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 H2∝ρTn in the presence of a scalar field φ and a barotropic perfect fluid, where H is a Hubble rate and ρT is a total energy density. The condition for the existence of scaling solutions restricts the form of Lagrangian to be p=X1/ng(Xenλφ), where X=-gμν-bar μφ-bar νφ/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 wφ and the fractional density Ωφ and apply it to a number of dark energy models