  We investigate the formation of rashbon bound states and strong-coupling
effects in an ultracold Fermi gas with a spherical spin-orbit interaction,
$H_{\rm so}=\lambda{\bf p}\cdot{\bf \sigma}$ (where ${\bf
\sigma}=(\sigma_x,\sigma_y,\sigma_z)$ are Pauli matrices). Extending the
strong-coupling theory developed by Nozi\`eres and Schmitt-Rink (NSR) to
include this spin-orbit coupling, we determine the superfluid phase transition
temperature $T_{\rm c}$, as functions of the strength of a pairing interaction
$U_s$, as well as the spin-orbit coupling strength $\lambda$. Evaluating poles
of the NSR particle-particle scattering matrix describing fluctuations in the
Cooper channel, we clarify the region where rashbon bound states dominate the
superfluid phase transition in the $U_{s}$-$\lambda$ phase diagram. Since the
antisymmetric spin-orbit interaction $H_{\rm so}$ breaks the inversion symmetry
of the system, rashbon bound states naturally have, not only a spin-singlet and
even-parity symmetry, but also a spin-triplet and odd-parity symmetry. Thus,
our results would be also useful for the study of this parity mixing effect in
the BCS-BEC crossover regime of a spin-orbit coupled Fermi gas.
