  We consider an extension of the Standard Model with a singlet sector
consisting of a real (pseudo)scalar and a Dirac fermion coupled with the
Standard Model only via the scalar portal. We assume that the portal coupling
is weak enough for the singlet sector not to thermalize with the Standard Model
allowing the production of singlet particles via the freeze-in mechanism. If
the singlet sector interacts with itself sufficiently strongly, it may
thermalize within itself, resulting in dark matter abundance determined by the
freeze-out mechanism operating within the singlet sector. We investigate this
scenario in detail. In particular, we show that requiring the absence of
inflationary isocurvature fluctuations provides lower bounds on the magnitude
of the dark sector self-interactions and in parts of the parameter space favors
sufficiently large self-couplings, supported also by the features observed in
the small-scale structure formation.
