Probing correlations of early magnetic fields using μ-distortion

The damping of a non-uniform magnetic field between the redshifts of about 10 ⁴ and 10 ⁶ 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 〈μ T〉 of this distortion with the temperature anisotropy T of the CMB to search for a correlation 〈 B ²ζ〉 between the magnetic field B and the curvature perturbation ζ; knowing the 〈 B ²ζ〉 correlation would help us distinguish between different models of magnetogenesis. 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 〈 B ²ζ〉, which is naturally parameterized by b _NL (a parameter defined analogously to f _NL). We find that a PIXIE-like CMB experiments has a signal to noise S/N≈ 1.0 × b _NL ( _μ/10nG) ², where _μ 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 _NL=(1). However, if the field is of inflationary origin, we generically expect it to be accompanied by a curvature bispectrum 〈ζ ³〉 induced by the magnetic field. For sufficiently small magnetic fields, the signal 〈 B ² ζ〉 will dominate, but for _μ≳ 1 nG, one would have to consider the specifics of the inflationary magnetogenesis model. We also discuss the potential post-magnetogenesis sources of a 〈 B ²ζ〉 correlation and explain why there will be no contribution from the evolution of the magnetic field in response to the curvature perturbation.