  Studying the collective pairing phenomena in a two-component Fermi gas, we
predict the appearance near the transition temperature $T_c$ of a well-resolved
collective mode of quadratic dispersion. The mode is visible both above and
below $T_c$ in the system's response to a driving pairing field. When
approaching $T_c$ from below, the phononic and pair-breaking branches,
characteristic of the zero temperature behavior, reduce to a very low
energy-momentum region when the pair correlation length reaches its critical
divergent behavior $\xi_{\rm pair}\propto|T_c-T|^{-1/2}$; elsewhere, they are
replaced by the quadratically-dispersed pairing resonance, which thus acts as a
precursor of the phase transition. In the strong-coupling and Bose-Einstein
Condensate regime, this mode is a weakly-damped propagating mode associated to
a Lorentzian resonance. Conversely, in the BCS limit it is a relaxation mode of
pure imaginary eigenenergy. At large momenta, the resonance disappears when it
is reabsorbed by the lower-edge of the pairing continuum. At intermediate
temperatures between 0 and $T_c$, we unify the newly found collective phenomena
near $T_c$ with the phononic and pair-breaking branches predicted from previous
studies, and we exhaustively classify the roots of the analytically continued
dispersion equation, and show that they provided a very good summary of the
pair spectral functions.
