  Diffusion damping of the cosmic microwave background (CMB) power spectrum
results from imperfect photon-baryon coupling in the pre-recombination plasma.
At redshift $5 \times 10^4 < z < 2 \times 10^6$, the plasma acquires an
effective chemical potential, and energy injections from acoustic damping in
this era create $\mu$-type spectral distortions of the CMB. These $\mu$
distortions trace the underlying photon density fluctuations, probing the
primordial power spectrum in short-wavelength modes $k_\mathrm{S}$ over the
range $50 \ \mathrm{Mpc}^{-1} \lesssim k \lesssim 10^4 \ \mathrm{Mpc}^{-1}$.
Small-scale power modulated by long-wavelength modes $k_\mathrm{L}$ from
squeezed-limit non-Gaussianities introduces cross-correlations between CMB
temperature anisotropies and $\mu$ distortions. Under single-field inflation
models, $\mu \times T$ correlations measured from an observer in an inertial
frame should vanish up to a factor of $(k_\mathrm{L}/k_\mathrm{S})^2 \ll 1$.
Thus, any measurable correlation rules out single-field inflation models. We
forecast how well the next-generation ground-based CMB experiment CMB-S4 will
be able to constrain primordial squeezed-limit non-Gaussianity, parameterized
by $f_\mathrm{NL}$, using measurements of $C_{\ell}^{\mu T}$ as well as
$C_{\ell}^{\mu E}$ from CMB $E$ modes. Using current experimental
specifications and foreground modeling, we expect $\sigma(f_\mathrm{NL})
\lesssim 1000$. This is roughly four times better than the current limit on
$f_\mathrm{NL}$ using $\mu \times T$ and $\mu \times E$ correlations from
Planck and is comparable to what is achievable with LiteBIRD, demonstrating the
power of the CMB-S4 experiment. This measurement is at an effective scale of $k
\simeq 740 \ \text{Mpc}^{-1}$ and is thus highly complementary to measurements
at larger scales from primary CMB and large-scale structure.
