Description
During inflation, quantum vacuum fluctuations are stretched beyond the Hubble radius, modifying the large-scale dynamics of the Universe. While their backreaction is negligible in the perturbative regime, it can become important in scenarios leading to primordial black hole (PBH) formation, where large density fluctuations are generated. The combination of stochastic inflation and the 𝛿N formalism provides an efficient framework to track these large fluctuations non-perturbatively. After reviewing the emergence of heavy non-Gaussian tails in the curvature perturbation distribution, I will show how large fluctuations are spatially correlated by computing real-space correlation functions within the stochastic-𝛿N approach. This can be used to characterise the spatial distribution of PBHs, revealing a universal clustering profile at their formation.
I will then introduce a novel framework that implements stochastic inflation on stochastic trees, modelling inflationary expansion as a branching process. Notably, stochastic trees do not operate on a fixed background, instead new spacetime units dynamically emerge as the trees unfold, naturally incorporating metric fluctuations. The tree structure naturally encodes the statistics of curvature perturbations and other cosmological fields, providing a direct tool to study PBH formation. Within this picture, PBHs emerge at unbalanced nodes of the tree, and their mass function can be derived while automatically accounting for the “cloud-in-cloud” effect. Finally, I will discuss ongoing work on extracting compaction-function criteria for PBH formation and probing type I and type II perturbations directly from the stochastic tree population.
