Feynman diagrams for stochastic inflation and quantum field theory in de Sitter space

We consider a massive scalar field with quartic self-interaction $\lambda /4!{\varphi }^4$ in de Sitter spacetime and present a diagrammatic expansion that describes the field as driven by stochastic noise. This is compared with the Feynman diagrams in the Keldysh basis of the amphichronous (closed-time-path) field theoretical formalism. For all orders in the expansion, we find that the diagrams agree when evaluated in the leading infrared approximation, i.e. to leading order in $m^2/H^2$, where $m$ is the mass of the scalar field and $H$ is the Hubble rate. As a consequence, the correlation functions computed in both approaches also agree to leading infrared order. This perturbative correspondence shows that the stochastic theory is exactly equivalent to the field theory in the infrared. The former can then offer a nonperturbative resummation of the field theoretical Feynman diagram expansion, including fields with $0\le m^2\ll \sqrt{\lambda }{H}^2$ for which the perturbation expansion fails at late times.

Figure 1

The elements out of which stochastic diagrams are constructed. The choice of vertex factor implies that the assembled diagrams should be divided by their symmetry factor.

Figure 2

The elements out of which the QFT Feynman diagrams are constructed. The choice of vertex factor implies that the assembled diagrams should be divided by their symmetry factor. Note the similarity with the stochastic diagrams of Sec. 2 as well as the obvious differences: Integrations extend over all of spacetime and the extra ${\psi }^3\varphi$ vertex appears.

Figure 3

For a correlation function $⟨\varphi (x_o)\varphi (x_o)⟩$, the support of the spacetime integration is given by the past light cone of $x_o$ because of causality. The diagram illustrates that this implies a maximal spatial distance for any two points with given ${\eta }_a$ and ${\eta }_b$.

Figure 4

The vertex integration in Eq. (35) is confined to the past light cone of $x_1$. The large ${\delta }^2$ region for $y(x^{\prime },x_2)$ is the narrow strip between the red dashed lines.