Faraday Rotation From a Bounced Primordial Field: Present Signals and the Bound They Actually Set

Faraday rotation of polarised radio light is the most direct probe of magnetic fields along the line of sight, and it is often assumed to be the sharpest tool for testing primordial fields. This paper computes what it actually delivers for a field produced through a bounce. A field of 1 nanogauss, coherent over a gigaparsec path at the mean cosmic electron density, produces a rotation measure of only 1.78e-01 radians per square metre, far below the scatter of measured extragalactic rotation measures. Even the microwave-background ceiling of 4.4 nanogauss gives 7.84e-01 radians per square metre on a fully coherent path, and tangling the field on megaparsec scales reduces the signal by the square root of the number of cells crossed. Present rotation-measure data bound a coherent gigaparsec field only at 5.61e+01 nanogauss, much weaker than the microwave limit. A future grid reaching 0.1 radians per square metre of residual precision would improve that to 5.61e-01 nanogauss and make Faraday rotation competitive for the first time. The numbers are computed from the standard rotation integral, not measured.