Study of dynamical systems and large-scale structure

In this study, we employ dynamical systems methods to analyse the large-scale structure by considering two distinct interaction models (linear and non-linear) within the dark sector, associated with a specific dynamical dark energy model inspired by the Veneziano ghost theory in quantum chromodynamics (QCD). In these models, the dark energy density ($ρ_{DE}$) varies with the Hubble parameter ($H$), expressed as $ρ_{DE} = αH + βH^2$. After defining the dimensionless parameters, we present autonomous equations that allow us to find the trace $\text{Tr}(J)$ and the determinant $D(J)$. With these solutions, we demonstrate the presence of unstable, saddle, and stable fixed points, corresponding to the radiation-, matter-, and dark-energy-dominated eras, respectively. Our results suggest that these models are theoretically viable for representing the interaction between dark sector fluids.

💡 Research Summary

This paper investigates the cosmological dynamics of interacting dark sector models by employing dynamical‑system techniques. The authors adopt a dark‑energy (DE) density inspired by the Veneziano ghost model of QCD, namely ρ_DE = α H + β H², where α and β are constants with dimensions of energy³ and energy² respectively, and H is the Hubble parameter. This functional form allows the DE component to evolve with the expansion rate, addressing fine‑tuning and coincidence problems that plague the standard ΛCDM paradigm.

Two interaction prescriptions between dark matter (DM) and DE are considered. Model I (linear interaction) uses Q = 3 b² H ρ_m, where b is a dimensionless coupling constant and ρ_m is the DM density. Model II (non‑linear interaction) adopts Q = 3 b² H ρ_m ρ_DE / ρ_tot, introducing a product of the two dark sector densities normalized by the total density. In both cases Q > 0 corresponds to energy transfer from DE to DM, a direction that can alleviate the cosmic coincidence problem.

The continuity equations for radiation, DM and DE are rewritten in terms of dimensionless density parameters Ω_m, Ω_DE and an auxiliary interaction variable Ω_Q. Using the Friedmann equation (H² = 8πG ρ_tot/3) and the Raychaudhuri equation, the authors derive expressions for the effective equation‑of‑state parameter ω_eff and the deceleration parameter q as functions of Ω_m, Ω_DE, the early‑DE parameter ξ (related to β) and the coupling b.

The core of the analysis is the construction of an autonomous system for Ω_m and Ω_DE (Eqs. 8a‑8b). Linear stability around the critical points is examined by computing the Jacobian matrix J, its trace Tr(J), and determinant D(J). The eigenvalues λ are obtained from λ = ½