  Strongly interacting, dilute Fermi gases exhibit a scale-invariant, universal
thermodynamic behaviour. This is notoriously difficult to understand
theoretically because of the absence of a small interaction parameter. Here we
present a systematic comparison of theoretical predictions from different
quantum many-body theories with recent experimental data of Nascimbene et. al.
(arXiv:0911.0747v1). Our comparisons have no adjustable parameters, either
theoretically or experimentally. A simple Gaussian pair fluctuation theory is
shown to give the best quantitative agreement in the superfluid state below
threshold. In the normal state, we also calculate the equation of state by
using a quantum cluster expansion theory and explore in detail its
applicability to low temperatures. Using the accurate experimental result for
the thermodynamic function $S(T)$, we determine the temperature $T$ of a
trapped Fermi gas at unitarity as a function of a non-interacting temperature
$T_{i}$ which can be obtained by an adiabatic sweep to the free gas limit. By
analyzing the recent experimental data, we find a normal-superfluid transition
temperature $(T/T_{F})_{c}=0.19\pm0.02$ or $(T_{i}/T_{F})_{c}=0.16\pm0.02$ in a
harmonic trap, where $T_{F}$ is the Fermi temperature for a trapped ideal,
non-interacting Fermi gas.
