  We study the unitary Fermi gas in a harmonic trapping potential starting from
a microscopic theory in the limit of large charge and large number of fermion
flavors N. In this regime, we present an algorithmic procedure for extracting
data from perturbation theory, order-by-order, without the need for other
assumptions. We perform a gradient expansion in the interior of the particle
cloud, sufficiently far from the cloud edge where the particle density drops
rapidly to zero. In this latter region we present the first microscopic
computation characterizing the contribution of the edge terms. The microscopic
theory reproduces the predictions of the superfluid EFT, including the action,
the form of the gap equation, and the energy of the system in a harmonic trap
(which maps, via the non-relativistic state-operator correspondence, to the
scaling dimension of the lowest operator of charge Q). We additionally give the
Wilsonian coefficients at leading order in N up to NNLO in the large-charge
expansion.
