We explore the paradigm according to which inflation is driven by a four-dimensional strongly coupled dynamics coupled non-minimally to gravity. We start by introducing the general setup, both in the metric and Palatini formulation, for generic models of composite inflation. We then analyze the relevant example where the inflaton is identified with the glueball field of a pure Yang-Mills theory. We introduce the dilatonic-like glueball action which is obtained by requiring saturation of the underlying Yang-Mills trace anomaly at the effective action level. We couple the resulting action non-minimally to gravity. We demonstrate that it is possible to achieve successful inflation with the confining scale of the underlying Yang-Mills theory naturally of the order of the grand unified energy scale. We also argue that within the metric formulation models of composite inflation lead to a more consistent picture than within the Palatini one. Finally we show that, in the metric formulation, the model nicely respects tree-level unitarity for the scattering of the inflaton field all the way to the Planck scale.

16 pages, RevTeX. 2 new sessions have been added, one on graviton scattering and the other on the relevant energy scales of the theory. References added. Main results are unchanged. Version to match the one to appear in Phys.Rev. D