Global stability analysis for cosmological models with nonminimally coupled scalar fields

We explore dynamics of cosmological models with a nonminimally coupled scalar field evolving on a spatially flat Friedmann-Lemaître-Robertson-Walker background. We consider cosmological models including the Hilbert-Einstein curvature term and the $N$ degree monomial of the scalar field nonminimally coupled to gravity. The potential of the scalar field is the $n$ degree monomial or polynomial. We describe several qualitatively different types of dynamics depending on values of power indices $N$ and $n$. We identify that three main possible pictures correspond to $n<N$, $N<n<2N$ and $n>2N$ cases. Some special features connected with the important cases of $N=n$ (including the quadratic potential with quadratic coupling) and $n=2N$ (which shares its asymptotics with the potential of the Higgs-driven inflation) are described separately. A global qualitative analysis allows us to cover the most interesting cases of small $N$ and $n$ by a limiting number of phase-space diagrams. The influence of the cosmological constant to the global features of dynamics is also studied.

Figure 1

The effective potentials $V_{\text{eff}}=\frac{V_0{\phi }^2+\mathrm{\Lambda }}{{(1-6\xi {\phi }^2)}^2}$ for $N=2$ and $n=2$ (left) and the corresponding phase portrait at $\mathrm{\Lambda }=0$. In both parts $V_0=2$, $\xi =-\frac{1}{4}$. In the left part $\mathrm{\Lambda }=0$ (black curve); $2/3$ (red curve); $4/3$ (blue curve); and 2 (green curve) [from the bottom upwards]. In the right part points $A$ correspond to power-law solutions (20), points $B$ are exponential solutions (22), and points $C$ are de Sitter solutions (16).

Figure 2

The basin of attraction of the oscillations (black bold points) and the exponential solution (22) (green points) for $V={\phi }^2$, $B={\phi }^2$. We take $\xi =-\frac{1}{4}$ (left), and $\xi =-1$ (right).

Figure 3

The solution of system (31) with $V=V_0{\phi }^2$ and $B={\phi }^4$. We put $\xi =-1$, $V_0=1$, and $\mathrm{\Lambda }=0$ (left). The effective potential $V_{\text{eff}}=\frac{V_0{\phi }^2+\mathrm{\Lambda }}{{(1-6\xi {\phi }^4)}^2}$ for $V=V_0{\phi }^2+\mathrm{\Lambda }$ and $B={\phi }^4$, $\xi =-1/4$, $V_0=4$ and $\mathrm{\Lambda }=0$ (black curve); $10/3$ (red curve); $20/3$ (blue curve); and 10 (green curve) [from the bottom upwards].

Figure 4

The effective potential $V_{\text{eff}}=\frac{V_0{\phi }^n+\mathrm{\Lambda }}{{(1-6\xi {\phi }^N)}^2}$ for $N=2$, $n=4$, $\xi =-\frac{1}{4}$, $V_0=4$ and $\mathrm{\Lambda }=0$ (black curve); $4/3$ (red curve); $8/3$ (blue curve); and 4 (green curve) [from the bottom upwards].

Figure 5

The phase portrait of system (31) with $B={\phi }^2$, $\xi =-1/4$ and $\mathrm{\Lambda }=0$. The potentials are $V={\phi }^4$ (left), $V={\phi }^6$ (right).

Figure 6

In left picture there is the effective potential $V_{\text{eff}}=\frac{V_0{\phi }^n+\mathrm{\Lambda }}{{(1-6\xi {\phi }^N)}^2}$ for $N=4$, $n=6$, $\xi =-\frac{1}{4}$, $V_0=4$ and $\mathrm{\Lambda }=0$ (black curve); 0.8 (blue curve); 1.0887 (red curve); 1.4 (green curve) [from the bottom upwards]. In right picture there is the phase portrait of system (31) with $V=5{\phi }^6$, $B={\phi }^4$, $\xi =-10$.

Figure 7

The effective potential $V_{\text{eff}}=\frac{V_0{\phi }^2+\lambda {\phi }^6}{{(1-6\xi {\phi }^N)}^2}$ for $N=2$, $\xi =-\frac{1}{4}$, $V_0=2$ and $\lambda =0$ (black curve); 0.1 (blue curve); 0.25 (red curve); 0.4 (green curve) [from the bottom upwards]; and the phase portrait of system (31) with the corresponding potential $V$ at $\lambda =0.1$.

Figure 8

The solution of system (31) with $B={\phi }^2$ and $V=V_0{\phi }^2+\mathrm{\Lambda }$. We choose $\xi =-1/4$, $V_0=1/2$ and $\mathrm{\Lambda }=0.05$ (left), $\mathrm{\Lambda }=0.2$ (middle) or $\mathrm{\Lambda }=1$ (right).

Figure 9

The solution of system (31) with $V={\phi }^4+\mathrm{\Lambda }$, $B={\phi }^2$, $\xi =-1/4$. We choose $\mathrm{\Lambda }=0.001$, $\mathrm{\Lambda }=0.01$, $\mathrm{\Lambda }=0.1$, and $\mathrm{\Lambda }=1$ (from left to right).