IR finite correlation functions in de Sitter space, a smooth massless limit, and an autonomous equation

We explore two-point and four-point correlation functions of a massive scalar feld on the fat de Sitter background in the long-wavelength approximation. By employing the Yang-Feldman-type equation, we compute the two-point correlation function up to the λ 3 order and the four-point correlation function up to the λ 2 one. In contrast to the standard theory of a massive scalar feld based on the de Sitter-invariant vacuum, we develop the vacuum-independent reasoning that may not possess de Sitter invariance but results in a smooth massless limit of the correlation function’s infrared part. Our elaboration afords to calculate correlation functions of a free massive scalar feld and to proceed with quantum corrections, relying only on the known two-point correlation function’s infrared part of a free massless one. Remarkably, the two-point correlation function of a free massive scalar feld coincides with the Ornstein-Uhlenbeck stochastic process’s one and has a clear physical interpretation. We compared our results with those obtained with the SchwingerKeldysh diagrammatic technique, Starobinsky’s stochastic approach, and the Hartree-Fock approximation. At last, we have constructed a renormalization group-inspired autonomous equation for the two-point correlation function. Integrating its approximate version, one obtains the non-analytic expression with respect to a self-interaction coupling constant λ. That solution reproduces the correct perturbative series up to the two-loop level. At the late-time limit, it almost coincides with the result of Starobinsky’s stochastic approach in the whole interval of a new dimensionless parameter 0 ≤ π 2m4 3λH4 < ∞.