  We present a comprehensive derivation of a set of universal relations for
spin-orbit-coupled Fermi gases in three or two dimension, which follow from the
short-range behavior of the two-body physics. Besides the adiabatic energy
relations, the large-momentum distribution, the grand canonical potential and
pressure relation derived in our previous work for three-dimensional systems
{[}Phys. Rev. Lett. 120, 060408 (2018){]}, we further derive high-frequency
tail of the radio-frequency spectroscopy and the short-range behavior of the
pair correlation function. In addition, we also extend the derivation to
two-dimensional systems with Rashba type of spin-orbit coupling. To simply
demonstrate how the spin-orbit-coupling effect modifies the two-body
short-range behavior, we solve the two-body problem in the sub-Hilbert space of
zero center-of-mass momentum and zero total angular momentum, and
perturbatively take the spin-orbit-coupling effect into account at short
distance, since the strength of the spin-orbit coupling should be much smaller
than the corresponding scale of the finite range of interatomic interactions.
The universal asymptotic forms of the two-body wave function at short distance
are then derived, which do not depend on the short-range details of interatomic
potentials. We find that new scattering parameters need to be introduced
because of spin-orbit coupling, besides the traditional $s$- and $p$-wave
scattering length (volume) and effective ranges. This is a general and unique
feature for spin-orbit-coupled systems. We show how these two-body parameters
characterize the universal relations in the presence of spin-orbit coupling.
This work probably shed light for understanding the profound properties of the
many-body quantum systems in the presence of the spin-orbit coupling.
