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We study the simple gauge invariant model f 2 F F as a way to generate primordial magnetic fields (PMF) in Natural Inflation (NI). We compute both magnetic and electric spectra generated by the f 2 F F model in NI for different values of model parameters and find that both de Sitter and power law expansion lead to the same results at sufficiently large number of e-foldings. We also find that the necessary scale invariance property of the PMF cannot be obtained in NI in first order of slow roll limits under the constraint of inflationary potential, V (0) 0. Furthermore, if this constraint is relaxed to achieve scale invariance, then the model suffers from the backreaction problem for the co-moving wave number, k 8.0×10 −7 Mpc −1 and Hubble parameter, H i 1.25 × 10 −3 M Pl . The former can be considered as a lower bound of k and the later as an upper bound of H i for a model which is free from the backreaction problem. Further, we show that there is a narrow range of the height of the potential Λ around Λ min ≈ 0.00874M Pl and of k around k min ∼ 0.0173Mpc −1 , at which the energy of the electric field can fall below the energy of the magnetic field. The range of k lies within some observable scales. However, the relatively short range of k presents a challenge to the viability of this model.

We use the f 2 FF model to study the generation of primordial magnetic fields (PMF) in the context of large field inflation (LFI), described by the potential, V ∼ Mφ p . We compute the magnetic and electric spectra for all possible values of the model parameters under de Sitter and power law expansion. We show that scale invariant PMF are not obtained in LFI to first order in the slow roll approximation, if we impose the constraint V(φ = 0) ∼ 0. Alternatively, if these constraints are relaxed, the scale invariant PMF can be generated. The associated electric field energy can fall below the energy density of inflation, ρ Inf for the ranges of comoving wavenumbers, k > 8×10 −7 Mpc −1 and k > 4×10 −6 Mpc −1 in de Sitter and power law (PL) expansion. Further, it can drop below ρ Inf on the ranges, e-foldings N > 51, p < 1.66, p > 2.03, l 0 > 3×10 5 M Pl −1 (H i < 3.3×10 −6 M Pl ), and M > 2.8×10 −3 M Pl . All of the above ranges fit with the observational constraints.

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