The existence of primordial black holes (PBHs), which may form from the collapse of matter overdensities shortly after the Big Bang, is still under debate. Among the potential signatures of PBHs are gravitational waves (GWs) emitted from binary black hole (BBH) mergers at redshifts $zgtrsim 30$, where the formation of astrophysical black holes is unlikely. Future ground-based GW detectors, Cosmic Explorer and Einstein Telescope, will be able to observe equal-mass BBH mergers with total mass of $mathcal{O}(10-100)~M_{odot}$ at such distances. In this work, we investigate whether the redshift measurement of a single BBH source can be precise enough to establish its primordial origin. We simulate BBHs of different masses, mass ratios and orbital orientations. We show that for BBHs with total masses between $20~M_{odot}$ and $40~M_{odot}$ merging at $z geq 40$ one can infer $z>30$ at up to 97% credibility, with a network of one Einstein Telescope, one 40-km Cosmic Explorer in the US and one 20-km Cosmic Explorer in Australia. A smaller network made of one Einstein Telescope and one 40-km Cosmic Explorer in the US measures $z>30$ at larger than 90% credibility for roughly half of the sources than the larger network. We then assess the dependence of this result on the Bayesian redshift priors used for the analysis, specifically on the relative abundance of the BBH mergers originated from the first stars, and the primordial BBH mergers.
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