  We study the contacts, large-momentum tail, radio-frequency spectroscopy, and
some other universal relations for an ultracold one-dimensional (1D)
two-component Fermi gas with spin-orbit coupling (SOC). Different from previous
studies, we find that the $q^{-8}$ tail in the spin-mixing (off-diagonal) terms
of the momentum distribution matrix is dependent on the two SOC parameters in
the laboratory frame for 1D systems, where $q$ is the relative momentum. This
tail can be observed through time-of-flight measurement as a direct
manifestation of the SOC effects on the many-body level. Besides the
traditional 1D even-wave scattering length, we find that two new physical
quantities must be introduced due to the SOC. Consequently, two new adiabatic
energy relations with respect to the two SOC parameters are obtained.
Furthermore, we derive the pressure relation and virial theorem at short
distances for this system. To find how the SOC modifies the large-momentum
behavior, we take the SOC parameters as perturbations since the strength of the
SOC should be much smaller than the corresponding strength scale of the
interatomic interactions. In addition, by using the operator product expansion
method, we derive the asymptotic behavior of the large-momentum distribution
matrix up to the $q^{-8}$ order and find that the diagonal terms of the
distribution matrix include the contact of traditional 1D even-wave scattering
length as the leading term and the SOC modified terms beyond the leading term,
the off-diagonal term is beyond the subleading term and is corrected by the SOC
parameters. We also find that the momentum distribution matrix shows
spin-dependent and anisotropic features. Furthermore, we calculate the momentum
distribution matrix in the laboratory frame for the experimental implication.
