The interest in primordial black holes (PBHs) has recently risen up
again after the discovery of around 30 solar mass black holes through
the LIGO/Virgo gravitational wave events. PBHs are black holes produced
by gravitational collapse of the over-dense region in the early universe.
An attractive origin of the overdensity is primordial perturbations
generated during inflation. In this talk, I will first explain the
relation between the power spectrum of the perturbations predicted by
inflation models and the estimated PBH abundance. This relation has
various theoretical uncertainties including the choice of the window
function (https://arxiv.org/abs/1802.06393). Next, as a concrete model,
I will briefly introduce the so-called axion-like curvaton model, where
the phase direction of a complex scalar field different from inflaton is
responsible for small-scale perturbations (
https://arxiv.org/abs/1711.08956, https://arxiv.org/abs/1805.07757). We
found that PBHs produced in this model can successfully reproduce the
LIGO/Virgo gravitational wave events. Finally, I would like to discuss
the case where a single-field inflaton potential has an inflection point.
There, a slow-roll violation enhances quantum fluctuation. We should
use stochastic inflation formalism since the quantum back reaction could
become significant. However, stochastic inflation beyond slow-roll
regimes is nontrivial and worth investigating by itself. I will show a
numerical result obtained so far under a simplified setup.