  In open system approaches with non-Markovian environments, the process of
inserting an individual mode (denoted as "pseudomode") into the bath or
extracting it from the bath is widely employed. This procedure, however, is
typically performed on basis of the spectral density (SD) and does not
incorporate temperature. Here, we show how the - temperature-dependent - bath
correlation function (BCF) transforms in such a process. We present analytic
formulae for the transformed BCF and numerically study the differences between
factorizing initial state and global thermal (correlated) initial state of mode
and bath, respectively. We find that in the regime of strong coupling of the
mode to both system and bath, the differences in the BCFs give rise to
pronounced differences in the dynamics of the system.
