Study of dynamical systems and large-scale structure

Technical Summary: Dynamical Systems Analysis of Interacting Dark Sector Models in Veneziano Ghost Dark Energy

Problem Statement
The standard $\Lambda$CDM cosmological model, while successful in describing large-scale structure, faces persistent theoretical challenges, including the fine-tuning and coincidence problems, as well as tensions such as $H_0$ and $S_8$. These issues motivate the exploration of alternatives, specifically dynamical dark energy models and interactions within the dark sector (between dark matter and dark energy). This study investigates the viability of a specific dark energy model inspired by the Veneziano ghost theory in Quantum Chromodynamics (QCD), where the dark energy density ($\rho_{DE}$) is defined as $\rho_{DE} = \alpha H + \beta H^2$. The authors aim to determine if this model, when coupled with dark matter via linear and non-linear interaction terms, can successfully reproduce the observed cosmological evolution (radiation, matter, and dark energy-dominated eras) using dynamical systems analysis.

Methodology
The authors employ the dynamical systems approach to analyze the evolution of a flat universe containing radiation ($\rho_r$), dark matter ($\rho_m$), and the specified ghost dark energy ($\rho_{DE}$).

1. Framework: The analysis is grounded in the Friedmann and Raychaudhuri equations. The continuity equations for the energy densities include an interaction term $Q(t)$ representing energy exchange between dark matter and dark energy.
2. Dimensionless Variables: The system is transformed into an autonomous system of first-order differential equations using dimensionless density parameters: $\Omega_m$, $\Omega_{DE}$, and a specific interaction parameter $\Omega_Q$. A key parameter, $\xi$, is introduced to represent the early dark energy density contribution derived from the $\beta H^2$ term.
3. Interaction Models: Two distinct interaction scenarios are analyzed:
   • Model I (Linear): $Q = 3b^2 H \rho_m$, where energy transfer is proportional to dark matter density.
   • Model II (Non-linear): $Q = 3b^2 H \frac{\rho_m \rho_{DE}}{\rho_{tot}}$, where the transfer depends on the product of densities normalized by the total density.
4. Stability Analysis: The authors identify fixed points (equilibrium states) of the autonomous equations. The stability of these points is determined by calculating the trace ($Tr(J)$) and determinant ($D(J)$) of the Jacobian matrix to find eigenvalues. This classification determines whether a fixed point represents an unstable, saddle, or stable cosmological epoch.

Key Contributions and Results
The study provides a phase-space analysis of the two interaction models, demonstrating that both can support a viable cosmological history:

• Fixed Point Identification: For both models, the analysis identifies three critical fixed points corresponding to the standard cosmological eras:
  • Radiation-dominated era: Represented by an unstable fixed point (Points A and D). Trajectories diverge from this state, consistent with the early universe.
  • Matter-dominated era: Represented by a saddle fixed point (Points B and E). Trajectories approach this state temporarily before moving toward the late-time attractor.
  • Dark energy-dominated era: Represented by a stable fixed point (Points C and F). All trajectories in the viable region converge to this state, indicating a late-time accelerated expansion.
• Model Viability:
  • Model I: Under the assumption $0 < b^2$ and $\xi \ll 1$, the model successfully transitions from radiation to matter to dark energy domination. The authors note that while previous studies on linear interactions in ghost dark energy suggested infeasibility for radiation domination, this specific coupling ($Q \propto \rho_m$) resolves that issue.
  • Model II: Similarly, the non-linear interaction yields a stable evolution through the three eras. The results are constrained by observational data (Planck 2018 for Model I and SNIa data for Model II) to ensure physical consistency.
• Stability Conditions: The stability of the late-time dark energy attractor is confirmed when the trace of the Jacobian is negative and the determinant is positive, ensuring the universe settles into a stable accelerated expansion phase.

Significance and Claims
The paper claims that the Veneziano ghost dark energy model, when coupled with dark matter via either linear or non-linear interaction terms, is theoretically viable. Specifically:

• It demonstrates that the interaction term $Q$ can alleviate the coincidence problem by facilitating a smooth transition between cosmological eras.
• It counters previous assertions that linear interactions in ghost dark energy theories preclude a radiation-dominated epoch; the authors show that with the specific form $Q = 3b^2 H \rho_m$, a radiation era is preserved.
• The study concludes that both models evolve within a theoretically valid region consistent with cosmological parameter constraints.

The authors maintain a modest stance regarding their findings, noting that while the models are theoretically robust, their consistency with broader observational data requires further investigation, which they intend to address in future work. The paper does not claim to solve the fundamental microphysical basis of the interactions but rather provides a phenomenological framework that fits the dynamical requirements of the universe's evolution.