Study of dynamical systems and large-scale structure

TL;DR

This work addresses whether ghost-inspired dynamical dark energy, with $\rho_{DE} = \alpha H + \beta H^2$, can coherently interact with dark matter in a flat universe. It adopts a dynamical-systems approach and analyzes two dark-sector coupling forms: a linear coupling $Q=3 b^2 H \rho_m$ and a nonlinear coupling $Q=3 b^2 H \frac{\rho_m \rho_{DE}}{\rho_{tot}}$, deriving autonomous equations for the dimensionless densities $\Omega_m$, $\Omega_{DE}$, and $\Omega_Q$. The study identifies fixed points corresponding to radiation-, matter-, and dark-energy-dominated epochs (A,B,C for the linear case; D,E,F for the nonlinear case) and demonstrates their stability patterns within a theoretically viable region $0<\xi\ll1$ with small coupling $b^2$. The results show that both interactions yield a plausible cosmic evolution from radiation through matter to dark-energy domination, supporting the theoretical viability of dark-sector interactions in ghost-dark-energy scenarios and laying groundwork for future observational tests conducted in follow-up work.

Abstract

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.

Figures (2)

• Figure 1: (left) The phase portrait plot illustrates the trajectories of $\Omega_{\text{DE}}$ vs $\Omega_{m}$ for Model I with parameters $(\xi, b) = (0.02, \sqrt{0.40})$. The blue arrows show the direction of the trajectories. Fixed point $A$ behaves almost as an unstable radiation-dominated era, with all trajectories diverging from it. Fixed point $B$ is a saddle point behaving as a dark matter-dominated era, where some trajectories approach while others diverge. Fixed point $C$ signifies a stable dark energy-dominated epoch, where all trajectories converge. (right) This plot depicts the evolution of the universe's energy densities as a function of redshift for Model I, using parameters $(\Omega_{m,0}, \Omega_{DE,0}, \xi, b) = (0.315, 0.685, 0.02, \sqrt{0.40})$. The plot tracks the transition from the radiation-dominated era through the dark matter-dominated epoch to the current dark energy-dominated state, all within the viable region consistent with the phase portrait plot.
• Figure 2: (left) The phase portrait for Model II with parameters $(\xi, b) = (0.219, \sqrt{0.27})$ shows the trajectories of $\Omega_{\text{DE}}$ vs $\Omega_{m}$. The fixed point $D$ is unstable and behave almost as the radiation-dominated era, with trajectories diverging from it. The fixed point $E$ is a saddle point that behaves almost as a dark-matter-dominated era, where some trajectories approach it while others move away. The fixed point $F$ is a stable dark energy-dominated era, where all trajectories converge. (right) This plot illustrates the evolution of the universe's energy densities with redshift for Model I, using the parameters $(\Omega_{m,0}, \Omega_{DE,0}, \xi, b) = (0.219, 0.779, 0.22, \sqrt{0.27})$. It shows the transition from the radiation-dominated era, through the dark-matter-dominated phase, to the current dark-energy-dominated state, consistent with the phase portrait plot.