  Rotating proto-neutron stars can be important sources of gravitational waves
to be searched for by present-day and future interferometric detectors. It was
demonstrated by Imshennik that in extreme cases the rapid rotation of a
collapsing stellar core may lead to fission and formation of a binary
proto-neutron star which subsequently merges due to gravitational wave
emission. In the present paper, we show that such dynamically unstable
collapsing stellar cores may be the product of a former merger process of two
stellar cores in a common envelope. We applied population synthesis
calculations to assess the expected fraction of such rapidly rotating stellar
cores which may lead to fission and formation of a pair of proto-neutron stars.
We have used the BSE population synthesis code supplemented with a new
treatment of stellar core rotation during the evolution via effective
core-envelope coupling, characterized by the coupling time, $\tau_c$. The
validity of this approach is checked by direct MESA calculations of the
evolution of a rotating 15 $M_\odot$ star. From comparison of the calculated
spin distribution of young neutron stars with the observed one, reported by
Popov and Turolla, we infer the value $\tau_c \simeq 5 \times 10^5$ years. We
show that merging of stellar cores in common envelopes can lead to collapses
with dynamically unstable proto-neutron stars, with their formation rate being
$\sim 0.1-1\%$ of the total core collapses, depending on the common envelope
efficiency.
