The dynamical systems observability properties of barotropic bosonic and
fermionic FRW cosmological oscillators are investigated. Nonlinear techniques
for dynamical analysis have been recently developed in many engineering areas
but their application has not been extended beyond their standard field. This
paper is a small contribution to an extension of this type of dynamical systems
analysis to FRW barotropic cosmologies. We find that determining the Hubble
parameter of barotropic FRW universes does not allow the observability, i.e.,
the determination of neither the barotropic FRW zero mode nor of its derivative
as dynamical cosmological states. Only knowing the latter ones correspond to a
rigorous dynamical observability in barotropic cosmologyComment: 10 pages, 0 figure

Ibarra-Junquera, V.

Rosu, H. C.

English

arXiv

"The dynamical systems observability properties of barotropic bosonic and fermionic FRW cosmological oscillators are investigated. Nonlinear techniques for dynamical analysis have been recently developed in many engineering areas but their application has not been extended beyond their standard ﬁeld. This paper is a small contribution to an extension of this type of dynamical systems analysis to FRW barotropic cosmolo-gies. We ﬁnd that determining the Hubble parameter of barotropic FRW universes does not allow the observability, i.e., the determination of neither the barotropic FRW zero mode nor of its derivative as dy-namical cosmological states. Only knowing the latter ones correspond to a rigorous dynamical observability in barotropic cosmology.

