  In the early universe, the potential of a scalar field can be significantly
modified, and the scalar field may be trapped for a long time in a different
location than the current vacuum. The trapping effect can increase or decrease
the scalar abundance. For instance, in thermal inflation, a scalar field is
trapped at the top of the potential by a thermal effect and dominates the
universe to drive inflation for a short period of time. On the other hand, a
scalar abundance can be exponentially suppressed in the adiabatic suppression
mechanism, where a scalar field moves adiabatically by a time-dependent
trapping potential. In this study, we investigate such a trapping effect on the
abundance of scalar fields. Specifically, we investigate how the abundance of a
scalar field depends on its initial position in the case of a double well
potential and identify the physical quantity that controls the abundance. Then,
we study the QCD axion abundance for various values of the misalignment angle,
where the axon potential receives a large temporal mass due to the Witten
effect. We find that the axion abundance is suppressed due to the adiabatic
suppression mechanism even when it is trapped near the maximum of the
potential, if the trapping effect is sufficiently large.
