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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.
Why this matters: it opens a route by which PBHs that were thought to be excluded by evaporation constraints could nonetheless account for all dark matter. If clusterogenesis happened, the ET could detect the resulting stochastic gravitational‑wave background and provide an indirect hint about dark matter’s origin. The paper gives concrete estimates. For example, to have PBHs make up all dark matter around 10^(−19) solar masses, the Gaussian threshold parameter they use corresponds roughly to νcg ≃ 9. The authors also give an estimate for PBH number density, nPBH ≃ 30 fPBH (MPBH/M⊙)^(−1) kpc^(−3), which helps quantify how clustered the seeds would be.
Important caveats and uncertainties are stressed. The scenario requires PBHs to be formed much more clustered than the simple Poisson expectation. It also depends on details of the primordial perturbation statistics. The paper treats non‑Gaussian models by mapping them to a Gaussian proxy and often uses approximations valid for rare peaks (large ν). The overall picture depends on the shape and breadth of the perturbation spectrum, on the collapse efficiency parameter (γ), and on whether cluster collapse indeed proceeds as assumed. Detection with ET would be indirect: seeing a flat SIGW background would be suggestive, not definitive proof, and the paper’s results rely on theoretical assumptions that need further study and numerical tests.
In short, the authors present a plausible mechanism that could rescue very light PBHs as dark matter by having them form in dense clusters that later collapse. That mechanism has an observable consequence—a flat stochastic gravitational‑wave background in the ET band—so future observations could help test this idea. The paper outlines the theoretical steps and points to the model ingredients that are most uncertain and in need of follow up.