Gravitational-wave (GW) events and galaxies both trace the cosmic matter distribution, but the mergers of astrophysical black holes and primordial black holes (PBHs) are expected to populate different environments and therefore to cluster with different biases. The GW clustering bias is thus a statistical observable that can separate the two populations. We assess how well this can be done by cross-correlating the photometric galaxy survey of the Chinese Space-station Survey Telescope (CSST) with mock GW catalogs from two future detector networks: the third-generation ET2CE network (the Einstein Telescope and two Cosmic Explorer detectors) and the multiband BDET2CE network, which adds the space-based baseline Decihertz Interferometer Gravitational-Wave Observatory. We find that CSST combined with 10 years of ET2CE observations can reveal a PBH contribution once its fraction in the local total merger rate exceeds about 40%, while the much sharper sky localization of BDET2CE lowers this threshold to about 20%. The improvement comes from recovering the small-scale clustering information that localization errors would otherwise erase. These results show that combining future GW detector networks with CSST galaxy clustering offers a promising and largely independent route to identifying PBHs statistically.

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• Gravitational waves
• Large scale structure of the Universe
• Primordial black holes