Regulating the infrared by mode matching: A massless scalar in expanding
  spaces with constant deceleration
  In this paper we consider a massless scalar field, with a possible coupling
$\xi$ to the Ricci scalar in a $D$ dimensional FLRW spacetime with a constant
deceleration parameter $q=\epsilon-1$, $\epsilon=-{\dot{H}}/{H^2}$. Correlation
functions for the Bunch-Davies vacuum of such a theory have long been known to
be infrared divergent for a wide range of values of $\epsilon$. We resolve
these divergences by explicitly matching the spacetime under consideration to a
spacetime without infrared divergencies. Such a procedure ensures that all
correlation functions with respect to the vacuum in the spacetime of interest
are infrared finite. In this newly defined vacuum we construct the coincidence
limit of the propagator and as an example calculate the expectation value of
the stress energy tensor. We find that this approach gives both in the
ultraviolet and in the infrared satisfactory results. Moreover, we find that,
unless the effective mass due to the coupling to the Ricci scalar $\xi R$ is
negative, quantum contributions to the energy density always dilute away
faster, or just as fast, as the background energy density. Therefore, quantum
backreaction is insignificant at the one loop order, unless $\xi R$ is
negative. Finally we compare this approach with known results where the
infrared is regulated by placing the Universe in a finite box. In an
accelerating universe, the results are qualitatively the same, provided one
identifies the size of the Universe with the physical Hubble radius at the time
of the matching. In a decelerating universe however, the two schemes give
different late time behavior for the quantum stress energy tensor. This happens
because in this case the length scale at which one regulates the infrared
becomes sub-Hubble at late times.
