Natal kicks are one of the most debated issues about double neutron star ( DNS ) formation .
Several observational and theoretical results suggest that some DNSs have formed with low natal kicks ( \lesssim { } 50 km s ^ { -1 } ) , which might be attributed to electron-capture supernovae ( ECSNe ) .
We investigate the impact of ECSNe on the formation of DNSs by means of population synthesis simulations .
In particular , we assume a Maxwellian velocity distribution for the natal kick induced by ECSNe with one dimensional root-mean-square \sigma _ { ECSN } = 0 , 7 , 15 , 26 , 265 km s ^ { -1 } .
The total number of DNSs scales inversely with \sigma _ { ECSN } and the number of DNS mergers is higher for relatively low kicks .
This effect is particularly strong if we assume low efficiency of common-envelope ejection ( described by the parameter \alpha = 1 ) , while it is only mild for high efficiency of common-envelope ejection ( \alpha { } = 5 ) .
In most simulations , more than 50 per cent of the progenitors of merging DNSs undergo at least one ECSN and the ECSN is almost always the first SN occurring in the binary system .
Finally , we have considered the extreme case in which all neutron stars receive a low natal kick ( \lesssim { } 50 km s ^ { -1 } ) .
In this case , the number of DNSs increases by a factor of ten and the percentage of merging DNSs which went through an ECSN is significantly suppressed ( < 40 per cent ) .