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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 gamma-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 rho_{inf}^{1/4} < 2.5 times 10^{-7}M_{Pl} times (B_{obs}/10^{-15}G)^{-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} times (B_{obs}/10^{-15}G)^{-8}. Therefore, if the reported strength B_{obs} \geq 10^{-15}G 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.