Rosu Barbus, Haret-Codratian

Ibarra Junquera, Vrani

Repositorio IPICYT

Applied Mathematical Sciences, Vol. 1, 2007, no. 17, 843 - 852FRW Barotropic Zero Modes:Dynamical Systems ObservabilityH. C. RosuInstituto Potosino de Investigación Cient́ıfica y TecnológicaApdo Postal 3-74 Tangamanga, 78231 San Luis Potośı, S.L.P., MexicoV. Ibarra-JunqueraFaculty of Chemical Sciences, University of Colima28400 Coquimatlán, Col., MexicoAbstractThe dynamical systems observability properties of barotropic bosonicand fermionic FRW cosmological oscillators are investigated. Nonlineartechniques for dynamical analysis have been recently developed in manyengineering areas but their application has not been extended beyondtheir standard field. This paper is a small contribution to an extensionof this type of dynamical systems analysis to FRW barotropic cosmolo-gies. We find that determining the Hubble parameter of barotropicFRW universes does not allow the observability, i.e., the determinationof neither the barotropic FRW zero mode nor of its derivative as dy-namical cosmological states. Only knowing the latter ones correspondto a rigorous dynamical observability in barotropic cosmology.Mathematics Subject Classification: 83F05, 93B07Keywords: supersymmetry, FRW barotropic cosmology, (nonlinear) dy-namical systems observability1 IntroductionOver the years, modern dynamical systems theory has been applied with con-siderable success to the evolution of cosmological models. Many results con-cerning the possible asymptotic cosmological states (at both early and latetimes) have been obtained and compared with Hamiltonian methods and nu-merical studies in the authoritative book of Wainwright and Ellis [6].844 H. C. Rosu and V. Ibarra-JunqueraDynamical systems observability (DSO henceforth) is a rigorous mathemat-ical concept used to investigate if it is possible to know the internal functioningof a given dynamical system. In the case of cosmology, DSO can be considereda criterion of whether the possible states of the dynamical universe as definedby some intelligent observers can be indeed estimated. More precisely, one saysthat a dynamical system is observable if it allows the usage of the availableinformation about the input u(t) and the output y(t) of the system to estimatethe states X(t) of the system. Thus, the main idea behind (engineering) ob-servability is to find the dynamical states of a system based on the knowledgeof its output.In this paper we will investigate a cosmological application of this type ofobservability analysis. In the context of FRW barotropic cosmologies one im-portant measured output is the Hubble parameter and an interesting problemis if we can determine the so-called barotropic zero modes since these modescan be identified as the normal dynamical cosmological states. In the con-formal time variable, the logderivative of these modes determine the Hubbleparameter and thus they are equivalent to the comoving time scale factors.The answer that we get according to the (engineering) observability analysis isthat we cannot determine the FRW zero modes and therefore the scale factor ifthe measurable output is the conformal Hubble parameter. However, knowingthe latter ones or their derivatives leads to a rigorous dynamical observabilityin barotropic cosmology. We choose to present in the first sections of the pa-per the physical part of the problem and the direct mathematical application,whereas in the last section we collect the required mathematical results. Weend up with a short conclusion section.equations:2 FRW barotropic cosmologyBarotropic cosmological zero modes are simple trigonometric and/or hyper-bolic solutions of second order differential equations of the oscillator type towhich the FRW system of equations can be reduced when it is passed to theconformal time variable. They have been discussed in a pedagogical way byFaraoni [1] and also were the subject of several recent papers [4]. In thissection, we briefly review their mathematical scheme.Barotropic FRW cosmologies in comoving time t obey the Einstein-Friedmanndynamical equations for the scale factor a(t) of the universe supplemented bythe (barotropic) equation of state of the cosmological fluidäa= −4πG3(ρ + 3p) ,H20(t) ≡(ȧa)2= 8πGρ3− κa2,p = (γ − 1)ρ ,FRW barotropic zero modes 845where ρ and p are the energy density and the pressure, respectively, of theperfect fluid of which a classical universe is usually assumed to be made of,κ = 0,±1 is the curvature index of the flat, closed, open universe, respectively,and γ is the constant adiabatic index of the cosmological fluid.2.1 FRW barotropic bosonic zero modesPassing to the conformal time variable η, defined through dt = a(η)dη, onecan combine the three equations in a single Riccati equation for the Hubbleparameter H0(η) (we shall use eitherddηor ′ for the derivative with respect toη in the following)H′0 + cH20 + κc = 0 , (1)where c = 32γ − 1.Employing now H0(η) =1cw′wone gets the very simple (harmonic oscillator)second order differential equationw′′ − c · cκ,bw = 0 , (2)where cκ,b = −κc. The solutions are the following:For κ = 1 : w1,b ∼ cos(cη + φ) → w1,b ∼ [a1,b(η)]c ,where φ is an arbitrary