Abstract
Abstract
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-δ 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(ζ
R
1, ζ
R
2) = F(R
1,R
2, r) P(ζ
R
1)P( ζ
R
2), where ζ
R
1 and ζ
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(ζ
R
1 > ζc, ζ
R
2 > ζc)/[P(ζ
R
1 > ζc) P(ζ
R
2 > ζc)]-1, is independent of the threshold value ζc. This contrasts with Gaussian statistics where the same quantity strongly decays with ζc, 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.