Leverage on small-scale primordial non-Gaussianity through
  cross-correlations between CMB $E$-mode and $\mu$-distortion anisotropies
  Multi-field inflation models and non-Bunch-Davies vacuum initial conditions
both predict sizeable non-Gaussian primordial perturbations and anisotropic
$\mu$-type spectral distortions of the cosmic microwave background (CMB)
blackbody. While CMB anisotropies allow us to probe non-Gaussianity at
wavenumbers $k\simeq 0.05\,{\rm Mpc^{-1}}$, $\mu$-distortion anisotropies are
related to non-Gaussianity of primordial perturbation modes with much larger
wavenumbers, $k\simeq 740\,{\rm Mpc^{-1}}$. Through cross-correlations between
CMB and $\mu$-distortion anisotropies, one can therefore shed light on the
aforementioned inflation models. We investigate the ability of a future CMB
satellite imager like LiteBIRD to measure $\mu T$ and $\mu E$ cross-power
spectra between anisotropic $\mu$-distortions and CMB temperature and $E$-mode
polarization anisotropies in the presence of foregrounds, and derive LiteBIRD
forecasts on ${f_{\rm NL}^\mu(k\simeq 740\,{\rm Mpc^{-1}})}$. We show that $\mu
E$ cross-correlations with CMB polarization provide more constraining power on
$f_{\rm NL}^\mu$ than $\mu T$ cross-correlations in the presence of
foregrounds, and the joint combination of $\mu T$ and $\mu E$ observables adds
further leverage to the detection of small-scale primordial non-Gaussianity. We
find that LiteBIRD would detect ${f_{\rm NL}^\mu}=4500$ at $5\sigma$
significance after foreground removal, and achieve a minimum error of
${\sigma(f_{\rm NL}^\mu=0) \simeq 800}$ at 68\% CL by combining CMB temperature
and polarization. Due to the huge dynamic range of wavenumbers between CMB and
$\mu$-distortion anisotropies, such large $f^\mu_{\rm NL}$ values would still
be consistent with current CMB constraints in the case of very mild
scale-dependence of primordial non-Gaussianity. Anisotropic spectral
distortions thus provide a new path, complementary to CMB $B$-modes, to probe
inflation with LiteBIRD.