phase.For κ = −1 : w−1,b ∼ sinh(cη) → w−1,b ∼ [a−1,b(η)]c .parameters.Moreover, the particular Riccati solutions particular solution, i.e., H+0 = − tan cηand H−0 = coth cη for κ = ±1, respectively, are related to the common factor-izations of the equation (2) FACTORIZATION(ddη+ cH0) (ddη− cH0)w = w′′ − c(H′0 + cH20)w = 0 . (3)Borrowing a terminology from supersymmetric quantum mechanics, we callthe solutions w as bosonic zero modes. As one can see, they are actually somepowers of the scale factors of the barotropic universes.2.2 FRW barotropic fermion zero modesA class of barotropic FRW cosmologies with inverse scale factors with respectto the bosonic ones can be obtained by considering the supersymmetric partner846 H. C. Rosu and V. Ibarra-Junquera(or fermionic) equation of Eq. (3) which is obtained by applying the factoriza-tion brackets in reverse order(ddη− cH0) (ddη+ cH0)w = w′′ − c(−H′0 + cH20)w = 0 . (4)Thus, one can writew′′ − c · cκ,fw = 0 , (5)wherecκ,f(η) = −H′0 + cH20 ={c(1 + 2tan2cη) if κ = 1c(−1 + 2coth2cη) if κ = −1denotes the supersymmetric partner adiabatic index of fermionic type associ-ated through the mathematical scheme to the constant bosonic index. Noticethat the fermionic adiabatic index is time dependent. The fermionic w solu-tions arew1,f =ccos(cη + φ)→ a1,f(η) ∼ [cos(cη + φ)]−1/c ,andw−1,f =csinh(cη)→ a−1,f(η) ∼ [sinh(cη)]−1/c ,for κ = 1 and κ = −1, respectively.We can see that the bosonic and fermionic barotropic cosmologies are re-ciprocal to each other, in the sense thata±,ba±,f = const .Thus, bosonic expansion corresponds to fermionic contraction and viceversa.A matrix formulation of the previous results is possible employing two Paulimatrices [5]. The following spinor zero mode is obtained:W =(wκ,fwκ,b),showing that the two reciprocal barotropic zero modes enter on the same foot-ing as the two components of the spinor W.3 DSO Analysis of the FRW Barotropic ZeroModesThe DSO analysis is based on the concept of observability space O, which isa smooth codistribution defined in an algorithmic way by Isidori [3]. All theFRW barotropic zero modes 847basic mathematical material related to this concept is presented in section 5of this paper.Frequently, the following form is used in controlled systemsΓ :{Ẋ = f(X) + g(X)uy = h(X)(6)where u ∈ Ω ⊂ R1, X ∈ S ⊂ R2 and f and g are C∞ functions. Note that ingeneral f(X) = f(x) + g(X)u. For the sake of simplicity we denote by Γ theset of equations 6.In this section, we write the conformal time oscillator equations in the abovedynamical systems form and introduce the corresponding nonlinear outputs forwhich we apply the algorithmic observability criterion of Isidori according tothe basic concepts in section 5.BOSONIC CASE3.1 The Bosonic CaseThe dynamical system for this case is based on Eq. (2)Γb :{Ẋ1 = X2Ẋ2 = − c (−κ · c) X1 (7)In matrix formẊ = fj(X) , (8)where X ∈ R2, with j = 1 when κ = 1 and j = 2 when κ = −1.Suppose that we take as outputsh1 =X2X1≡ Ẋ1X1≡ c H0(η) h2 = X1X2≡ 1/h1 (9)or justh3 = X1 ≡ w h4 = X2 ≡ w′ . (10)Then, we can write the codistributionOj,i =[hi(X), Lfjhi(X)](11)Ξj,i = d Oj,i (12)This gives for j = 1:Ξb,1,1 = −⎡⎣(−X2X1)1X11X1(−X2X1)2X2X122X2X12⎤⎦ , Ξb,1,2 = −⎡⎣ 1X2(−X1X2)1X22X1 c2X22(−X1X2)2X1 c2X22⎤⎦ .(13)848 H. C. Rosu and V. Ibarra-JunqueraThus, for this case there exits a linear dependence in the columns of theabove matrices, therefore the rank is less than the number of states. As con-sequence the outputs h1 and h2 make the systems unobservable.For j = 2:Ξb,2,1 = −⎡⎣(−X2X1)1X11X1(−X2X1)2X2X122X2X12⎤⎦ , Ξb,2,2 =⎡⎣ 1X2(−X1X2)1X22X1 c2X22(−X1X2)2X1 c2X22⎤⎦ .(14)Again, it is possible to notice the linear dependence in the columns ofthe above matrices, therefore the rank is less than the number of states. Asconsequence the outputs h1 and h2 make the systems unobservable.However, for the outputs h3 and h4, i.e., if one can know directly the zeromodes or their derivatives, respectively, thenΞb,1,3 =[1 00 1], Ξb,1,4 =[0 1+c2 0]. (15)andΞb,2,3 =[1 00 1], Ξb,2,4 =[0 1−c2 0]. (16)for j = 1 and j = 2, respectively.Both pairs of matrices have linear independent columns in an obvious wayand therefore the cosmological system is observable.3.2 The Fermionic CaseWe write the cosmological dynamical system derived from Eq. (5) in the formΓf :{dX1dη= X2dX2dη= − c Fj(η) X1 (17)where we definedF1(η) = c(1 + 2 tan2 cη)(18)F2(η) = c(−1 + 2 coth2 cη) (19)In matrix formdXdη= fj(X) , (20)FRW barotropic zero modes 849where X ∈ R2. Using the same outputs as in the bosonic case and definingthe codistribution in the same way we getΞf,1,1 = −⎡⎣(−X2X1)1X11X1(−X2X1)2X2X122X2X12⎤⎦ , Ξf,1,2 = c⎡⎣ 1X2(−X1X2)1X22X1 F1(η)X22(−X1X2)2X1 F1(η)X22⎤⎦ .