[Paper Review] Dynamical analysis for a scalar-tensor model with Gauss-Bonnet and non-minimal couplings

L. N. Granda, D. F. Jimenez|arXiv (Cornell University)|Oct 13, 2017

Cosmology and Gravitation Theories29 references3 citations

TL;DR

This paper presents a dynamical systems analysis of a scalar-tensor cosmological model with non-minimal coupling to curvature and Gauss-Bonnet (GB) invariants. By studying critical points under power-law and exponential forms of coupling and potential functions, it identifies stable attractor solutions for quintessence, phantom, and de Sitter phases—achieving constant effective Newtonian coupling in the asymptotic limit for exponential couplings, with phantom behavior realized without ghost degrees of freedom.

ABSTRACT

We study the autonomous system for a scalar-tensor model of dark energy with Gauss-Bonnet and non-minimal couplings. The critical points describe important stable asymptotic scenarios including quintessence, phantom and de Sitter attractor solutions. Two functional forms for the coupling functions and the scalar potential were considered: power-law and exponential functions of the scalar field. For the exponential functions the existence of stable quintessence, phantom or de Sitter solutions, allows an asymptotic behavior where the effective Newtonian coupling becomes constant. The phantom solutions could be realized without appealing to ghost degrees of freedom. Transient inflationary and radiation dominated phases can also be described.

Motivation & Objective

• To investigate the late-time cosmological dynamics of a scalar-tensor model with non-minimal coupling to Ricci scalar and coupling to Gauss-Bonnet invariant.
• To analyze the stability and asymptotic behavior of critical points in the autonomous dynamical system derived from the model’s field equations.
• To determine whether stable quintessence, phantom, or de Sitter solutions can emerge without introducing ghost degrees of freedom.
• To compare the cosmological implications of power-law versus exponential functional forms for coupling and potential functions.
• To examine the asymptotic behavior of the effective Newtonian coupling (defined as $ F(\phi)^{-1} $) in different late-time scenarios.

Proposed method

• Formulate the action for a scalar-tensor model with non-minimal coupling $ F(\phi) $, GB coupling $ \eta(\phi) $, and scalar potential $ V(\phi) $, including matter Lagrangian.
• Derive the Friedmann-Robertson-Walker (FRW) field equations from the action and reduce them to an autonomous dynamical system using dimensionless variables.
• Define critical points by setting time derivatives of dynamical variables to zero and solve for fixed points in terms of model parameters.
• Perform linear stability analysis by computing the Jacobian matrix at each critical point and evaluating eigenvalues to determine stability (attractor, saddle, etc.).
• Analyze the asymptotic behavior of the effective Newtonian coupling $ F(\phi)^{-1} $ in the limit $ t \to \infty $, particularly for exponential and power-law forms.
• Compare results between two functional forms: power-law (e.g., $ \eta \propto \phi^b $, $ V \propto \phi^c $) and exponential (e.g., $ \eta \propto e^{b\phi} $, $ V \propto e^{c\phi} $), focusing on attractor solutions and coupling behavior.

Experimental results

Research questions

• RQ1 What stable late-time cosmological attractor solutions (e.g., quintessence, phantom, de Sitter) emerge in a scalar-tensor model with non-minimal and Gauss-Bonnet couplings?
• RQ2 How does the asymptotic behavior of the effective Newtonian coupling $ F(\phi)^{-1} $ depend on the functional form (power-law vs. exponential) of the coupling and potential functions?
• RQ3 Can phantom-like behavior be realized without introducing ghost degrees of freedom in this model?
• RQ4 What role do the parameters $ b $, $ c $, and $ d $ play in determining the stability and nature of de Sitter, quintessence, or radiation-dominated fixed points?
• RQ5 Under what conditions do the critical points correspond to constant effective Newtonian coupling in the late-time limit, and how does this differ between power-law and exponential models?

Key findings

• Stable de Sitter attractor solutions exist when $ d = b $ in the exponential model (point B5), with stability depending on the sign of $ b - c $, where $ b > c $ ensures stability.
• For the exponential model, the effective Newtonian coupling $ F(\phi)^{-1} $ tends to a constant value at late times when $ b < d $, and vanishes when $ b > d $, indicating a transition to a constant or diverging gravitational strength.
• The Higgs-like potential $ V \propto \phi^4 $ and constant potential both lead to stable de Sitter solutions in the power-law model, particularly at large non-minimal coupling limits.
• Phantom behavior is realized without ghost degrees of freedom, as confirmed by the absence of ghost modes in the second-order field equations due to the GB coupling.
• The critical point B7 describes a transient radiation-dominated phase with $ w_{\text{eff}} = 1/3 $, though it is a saddle point, indicating it is not a late-time attractor.
• In the exponential model, the parameters $ b $, $ c $, and $ d $ can be tuned to achieve asymptotic dark energy equation of state $ w_{\text{eff}} $ arbitrarily close to $-1$, enabling precise phenomenological fits to observational data.