  We further progress along the line of Ref. [Phys. Rev. {\bf A 94}, 043614
(2016)] where a functional for Fermi systems with anomalously large $s$-wave
scattering length $a_s$ was proposed that has no free parameters. The
functional is designed to correctly reproduce the unitary limit in Fermi gases
together with the leading-order contributions in the s- and p-wave channels at
low density. The functional is shown to be predictive up to densities
$\sim0.01$ fm$^{-3}$ that is much higher densities compared to the Lee-Yang
functional, valid for $\rho < 10^{-6}$ fm$^{-3}$. The form of the functional
retained in this work is further motivated. It is shown that the new functional
corresponds to an expansion of the energy in $(a_s k_F)$ and $(r_e k_F)$ to all
orders, where $r_e$ is the effective range and $k_F$ is the Fermi momentum. One
conclusion from the present work is that, except in the extremely low--density
regime, nuclear systems can be treated perturbatively in $-(a_s k_F)^{-1}$ with
respect to the unitary limit. Starting from the functional, we introduce
density--dependent scales and show that scales associated to the bare
interaction are strongly renormalized by medium effects. As a consequence, some
of the scales at play around saturation are dominated by the unitary gas
properties and not directly to low-energy constants. For instance, we show that
the scale in the s-wave channel around saturation is proportional to the
so-called Bertsch parameter $\xi_0$ and becomes independent of $a_s$. We also
point out that these scales are of the same order of magnitude than those
empirically obtained in the Skyrme energy density functional. We finally
propose a slight modification of the functional such that it becomes accurate
up to the saturation density $\rho\simeq 0.16$ fm$^{-3}$.
