Slow-roll Analysis of Double Field Axion Inflation

We adopt the double field natural inflation model motivated by the non-perturbative effects of supergravity and superstring theory to do the slow roll analysis. We show that when the parameters are suitably chosen, there exist ranges of initial values of fields that can satisfy the constraints of Planck observations. This implies less fine tuning of field values can be allowed with tolerance of $4M_{\text{pl}}$ for ${\varphi }_1$ and $5 {M}{$\text{pl}$}$for${\phi}{2}$ respectively, which become more physical for field fluctuation in quantum era. We also show that the spectral index $n_{\mathcal{R}\mathcal{R}}$, the fraction of entropic power spectrum and the fraction of power spectra (so-called ${\beta }_{\text{iso}}$) and $\cos \mathrm{\Delta }$ can satisfy the constraints of Planck observation. This implies that double field natural inflation is a valid model to describe cosmological inflation.