Friedmann equations as n-dimensional dynamical system

Branković, Danijela

doi:10.2478/auom-2023-0017

Abstract

In this paper we study dynamics of the standard cosmological model of the universe assuming that it is filled with n types of non-interacting barotropic perfect fluids. For that purpose, a dynamical system of a class of Lotka-Volterra dynamical systems is derived, that consists of n nonlinear differential equations of the first order, whose dependent variables are density parameters of the material in the universe. Analytical solution of that system represents new parametrization of density parameters. Moreover, we perceive the evolution of the universe in the frame of the linear stability theory.

References

V. Aboites, J. F. Bravo-Avilés, J. H. García-López, R. Jaimes-Reategui and G. Huerta-Cuellar, Interpretation and Dynamics of the Lotka-Volterra Model in the Description of a Three-Level Laser. Photonics 9 (2022), 16.
Search in Google Scholar Back to article
S. Bahamonde, C. G. Böhmer, S. Carloni, E. J. Copeland, W. Fang and N. Tamanini, Dynamical systems applied to cosmology: Dark energy and modified gravity. Phys. Rep. 775-777 (2018), 1-122.
Search in Google Scholar Back to article
D. Branković, Cosmic Time for Multi-Component Universe. Serb. Astron. J. 201 (2020), 15-23.
Search in Google Scholar Back to article
D. Branković and Ž. Mijajlović, Dynamics of the CDM model of the universe from the aspect of the dynamical systems theory. arXiv preprint (2022). https://arxiv.org/abs/2209.02605.
Search in Google Scholar Back to article
Y. Chen, J. Ni and Y. C. Ong, Lotka-Volterra models for extraterrestrial self-replicating probes. Eur. Phys. J. Plus 137 (2022), 1109.
Search in Google Scholar Back to article
U. Foryś, Multi-dimensional Lotka-Volterra systems for carcinogenesis mutations. Math. Meth. Appl. Sci. 32 (2009), 2287-2308.
Search in Google Scholar Back to article
A. Friedmann, Über die Möglichkeit einer Welt mit konstanter negativer Krümmung des Raumes. Z. Phys. 21 (1924), 326-332.
Search in Google Scholar Back to article
R. García-Salcedo, T. Gonzalez, F. A. Horta-Rangel, I. Quiros and D. Sanchez-Guzmán, Introduction to the application of dynamical systems theory in the study of the dynamics of cosmological models of dark energy. Eur. J. Phys. 36 (2015), 025008.
Search in Google Scholar Back to article
M. Goliath and G. F. R. Ellis, Homogeneous cosmologies with a cosmological constant. Phys. Rev. D 60 (1999), 023502.
Search in Google Scholar Back to article
M. P. Hobson, G. P. Efstathiou and A. N. Lasenby, General Relativity: An Introduction for Physicists. Cambridge University Press, 2006.
Search in Google Scholar Back to article
A. R. Liddle and D. H. Lyth, Cosmological Inflation and Large-Scale Structure. Cambridge University Press, 2000.
Search in Google Scholar Back to article
A. J. Lotka, UNDAMPED OSCILLATIONS DERIVED FROM THE LAW OF MASS ACTION. J. Am. Chem. Soc. 42 (1920), 1595-1599.
Search in Google Scholar Back to article
Ž. Mijajlović and D. Branković, Algebraic dependencies and representations of cosmological parameters. Publ. Astron. Obs. Belgrade 100 (2021), 295-300.
Search in Google Scholar Back to article
J. Perez, A. Füzfa, T. Carletti, L. Mélot and L. Guedezounme, The Jungle Universe: coupled cosmological models in a Lotka-Volterra framework. Gen. Relativ. Gravit. 46 (2014), 1753.
Search in Google Scholar Back to article
A. M. Stuart and A. R. Humphries, Dynamical Systems and Numerical Analysis. Cambridge Monographs on Applied and Computational Mathematics. Cambridge University Press, 1996.
Search in Google Scholar Back to article
J.-P. Uzan and R. Lehoucq, A dynamical study of the Friedmann equations. Eur. J. Phys. 22 (2001), 371.
Search in Google Scholar Back to article
V. Volterra, Fluctuations in the Abundance of a Species considered Mathematically. Nature 118 (1926), 558-560.
Search in Google Scholar Back to article

Friedmann equations as n-dimensional dynamical system

Abstract

Paradigm

My account