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We study how large fluctuations are spatially correlated in the presence of quantum diffusion during inflation. This is done by computing real-space correlation functions in the stochastic-$\delta N$ formalism. We first derive an exact description of physical distances as measured by a local observer at the end of inflation, improving on previous works. Our approach is based on recursive algorithmic methods that consistently include volume-weighting effects. We then propose a "large-volume'' approximation under which calculations can be done using first-passage time analysis only, and from which a new formula for the power spectrum in stochastic inflation is derived. We then study the full two-point statistics of the curvature perturbation. Due to the presence of exponential tails, we find that the joint distribution of large fluctuations is of the form $P(\zeta_{R_1}, \zeta_{R_2}) = F(R_1,R_2,r) P(\zeta_{R_1})P( \zeta_{R_2})$, where $\zeta_{R_1}$ and $\zeta_{R_2}$ denote the curvature perturbation coarse-grained at radii $R_1$ and $R_2$, around two spatial points distant by $r$. This implies that, on the tail, the reduced correlation function, defined as $P(\zeta_{R_1}>\zeta_{\rm{c}}, \zeta_{R_2}>\zeta_{\rm{c}})/[P(\zeta_{R_1}>\zeta_{\rm{c}}) P(\zeta_{R_2}>\zeta_{\rm{c}})]-1$, is independent of the threshold value $\zeta_{\rm{c}}$. This contrasts with Gaussian statistics where the same quantity strongly decays with $\zeta_{\rm{c}}$, and shows the existence of a universal clustering profile for all structures forming in the exponential tails. Structures forming in the intermediate (i.e. not yet exponential) tails may feature different, model-dependent behaviours., Comment: 52 pages, 14 figures. Discussion around Eq.(3.6) expanded, a few minor changes and typos fixed. Some references added. Matches published version in JCAP

If the inflaton gets trapped in a local minimum of its potential shortly before the end of inflation, it escapes by building up quantum fluctuations in a process known as stochastic tunnelling. In this work we study cosmological fluctuations produced in such a scenario, and how likely they are to form Primordial Black Holes (PBHs). This is done by using the stochastic-$\delta N$ formalism, which allows us to reconstruct the highly non-Gaussian tails of the distribution function of the number of $e$-folds spent in the false-vacuum state. We explore two different toy models, both analytically and numerically, in order to identify which properties do or do not depend on the details of the false-vacuum profile. We find that when the potential barrier is small enough compared to its width, $\Delta V/V < \Delta\phi^2/M_{\text{Pl}}^2$, the potential can be approximated as being flat between its two local extrema, so results previously obtained in a "flat quantum well'' apply. Otherwise, when $\Delta V/V < V/M_{\text{Pl}}^4$, the PBH abundance depends exponentially on the height of the potential barrier, and when $\Delta V/V > V/M_{\text{Pl}}^4$ it depends super-exponentially ( i.e. as the exponential of an exponential) on the barrier height. In that later case PBHs are massively produced. This allows us to quantify how much flat inflection points need to be fine-tuned. In a deep false vacuum, we also find that slow-roll violations are typically encountered unless the potential is close to linear. This motivates further investigations to generalise our approach to non-slow-roll setups., Comment: 25 pages without appendix ( 36 pages in total ), 13 figures. Matches the published version in JCAP

We introduce a new approach to renormalize physical quantities in curved space-time by adiabatic subtraction. We use a comoving infrared cut-off in defining the adiabatic counterpart of the physical quantity under consideration, building on the fact that the adiabatic approximation is ill-defined in the infrared tail of the spectrum. We show how this infrared cut-off should be used to obtain a completely well-defined renormalization scheme and how it is fundamental to avoid unphysical divergences that can be generated by a pathological behavior of the adiabatic subtraction extended to the infrared tail. The infrared cut-off appears as a new degree of freedom introduced in the theory and its actual value has to be consistently fixed by a physical prescription. As an example, we show how such degree of freedom can be set to obtain the correct value of the conformal anomaly in the symptomatic case of an inflationary model with gauge fields coupled to a pseudo-scalar inflaton., Comment: 16 pages, 1 figure. Few comments and clarifications added, version accepted for publication in JCAP