  It has been recently suggested \cite{Bezrukov:2017ike,Bezrukov:2018wvd} that
a cosmic scalar field can completely change the keV-scale sterile neutrino
production in the early Universe. Its effect may, for various parameter
choices, either suppress sterile neutrino production and make moderate
active-sterile mixing cosmologically acceptable, or increase the production and
generate considerable dark matter component out of sterile neutrino with
otherwise negligible mixing with SM. In this paper we provide analytic
estimates complementing and providing details of the numerical calculations
performed in \cite{Bezrukov:2018wvd} in the case of resonant amplification of
the sterile neutrino production. We also discuss phenomenological and
theoretical issues related to the successful implementation of this idea in
fully realistic extensions of the Standard Model of particle physics.
