Generalized hybrid metric-Palatini gravity

We introduce a new approach to modified gravity which generalizes the recently proposed hybrid metric-Palatini gravity. The gravitational action is taken to depend on a general function of both the metric and Palatini curvature scalars. The dynamical equivalence with a nonminimally coupled biscalar field gravitational theory is proved. The evolution of cosmological solutions is studied using dynamical systems techniques.

Figure 1

Phase space for model 1 with $\lambda =1$. The global attractor represents an accelerating solution because it lies in the region above the dashed/red line.

Figure 2

Phase space for model 1 with $\lambda =4$. The global attractor does not represent an accelerating solution because it lies in the region below the dashed/red line.

Figure 3

Phase space for model 1 with $\lambda =5.9$. The global attractor is now point $B$, while $C$ is a saddle point.

Figure 4

Phase space for model 1 with $\lambda =8$. The global attractor is point $B$, while $A_{+}$ is now a saddle point.

Figure 5

Phase space for model 2 with $\lambda =1$ and $\beta =0.01$. The global attractor is point $D_1$, representing an accelerating solution whenever $\lambda <2\sqrt{3}$ because it lies in the region marked by the dashed/red line.

Figure 6

Phase space for model 2 with $\lambda =5.9$ and $\beta =0.01$. The global attractor is still point $D_1$ but now the trajectories are first attracted by point $C_1$, which is not a critical point.

Figure 7

Phase space for model 2 with $\lambda =8$ and $\beta =0.01$.

Figure 8

Phase space for model 3 with $\lambda =1$ and $w=0$. The global attractor is point $C$, which represents an accelerating solution when $\lambda <2\sqrt{3}$, while point $D$ is a matter-dominated saddle point.

Figure 9

Evolution of the effective equation of state parameter for the dashed/red trajectories in Fig. 8. After reaching a long-lasting matter-dominated evolution according to observations, the universe eventually ends in an accelerated expansion, representing the final cosmological stage.

Figure 10

Phase space for model 3 with $\lambda =5$ and $w=0$. The global attractor is now point $E$, where the universe evolves according to the matter equation of state.

Figure 11

Phase space for model 3 with $\lambda =5.9$ and $w=1/3$. The global attractor is still point $E$, but now the trajectories starting from the $z=0$ plane are first attracted by saddle point $B$.

Figure 12

Phase space for model 3 with $\lambda =8$ and $w=1/3$. The global attractor is point $E$, but now $A_{+}$ is a saddle point attracting the $z=0$ trajectories.