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Recent experimental progresses allow for exploring some important physicalquantities of ultracold Fermi gases, such as the compressibility, spinsusceptibility, viscosity, optical conductivity and spin diffusivity.Theoretically, these quantities can be evaluated from suitable linear responsetheories. For BCS superfluid, it has been found that the gauge invariant linearresponse theories can be fully consistent with some stringent consistencyconstraints. When the theory is generalized to stronger-than-BCS regime, onemay meet serious difficulties to satisfy the gauge invariance conditions. Inthis paper, we try to construct density and spin linear response theories whichare formally gauge invariant for a Fermi gas undergoing BCS-Bose-EinsteinCondensation (BEC) crossover, especially below the superfluid transitiontemperature $T_c$. We adapt a particular $t$-matrix approach which is close tothe $G_0G$ formalism to incorporate non-condensed pairing in the normal state.We explicitly show that the fundamental constraints imposed by the Wardidentities, $Q$-limit Ward identity are indeed satisfied.

Gauge Invariant Linear Response Theories for Ultracold Fermi Gases with Pseudogap