Clustering of primordial black holes from quantum diffusion during inflation
arxiv(2024)
摘要
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>ζ_c,
ζ_R_2>ζ_c)/[P(ζ_R_1>ζ_c)
P(ζ_R_2>ζ_c)]-1, is independent of the threshold value
ζ_c. This contrasts with Gaussian statistics where the same
quantity strongly decays with ζ_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.
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