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.
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