pseudo-scalar coupled to a non-Abelian gauge field. When the pseudo-scalar can be integrated out, one recovers the model of Gauge-flation plus corrections. It was shown that this identification is very accurate at the background level, but differences emerged in the literature concerning the perturbations of the two models. On the contrary, our results show that the analogy between the two models continues to be accurate also at the perturbative level.

Introduction {#sec:intro}
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The paradigm of inflation offers a successful theoretical framework for the physics of the Early Universe [@Linde:2005ht]. In most of its realizations, inflation is driven by scalar fields, a feature which can easily result in an isotropic expansion. Quantum fluctuations generated from scalar fields during or at the end of inflation can account both for the cosmological fluctuations in the Cosmic Microwave Background (CMB) Radiation and for the formation of Large Scale Structures. A large number of inflationary models with scalar fields provide results in excellent agreement with observations [@Ade:2013uln]. Nevertheless, models of inflation where vector fields play a major role have been receiving quite some attention, inspired both by theoretical and observationally-related motivations (see e.g. [@review1; @review2; @review3; @review4; @review5] for some recent reviews).

In this 