  Whilst current observational evidence favors a close-to-Gaussian spectrum of
primordial perturbations, there exist many models of the early Universe that
predict this distribution to have exponentially enhanced or suppressed tails.
In this work, we generate realizations of the primordial potential with
non-Gaussian tails via a phenomenological model; these are then evolved
numerically to obtain maps of the cosmic microwave background (CMB) and
large-scale structure (LSS). In the CMB maps, our added non-Gaussianity
manifests as a localized enhancement of hot and cold spots, which would be
expected to contribute to $N$-point functions up to large $N$. Such models are
indirectly constrained by \textit{Planck} trispectrum bounds, which restrict
the changes in the temperature fluctuations to $O(10\mu\mathrm{K})$. In the
late-time Universe, we find that tailed cosmologies lead to a halo mass
function enhanced at high masses, as expected. Furthermore, significant
scale-dependent bias in the halo-halo and halo-matter power spectrum is also
sourced, which arises from the squeezed limit of large $N$-point functions that
are implicitly generated through the enhancement of the tails. These results
underscore that a detection of scale-dependent bias alone cannot be used to
rule out single field inflation, but can be used together with other statistics
to probe a wide range of primordial processes.
