Dynamics of interacting dark energy

Dark energy and dark matter are only indirectly measured via their gravitational effects. It is possible that there is an exchange of energy within the dark sector, and this offers an interesting alternative approach to the coincidence problem. We consider two broad classes of interacting models where the energy exchange is a linear combination of the dark sector densities. The first class has been previously investigated, but we define new variables and find a new exact solution, which allows for a more direct, transparent, and comprehensive analysis. The second class has not been investigated in general form before. We give general conditions on the parameters in both classes to avoid unphysical behavior (such as negative energy densities).

Figure 1

The shaded (red) region contains the allowed non-negative values of ${\tilde{\alpha }}_c$ and ${\tilde{\alpha }}_x$ satisfying the reality constraint (23). The dotted (cyan) line represents the equality in Eq. (34) for $R_i=0.1$, the dashed (blue) line represents the equality in Eq. (36) for $R_0=3.4$, and the dash-dotted (green) line is Eq. (37) with $R_{\infty ,m}=10$. The solid (yellow) line corresponds to the particular case ${\alpha }_c={\alpha }_x$ [5]. The only region that provides reasonable values of the DE-DM ratio at both early and late times is the one surrounded by the solid (thick black) lines. The value $R_i=0.1$ was chosen for presentation purposes; the allowed region would become much smaller for a more realistic value $R_i\ll 0.1$.

Figure 2

Evolution of the DE-DM ratio $R(N)$ according to Eqs. (16, 27). The chosen values of the various parameters are $R_0=3.4$, $R_{\infty }=10$, whereas the value of ${\tilde{\alpha }}_c$ was determined from Eq. (35) for the given values of ${\tilde{\alpha }}_x$. For comparison, the dash-dotted (green) line represents the standard $\Lambda \mathrm{CDM}$ case. The dotted line corresponds to exactly ${\tilde{\alpha }}_c=0$, for which case $R_{-\infty }=0$ and the earlier evolution is very similar to that of standard $\Lambda \mathrm{CDM}$.

Figure 3

The phase space of model I in the case ${\tilde{\alpha }}_x=0.15$, and ${\tilde{\alpha }}_c=0.05$. The circles (red) are the critical points $A$ and $B$ (see Table ), and the dashed (green) line is the (heteroclinic) constraint $x+y=1$ that connects them. The dot-dashed (green) line that connects the saddle point $A$ to the unstable critical point at the origin is a good approximation to the heteroclinic trajectory between the two points. The long-dashed (red) line represents the ratio $x/y=0.1$. The dotted (blue) line is the approximate Friedmann constraint (39). The solid (black) lines are numerical solutions of the equations of motion for different initial conditions. The trajectories with initial conditions on the right of the heteroclinic line end up at point $B$; if the initial conditions are on the left, the DE component becomes negative and the DM one grows without bound.

Figure 4

Examples of the DE-DM ratio $R$ as obtained from the numerical solutions of Eqs. (45), under the condition that all cases have $x_0=0.7$ and $y_0=0.24$ at $N=0$. We took a fixed value of ${\gamma }_x=0.2$ for the case $w_x=-1$, and the values of ${\gamma }_c$ are as indicated on the plot. Negative (positive) values of DE at early times appear for negative (positive) values of ${\gamma }_c$ (see the inset), and a finite late-time attractor appears only if the condition $({\gamma }_x-{\gamma }_c)>0$ holds. The values for which $({\gamma }_x-{\gamma }_c)\le 0$ lead to $x\to -\infty$, and may also break the constraint (60).

Figure 5

The same as in Fig. 4, but now for the fixed value ${\gamma }_c=0.2$. As expected, early evolution is positive because ${\gamma }_c>0$ (see the inset), but point $C$ is unstable in all cases. The DE-DM ratio first diverges for $x_0<x_{*}^{+}$, but asymptotes to $R\to \pm -1$ at late times whatever the case $x_0>x_{*}^{+}$ or $x_0<x_{*}^{+}$; see Sec. 4b and Eqs. (49).