Interacting Cosmological Fluids and the Coincidence Problem

Abstract

We examine the evolution of a universe comprising two interacting fluids, which interact via a term proportional to the product of their densities. In the case of two matter fluids it is shown that the ratio of the densities tends to a constant after an initial cooling-off period. We then obtain a complete solution for the cosmological constant (w = -1) scenario. Finally, we investigate the general case in which the dark energy equation of state is p = w*rho, where w is a constant, and show that periodic solutions can occur if w < -1. We further demonstrate that the ratio of the dark matter to dark energy densities is confined to a bounded interval, and that this ratio can be O(1) at infinitely many times in the history of the universe, thus solving the coincidence problem.

Figures (8)

• Figure 1: Phase-plane diagram showing the evolution of the dark matter density $\rho_m$ and the dark energy density $\rho_{\Lambda}$, for an interaction term $\rho_m\rho_{\Lambda}$ ($\gamma = 1$). In this case, the dark energy is assumed to behave like a cosmological constant, and obeys the equation of state $p = -\rho$. The straight line represents universes with zero acceleration. The region of phase space above the line corresponds to an accelerating universe, and the region below corresponds to a decelerating universe. Note that this figure (and the next four) are intended only to display the qualitative behaviour of the model, so the units on the axes are arbitrary.
• Figure 2: As in Figure 1, with the same interaction term $\rho_m\rho_{\Lambda}$ ($\gamma = 1$), but now assuming a dark energy component whose equation of state is $p = -\rho/2$.
• Figure 3: As in Figure 1, but with an interaction term $-\rho_m\rho_{\Lambda}$ ($\gamma = -1$) and a dark energy component whose equation of state is $p = -\rho/2$.
• Figure 4: As in Figure 1, but with an interaction term $-\rho_m\rho_{\Lambda}$ ($\gamma = -1$) and a phantom dark energy component with equation of state $p = -3\rho/2$.
• Figure 5: As in Figure 1, with an interaction term $\rho_m\rho_{\Lambda}$ ($\gamma = 1$) but a phantom dark energy component with equation of state $p = -3\rho/2$. Note that the system now exhibits periodic orbits, and that the ratio of the dark densities for any particular orbit is constrained to lie within a bounded interval.