Classes of nonminimally coupled scalar fields in spatially curved FRW spacetimes

In this work we perform a dynamical analysis of a broad class of nonminimally coupled real scalar fields in the Friedmann-Robertson-Walker (FRW) spacetime framework. The first part of our study concerns the dynamics of an unspecified positive potential in a spatially curved FRW spacetime, for which we define a new set of dimensionless variables and a new evolution parameter. In the framework of this general setup, we have recognized several general features of the system, like symmetries, invariant subsets and critical points, and provide their cosmological interpretation. The second part of our work focuses on flat FRW cases for which the tracker parameter is constant; i.e., we examine specific classes of potentials. After analyzing these cases dynamically, we discuss their physical interpretation.

Figure 1

Invariant subsets constrained by the Friedmann equations for $\xi =1/10$. Left panel: Positive curvature equation (21) for ${\mathrm{\Omega }}_V=0$. Middle panel: Spatially flat equations (21) and (24) for ${\mathrm{\Omega }}_H=1$. Right panel: Nonpositive curvature equation (24) for ${\mathrm{\Omega }}_V=0$.

Figure 2

Some invariant subsets ${\mathrm{\Omega }}_{\partial V}=\text{const}$ for $\xi =\frac{1}{10}$. The left column of panels shows ${\mathrm{\Omega }}_H=-1$ cases, while the right column shows the ${\mathrm{\Omega }}_H=1$ cases. Upper panels show ${\mathrm{\Omega }}_{\partial V}=-1$, the middle ones ${\mathrm{\Omega }}_{\partial V}=1$, and the lower ones ${\mathrm{\Omega }}_{\partial V}=0$. Blue dots identify sources, red dots are sinks, and black ones are saddle points. The green areas denote the phase of accelerated expansion $q<0$.

Figure 3

Global view of the parameter space for the system with $\mathrm{\Gamma }=1$, ${\mathrm{\Omega }}_H=1$, and $\xi =1/10$. The dynamics takes place inside every horizontal plane with constant ${\mathrm{\Omega }}_{\partial V}$; the case ${\mathrm{\Omega }}_{\partial V}=-1$ is shown as representative. For a better view, the closed boundary surface corresponding to ${\mathrm{\Omega }}_V=0$ is cut along the plane ${\mathrm{\Omega }}_{\psi }=0$. The blue dashed line is one set of sources, the black lines are the saddles, and the red line is the set of future attractors.

Figure 4

Global view of the parameter space for the system with $\mathrm{\Gamma }\ne 1$ and constant, ${\mathrm{\Omega }}_H=1$, $\xi =1/10$, and $c_1=0$. The dynamics takes place inside every horizontal plane with constant $\mathrm{\Gamma }$; the cases $\mathrm{\Gamma }=1/2$ and $\mathrm{\Gamma }=3/2$ are shown as representative. For a better view, the closed boundary surface corresponding to ${\mathrm{\Omega }}_V=0$ is cut along the plane ${\mathrm{\Omega }}_{\psi }=0$. The blue dashed line is the set of sources, the black lines are the saddles, and the red lines are the sets of future attractors.

Figure 5

For the case $\xi =\frac{1}{10}$ invariant subsets for $\mathrm{\Gamma }=-1$. The upper panel shows $c_1=\sqrt{\frac{(1-\mathrm{\Gamma }{)}^2}{\xi }}+5$, the middle panel $c_1=\sqrt{\frac{(1-\mathrm{\Gamma }{)}^2}{\xi }}$, and the bottom panel $c_1=\sqrt{\frac{(1-\mathrm{\Gamma }{)}^2}{\xi }}-2$. The green areas denote the phase of accelerated expansion $q<0$.

Figure 6

For the case $\xi =\frac{1}{10}$ invariant subsets for $\mathrm{\Gamma }=3/2$. The upper left panel shows $c_1=\sqrt{\frac{(1-\mathrm{\Gamma }{)}^2}{\xi }}+1$, the upper right panel $c_1=\sqrt{\frac{(1-\mathrm{\Gamma }{)}^2}{\xi }}$, and the bottom ones $c_1=\sqrt{\frac{(1-\mathrm{\Gamma }{)}^2}{\xi }}-1$. The green areas denote the phase of accelerated expansion $q<0$. The bottom right panel shows the effective equation of state parameter corresponding to the black-dashed trajectory in the left bottom panel, with initial conditions given by $\{\mathrm{\Omega }=5/2,{\mathrm{\Omega }}_{\psi }=-1-1/\sqrt{3}\}$.

Figure 7

Top panel: Invariant subset ${\mathrm{\Omega }}_H=1$ for $\mathrm{\Gamma }=3/4$ ($V\propto {\psi }^4$), $c_1=0$ and $\xi =1/10$. Middle panel: Invariant subset ${\mathrm{\Omega }}_H=1$ for $\mathrm{\Gamma }=3/4$, $c_1=\sqrt{\frac{(1-\mathrm{\Gamma }{)}^2}{\xi }}+1$ and $\xi =1/10$. Bottom panel: Evolution of the effective equation of state parameter for the black-dashed trajectory showed in the middle panel, with initial conditions $\{\mathrm{\Omega }=-2.63,{\mathrm{\Omega }}_{\psi }=0.9\}$.