  We investigate the potential of CO rotational lines at redshifts $z\sim 0-6$
being an appreciable source of extragalactic foreground anisotropies in the
cosmic microwave background. Motivated by previous investigations, we
specifically focus on the frequency bands and small scales probed by
ground-based surveys. Using an empirical parameterization for the relation
between the infrared luminosity of galaxies and their CO line luminosity,
conditioned on sub-mm observations of CO luminosity functions from $J=1$ to
$J=7$ at $\nu = \{100,250\}$ GHz, we explore how uncertainty in the CO
luminosity function translates into uncertainty in the signature of CO emission
in the CMB. We find that at $\ell = 3000$ the amplitude of both CO
autocorrelation and cross-correlation with the CIB could be detectable in an
ACT-like experiment with 90, 150 and 220 GHz bands, even in the scenarios with
the lowest amplitude consistent with sub-mm data. We also investigate, for the
first time, the amplitude of the CO$\times$CIB correlation between different
frequency bands and find that our model predicts that this signal could be
comparable to the amplitude of the cosmic infrared background frequency
cross-correlation at certain wavelengths. This implies current observations can
potentially be used to constrain the bright end of CO luminosity functions,
which are difficult to probe with current sub-mm telescopes due to the small
volumes they survey. Our findings corroborate past results and have significant
implications in template-based searches for CMB secondaries, such as the
kinetic Sunyaev Zel'dovich effect, using the frequency-dependent high-$\ell$ TT
power spectrum.
