Probing correlations of early magnetic fields using mu-distortion

Abstract

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 $\mu$-distortion. We can calculate the correlation $⟨\mu T⟩$ of this distortion with the temperature anisotropy $T$ of the CMB to search for a correlation $⟨B^2\zeta ⟩$ between the magnetic field $B$ and the curvature perturbation $\zeta$; knowing the $⟨B^2\zeta ⟩$ correlation would help us distinguish between different models of magnetogenesis. Since the perturbations which produce the $\mu$-distortion will be much smaller scale than the relevant density perturbations, the observation of this correlation is sensitive to the squeezed limit of $⟨B^2\zeta ⟩$, 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 $S/N\approx 1.0\times b_{\text{NL}}({\tilde{B}}_{\mu }/10\text{ nG}{)}^2$, where ${\tilde{B}}_{\mu }$ is the magnetic field's strength on $\mu$-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 $⟨{\zeta }^3⟩$ induced by the magnetic field. For sufficiently small magnetic fields, the signal $⟨B^2\zeta ⟩$ will dominate, but for ${\tilde{B}}_{\mu }\gtrsim 1$ nG, one would have to consider the specifics of the inflationary magnetogenesis model. We also discuss the potential post-magnetogenesis sources of a $⟨B^2\zeta ⟩$ correlation and explain why there will be no contribution from the evolution of the magnetic field in response to the curvature perturbation.

arXiv comment23 pages, 1 figure. v2: Noted that a competing effect could potentially be smaller than originally stated. Fixed references. Matches JCAP version

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