Conformal anomaly and primordial magnetic fields

Abstract

The conformal symmetry of the quantized electromagnetic field breaks down in curved space-time. We point out that this conformal anomaly is able to generate a sizable magnetic field during a phase of slow-roll inflation. Such primordial magnetism is characterized by the expectation value of the squared of the magnetic field for comoving observers, which at leading order in slow-roll takes the value $\ < \vec B^2\ > =\frac{8}{15(4\pi)^2}\, H^4\epsilon$, where $\epsilon$ is the standard slow-roll parameter. This result is insensitive to the intrinsic ambiguities of renormalization in curved space-times. The information in the quantum state gets diluted during inflation and does not affect the prediction. A primordial field of this strength is able to seed the observed cosmic magnetism.