  We employ population synthesis method to model the double neutron star (DNS)
population and test various possibilities on natal kick velocities gained by
neutron stars after their formation. We first choose natal kicks after standard
core collapse SN from a Maxwellian distribution with velocity dispersion of
sigma=265 km/s as proposed by Hobbs et al. (2005) and then modify this
distribution by changing the velocity dispersion towards smaller and larger
kick values. We also take into account the possibility of NS formation through
electron capture supernova. In this case we test two scenarios: zero natal kick
or small natal kick, drawn from Maxwellian distribution with sigma = 26.5 km/s.
We calculate the present-day orbital parameters of binaries and compare the
resulting eccentricities with those known for observed DNSs. As an additional
test we calculate Galactic merger rates for our model populations and confront
them with observational limits. We do not find any model unequivocally
consistent with both observational constraints simultaneously.The models with
low kicks after CCSN for binaries with the second NS forming through core
collapse SN are marginally consistent with the observations. This means that
either 14 observed DNSs are not representative of the intrinsic Galactic
population, or that our modeling of DNS formation needs revision.
