Dark matter, dark energy, and dark radiation coupled with a transversal interaction

We investigate a cosmological scenario with three interacting components that includes dark matter, dark energy, and radiation in the spatially flat Friedmann-Robertson-Walker universe. We introduce a 3-dimensional internal space, the interaction vector $\mathbf{Q}=(Q_x,Q_m,Q_r)$ satisfying the constraint plane $Q_x+Q_m+Q_r=0$, the barotropic index vector $\text{boldmath}${\gamma}$=({\text{ga}}_x,{\text{ga}}_m,{\text{ga}}_r)$ and select a transversal interaction vector ${\mathbf{Q}}_{\mathbf{t}}$ in a sense that ${\mathbf{Q}}_{\mathbf{t}}\cdot \text{boldmath}${\gamma}$=0$. We exactly solve the source equation for a linear ${\mathbf{Q}}_{\mathbf{t}}$, that depends on the total energy density and its derivatives up to third order, and find all the component energy densities. We obtain a large set of interactions for which the source equation admits a power law solution and show its asymptotic stability by constructing the Lyapunov function. We apply the ${\chi }^2$ method to the observational Hubble data for constraining the cosmic parameters, and analyze the amount of dark energy in the radiation era for the above linear ${\mathbf{Q}}_{\mathbf{t}}$. It turns to be that our model fulfills the severe bound of ${\mathrm{\Omega }}_x(z\simeq 1100)<0.1$ and is consistent with the future constraints achievable by Planck and CMBPol experiments.