  We study the distribution of quantum steerability for continuous variables
between two causally disconnected open charts in de Sitter space. It is shown
that quantum steerability suffers from "sudden death" in de Sitter space, which
is quite different from the behaviors of entanglement and discord because the
latter always survives and the former vanishes only in the limit of infinite
curvature. In addition, we find that the attainment of maximal steerability
asymmetry indicates a transition between unidirectional steerable and
bidirectional steerable. Unlike in the flat space, the asymmetry of quantum
steerability can be completely destroyed in the limit of infinite curvature for
the conformal and massless scalar fields in de Sitter space.
