Primordial Magnetic Field Generated in Natural Infation

Abstract

We study the simple gauge invariant model $f^{2}FF$ as a way to generate primordial magnetic fields (PMF) in Natural Inflation (NI). We compute both magnetic and electric spectra generated by the $f^{2}FF$ 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, $V\left(0\right)\simeq 0$. Furthermore, if this constraint is relaxed to achieve scale invariance, then the model suffers from the backreaction problem for the co-moving wave number, $k\lesssim 8.0\times 10^{-7}\mathrm{Mp}{\mathrm{c}}^{-1}$ 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 ${\Lambda }_{\min }\approx 0.00874M_{\mathrm{Pl}}$ and of $k$ around $k_{\min }\sim 0.0173\mathrm{Mp}{\mathrm{c}}^{-1}$, 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.

8 pages, 6 figures