Preprint number: CP3-Origins-2014-16 DNRF90 and DIAS-2014-16
The damping of a non-uniform magnetic field between the redshifts of about 104 and 106 injects energy into the photon-baryon plasma and causes the CMB to deviate from a perfect blackbody spectrum, producing a so-called μ-distortion. We can calculate the correlation 〈μT〉 of this distortion with the temperature anisotropy T of the CMB to search for a correlation 〈B2ζ〉 between the magnetic field B and the curvature perturbation ζ. Since the perturbations which produce the μ-distortion will be much smaller scale than the relevant density perturbations, the observation of this correlation is sensitive to the squeezed limit of 〈B2ζ〉, which is naturally parameterized by b\textNL (a parameter defined analogously to f\textNL). We find that a PIXIE-like CMB experiments has a signal to noise S/N ≈ 1.0 ×b\textNL (~Bμ/10\text nG)2, where ~Bμ is the magnetic field’s strength on μ-distortion scales normalized to today’s redshift; thus, a 10 nG field would be detectable with b\textNL=O(1). However, if the field is of inflationary origin, we generically expect it to be accompanied by a curvature bispectrum 〈ζ3〉; for field strengths Bμ >~1 nG, the signal of this bispectrum in 〈μT〉 would dominate over the signal from b\textNL.
We also discuss the potential post-magnetogenesis sources of a 〈B2ζ〉 correlation and explain why there will be no contribution from the evolution of the magnetic field in response to the curvature perturbation.