  We examine a quantised massive scalar field in $(1+1)$-dimensional spatially
compact cosmological spacetimes in which the early time and late time expansion
laws provide distinguished definitions of Fock "in" and "out" vacua, with the
possible exception of the spatially constant sector, which may become
effectively massless at early or late times. We show, generalising the work of
Ford and Pathinayake, that when such a massive zero mode occurs, the freedom in
the respective "in" and "out" states is a family with two real parameters. As
an application, we consider massive untwisted and twisted scalar fields in the
$(1+1)$-dimensional spatially compact Milne spacetime, where the untwisted
field has a massive "in" zero mode. We demonstrate, by a combination of
analytic and numerical methods, that the choice of the massive "in" zero mode
state has a significant effect on the response of an inertial Unruh-DeWitt
detector, especially in the excitation part of the spectrum. The detector's
peculiar velocity with respect to comoving cosmological observers has the
strongest effect in the "in" vacuum of the untwisted field, where it shifts the
excitation and de-excitation resonances towards higher values of the detector's
energy gap.
