Universal upper limit on inflation energy scale from cosmic magnetic field

Recently observational lower bounds on the strength of cosmic magnetic fields were reported, based on γ-ray flux from distant blazars. If inflation is responsible for the generation of such magnetic fields then the inflation energy scale is bounded from above as ρ _inf ^1/4 < 2.5 × 10 ^-7M _Pl × (B _obs/10 ^-15G) ^-2 in a wide class of inflationary magnetogenesis models, where B _obs is the observed strength of cosmic magnetic fields. The tensor-to-scalar ratio is correspondingly constrained as r < 10 ^-19 × (B _obs/10 ^-15G) ^-8. Therefore, if the reported strength B _obs 10 ^-15G is confirmed and if any signatures of gravitational waves from inflation are detected in the near future, then our result indicates some tensions between inflationary magnetogenesis and observations.