Free scalar correlators in de Sitter space via the stochastic approach beyond the slow-roll approximation

The stochastic approach to calculating scalar correlation functions in de Sitter spacetime is extended beyond the overdamped “slow-roll” approximation. We show that with the correct noise term, it reproduces the exact asymptotic long-distance behavior of field correlators in free field theory, thereby demonstrating the viability of the technique. However, we also show that the naïve way of calculating the noise term by introducing a cutoff at the horizon does not give the correct answer unless the cutoff is chosen specifically to give the required result. We discuss the implications of this for interacting theories.

Figure 1

The quantum (blue) ${\sigma }_{Q,qq}^2$ and the cutoff (red) noises ${\sigma }_{cut,qq}^2$ with $\epsilon =0$ (dot-dashed), $\epsilon =0.01$ (solid), $\epsilon =0.5$ (dashed), and $\epsilon =0.99$ (dotted) is plotted as a function of $\nu$. We see that the two approaches do not agree for any value of $\epsilon$ and therefore the cutoff procedure is unable to reproduce the quantum field correlator for all masses.