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 Q=(Q x,Q m,Q r) satisfying the constraint plane Q x+Q m+Q r=0, the barotropic index vector γ=(γ x,γ m, γ r) and select a transversal interaction vector Q t in a sense that Q t•γ=0. We exactly solve the source equation for a linear Q 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 χ2 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 Q t. It turns to be that our model fulfills the severe bound of Ω x(z1100)