Generalized Fibonacci numbers, cosmological analogies, and an invariant
aa r X i v : . [ g r- q c ] J a n Generalized Fibonacci numbers, cosmological analogies, and an invariant
Valerio Faraoni and Farah Atieh Department of Physics & Astronomy, Bishop’s University,2600 College Street, Sherbrooke, Qu´ebec, Canada J1M 1Z7
Continuous generalizations of the Fibonacci sequence satisfy ODEs that are formal analogues ofthe Friedmann equation describing spatially homogeneous and isotropic cosmology in general rela-tivity. These analogies are presented, together with their Lagrangian and Hamiltonian formulationsand with an invariant of the Fibonacci sequence.
I. INTRODUCTION
Fibonacci, also known as Leonardo Pisano or Leonardo Bonacci, introduced Hindu-Arabic numerals to Europe withhis book
Liber Abaci in 1202 [1]. He posed and solved a well-known problem involving the growth of a population ofrabbits in idealized situations. The solution, now known as the Fibonacci sequence 0 , , , , , , , , ... is written as F n = F n − + F n − , (1.1)where F n is the n -th Fibonacci number and F = 0 , F = 1. Furthermore, the ratio of two consecutive terms F n +1 /F n approaches the golden ratio ϕ ≡ √ ≈ . ... (1.2)as n → + ∞ .The Fibonacci sequence can be generalized to the continuum using Binet’s formula F n = ϕ n − ( − ϕ ) − n √ . (1.3)Furthermore, the analytic function F ( e ) ( x ) = ϕ x − ϕ − x √ √ x ln ϕ ) (1.4)reproduces part of the Fibonacci numbers, F n = F ( e ) ( n ) for even x = n ∈ N , while F ( o ) ( x ) = ϕ x + ϕ − x √ √ x ln ϕ ) (1.5)reproduces the other Fibonacci numbers for odd x = n ∈ N . The function F ( x ) = ϕ x − cos ( πx ) ϕ − x √ x = n ∈ N . Here we focus on F ( e,o ) ( x ), which admit analogies withrelativistic cosmology, while no such analogy exists for F ( x ).The functions F ( e,o ) satisfy the dual relations (where a prime denotes differentiation with respect to x ) F ′ ( e ) ( x ) = (ln ϕ ) F ( o ) ( x ) , (1.7) F ′ ( o ) ( x ) = (ln ϕ ) F ( e ) ( x ) , (1.8)and the second order ODE F ′′ ( e,o ) − (cid:0) ln ϕ (cid:1) F ( e,o ) = 0 , (1.9)of which F ( e ) and F ( o ) are two linearly independent solutions. In physics, this equation describes the one-dimensionalmotion of a particle of position F in the inverted harmonic oscillator potential V ( F ) = − kF / K = 2 ln ϕ ),which is used as an example of an unstable mechanical system (this property corresponds to the fact that the Fibonaccinumbers F n increase without bound as n → ∞ ).We will also use f ( x ) = ϕ x , (1.10)which is a Fibonacci function according to the definition of Ref. [2], i.e. , f ( x + 2) = f ( x + 1) + f ( x ) ∀ x ∈ R . (1.11)However, F ( e,o ) ( x ) are not Fibonacci functions since they contain ϕ − x , which is not a Fibonacci function (it is easyto prove [2] that, among power-law functions y ( x ) = b x , the only Fibonacci function is the one with base equal to thegolden ratio, b = ϕ ).In the following section, we briefly recall the basics of spatially homogeneous and isotropic cosmology in generalrelativity, and then we present the formal analogy with Eqs. (1.7)-(1.9) in Sec. III. We then deduce Lagrangian andHamiltonian formulations for the Fibonacci ODEs and derive from these an invariant of the (discrete) Fibonaccisequence. II. FLRW COSMOLOGY
In the context of Einstein’s relativistic theory of gravity, the most basic assumption of cosmology is the Copernicanprinciple stating that, on average ( i.e. , on scales larger than a few tens of Megaparsecs), the universe is spatiallyhomogeneous and isotropic [3–8]. The stringent symmetry requirements of spatial homogeneity and isotropy forcethe geometry to have constant spatial curvature [9]. The spacetime metric is necessarily given by the Friedmann-Lemaˆıtre-Robertson-Walker (FLRW) line element, written in polar comoving coordinates ( t, r, ϑ, φ ) as [3, 4] ds = − dt + a ( t ) " dr − Kr + r d Ω (2.1)where d Ω = dϑ + sin ϑ dφ is the line element on the unit 