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

Quantum Gases;Superconductivity