  Primordial black holes (PBHs) are one of the candidates to explain the
gravitational wave (GW) signals observed by the LIGO detectors. Among several
phenomena in the early Universe, cosmic inflation is a major example to
generate PBHs from large primordial density perturbations. In this paper, we
discuss the possibility to interpret the observed GW events as mergers of PBHs
which are produced by cosmic inflation. The primordial curvature perturbation
should be large enough to produce a sizable amount of PBHs and thus we have
several other probes to test this scenario. We point out that the current
pulsar timing array (PTA) experiments already put severe constraints on GWs
generated via the second-order effects, and that the observation of the cosmic
microwave background (CMB) puts severe restriction on its $\mu$ distortion. In
particular, it is found that the scalar power spectrum should have a very sharp
peak at $k \sim 10^{6}$ Mpc$^{-1}$ to fulfill the required abundance of PBHs
while evading constraints from the PTA experiments together with the $\mu$
distortion. We propose a mechanism which can realize such a sharp peak. In the
future, simple inflation models that generate PBHs via almost Gaussian
fluctuations could be probed/excluded.
