Asymptotic behavior of Cardassian cosmologies with exponential potentials

In this paper we analyze the asymptotic behavior of Cardassian cosmological models filled with a perfect fluid and a scalar field with an exponential potential. Cardassian cosmologies arise from modifications of the Friedmann equation, and among the different proposals within that framework we will choose those of the form $3H^2-\rho \propto {\rho }^n$ with $n<1$. We construct two three-dimensional dynamical systems arising from the evolution equations, respectively, adapted for studying the high and low energy limits. Using standard dynamical systems techniques we find the fixed points and characterize the solutions they represent. We pay especial attention to the properties inherent to the modifications and compare with the (standard) unmodified scenario. Among other interesting results, we find there are no late-time tracking attractors.

Figure 1

Phase-space trajectories lying on the $y=0$ hypersurface for $\gamma =1$, $n=0.2$ and $\lambda =1$.

Figure 2

Phase-space trajectories lying on the $w=0$ hypersurface for $\gamma =1$, $n=0.2$ and an arbitrary $\lambda$.

Figure 3

XY projection of some phase-space trajectories for $\gamma =\lambda =1$ and $n=0.2$. All trajectories cross the hypersurface $w=0.69$.

Figure 4

$XW$ projection of some phase-space trajectories for $\gamma =\lambda =1$ and $n=0.2$. All trajectories cross the hypersurface $w=0.69$.

Figure 5

Phase-space trajectories lying on the $\tilde{y}=0$ hypersurface for $\gamma =1$, $n=0.2$ and $\lambda =1$.

Figure 6

Phase-space trajectories lying on the $\tilde{w}=0$ hypersurface for $\gamma =1$, $n=0.2$ and an arbitrary $\lambda$.

Figure 7

$\tilde{X}\tilde{Y}$ projection of some phase-space trajectories for $\gamma =\lambda =1$ and $n=0.2$. All trajectories cross the hypersurface $\tilde{w}=0.1$.

Figure 8

$\tilde{X}\tilde{W}$ projection of some phase-space trajectories for $\gamma =\lambda =1$ and $n=0.2$. All trajectories cross the hypersurface $\tilde{y}=0.01$.