  Since inflationary perturbations must generically couple to all degrees of
freedom present in the early Universe, it is more realistic to view these
fluctuations as an open quantum system interacting with an environment. Then,
on very general grounds, their evolution can be modelled with a Lindblad
equation. This modified evolution leads to quantum decoherence of the system,
as well as to corrections to observables such as the power spectrum of
curvature fluctuations. On one hand, current cosmological observations
constrain the properties of possible environments and place upper bounds on the
interaction strengths. On the other hand, imposing that decoherence completes
by the end of inflation implies lower bounds on the interaction strengths.
Therefore, the question arises of whether successful decoherence can occur
without altering the power spectrum. In this paper, we systematically identify
all scenarios in which this is possible. As an illustration, we discuss the
case in which the environment consists of a heavy test scalar field. We show
that this realises the very peculiar configuration where the correction to the
power spectrum is quasi scale invariant. In that case, the presence of the
environment improves the fit to the data for some inflationary models but
deteriorates it for others. This clearly demonstrates that decoherence is not
only of theoretical importance but can also be crucial for astrophysical
observations.
