In this thesis the infrared properties of massless scalar fields, with a possible coupling to the Ricci scalar on a cosmological background are studied. Our background space-time is a homogeneous, flat FLRW space-time, with the additional constraint that the deceleration parameter is constant. It has been known for a long time that the propagator of such a scalar field diverges at the lower end of the integration over the momenta when evaluated for the Bunch-Davies vacuum. The resolution we propose to this problem is to work on a spatially compact universe. This effectively generates an infrared cut-off in the integral over the mode functions. To see what the physical effect of this approaches is, we calculated the expectation value of the stress energy tensor. Using the infrared regulated propagator, we find in the ultraviolet that we correctly reproduce the Bunch-Davies result. In the infrared the discussion can be split in two parts. If the universe is accelerating we found that, although we see a growth in energy due to the production of infrared particles, this energy will always dilute away faster than the energy present in the background space-time. In a decelerating universe the situation is quite different. We find that, since in a decelerating universe, the Hubble radius grows faster then physical scales, the cut-off scale actually enters the Hubble volume. This results in a large effect which is thus not due to particle creation, but it is due to the fact that the influence of the cut-off grows more and more profound as time goes on. We also calculated the one loop effective action in a theory where both gravity and a scalar field produce quantum fluctuations. The mixing between the dynamical degrees of freedom in such a case leads to more complicated expressions, but in the end the final answer is very similar to the obtained expectation value of the stress energy tensor for just a single massless scalar field. Also for the graviton we find that at one loop order all growth in energy due to the creation of infrared modes is cancelled by the redshift of these modes. As a second application we used the cut-off regulated propagator to calculate the one-loop effective potential for a massless scalar field, with a coupling to the Ricci scalar and a quartic self interaction. We calculated the effective action by assuming that the background field has the same time dependence as the Hubble parameter. While this is true at the classical level, this scaling turns out to be broken by the quantum corrections. Furthermore we find that initially, the quantum corrections can induce a phase transition, by generating minima in the potential for nonzero field values. However the growing infrared contributions are such that, after a sufficient amount of time, the effective potential will always end up with only one minimum at the origin. Thus the infrared modes have restored the symmetry of the potential
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