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

Constraints on the formation of double neutron stars from the observed eccentricities and current limits on merger rates