Cosmological constraints from $\mu E$ cross-correlations
  We derive general expressions of the $C^{\mu X}_l$, the cross correlation
function between the cosmic microwave background spectral $\mu$ distortion and
the linear perturbations in the cosmic microwave background such as the
temperature perturbations and the polarizations. The cross correlations are
known as new tests for the extremely squeezed shape of primordial
non-Gaussianity, which is inaccessible through the direct observations of the
temperature 3-point functions. Our formulae are applicable to the arbitrary
combinations of the scalar and the tensor perturbations, and we discuss the
potential for detecting these quantities. We provide signal-to-noise ratio of
the $\mu E$ as well as $\mu T$, based on an experiment like the Primordial
Inflation Explorer. We also find the signal-to-noise ratio from the scale
dependent nonlinearity. For instance, we show that $f^{\rm loc}_{\rm
NL}(k_1,k_2,k_3)=\left(k_1k_2k_3\right)^{1/3}k_0^{-1}F_0$ with $k_0=0.05{\rm
Mpc}^{-1}$ can be detectable at 1$\sigma$ level even for $F_0 \sim \mathcal
O(1)$.
