The formation of primordial black holes is studied using superconformal inflationary α-attractors. An inflaton potential is constructed with a plateau-like region which brings about the onset of an ultra slow-roll region where the required enhancement in the curvature power spectrum takes place. This is accomplished by carefully finetuning the parameters in the potential. For the parameter sets, the amount of observable inflation is such that the curvature perturbation P R peaks at ∼ 10 14 Mpc −1 , producing PBHs in the mass window 10 16 g ≤ M ≤ 10 18 g. The reheating period after inflation is taken into account to determine whether or not PBH formation takes place in such a phase. Finally, the spectrum of second order gravitational waves, that can arise due to the curvature perturbation enhancement, is calculated.

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We consider stochastic inflation coarse-grained using a general class of exponential filters. Such a coarse-graining prescription gives rise to inflaton-Langevin equations sourced by colored noise that is correlated in e-fold time. The dynamics are studied first in slow-roll for simple potentials using first-order perturbative, semi-analytical calculations which are later compared to numerical simulations. Subsequent calculations are performed using an exponentially correlated noise which appears as a leading order correction to the full slow-roll noise correlation functions of the type 〈ξ(N)ξ(N')〉(n) ∼ (cosh[n(N-N'+1])-1. We find that the power spectrum of curvature perturbations 𝒫 ζ is suppressed at small e-folds, with the suppression controlled by n. Furthermore, we use the leading order, exponentially correlated noise and perform a first passage time analysis to compute the statistics of the stochastic e-fold distribution 𝒩 and derive an approximate expression for the mean number of e-folds 〈𝒩〉. Comparing analytical results with numerical simulations of the inflaton dynamics, we show that the leading order noise correlation function can be used as a very good approximation of the exact noise, the latter being more difficult to simulate.

We consider stochastic inflation coarse-grained using a general class of exponential filters. Such a coarse-graining prescription gives rise to inflaton-Langevin equations sourced by colored noise that is correlated in e-fold time. The dynamics are studied first in slowroll for simple potentials using first-order perturbative, semi-analytical calculations which are later compared to numerical simulations. Subsequent calculations are performed using an exponentially correlated noise which appears as a leading order correction to the full slow-roll noise correlation functions of the type ξ(N )ξ(N ) (n) ∼ (cosh [n(N − N ) + 1]) −1 . We find that the power spectrum of curvature perturbations P ζ is suppressed at early e-folds, with the suppression controlled by n. Furthermore, we use the leading order, exponentially correlated noise and perform a first passage time analysis to compute the statistics of the stochastic efold distribution N and derive an approximate expression for the mean number of e-folds N . Comparing analytical results with numerical simulations of the inflaton dynamics, we show that the leading order noise correlation function can be used as a very good approximation of the exact noise, the latter being more difficult to simulate.