Generalized slow-roll inflation

The slow-roll approximation to inflation is ultimately justified by the presence of inflationary attractors for the orbits of the solutions of the dynamical equations in phase space. There are many indications that the inflaton field couples nonminimally to the spacetime curvature: the existence of attractor points for inflation with nonminimal coupling is demonstrated, subject to a condition on the inflaton potential and the value of the coupling constant. A period of inflationary expansion of the early universe has come to be widely accepted in order to solve the horizon, flatness and monopole problems that plague the standard big bang cosmology. As a bonus, inflation provides quantum fluctuations of the inflaton field as a much needed mechanism for generating density perturbations, the seeds of structures observed in the universe today [1, 2]. Most of the proposed inflationary scenarios are based on the slow-roll approximation, which assumes that the kinetic energy of the inflaton is much smaller than its potential energy, and that the cosmic expansion is nearly exponential; the slow-roll approximation allows one to predict amplitudes and spectra of scalar and tensor perturbations and the features of the Doppler peaks in the cosmic microwave background (see [2] for a recent review). There are many indications (summarized in [3]) that the inflaton couples explicitly to the Ricci curvature of spacetime R, as described by the action S = ∫ dx √−g [ R 2κ − ξφ R 2 − 1 2 ∇φ∇μφ− V (φ) ] , (1) where κ ≡ 8πG, G is Newton’s constant, φ the inflaton field and V (φ) its potential. Recently, much attention has been paid to nonminimally coupled models of inflation [4, 5, 6], dark matter [7], and quintessence [8]. The slow-roll approximation that allows to solve for the dynamics of inflation requires the existence of inflationary attractors for the orbits of the solutions in phase space [9, 10]. While such attractors are well known to exist for minimal coupling (ξ = 0), their presence is assumed, explicitly [11] or tacitly, for nonminimal coupling (ξ 6= 0). Although slow-roll parameters have been introduced, and a Hubble slow-roll formalism developed, for ξ 6= 0 [12, 13, 11], scenarios based on this approximation would be empty theories if inflationary attractors in phase space did not exist. In this Letter general potentials V (φ) and values of ξ are considered and it is demonstrated that, subject to the condition (23), such attractors exist in the general nonminimally coupled case. In a spatially flat Friedmann-Robertson-Walker universe with line element ds = −dt + a(t) (