  Due to the fermion sign problem, standard lattice Monte-Carlo method for QCD
fail at small temperatures and high baryon densities. $G_2$-QCD, QCD with the
gauge group $SU(3)$ replaced by the exceptional Lie group $G_2$, can be
simulated using lattice techniques at these densities, and can therefore
provide an illustration of the possible phase structure. Here we present a
systematic investigation of the ground-state hadronic spectrum using lattice
simulations for different quark masses in several hadronic sectors. We then
show that the different hadronic scales of Goldstone bosons, intermediate
bosons, and baryons is reflected in the phase structure at finite density.
