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Primordial magnetic field generated in natural inflation. (English) Zbl 1329.83201
Summary: We study the simple gauge invariant model \(f^2F F\) as a way to generate primordial magnetic fields (PMF) in natural inflation (NI). We compute both magnetic and electric spectra generated by the \(f^2F F\) 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, \(V(0)\simeq0\). Furthermore, if this constraint is relaxed to achieve scale invariance, then the model suffers from the backreaction problem for the co-moving wave number, \(k \lesssim8 .0 \times 10^{-7}\mathrm{Mpc}^{-1}\) and Hubble parameter, \(H_i \gtrsim 1.25 \times 10^{-3}\mathrm{M}_{\mathrm{Pl}}\). 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 \(\lambda\) around \(\lambda_{\min}\approx 0.00874 \mathrm{M}_{\mathrm{Pl}}\) and of \(k\) around \(k_{\min}\sim 0.0173 \mathrm{Mpc}^{-1}\), 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.

MSC:
83F05 Cosmology
85A40 Cosmology
83C22 Einstein-Maxwell equations
78A25 Electromagnetic theory, general
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