Ultra-light primordial black holes, with masses m ^PBH < 10⁹g, evaporate before big-bang nucleosynthesis and can therefore not be directly constrained. They can however be so abundant that they dominate the universe content for a transient period (before reheating the universe via Hawking evaporation). If this happens, they support large cosmological fluctuations at small scales, which in turn induce the production of gravitational waves through second-order effects. Contrary to the primordial black holes, those gravitational waves survive after evaporation, and can therefore be used to constrain such scenarios. In this work, we show that for induced gravitational waves not to lead to a backreaction problem, the relative abundance of black holes at formation, denoted Ω_PBH,f, should be such that Ω_PBH,f < 10^-4(m ^PBH/10⁹g)^-1/4. In particular, scenarios where primordial black holes dominate right upon their formation time are all excluded (given that m ^PBH > 10 g for inflation to proceed at ρ^1/4 < 10¹⁶ GeV). This sets the first constraints on ultra-light primordial black holes.

超轻原始黑洞，质量m ^PBH < 10 ⁹ g，在大爆炸核合成之前蒸发，因此不能直接受到约束。然而，它们可能如此丰富，以至于它们在短暂的时期内主导着宇宙的内容（在通过霍金蒸发重新加热宇宙之前）。如果发生这种情况，它们会支持小尺度的大宇宙波动，进而通过二阶效应诱导引力波的产生。与原始黑洞相反，这些引力波在蒸发后仍然存在，因此可以用来限制这种情况。在这项工作中，我们表明，对于不导致反向反应问题的诱导引力波，形成时黑洞的相对丰度，表示为 Ω_PBH,f应满足 Ω _PBH,f < 10 ^-4 ( m ^PBH /10 ⁹ g) ^-1/4。特别是，原始黑洞在其形成时间占主导地位的情况都被排除在外（假设m ^PBH > 10 g 膨胀以 ρ ^1/4 < 10 ¹⁶ GeV 进行）。这为超轻原始黑洞设定了第一个限制条件。