[Paper Review] On coincidence problem and attractor solutions in ELKO dark energy model

H. Mohseni Sadjadi|arXiv (Cornell University)|Sep 9, 2011

Cosmology and Gravitation Theories44 references15 citations

TL;DR

This paper investigates the ELKO spinor field as a dark energy candidate in a spatially flat Friedmann-Robertson-Walker universe, analyzing critical points and attractor solutions without assuming specific potentials or interactions. It finds that no stable attractor solution exists that resolves the coincidence problem—where matter and dark energy densities are comparable today—due to instability in the dynamical system and violation of accelerated expansion conditions.

ABSTRACT

We study the critical points of a universe dominated by ELKO spinor field dark energy and a barotropic matter without considering a specific potential or interaction. The coincidence problem and attractor solutions are discussed at late time, and it is shown that the coincidence problem can not be solved in this model.

Motivation & Objective

• To examine whether the ELKO spinor field can resolve the coincidence problem in late-time cosmology.
• To analyze critical points and attractor solutions in a universe dominated by ELKO dark energy and barotropic matter.
• To assess the stability of the dynamical system under general potentials and interactions, without restricting to specific forms.
• To determine whether any critical point can yield a stable, late-time solution with matter-to-dark energy density ratio ≈1.
• To evaluate whether the model can support an accelerated expansion phase consistent with observations.

Proposed method

• Formulated a dynamical system using dimensionless variables: x (field velocity), y (potential energy), z (matter density), u (field amplitude) normalized by Hubble parameter and Planck mass.
• Derived autonomous equations from Friedmann and Raychaudhuri equations, incorporating a general interaction term C between matter and dark energy.
• Defined the effective equation of state parameter w = -1 + 2/3 ω, with ω = -Ḣ/H², to assess cosmic acceleration.
• Analyzed critical points (II–VIII) by linearizing the system around them and computing the Jacobian matrix M to evaluate stability via eigenvalues.
• Evaluated the interaction terms C = σHρ_m, C = ζH(ρ_m + ρ_d), and C = αρ_m/M_p ⋅ ẟφ to determine which yield well-defined critical points.
• Used the condition that all eigenvalues of the stability matrix M must have negative real parts for a stable attractor solution.

Experimental results

Research questions

• RQ1 Can the ELKO spinor field model produce a stable attractor solution that naturally explains the observed coincidence of matter and dark energy densities?
• RQ2 Under what conditions on the potential and interaction can the system reach a critical point with ρ_m / ρ_d ≈ 1?
• RQ3 Is there any critical point in the ELKO dark energy model that supports an accelerated expansion phase (w < -1/3) and is dynamically stable?
• RQ4 How do different interaction forms (C = σHρ_m, C = ζH(ρ_m + ρ_d), C = αρ_m/M_p ⋅ ẟφ) affect the existence and stability of critical points?
• RQ5 Why does the model fail to resolve the coincidence problem despite the inclusion of general interactions and potentials?

Key findings

• No stable attractor solution exists in the ELKO dark energy model that resolves the coincidence problem, as the system lacks a critical point with all negative eigenvalues in the stability matrix.
• Among all critical points, only case IV yields a matter-to-dark energy density ratio r ≈ 3/7 ≈ O(1), but it fails due to a positive eigenvalue λ = 3γ/2, indicating instability.
• The critical point IV predicts ω̄ = 3γ/2 > 1, implying w̄ = -1 + 2/3 ω̄ > -1/3, which corresponds to a decelerating universe, contradicting observed cosmic acceleration.
• Interaction terms C = σHρ_m and C = ζH(ρ_m + ρ_d) lead to singular or ill-defined C₁ at critical points (especially when x̄ = 0), invalidating attractor solutions.
• Only the interaction C = αρ_m/M_p ⋅ ẟφ yields a well-defined C₁, but even then, no stable attractor with r ≈ O(1) and w < -1/3 is found.
• The model cannot alleviate the coincidence problem in general, regardless of the choice of potential or interaction, due to inherent dynamical instability and incorrect expansion behavior.