2-sphere and the sign of the constant curvature index K classifies the 3-dimensional spatial sections t = const. K > K = 0to Euclidean flat sections, and K < a ( t ). In FLRW cosmology, the matter source causing the spacetime to curve is usually (but notnecessarily) taken to be a perfect fluid with energy density ρ ( t ) and pressure P ( t ) = wρ ( t ), where w is a constantequation of state parameter. The evolution of a ( t ) and ρ ( t ) is ruled by the Einstein equations adapted to the highdegree of symmetry, the Einstein-Friedmann equations. They comprise [3–5] the Friedmann equation H = 8 πG ρ − Ka + Λ3 (2.2)(a first order constraint), the acceleration or Raychaudhuri equation¨ aa = − πG ρ + 3 P ) + Λ3 , (2.3)and the covariant conservation equation ˙ ρ + 3 H ( P + ρ ) = 0 (2.4)expressing energy conservation for the cosmic fluid. Here G is Newton’s gravitational constant, Λ is Einstein’s famouscosmological constant, an overdot denotes differentiation with respect to the cosmological time t , H ≡ ˙ a/a is theHubble function [4, 5], and units are used in which the speed of light is unity. For a generic cosmic fluid, only two ofthe three equations (2.2)-(2.4) are independent. When there is no cosmic fluid and the cosmological constant Λ (whichcan be treated as an effective fluid with energy density ρ = Λ8 πG = − P ) is the only energy content, the conservationequation (2.4) is satisfied identically. III. THE COSMOLOGICAL ANALOGY
Let us consider first the function F ( e ) ( x ), which satisfies F ′ ( e ) = 2 √ ϕ cosh ( x ln ϕ ) = 2 √ ϕ s F e ) . (3.1)Dividing this equation by F ( e ) and squaring, one obtains F ′ ( e ) F ( e ) ! = 4 ln ϕ ϕ F e ) , (3.2)which is formally analogous to the Friedmann equation (2.2), provided that (cid:0) x, F ( e ) ( x ) (cid:1) → ( t, a ( t )) and ρ = P = 0 , (3.3)Λ = 3 ln ϕ , (3.4) K = − ϕ We follow the notations of Refs. [4, 5]. (contrary to FLRW cosmology, these quantities are dimensionless in the Fibonacci side of the analogy).
A priori theformal equivalence with the Friedmann equation (2.2) is not sufficient for the analogy to hold and one must check thatalso Eqs. (2.3) and (2.4) are satisfied: this is straightforward to do using Eq. (1.9), while the conservation equationis trivially satisfied with ρ = P = 0 (and also by the Λ-effective fluid with P = − ρ = const.). The universe withscale factor analogous to the function F ( e ) ( x ) has hyperbolic 3-D spatial sections and is empty, but expands due tothe repulsive cosmological constant.By squaring Eq. (3.1), one has introduced the possibility of solutions with F ′ ( e ) <
0, corresponding to a contracting,instead of expanding, universe. In fact, Eq. (2.2) then gives˙ a = ± r Λ a | K | , (3.6)which integrates to ln h C (cid:16)p Λ a + 3Λ | K | + Λ a (cid:17)i = ± r Λ3 ( t − t ) (3.7)where C and t are integration constants ( C serves the purpose of making the argument of the logarithm dimensionlesssince, in the units used [3–5], the scale factor a carries the dimensions of a length and Λ those of an inverse lengthsquared). By exponentiating both sides, squaring, and collecting similar terms, one is left with a ( t ) = a h e ± √ Λ3 ( t − t ) − | K | Λ C e ∓ √ Λ3 ( t − t ) i , (3.8)with a constant. In practice, in FLRW cosmology the dimensionless radial coordinate r can be rescaled to normalizethe curvature index K (usually to ± | K | Λ C = 1 / a ( t ) = ± a sinh "r Λ3 ( t − t ) (3.9)where, in order to keep the scale factor non-negative, the upper sign applies for t ≥ t and the lower one for t ≤ t .At late times t → + ∞ , this metric is asymptotic to the metric of de Sitter spacetime a ( dS ) ( t ) = a e √ Λ3 ( t − t ) .Let us focus now on the function F ( o ) ( x ): proceeding as we did for F ( e ) ( x ), one obtains F ′ ( o ) F ( o ) ! = − ln ϕ + 4 ln ϕ F o ) , (3.10)which is formally analogous to the Friedmann equation (2.2) with ρ = P = 0 , (3.11)Λ = 3 ln ϕ , (3.12) K = 4 ln ϕ . (3.13)The analogous