Universal Properties of the Ultra-Cold Fermi Gas

We present some general considerations on the properties of a two-component ultra-cold Fermi gas along the BEC-BCS crossover. It is shown that the interaction energy and the ground state energy can be written in terms of a single dimensionless function $h(\xi ,\tau )$, where $\xi =-(k_F{a}_s{)}^{-1}$ and $\tau =T\mathrm{/}{T}_F$. The function $h(\xi ,\tau )$ incorporates all the many-body physics and naturally occurs in other physical quantities as well. In particular, we show that the RF-spectroscopy shift $\overline{\text{d}\text{o}}(\xi ,\tau )$ and the molecular fraction $f_c(\xi ,\tau )$ in the closed channel can be expressed in terms of $h(\xi ,\tau )$ and thus have identical temperature dependence. The conclusions should have testable consequences in future experiments.