Phase space analysis of an exponential model in $f(Q)$ gravity including linear dark-sector interactions

Technical Summary: Phase Space Analysis of an Exponential Model in $f(Q)$ Gravity Including Linear Dark-Sector Interactions

Problem Statement
The standard $\Lambda$CDM cosmological model, while observationally successful, faces theoretical challenges such as the cosmological constant problem and the coincidence problem (the apparent fine-tuning of dark energy and dark matter densities at the present epoch). Recent observations, including data from the Dark Energy Spectroscopic Imaging (DESI) collaboration, suggest the possibility of a dynamical dark energy fluid rather than a strict cosmological constant. This motivates the exploration of modified gravity theories, specifically $f(Q)$ gravity (Extended Symmetric Teleparallel Gravity), where the gravitational action is a function of the non-metricity scalar $Q$. While various $f(Q)$ models have been proposed, there is a need to analyze specific exponential models that act as smooth perturbations around $\Lambda$CDM to determine their viability in reproducing the full history of cosmic expansion, including radiation, matter, and dark energy domination epochs. Furthermore, the introduction of interactions between dark matter (DM) and dark energy (DE) offers a potential mechanism to address the coincidence problem, necessitating a dynamical analysis of such coupled systems within the $f(Q)$ framework.

Methodology
The authors employ the dynamical systems formalism to analyze an exponential $f(Q)$ model defined by the functional form $F(Q) = 2\Lambda e^{-(b\Lambda/Q)^n}$, where $b$ and $n$ are real parameters. The study follows the strategy proposed by Böhm er et al. to reduce the modified Friedmann equations into an autonomous system of differential equations.

1. Variable Definition: To overcome the non-autonomous nature of the standard normalization method (where variables depend on the Hubble parameter $H$), the authors introduce a set of dimensionless dynamical variables:
   • $x = 8\pi\rho_m / 3H^2$ (matter density parameter)
   • $y = 8\pi\rho_r / 3H^2$ (radiation density parameter)
   • $m = H^2 / (\Lambda + H^2)$ (a variable related to the Hubble rate and cosmological constant)
   • The variable $Z$ (related to the geometric modification) is expressed in terms of $m$ to close the system.
2. Autonomous System Construction: By rewriting the Friedmann equations in terms of the evolution parameter $\eta = \ln a$, the authors derive a closed system of equations for the pair $\{x, m\}$ (with $y$ determined by the constraint equation).
3. Critical Point Analysis: The system is analyzed by finding equilibrium points where $\dot{x} = 0$ and $\dot{m} = 0$. The stability of these points is determined via linear stability analysis using the Jacobian matrix and its eigenvalues.
4. Interaction Extension: The model is extended to include a linear interaction kernel $Q = \xi H \rho_{DE}$ between dark matter and dark energy. The continuity equations are modified to include this interaction term, and the dynamical system is re-analyzed to find new critical points and stability conditions dependent on the coupling parameter $\xi$.

Key Contributions and Results

• Non-Interacting Model Analysis:

  • The autonomous system yields three critical points corresponding to the standard cosmological epochs:
    1. Radiation-dominated ($R$): Coordinates $\{0, 1\}$. Eigenvalues $\{1, 4\}$ indicate this is an unstable repeller.
    2. Matter-dominated ($M$): Coordinates $\{1, 1\}$. Eigenvalues $\{-1, 3\}$ indicate this is a saddle point, serving as a transition between radiation and dark energy domination.
    3. de Sitter Attractor ($P_{dS}$): Coordinates $\{0, m_{DE}\}$. The value $m_{DE}$ is a solution to a transcendental equation dependent on the parameter $b$. Linear stability analysis shows eigenvalues are negative for both $b > 0$ and $b < 0$, confirming $P_{dS}$ is a stable attractor.
  • Cosmological Parameters: The model successfully reproduces the effective equation of state ($w_{eff}$) and deceleration parameter ($q$) for each epoch: $w_{eff} = 1/3, q=1$ for radiation; $w_{eff} = 0, q=1/2$ for matter; and $w_{eff} = -1, q=-1$ for the de Sitter phase.
  • Parameter $b$ Influence: The parameter $b$ controls the timing of the transition to the de Sitter epoch. For $b < 0$, the model exhibits phantom-like behavior ($w_{eff} < -1$), while $b > 0$ corresponds to quintessence-like behavior. The de Sitter point exists only for $b < 1/6$; for larger values, the point becomes complex and disappears from the phase space.

• Interacting Model Analysis:

  • With the inclusion of the linear interaction $Q = \xi H \rho_{DE}$, the radiation and matter points remain unchanged in stability (unstable and saddle, respectively).
  • A new interacting de Sitter point ($P_{int}$) emerges. Its coordinates depend on both $b$ and the coupling $\xi$.
  • Stability: The interacting point $P_{int}$ is found to be a stable attractor for the range of parameters analyzed.
  • Physical Implications of Interaction:
    • For $\xi > 0$ (energy transfer from DE to DM), the matter density does not vanish completely at the attractor ($x_{int} \neq 0$), implying a universe where dark matter persists alongside dark energy.
    • For $\xi < 0$ (energy transfer from DM to DE), the model predicts negative matter density parameters ($\Omega_m < 0$) at the attractor, which is unphysical.
    • The interaction induces a "phantom crossing" where the effective equation of state $w_{eff}$ crosses the $-1$ barrier, leading to values $w_{eff} < -1$ in the far future. This suggests a violation of the Null Energy Condition (NEC) in the late-time universe.

Significance and Conclusions
The paper demonstrates that the proposed exponential $f(Q)$ model is a viable cosmological scenario that qualitatively reproduces the standard sequence of cosmic domination epochs (radiation $\to$ matter $\to$ dark energy) without requiring a strict cosmological constant, but rather emerging from the geometric modifications of gravity. The model behaves as a perturbative expansion around $\Lambda$CDM, consistent with previous statistical constraints using $H(z)$ datasets.

The inclusion of linear dark-sector interactions reveals that while such models can alter the late-time evolution and potentially address the coincidence problem, they introduce complexities such as the potential for unphysical negative densities or violations of energy conditions (NEC) depending on the sign and magnitude of the coupling. The authors conclude that while these dynamical dark energy models in $f(Q)$ gravity are motivated by recent observational hints (e.g., DESI), the expansion of the parameter space (introducing $b$ and $\xi$) makes statistical constraints more challenging. The work underscores the need for further constraints using upcoming data releases to determine the viability of these extended gravity models.