Stochastic Inflation with Interacting Noises

Abstract

Stochastic $\delta N$ formalism is a powerful tool to calculate the cosmological correlators non-perturbatively. However, it requires the initial data for the amplitude of the noise on the initial flat hypersurface which for a free theory during inflation is fixed to be $\frac{H}{2\pi }$. In this work, we study the setups where the underlying theory involves interactions and the stochastic noises inherit these interactions. We extend the stochastic $\delta N$ formalism to these setups and rewrite the corresponding Langevin and Fokker-Planck equations in which the QFT corrections in the amplitude of the noises are taken into account. As an example, in the three-phase SR-USR-SR setup which is employed for PBHs formation, the modification in the amplitude of noise is calculated from the one-loop corrections in power spectrum via in-in formalism. We show that in these setups the amplitude of the stochastic noise is modified to $\frac{H}{2\pi }(1+\frac{\Delta {\mathcal{P}}_{\mathcal{R}}}{{\mathcal{P}}_{\mathcal{R}}^{(0)}}{)}^{\frac{1}{2}}$ in which $\frac{\Delta {\mathcal{P}}_{\mathcal{R}}}{{\mathcal{P}}_{\mathcal{R}}^{(0)}}$ is the fractional one-loop correction in power spectrum.

27 pages, 2 figures; updated to match the JCAP version

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