  We consider a two-component Fermi gas with a contact interaction from the BCS
regime to the unitary limit. Starting from the idea that many-body effects
should not depend on short-distance or high-momentum physics which is encoded
in the s-wave scattering length, but only on momentum scales of the order of
the Fermi momentum, we build effective low-momentum interactions that reproduce
the scattering phase shifts of the contact interaction below some momentum
cutoff. Inspired from recent successes of this method in nuclear structure
theory, we use these interactions to describe the equation of state of the
Fermi gas in the framework of Hartree-Fock-Bogliubov theory with perturbative
corrections. In the BCS regime, there is a range of cutoffs where we obtain
fully converged results. Near unitarity, convergence is not yet reached, but we
obtain promising results for the ground-state energies close to the
experimental ones. Limitations and possible extensions of the approach are
discussed.
