Magnetogenesis in the Starobinsky Inflationary Model by Axial Coupling.

Abstract

Magnetogenesis in the Starobinsky inflationary model by axial coupling \( I\left( \phi \right) F_{\mu \nu }\tilde{F}^{\mu \nu } \) is investigated. Three coupling functions are used. The simple coupling function \(I\left( \phi \right) =\chi _{1}\frac{\phi }{M_{p}} \) and non-minimal coupling function to gravity \( I\left( \phi \right) = 12\chi _{1}e^{\left( \sqrt{\frac{2}{3}}\frac{\phi }{M_{p}}\right) }\left[ \frac{1}{3M_{p}^{2}}\left( 4V\left( \phi \right) \right) +\right. \) \(\left. \frac{\sqrt{2}}{\sqrt{3}M_{p}}\left( \frac{dV}{d\phi }\right) \right] \) and finally the exponential coupling function \( I\left( \phi \right) =\exp \left( \chi _{1}\frac{\phi }{M_{p}}\right) \). It is found that only for exponential coupling function magnetogenesis is considerable and for other coupling functions is quite negligible. The effect of non-minimal function to gravity in comparison to the simple coupling function is appeared by increasing the values of coupling constants by order \( 10^{7} \) but does not alter the conclusion. Also it is found the strength of magnetic fields B for exponential coupling function at present epoch is compatible with observations \( 10^{-17} G\le B\le 10^{-9 } G \). Therefore, it seems these conclusions are consistent. The problem is, one should consider this fact that the strength of magnetic field is strongly depends on coupling functions and also coupling constants. By virtue of obtained results one must decide which coupling function naturally describes the interaction term. The most important results are, for coupling constants between \( \chi _{1}=1 \) to \( \chi _{1}=1.50 \) the strength of magnetic fields are quite consistent with observed magnetic field for present epoch. However for exponential coupling function only for \(\chi _{1}=2.45 \) there is strong back-reaction at the end of inflation in magnetic energy density.