Gravitational wave constraints on the primordial black hole dominated
  early universe
  We calculate the gravitational waves (GWs) induced by the density
fluctuations due to inhomogeneous distribution of primordial black holes (PBHs)
in the case where PBHs eventually dominate and reheat the universe by Hawking
evaporation. The initial PBH density fluctuations are isocurvature in nature.
We find that most of the induced GWs are generated right after evaporation,
when the universe transits from the PBH dominated era to the radiation
dominated era and the curvature perturbation starts to oscillate wildly. The
strongest constraint on the amount of the produced GWs comes from the big bang
nucleosynthesis (BBN). We improve previous constraints on the PBH fraction and
find that it cannot exceed $10^{-3}$. Furthermore, this maximum fraction
decreases as the mass increases and reaches $10^{-9}$ for $M_{\rm PBH}\sim
5\times10^8 {\rm g}$, which is the largest mass allowed by the BBN constraint
on the reheating temperature. Considering that PBH may cluster above a given
clustering scale, we also derive a lower bound on the scale of clustering.
Interestingly, the GW spectrum for $M_{\rm PBH}\sim 10^4 -10^8 {\rm g}$ enters
the observational window of LIGO and DECIGO and could be tested in the future.
Although we focus on the PBH dominated early universe in this paper, our
methodology is applicable to any model with early isocurvature perturbation.
