  Recently we have proposed a novel method to probe primordial gravitational
waves from upper bounds on the abundance of primordial black holes (PBHs). When
the amplitude of primordial tensor perturbations generated in the early
Universe is fairly large, they induce substantial scalar perturbations due to
their second-order effects. If these induced scalar perturbations are too large
when they reenter the horizon, then PBHs are overproduced, their abundance
exceeding observational upper limits. That is, primordial tensor perturbations
on superhorizon scales can be constrained from the absence of PBHs. In our
recent paper we have only shown simple estimations of these new constraints,
and hence in this paper, we present detailed derivations, solving the Einstein
equations for scalar perturbations induced at second order in tensor
perturbations. We also derive an approximate formula for the probability
density function of induced density perturbations, necessary to relate the
abundance of PBHs to the primordial tensor power spectrum, assuming primordial
tensor perturbations follow Gaussian distributions. Our new upper bounds from
PBHs are compared with other existing bounds obtained from big bang
nucleosynthesis, cosmic microwave background, LIGO/Virgo and pulsar timing
arrays.
