We calculate the gravitational waves (GWs) induced by the density fluctuations due to the 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^-4. Furthermore, this maximum fraction decreases as the mass increases and reaches 10^-12 for M_PBH∼ 510⁸ 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_PBH∼ 10⁴ -10⁸ 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.

我们计算了由于原始黑洞 (PBH) 的不均匀分布导致的密度波动引起的引力波 (GW)，因为原始黑洞 (PBH) 最终主导并通过霍金蒸发重新加热了宇宙。初始 PBH 密度波动本质上是等曲率的。我们发现大部分诱导引力波是在蒸发之后产生的，当时宇宙从 PBH 主导时代过渡到辐射主导时代，曲率扰动开始剧烈振荡。对产生的 GW 数量的最大限制来自大爆炸核合成 (BBN)。我们改进了之前对 PBH 分数的限制，发现它不能超过 10 ^-4。此外，这个最大分数随着质量的增加而减少并达到 10 ^-12对于 M _PBH ∼ 510 ⁸ g，这是 BBN 对再加热温度的约束所允许的最大质量。考虑到 PBH 可能会在给定的聚类规模以上聚类，我们还推导出了聚类规模的下限。有趣的是，M _PBH ∼ 10 ⁴ -10 ⁸ g的 GW 光谱进入了 LIGO 和 DECIGO 的观测窗口，可以在未来进行测试。尽管在本文中我们专注于 PBH 主导的早期宇宙，但我们的方法适用于任何具有早期等曲率扰动的模型。