Recently we have proposed a novel method to probe primordial gravitationalwaves from upper bounds on the abundance of primordial black holes (PBHs). Whenthe amplitude of primordial tensor perturbations generated in the earlyUniverse is fairly large, they induce substantial scalar perturbations due totheir second-order effects. If these induced scalar perturbations are too largewhen they reenter the horizon, then PBHs are overproduced, their abundanceexceeding observational upper limits. That is, primordial tensor perturbationson superhorizon scales can be constrained from the absence of PBHs. In ourrecent paper we have only shown simple estimations of these new constraints,and hence in this paper, we present detailed derivations, solving the Einsteinequations for scalar perturbations induced at second order in tensorperturbations. We also derive an approximate formula for the probabilitydensity function of induced density perturbations, necessary to relate theabundance of PBHs to the primordial tensor power spectrum, assuming primordialtensor perturbations follow Gaussian distributions. Our new upper bounds fromPBHs are compared with other existing bounds obtained from big bangnucleosynthesis, cosmic microwave background, LIGO/Virgo and pulsar timingarrays.
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