This paper studies a way very light primordial black holes (PBHs) could still make up all of dark matter. PBHs much lighter than an asteroid would normally evaporate and are tightly constrained by observations such as distortions of the cosmic microwave background. The authors point out that if those tiny PBHs are born strongly clustered, the clusters can collapse into heavier black holes early on. That process would avoid the evaporation bounds and leave a flat background of gravitational waves that the planned Einstein Telescope (ET) might detect.

The researchers analyze how clustering arises from large peaks in the early Universe’s curvature perturbations. They work with the compaction function, a measure of mass excess inside a region, because it controls whether a region collapses to a PBH. They show that peaks of the curvature perturbation map to peaks of the compaction function. They then compute two‑point correlation functions (which quantify clustering) for fully non‑Gaussian curvature perturbations by relating them to a Gaussian counterpart through probability conservation. For broad perturbation spectra, they find the PBH clustering can be much larger than the basic Poisson expectation.

At a high level, the proposed chain is this: inflationary fluctuations create rare, high peaks that collapse to tiny PBHs. If those peaks are correlated, many PBHs form close together. Their mutual gravity can make the whole cluster collapse into a heavier black hole even during the radiation era. This “clusterogenesis” can move mass from dangerous, evaporating seeds into safer, longer‑lived black holes. The same initial fluctuations also produce a scalar‑induced gravitational‑wave (SIGW) background. The paper notes a useful relation between PBH mass and the peak frequency of those induced waves: f ≃ 2.7 nHz × (MPBH/M⊙)^(−1/2). The ET is sensitive near O(10) Hz, which corresponds to seed PBHs of order 10^(−19) solar masses—masses that would otherwise have already evaporated.