arXiv 2510.045312025-10-10gr-qc

非最小耦合与非最小导数耦合引力理论中的宇宙学动力学

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

Institute of Physics, Kazan Federal University（喀山联邦大学物理研究所）

Ravil R. Fatykhov, Sergey V. Sushkov

备注：: 39 pages, 8 figures

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本文研究含非最小耦合与非最小导数耦合的标量-张量引力理论，通过动力系统分析揭示早期暴胀、幻影膨胀及晚期幂律、辐射和指数膨胀等稳定渐近行为，并给出数值相图。

摘要：

本文探讨了标量-张量引力理论中的宇宙学情景，该理论同时包含与标量曲率的形式为$Rϕ^2$的非最小耦合，以及在存在具有单项式依赖$V(ϕ) = V_0ϕ^n$的标量场势时，形式为$G^{μν}ϕ_{,μ}ϕ_{,ν}$的非最小导数耦合。我们获得并分析了系统的临界点。在没有标量场势的情况下，确定了这些点的稳定性条件。利用动力系统理论的方法，分析了模型的渐近行为。结果表明，在$V(ϕ)\equiv0$或$n < 2$的情况下，存在准德西特渐近行为，对应于早期暴胀宇宙。在$V_0 \rightarrow 0,\\ ξ\rightarrow 0$的近似下，该渐近行为与致力于非最小动力学耦合宇宙学模型的文献中得到的值$H = \frac{1}{\sqrt{9|η|}}$一致。对于$|ξ|\\ \rightarrow \infty$，该渐近行为趋向于值$H = \frac{1}{\sqrt{3|η|}}$。此外，对于模型的早期动力学，发现了具有幻影膨胀$w_{eff} < -1$的不稳定状态。对于晚期动力学，我们得到了以下稳定的渐近状态：具有$w_{eff} \ge 1$的幂律膨胀，具有$w_{eff} =\frac{1}{3}$的膨胀（$V(ϕ)\equiv0$），此时有效普朗克质量趋于零，以及当$n = 2$时具有$w_{eff} = 0$的指数膨胀。在这种情况下，哈勃参数的渐近值仅取决于$V_0 = \frac{1}{2}m^2$和$ξ$。针对理论参数的特定值，对模型动力学进行了数值积分。结果以相图形式呈现。

英文摘要：

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.

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