Primordial black hole formation in the inflationary Universe

Abstract

In this work, we consider the formation of primordial black holes (PBHs) using the theory of cosmic inflation. We construct an inflaton potential using $\alpha$-attractors, possessing a plateau-like region where the inflaton enters into a non-attractor phase, giving rise to the phenomenon of ultra slow-roll (USR). Exploiting the fact that the inflaton experiences a dramatic deceleration during USR, a sharp enhancement in the curvature perturbation $\mathcal{P}_\zeta$ can be shown to be characteristic of such models, subsequently allowing for copious production of PBHs in the post-inflationary, radiation epoch of the Universe. The parameters in our model are highly fine-tuned such that an $\mathcal{O}(10^7)$ enhancement in the power spectrum is produced for modes with comoving wavenumber $k\sim 10^{14}\:\text{Mpc}^{-1}$, allowing for abundant PBH formation of mass $M\sim 10^{17}\:\text{g}$. In the mass range $10^{17}-10^{19}\:\text{g}$, PBHs can constitute the entirety of cold dark matter (CDM) after pre-existing constraints in the range have been called into question. The formation fraction of PBHs $\beta(M)$ is the most important quantity that cosmologists compute. It relates the mass-energy contained in PBHs to that of the expanding background. The computation of $\beta(M)$ sometimes controversial due to the different methodologies one adopts for performing the calculations. In essence, this is the Press-Schechter vs. peaks theory debate. In this work, we address both issues. Firstly, we improve upon the simplest rendition of Press-Schechter -- which, with its simplicity set aside -- does not take into account the inherently nonlinear relationship between the primordial curvature perturbations and the post-inflationary overdensity perturbations. We refine the simple Press-Schechter formalism by taking into account such nonlinearities. One can then show that this unavoidably generates non-Gaussianities in the probability distribution of overdensities, with the suppression of PBH abundance arising as a consequence. Using this, we calculate the first two, nontrivial higher order moments, namely the skewness and kurtosis, which are subsequently used in a nonlinear redefinition of the PBH formation fraction. Furthermore, we perform the same calculations using optimized peaks theory, combining a more accurate calculation of PBH mass. Taking into account the detailed shape and profile of curvature perturbations, along with an accurate mass calculation, we show that the proper consideration of the effects of curvature generally predict PBHs to be more massive, by around a factor of 10, compared to na\"{i}ve estimates. Finally, we study the effects of enhanced effects of quantum diffusion in the USR region through the stochastic inflationary formalism. We aim at providing a generic framework with which such quantum diffusion effects can be numerically modeled without resorting to slow-roll approximations, commenting on how to skirt around the inherently non-Markovian nature of the noise terms arising in stochastic inflation. We show that, once quantum diffusion effects are taken into account, there is additional enhancement in $\mathcal{P}_\zeta$ during the USR phase, on top of the usual amplification, which can have important consequences not only for PBH formation, but also for inflationary model building and parameter fine-tuning.