if there is a homeomorphism Σ00f1 −→ Σ00f2 such that h(Σ00f1 ) = Σ00f2 . According to the following theorem, which was proved by Markus [6], Neumann [8] and Peixoto [9], it is sufficient to investigate the separatrix configuration of a differential system on a 2-dimensional manifold, for determining its phase portrait. Theorem 4. Two phase portraits on I0f1 and on I0f2 with finitely many separatri- ces are topologically equivalent if and only if the two separatrix configurations Σ00f1 and Σ00f2 are topologically equivalent. We note that the Hamiltonian system (3) restricted to every 2-dimensional man- ifold I0f has no separatrices because: First, these systems have no equilibrium points. Second, these systems have no periodic orbits because in the region limited by a periodic orbit a differential system in dimension two must have at least one equilibrium point (see for instance Theorem 1.31 of [3]), consequently these system have no limit cycles. Finally these systems have no hyperbolic sectors again because they do not have equilibrium points. In summary Σ0f is empty for all f ∈ R. Therefore, by Theorem 4 the phase portrait of the Hamiltonian system (3) restricted to an arbitrary connected com- ponent of a 2-dimensional manifold I0f is topologically the same for all f ∈ R, in other words a connected component of I0f1 and a connected component of I0f2 are topologically equivalent for all f1 , f2 ∈ R. Moreover, every connected component I0f is a canonical region. Since every one of these connected components are dif- feomorphic to R2 it follows that the flow on each of these connected components is strip. So their orbits come and go to the infinity of the phase space E. Therefore statement (e) of Theorem 1 is proved. This completes the proof of Theorem 1. 3. Proof of Theorem 2 From F (x, y, px , py ) = f we obtain that ex+y (a + b + by) − 2f + 2p2y (16) px = , 2py and from ! ex+y (a + b + by) − 2f + 2p2y H x, y, , py = 0, 2py we get that 2f − ex+y (a + b + by) (17) py = ± p . 2 2f − bex+y EINSTEIN-FRIEDMANN COSMOLOGICAL MODELS 7 Substituting py from (17) in (16) we obtain ex+y (a − b + by) + 2f (18) px = ∓ p . 2 2f − bex+y In order to prove Theorem 2 we must characterize the topology of the 2-dimensional manifolds I0f given by ( ! ) ex+y (a − b + by) + 2f 2f − ex+y (a + b + by) 2 (19) x, y, ∓ p ,± p : (x, y) ∈ R . 2 2f − bex+y 2 2f − bex+y Note that when in the expression of py there is a plus (respectively minus) in the expression of px there is a minus (respectively plus). From (19) it follows that I0f is empty if b < 0 and f ≤ 0, otherwise I0f is an invariant 2-dimensional manifold diffeomorphic to two copies of R2 . So statement (a) of Theorem 2 is proved. The proof of statement (b) of Theorem 2 is exactely the same than the proof of statement (e) of Theorem 1. This completes the proof of Theorem 2. Acknowledgements This work is supported by the Ministerio de Ciencia, Innovación y Universidades, Agencia Estatal de Investigación grants MTM2016-77278-P (FEDER), the Agència de Gestió d’Ajuts Universitaris i de Recerca grant 2017SGR1617, and the H2020 European Research Council grant MSCA-RISE-2017-777911. References [1] R. Abraham, J.E. Marsden and T. Ratiu, Manifolds, tensor analysis, and applications, Second edition, Applied Mathematical Sciences Vol. 75, Springer-Verlag, New York, 1988. [2] V.I. Arnold, Mathematical Methods of Classical Mechanics, Springer-Verlag, 1978. [3] F. Dumortier, J. Llibre and J.C. Artés, Qualitative Theory of Planar Differential Systems, Springer Verlag, New York, 2006. [4] J. Hietarinta, Direct methods for the search of the second invariant, Phys. Rept. 147 (1987), 87–154. [5] J. Llibre and A. Nunes, Separatrix surfaces and invariant manifolds of a class of in- tegrable Hamiltonian systems and their perturbations, Memoirs of the Amer. Math. Soc. Vol. 513, 1994. [6] L. Markus, Global structure of ordinary differential equations in the plane: Trans. Amer. Math Soc. 76 (1954), 127–148. [7] J. Martin, C. Ringeval and V. Vennin, Encyclopaedia Inflationaris, Phys. Dark Univ. 5–6 (2014), 75–235. [8] D. A. Neumann, Classification of continuous flows on 2–manifolds, Proc. Amer. Math. Soc. 48 (1975), 73–81. [9] M.M. Peixoto, On the classification of flows on 2–manifolds. Academic, New York, pages 389–419, 1973. Dynamical systems (Proc. Sympos., Univ. Bahia, Salvador, 1971). [10] V.V. Sokolov and A.S. Sorin, Integrable cosmological potential, Lett. Math. Phys. 107 (2017), no. 9, 1741–1768. 8 I. BASASK AND J. LLIBRE Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain Email address: innabasak@gmail.com Email address: jllibre@mat.uab.cat