FLRW universe is again empty and propelled by the positive cosmological constant, but the spatialsections are now closed 3-spheres.Again, by squaring one introduces the possibility of a negative sign for ˙ a . Proceeding in parallel with what hasbeen done for F ( e ) , one obtains a ( t ) = a h e ± √ Λ3 ( t − t ) + 3 K Λ C e ∓ √ Λ3 ( t − t ) i (3.14)and, normalizing K suitably, a ( t ) = a cosh "r Λ3 ( t − t ) . (3.15)This scale factor describes a universe contracting from a infinite size, bouncing at the minimum value a , and thenexpanding forever and asymptoting to the de Sitter space, a behaviour sought for in quantum cosmology to avoid theclassical Big Bang singularity a = 0.Finally, we can consider the Fibonacci function f ( x ) = ϕ x , which trivially satisfies (cid:18) f ′ f (cid:19) = ln ϕ (3.16)and is analogous to an empty ( ρ = P = 0), spatially flat ( K = 0) universe expanding exponentially, a ( dS ) = a e √ Λ3 t due to the positive cosmological constant Λ = 3 ln ϕ . This is the maximally symmetric de Sitter universe, whichis an attractor in inflationary models of the early universe [6, 7] and in dark energy-dominated models of the late(present-day) accelerating universe [10]. IV. LAGRANGIAN AND HAMILTONIAN
Both the mechanical analogy of Eq. (1.9) and the cosmological analogy suggest the Lagrangian for F ( e,o ) ( x ) L ( e,o ) (cid:16) F ( e,o ) ( x ) , F ′ ( e,o ) ( x ) (cid:17) = 12 ( F ′ ( e,o ) ) + ln ϕ F e,o ) . (4.1)The corresponding Euler-Lagrange equation ddx ∂L ( e,o ) ∂F ′ ( e,o ) ! − ∂L ( e,o ) ∂F ( e,o ) = 0 (4.2)reproduces Eq. (1.9).The associated Hamiltonian, expressed in terms of the canonical variables q ( e,o ) ≡ F ( e.o ) and p ( e,o ) ≡ ∂L ( e,o ) /∂F ′ ( e,o ) = F ′ ( e,o ) , is H ( e,o ) = p ( e,o ) F ′ ( e,o ) − L ( e,o ) = (cid:0) p ( e,o ) (cid:1) − ln ϕ F e,o ) , (4.3)and the Hamilton equations are q ′ ( e,o ) = ∂ H ( e,o ) ∂p ( e,o ) = p ( e,o ) = F ′ ( e,o ) = ln ϕ F ( o,e ) (4.4)(which reproduces Eqs. (1.7), (1.8)) and p ′ ( e,o ) = − ∂ H ( e,o ) ∂q ( e,o ) = ln ϕ F ( e,o ) . (4.5)Since L ( e,o ) does not depend explicitly on x , the Hamiltonian H ( e,o ) is conserved, (cid:0) p ( e,o ) (cid:1) − ln ϕ F e,o ) = E ( e,o ) , (4.6)where the constants E ( e,o ) have the meaning of energy of the system. Writing them explicitly, we have E ( e,o ) = ln ϕ (cid:0) F ( o,e ) + F ( e,o ) (cid:1) (cid:0) F ( o,e ) − F ( e,o ) (cid:1) = ± ϕ , (4.7)which gives the first integral (cid:16) F o,e ) − F e,o ) (cid:17) on the Fibonacci side of the analogy. In terms of the discrete Fibonaccisequence we have, therefore, the Proposition:
The quantity I = h ( F m + F m − ) − ( F m − + F m − ) i = F m + F m − − F m − − F m − + 2 ( F m F m − − F m − F m − ) (4.8)does not depend on m and is an invariant of the Fibonacci sequence.To the best of our knowledge, this invariant is not related to known invariants (for example, those of the Fibonacciconvolution sequences [11]). V. CONCLUSIONS
The functions F ( e,o ) associated with the continuum generalization of the Fibonacci sequence exhibit analogies withcertain spatially homogeneous and isotropic universes in FLRW cosmology, and also a mechanical analogy with aninverted harmonic oscillator, which we have presented. With the help of these analogies, it is easy to derive aLagrangian and a Hamiltonian associated with the ODEs satisfied by the functions F ( e,o ) . Then, the conservation ofthe Hamiltonian (or energy) yields a quantity I , given by Eq. (4.8) that is constant across the Fibonacci sequence, i.e. , does not depend on the index n . This invariant may turn out to be useful in view of the many applications ofthe Fibonacci numbers in mathematics and in the natural sciences ( e.g. , [12–18]). ACKNOWLEDGMENTS
We thank two referees for useful comments. This work is supported by the Natural Sciences and EngineeringResearch Council of Canada (Grant No. 2016-03803 to V.F.) and by Bishop’s University. [1] Sigler, L.E.
