Inflationary Magnetogenesis in $R^{2}$-Inflation after Planck 2015

Abstract: We study the primordial magnetic field generated by the simple model $f^2 FF$ in Starobinsky, $R^2$-inflationary, model. The scale invariant PMF is achieved at relatively high power index of the coupling function, $\left| \alpha \right| \approx 7.44$. This model does not suffer from the backreaction problem as long as, the rate of inflationary expansion, $H$, is in the order of or less than the upper bound reported by Planck ($\le 3.6 \times 10^{-5} M_\rm{Pl}$) in both de Sitter and power law expansion, which show similar results. We calculate the lower limit of the reheating parameter, $R_\rm{rad} > 6.888$ in $R^2$-inflation. Based on the upper limit obtained from CMB, we find that the upper limits of magnetic field and reheating energy density as, $\left(\rho_{B_\rm{end}} \right)\rm{CMB} < 1.184 \times 10^{-20} M\rm{Pl}^4$ and $\left(\rho_\rm{reh} \right)\rm{CMB} < 8.480 \times 10^{-22} M\rm{Pl}^4$. All of foregoing results are well more than the lower limit derived from WMAP7 for both large and small field inflation. By using the Planck inflationary constraints, 2015 in the context of ${R^2}$-inflation, the upper limit of reheating temperature and energy density for all possible values of $w \rm{reh}$ are respectively constrained as, $T\rm{reh} < 4.32 \times 10^{13} \rm{GeV}$ and $\rho_\rm{reh} < 3.259 \times 10^{-18} M_\rm{Pl}^4$ at $n_\rm{s} \approx 0.9674$. This value of spectral index is well consistent with Planck, 2015 results. Adopting $T_\rm{reh}$, enables us to constrain the reheating e-folds number, $N_\rm{reh}$ on the range $1 < N_\rm{reh} < 8.3$, for $- 1/3 < w_\rm{reh} < 1$. By using the scale invariant PMF generated by $f^2 FF$, we find that the upper limit of present magnetic field, $B_0 < 8.058 \times 10^{-9} \rm{G}$.