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Dynamical Systems in Cosmology: Reviewing An Alternative Approach

Nandan Roy, L. Arturo Ureña-López

Abstract

Dark energy is one of the deepest puzzles in modern cosmology, and mounting evidence suggests that it is not just a cosmological constant but a genuinely dynamical component. Although cosmology and dynamical systems theory emerged from different disciplines, dynamical systems methods have become essential tools to uncover the qualitative evolution of the universe. The equations governing homogeneous and isotropic cosmologies can be naturally written as systems of ordinary differential equations, making them an ideal arena for dynamical system analysis. This review begins with a sharp, streamlined introduction to the standard dynamical systems toolkit widely used in cosmology. We then move on to alternative formulations based on polar and hyperbolic variable transformations. These approaches unlock powerful new ways to probe a broad spectrum of scalar field dark energy models, to set and constrain initial conditions, and to analyze tracking behavior across wide classes of potentials. The review is self-contained, but consistently directs the reader to more specialized and in-depth treatments where needed.

Dynamical Systems in Cosmology: Reviewing An Alternative Approach

Abstract

Dark energy is one of the deepest puzzles in modern cosmology, and mounting evidence suggests that it is not just a cosmological constant but a genuinely dynamical component. Although cosmology and dynamical systems theory emerged from different disciplines, dynamical systems methods have become essential tools to uncover the qualitative evolution of the universe. The equations governing homogeneous and isotropic cosmologies can be naturally written as systems of ordinary differential equations, making them an ideal arena for dynamical system analysis. This review begins with a sharp, streamlined introduction to the standard dynamical systems toolkit widely used in cosmology. We then move on to alternative formulations based on polar and hyperbolic variable transformations. These approaches unlock powerful new ways to probe a broad spectrum of scalar field dark energy models, to set and constrain initial conditions, and to analyze tracking behavior across wide classes of potentials. The review is self-contained, but consistently directs the reader to more specialized and in-depth treatments where needed.

Paper Structure

This paper contains 30 sections, 56 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Basics of Dynamical Systems Analysis
  3. Linear Systems
  4. Non-Linear Systems
  5. Existence and Uniqueness Theorem
  6. The Stable Manifold Theorem
  7. The Hartman--Grobman Theorem
  8. The Center Manifold Theorem
  9. Lyapunov Functions
  10. Dynamical Systems Formalism in Cosmology
  11. Scalar Field Cosmology
  12. Autonomous System: the standard transformation
  13. Autonomous System: the polar transformation
  14. Critical Points and Physical Interpretation
  15. (a) Fluid-dominated solution.
  16. ...and 15 more sections

Figures (4)

  • Figure 1: (Left) The evolution of the EoS of the quintessence field with the choice of the tracker initial condition for the corresponding $\alpha_i$ parameter triplets shown in the labels of the plot. The lines in blue are shown for the tracker value of the EoS during matter and radiation domination. (Right) Evolution of the density parameters of the universe for the tracker quintessence for same choice of the $\alpha_i$ parameters.
  • Figure 2: The CMB anisotropies and the matter power spectrum (MPS) predicted by the tracker quintessence model, with the corresponding $\Lambda$CDM case also displayed for comparison in each panel.
  • Figure 3: (Left) The evolution of the EoS of the phantom field with the choice of the tracker initial condition for the corresponding $\alpha_i$ parameter triplets shown in the labels of the plot. The lines in blue are shown for the tracker value of the EoS during matter and radiation domination. (Right) Evolution of the density parameters of the universe for the tracker quintessence for same choice of the $\alpha_i$ parameters.
  • Figure 4: The CMB anisotropies and the matter power spectrum (MPS) predicted by the tracker phantom model, with the corresponding $\Lambda$CDM case also displayed for comparison in each panel.