Dynamical system analysis of cosmological evolution in the Aether scalar tensor theory

Technical Summary: Dynamical System Analysis of Cosmological Evolution in the Aether Scalar Tensor Theory

Problem Statement
The Aether Scalar Tensor (AeST) theory is a relativistic extension of General Relativity (GR) proposed to address galactic and cosmological observations without invoking dark matter. The theory introduces a scalar field $\phi$ and a vector field $A_a$, governed by an action containing an undetermined function $F(Y, Q)$. Previous work established that specific forms of this function could generate an effective fluid contribution to the cosmological evolution equations that approximates dust (cold dark matter) at late cosmic times. However, the functional freedom of $F$ allows for different phenomenological behaviors. This paper investigates an alternative class of functions where the effective fluid approximates dust at the earliest cosmic times, with deviations from dust-like behavior emerging gradually at later times. The central problem is to determine whether such models can reproduce the successful expansion history of the standard $\Lambda$CDM model while satisfying constraints from the weak-field quasistatic regime (Modified Newtonian Dynamics, or MOND).

Methodology
The authors employ the dynamical system formalism to analyze the cosmological evolution of the AeST theory within Friedmann-Lemaître-Robertson-Walker (FLRW) symmetry.

1. Variable Definition: The cosmological equations are recast into a dimensionless autonomous system. The dynamical variables include the curvature parameter $K$, density parameters for matter ($\Omega_m$), radiation ($\Omega_r$), and the cosmological constant ($\Omega_\Lambda$), alongside a variable $\Phi = Q/H$ representing the scalar field dynamics.
2. Function Classes: Three specific classes of the function $F(Q)$ are analyzed:
   • Class 1: $F(Q) \propto (Q-Q_0)^2$, which approximates dust at late times (previous work) but is re-examined here.
   • Class 2: $F(Q) \propto \sqrt{Q_0^2 - Q^2}$, approximating dust at early times.
   • Class 3: $F(Q) \propto \sqrt{Q_0 - Q}$, also approximating dust at early times.
3. Phase Space Analysis: The authors identify fixed points (equilibrium states) corresponding to radiation, matter, cosmological constant, and scalar-field dominated epochs. They analyze the stability of these points (attractors, repellers, saddles) and identify invariant submanifolds (e.g., $\Omega_m=0$, $\Omega_r=0$) which restrict the evolution of trajectories.
4. Numerical Integration: Full numerical integrations of the dynamical equations are performed for specific initial conditions consistent with Planck satellite data (spatial flatness, specific matter/radiation densities).
5. Generalization and Constraints: The models are generalized to include parameters $n$ (denoted as $\alpha$ and $\beta$ models). The authors derive constraints on these parameters by requiring the models to simultaneously:
   • Reproduce the observed dark matter abundance.
   • Satisfy the condition that the effective equation of state $w(\phi)$ remains small at early times (to match Generalized Dark Matter constraints on perturbations).
   • Satisfy the requirement that the theory recovers MOND phenomenology over astrophysical scales (imposing a limit on the mass term $\mu^2$).

Key Contributions and Results

• Phase Space Structure:

  • Class 1: The phase space features fixed points for matter, radiation, and cosmological constant domination. The scalar-field dominated points are repellers (unstable). Numerical integration shows that for small values of a deviation parameter $\Omega_B$, the model can reproduce a standard history (radiation $\to$ matter $\to$ $\Lambda$-domination), but large $\Omega_B$ leads to a direct transition from scalar-field domination to $\Lambda$-domination, skipping standard eras.
  • Class 2: The scalar-field dominated era is a future feature rather than a past one. The cosmological constant dominated fixed point changes stability from an attractor to a saddle. The model can reproduce the $\Lambda$CDM expansion history, with the scalar field effectively acting as dark energy in the future.
  • Class 3: No fixed point corresponds to a scalar-field dominated era; this domination is achieved asymptotically at the boundary of the phase space ($\Phi \to \pm \infty$). The model exhibits a radiation $\to$ matter $\to$ (temporary) $\Lambda$ $\to$ scalar-field dominated history. Notably, the asymptotic state corresponds to phantom matter ($w < -1$), leading to a "Big Rip" scenario.

• Cosmological Background Evolution:

  • All three classes can be tuned to closely approximate the $\Lambda$CDM model at the level of the cosmic background. By adjusting the parameter $\Omega_B$ (which quantifies the deviation from pure dust behavior), the models can reproduce the observed matter density and expansion history without a fundamental dark matter component.

• Constraints from Weak-Field Regime:

  • The paper analyzes the tension between cosmological success and astrophysical constraints. For the generalized models, the authors derive the effective equation of state $w$ and adiabatic sound speed $c_{ad}^2$.
  • Class 1 ($n=1$): The analysis suggests this model is likely inconsistent. To satisfy the MOND recovery constraint (limiting $\mu^2$) while maintaining a small $w$ at early times (required for perturbation stability), the parameter $n$ must be significantly larger than 1 (specifically $n \gtrsim 25$). Thus, the standard Class 1 model fails to satisfy both constraints simultaneously.
  • Class 2 and 3: Preliminary results indicate these models (and their generalizations with appropriate $n$) may satisfy the constraints. They can maintain a small $w$ and $c_{ad}^2$ at early times while allowing for MOND phenomenology on astrophysical scales, making them viable alternatives to dark matter at both the background and perturbation levels.

Significance and Claims
The paper claims that the dynamical system approach reveals that a wide class of functions $F(Q)$ in AeST theory can successfully reproduce the meaningful cosmological background evolution observed in the universe, even in the absence of a Cold Dark Matter (CDM) component.

However, the authors are modest regarding the viability of specific models. They emphasize that while the background evolution can be matched, the simultaneous satisfaction of cosmological perturbation constraints and weak-field MOND constraints is a stringent filter.

• The Class 1 model (previously considered in literature) is likely ruled out because it cannot simultaneously satisfy the requirement for a dust-like equation of state at early times and the recovery of MOND phenomenology on large scales.
• The Class 2 and Class 3 models (and their generalizations) appear more promising, potentially replicating the cosmological successes of CDM while satisfying astrophysical constraints.

The work concludes that while AeST theory offers a mechanism to replace dark matter, the specific functional form of the theory's free function $F$ is critical. Only specific forms (likely those approximating dust at early times with specific scaling behaviors) can survive the dual constraints of cosmological observations and local gravitational tests. The paper does not claim these models are definitively proven but identifies them as the most viable candidates within the current framework for further investigation.