A unified approach to scaling solutions in a general cosmological background

Our ignorance about the source of cosmic acceleration has stimulated study of a wide range of models and modifications to gravity. Cosmological scaling solutions in any of these theories are privileged because they represent natural backgrounds relevant to dark energy. We study scaling solutions in a generalized background $H^2 \propto \rho_T^n$ in the presence of a scalar field $\vp$ and a barotropic perfect fluid, where $H$ is a Hubble rate and $\rho_T$ is a total energy density. The condition for the existence of scaling solutions restricts the form of Lagrangian to be $p=X^{1/n}g(Xe^{n\lambda \vp})$, where $X=-g^{\mu\nu} \partial_\mu \vp \partial_\nu \vp /2$ and $g$ is an arbitrary function. This is very useful to find out scaling solutions and corresponding scalar-field potentials in a broad class of dark energy models including (coupled)-quintessence, ghost-type scalar field, tachyon and k-essence. We analytically derive the scalar-field equation of state $w_\vp$ and the fractional density $\Omega_\vp$ and apply it to a number of dark energy models.