Extracting foreground-obscured $\mu$-distortion anisotropies to
  constrain primordial non-Gaussianity
  Correlations between cosmic microwave background (CMB) temperature,
polarization and spectral distortion anisotropies can be used as a probe of
primordial non-Gaussianity. Here, we perform a reconstruction of
$\mu$-distortion anisotropies in the presence of Galactic and extragalactic
foregrounds, applying the so-called Constrained ILC component separation method
to simulations of proposed CMB space missions (PIXIE, LiteBIRD, CORE, PICO).
Our sky simulations include Galactic dust, Galactic synchrotron, Galactic
free-free, thermal Sunyaev-Zeldovich effect, as well as primary CMB temperature
and $\mu$-distortion anisotropies, the latter being added as correlated field.
The Constrained ILC method allows us to null the CMB temperature anisotropies
in the reconstructed $\mu$-map (and vice versa), in addition to mitigating the
contaminations from astrophysical foregrounds and instrumental noise. We
compute the cross-power spectrum between the reconstructed (CMB-free)
$\mu$-distortion map and the ($\mu$-free) CMB temperature map, after foreground
removal and component separation. Since the cross-power spectrum is
proportional to the primordial non-Gaussianity parameter, $f_{\rm NL}$, on
scales $k\simeq 740$ Mpc$^{-1}$, this allows us to derive $f_{\rm
NL}$-detection limits for the aforementioned future CMB experiments. Our
analysis shows that foregrounds degrade the theoretical detection limits (based
mostly on instrumental noise) by more than one order of magnitude, with PICO
standing the best chance at placing upper limits on scale-dependent
non-Gaussianity. We also discuss the dependence of the constraints on the
channel sensitivities and chosen bands. Like for $B$-mode polarization
measurements, extended coverage at frequencies $\nu\lesssim 40\,{\rm GHz}$ and
$\nu\gtrsim 400\,{\rm GHz}$ provides more leverage than increased channel
sensitivity.