(21)The linear dependence in the columns of the above matrices is manifestand therefore the rank is less than the number of states. We conclude that theoutputs h1 and h2 make the system unobservable.Using now F2(η), the matricesΞf,2,1 = −⎡⎣(−X2X1)1X11X1(−X2X1)2X2X122X2X12⎤⎦ , Ξf,2,2 = c⎡⎣ 1X2(−X1X2)1X22X1 F2(η)X22(−X1X2)2X1 F2(η)X22⎤⎦(22)are linear dependent at the level of their columns, so the rank is less than thenumber of states and the outputs h1 and h2 make the cosmological systemsunobservable.However, again, if one can know directly the zero modes or their derivatives,thenΞf,1,3 =[1 00 1], Ξf,1,4 =[0 1−cF1(η) 0](23)andΞf,2,3 =[1 00 1], Ξf,2,4 =[0 1−cF2(η) 0]. (24)All these four matrices have linear independent columns and therefore thecosmological system is observable.4 Nonlinear Observability: Definitions andBasic Theorem of ObservabilityThe strictly mathematical material of this section refers to the dynamicalsystem of equations Γ given in Eq. (6).Definition 4.1 (Indistinguishability and Observability) [2]Two states Xa, Xb ∈ S are said to be indistinguishable (denoted by XaIXb) if850 H. C. Rosu and V. Ibarra-Junquera(Γ, Xa) and (Γ, Xb) realize the same input-output map, i.e. for every admissibleinput u(t) ∈ u defined in [to, t1]ΓXa(u(t)) = ΓXb(u(t))Let I (Xo) be the set of points indistinguishable from Xo. The system Γ is saidto be observable at Xo if I (Xo) = {Xo} and is observable if I (X) = {X} forevery X ∈ SSince the observability is a global concept distinguishing between all thepoints in S is a difficult goal. Therefore, we need a local concept and further-more we merely need to distinguish Xo from its neighbors. For this purposewe define the concept of local weak observability.Definition 4.2 (Local Weak Observability) [2]We shall say that Γ is locally weakly observable at Xo if there exist an openneighborhood U of Xo such that for every open neighborhood V of Xo containedin U, IV (Xo) = {Xo} and is locally weakly observable if IV (X) = {X} forevery X ∈ S.The advantage of the local weak observability over the other concepts isthat its proof is just necessary a simple algebraic test.4Definition 4.3 (Lie Derivative of a Real Valued Function) [3]Consider a smooth vector field f and a function λ, defined in an open setU ⊂ Rn. The Lie derivative of λ along f is the function Lfλ : Rn −→ Rdefined by:Lfλ(X) =∂λ∂Xf(X) =n∑i=1∂λ∂Xifi(X) .Definition 4.4 (Observability Smooth Codistribution)The observability smooth codistribution can be generated incrementally in theIsidori algorithmic way [3] as followsΩ0 = span{h(X)}Ωk = Ωk−1 +q∑i=1span {LτiΩk−1} .This smooth codistribution is called the observability space O. LτiΩ is definedin Definition 4.3.FRW barotropic zero modes 851Definition 4.5 (Differential of f) [3]Let f : Rn −→ R be a smooth function. The differential of f is defined by:df =∂ f∂X=[∂ f∂X1∂ f∂X2· · · ∂ f∂Xn].Definition 4.6 (Observability Rank Condition)Γ is said to satisfy the observability rank condition at Xo if the dimension ofdO(Xo) is n where n is the dimension of the system and the differential d isgiven by Definition 4.5. Moreover, dO(X) satisfies the observability rankcondition if this is true for any X ∈ S.Theorem 4.7 (Basic Theorem of Observability)If Γ satisfies the observability rank condition in Definition 4.6 at Xo then Γsatisfies the observability rank condition for any X ∈ S if the dimension ofdO(X) is complete.The complete proof of this theorem is given in [2].5 ConclusionIn this paper we have treated the FRW barotropic zero modes as dynamicalstate(s) of the universe for which we applied the rigorous techniques of non-linear observability as defined in engineering by Hermann and Krener, andfurther discussed by Isidori in his textbook. We have shown that these zeromodes remain undetermined if only the Hubble parameter is known, which ingeneral is an expected result for cosmologists since it appears that the Hubbleparameter provides only partial though important cosmological information.In other words, the usual type of cosmological measurements relying on theHubble parameter are not the ones providing the complete dynamical informa-tion about the universe. On the other hand, we have obtained the interestingresult that having the chance to know directly the FRW zero modes gives usthe opportunity to know dynamically the whole system. This, indirectly, offersthe possibility that by fitting one can infer the initial conditions of the FRWuniverse.ACKNOWLEDGEMENTS.The second author wishes to thank IPICyT for a fellowship that allowed himto collaborate with the first author on the research of this paper.References[1] V. Faraoni, Solving for the dynamics of the universe, Am. J. Phys., 67(1999), 732-734, physics/9901006.852 H. C. Rosu and V. Ibarra-Junquera[2] R. Hermann and A.J. Krener, Nonlinear controllability and observability,IEEE Trans. Aut. Ctrl., AC-22 (1977) 728-740.[3] A. Isidori, Nonlinear control systems, Springer New York, 2nd edition,1989.[4] H.C. Rosu, Darboux class of cosmological fluids with time-dependent adi-abatic indices, Mod. Phys. Lett. A, 15(2000), 979-990, gr-qc/0003108; J.A.Sales Lima, Note on Faraoni’s paper, Am. J. Phys., 69 (2001) 1245-1247,astro-ph/0109215.[5] H.C. Rosu and P. Ojeda-May, Supersymmetry of FRW barotropic cos-mologies, Int. J. Theor. Phys., 45 (2006) 1191-1196, gr-qc/0510004.Springer New York,[6] J. Wainwright and G.F.R. Ellis, Dynamical systems in cosmology, Cam-bridge Univ. Press, 1997; See also, A.A. Coley, Dynamical systems incosmology, in Proc. Spanish Relativity Meeting, Bilbao, Sept. 1999, gr-qc/9910074.Received: August 31, 2006

FRW barotropic zero modes: Dynamical systems observability

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