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Probing correlations of early magnetic fields using mu-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 $\langleμT\rangle$ of this distortion with the temperature anisotropy $T$ of the CMB to search for a correlation $\langle B^2ζ\rangle$ between the magnetic field $B$ and the curvature perturbation $ζ$; knowing the $\langle B^2ζ\rangle$ 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 $\langle B^2ζ\rangle$, which is naturally parameterized by $b_{\text{NL}}$ (a parameter defined analogously to $f_{\text{NL}}$). We find that a PIXIE-like CMB experiments has a signal to noise $S/N\approx 1.0 \times b_{\text{NL}} (\tilde B_μ/10\text{ nG})^2$, where $\tilde B_μ$ is the magnetic field's strength on $μ$-distortion scales normalized to today's redshift; thus, a 10 nG field would be detectable with $b_{\text{NL}}=\mathcal{O}(1)$. However, if the field is of inflationary origin, we generically expect it to be accompanied by a curvature bispectrum $\langleζ^3\rangle$ induced by the magnetic field. For sufficiently small magnetic fields, the signal $\langle B^2 ζ\rangle$ will dominate, but for $\tilde B_μ\gtrsim 1$ nG, one would have to consider the specifics of the inflationary magnetogenesis model. We also discuss the potential post-magnetogenesis sources of a $\langle B^2ζ\rangle$ correlation and explain why there will be no contribution from the evolution of the magnetic field in response to the curvature perturbation.