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We address the thermodynamics, density profiles and superfluid density oftrapped fermions undergoing BCS-BEC crossover, with and without populationimbalance. Our approach represents a fully consistent treatment of "pseudogapeffects", which reflect the distinction between the pair formation temperature$T^*$ and the pair condensation temperature $T_c$ and must be accommodated bymodifying the fermionic excitation spectrum. This distinction has been omittedfrom all other many body approaches in the literature. We show how enforcingthis corollary implies that pairing fluctuation contributions enter into boththe gap and the number equations; this is necessary in order to be consistentwith a generalized Ward identity. Furthermore, we demonstrate that we obtainphysical results for the superfluid density $n_s(T)$ at all $T$. In contrast,previous work in the literature has led to ill-behaved $n_s(T)$. The superfluiddensity is a critical measure of the physicality of a given crossover theory.In order to properly address thermodynamic properties of a trapped Fermi gas,it is necessary to demonstrate that the particle density profiles areconsistent with experiment. The lack of demonstrable consistency betweentheoretical and experimental density profiles, along with problematic behaviorfound for $n_s(T)$, casts doubt on previous claims in the literature concerningquantitative agreement between thermodynamical calculations and experiment.

Thermodynamics and superfluid density in BCS-BEC crossover with and without population imbalance