Fibonacci’s Liber Abaci: A Translation into Modern English of Leonardo Pisano’s Book of Calculation ; Springer,New York, 2002.[2] Han, J.S., Kim, H.S., Neggers, J. “On Fibonacci functions with Fibonacci numbers”,
Advances in Difference Equations , , 126. DOI: https://doi.org/10.1186/1687-1847-2012-126[3] Hawking, S.W., Ellis, G.F.R. The Large Scale Structure Of Space-time ; Cambridge University Press, Cambridge, 1973.[4] Wald, R.M.
General Relativity ; Chicago University Press: Chicago, IL, USA, 1984.[5] Carroll, S.M.
Spacetime and Geometry: An Introduction to General Relativity ; Addison-Wesley, San Francisco, 2004.[6] Kolb, E.W., Turner, M.S.
The Early Universe ; Addison-Wesley, Redwood City, CA, 1990.[7] Liddle, A.R., Lyth, D.H.
Cosmological Inflation and Large-Scale Structure ; Cambridge University Press, Cambridge, 2000.[8] Liddle, A.R.
An Introduction to Modern Cosmology ; Wiley, New York, 2015.[9] Eisenhart, L.P.
Riemannian Geometry ; Princeton University Press, Princeton, 1949.[10] Amendola, L., Tsujikawa, S.
Dark Energy, Theory and Observations ; Cambridge University Press, Cambridge, 2010.[11] Hoggatt, V.E. Jr., Bicknell-Johnson, M. “Fibonacci Convolution Sequences”,
Fib. Quart. , , 117.[12] Baringhaus, L. “Fibonacci numbers, Lucas numbers and integrals of certain Gaussian processes”, Proc. Am. Math. Soc. , , 3875. DOI: https://doi.org/10.1090/S0002-9939-96-03691-X[13] Gardiner, J. “Fibonacci, quasicrystals and the beauty of flowers”, Plant Signaling & Behavior , , 1721. DOI:10.4161/psb.22417[14] Jacquod, Ph., Silvestrov, P.G., Beenakker, C.W.J. “Golden rule decay versus Lyapunov decay of the quantum Loschmidtecho”, Phys. Rev. E , , 055203(R). DOI: https://doi.org/10.1103/PhysRevE.64.055203[15] Affleck, I. “Golden mean seen in a magnet”, Nature , , 362. DOI: https://doi.org/10.1038/464362a[16] Luminet, J.-P., Weeks, J., Riazuelo, A., ehoucq, R., Uzan, J.-P. “Dodecahedral space topology as an explanationfor weak wide-angle temperature correlations in the cosmic microwave background”, Nature , , 593. DOI:https://doi.org/10.1038/nature01944[17] Boman, B.M., Dinh, T.N., Decker, K., Emerick, B., Raymond, C., Schleiniger, G. “Why do Fibonacci numbers appear inpatterns of growth in nature? A model for tissue renewal based on asymmetric cell division”, Fibonacci Quart. , ,30.[18] Boman, B.M., Ye, Y., Decker, K., Raymond, C., Schleiniger, G. “Geometric Branching Patterns based on p-FibonacciSequences: Self-similarity Across Different Degrees of Branching and Multiple Dimensions”, Fibonacci Quart. ,7