Constraints on $\mu$-distortion fluctuations and primordial
  non-Gaussianity from Planck data
  We use the Planck HFI channel maps to make an all sky map of $\mu$-distortion
fluctuations. Our $\mu$-type distortion map is dominated by the $y$-type
distortion contamination from the hot gas in the low redshift Universe and we
can thus only place upper limits on the $\mu$-type distortion fluctuations. For
the amplitude of $\mu$-type distortions on $10'$ scales we get the limit on
root mean square (rms) value $\mu_{rms}^{10'}< 6.4\times 10^{-6}$, a limit 14
times stronger than the COBE-FIRAS ($95\%$ confidence) limit on the mean of $<
\mu > <90\times 10^{-6}$. Using our maps we also place strong upper limits on
the auto angular power spectrum of $\mu$, $C_{\ell}^{\mu\mu}$ and the cross
angular power spectrum of $\mu$ with the CMB temperature anisotropies,
$C_{\ell}^{\mu T}$. The strongest observational limits are on the largest
scales, $\ell(\ell+1)/(2\pi)C_{\ell}^{\mu\mu}|_{\ell=2-26}<(2.3\pm 1.0)\times
10^{-12}$ and $\ell(\ell+1)/(2\pi)C_{\ell}^{\mu T}|_{\ell=2-26}<(2.6\pm
2.6)\times 10^{-12}~{K}$. Our observational limits can be used to constrain new
physics which can create spatially varying energy release in the early Universe
between redshifts $5\times 10^4\lesssim z\lesssim 2\times 10^6$. We
specifically apply our observational results to constrain the primordial
non-Gaussianity of the local type, when the source of $\mu$-distortion is Silk
damping, for very squeezed configurations with the wavenumber for the short
wavelength mode $46 \lesssim k_{S} \lesssim 10^4 ~{Mpc^{-1}}$ and for the long
wavelength mode $k_{L}\approx 10^{-3} ~{Mpc^{-1}}$. Our limits on the
primordial non-Gaussianity parameters are $f_{NL}<10^5, \tau_{NL}<1.4\times
10^{11}$ for $k_{S}/k_{L}\approx 5\times 10^4- 10^7$. We give a new derivation
of the evolution of the $\mu$-distortion fluctuations. We also introduce mixing
of Bose-Einstein spectra and $y^{BE}$-type distortions.
