  We consider Dark Matter composed of an oscillating singlet scalar field. On
top of the mass term, the scalar is equipped with a potential spontaneously
breaking Z_2-symmetry. This potential dominates at early times and leads to the
time-dependent expectation value of the scalar, which decreases in the
expanding Universe. As it drops below some critical value, the symmetry gets
restored, and the Dark Matter field starts to oscillate around zero. We arrange
the spontaneous symmetry breaking through the interaction of the scalar with
the Ricci curvature. In that way, superheavy Dark Matter can be produced at
very early times. Depending on its mass, the production takes place at
inflation (very large masses up to the Grand Unification scale), at preheating,
or at radiation-dominated stage (masses 10^{6}-10^{7} Gev).
