Cosmological dynamics in the theory of gravity both with non-minimal and non-minimal derivative coupling

TL;DR

The paper analyzes cosmological dynamics in a scalar-tensor theory featuring both a non-minimal curvature coupling $\xi\phi^2R$ and a non-minimal derivative coupling $\eta G^{\mu\nu}\phi_{,\mu}\phi_{,\nu}$, with a monomial potential $V(\phi)=V_0\phi^n$. By recasting the equations into a constrained dynamical system using dimensionless variables, the authors perform a thorough stability and asymptotic analysis, identifying multiple stationary regimes including quasi-de Sitter and phantom early behavior and various late-time attractors such as power-law and exponential expansions. For $V(\phi)\equiv0$ or $n<2$, the model admits early quasi-de Sitter behavior with $H$ approaching $\frac{1}{\sqrt{9|\eta|}}$ as $V_0\to0,\ \xi\to0$, and approaching $\frac{1}{\sqrt{3|\eta|}}$ as $|\xi|\to\infty$; with $n=2$ a transition to exponential expansion with $w_{eff}=0$ occurs depending on $V_0$ and $\xi$. The analysis reveals a rich structure of stationary points and their stability across parameter space, and numerical phase portraits illustrate the qualitative dynamics. The results extend previous work on non-minimal and non-minimal derivative couplings by showing how the combination of couplings shapes early and late-time cosmological evolution.

Abstract

This paper explores cosmological scenarios in a scalar-tensor theory of gravity, including both a non-minimal coupling with scalar curvature of the form $Rφ^2$ and a non-minimal derivative coupling of the form $G^{μν}φ_{,μ}φ_{,ν}$ in the presence of a scalar field potential with the monomial dependence $V(φ) = V_0φ^n$. Critical points of the system were obtained and analyzed. In the absence of a scalar field potential, stability conditions for these points were determined. Using methods of dynamical systems theory, the asymptotic behavior of the model was analyzed. It was shown that in the case of $V(φ)\equiv0$ or $n < 2$, a quasi-de Sitter asymptotic behavior exists, corresponding to an early inflationary universe. This asymptotic behavior in the approximation $V_0 \rightarrow 0,\ ξ\rightarrow 0$ coincides with the value $H = \frac{1}{\sqrt{9|η|}}$ obtained in works devoted to cosmological models with non-minimal kinetic coupling. For $|ξ|\ \rightarrow \infty$, this asymptotic behavior tends to the value $H = \frac{1}{\sqrt{3|η|}}$. Moreover, unstable regimes with phantom expansion $w_{eff} < -1$ were found for the early dynamics of the model. For the late dynamics, the following stable asymptotic regimes were obtained: a power-law expansion with $w_{eff} \ge 1$, an expansion with $w_{eff} =\frac{1}{3}$ ($V(φ)\equiv0$), at which the effective Planck mass tends to zero, and an exponential expansion with $w_{eff} = 0$ as $n = 2$. In this case, the asymptotic value of the Hubble parameter depends only on $V_0 = \frac{1}{2}m^2$ and $ξ$. Numerical integration of the model dynamics was performed for specific values of the theory parameters. The results are presented as phase portraits.

Figures (8)

• Figure 1: The $y=0$ curve (equation \ref{['eq:eq4_quad']}). The blue part of the curve corresponds to saddle points, the black part to nodal points. The lower right corner shows a zoomed-in region near $\xi=\frac{1}{6},\ \frac{3}{16}$.
• Figure 2: Left: The black and blue solid lines represent the roots $r_*$ of the equation \ref{['eq:eq4_cub']} as functions of $\xi$; the black dashed line represents the asymptotic behavior of $r_* = -12\xi+5$ as $\xi \rightarrow \infty$; the red line represents the curve $y=0$ ($H=0$). For $\xi=0$, there is a single value $r_1=6$. Right: An enlarged region is shown for the values $\xi = \frac{1}{6},\ \frac{3}{16}$ and $\xi_* \approx 0.19$. The region inside the hyperbola $y=0$ for $\eta < 0$ corresponds to the asymptotics with $H^2 < 0$ and has no physical meaning.
• Figure 3: The value of $9\eta H^2$ depending on the value of the parameter $\xi$. The blue dashed line shows the asymptote $9\eta H^2 = -3$ at $\xi \rightarrow \infty$. The black dot shows the point of self-intersection of the curve with coordinates $(0.1, 0.6)$. The red dots mark the extremum points. The value $\xi=0$ corresponds to a single point on the curve with the value $9\eta H^2 = -1$.
• Figure 4: Different classes of phase portraits for $V(\phi) \equiv 0$. Phase portraits of the system in $(\phi, \dot{\phi})$ coordinates for $\eta < 0$. White and gray areas correspond to expansion with deceleration (barotropic index $w \leq 1/3$ and $w > 1/3$, respectively), yellow and red areas correspond to accelerated ($w < -1/3$) and superaccelerated expansion ($w < -1,\ \dot{H} > 0$). The black dotted lines indicate the discontinuity of the dynamics (the $\ddot{\phi}$ singularity). The light blue regions correspond to the contraction $H < 0$. The black dots indicate stationary points $\pm 1/\sqrt{\xi}$. The white regions without arrows indicate unphysical regions.
• Figure 5: Equation \ref{['eq:eqz2']}.