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
$\mu$-distortion. We can calculate the correlation $\langle\mu T\rangle$ of
this distortion with the temperature anisotropy $T$ of the CMB to search for a
correlation $\langle B^2\zeta\rangle$ between the magnetic field $B$ and the
curvature perturbation $\zeta$; knowing the $\langle B^2\zeta\rangle$
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 $\langle
B^2\zeta\rangle$, 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 $S/N\approx 1.0 \times b_{\text{NL}}
(\tilde B_\mu/10\text{ nG})^2$, 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 $\langle\zeta^3\rangle$ induced by the magnetic
field. For sufficiently small magnetic fields, the signal $\langle B^2
\zeta\rangle$ 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 $\langle
B^2\zeta\rangle$ correlation and explain why there will be no contribution from
the evolution of the magnetic field in response to the curvature perturbation.
