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As potential candidates of dark matter, primordial black holes (PBHs) arewithin the core scopes of various astronomical observations. In light of theexplosive development of gravitational wave (GW) and radio astronomy, wethoroughly analyze a stochastic background of cosmological GWs, induced by overlarge primordial density perturbations, with several spikes that was inspiredby the sound speed resonance effect and can predict a particular pattern on themass spectrum of PBHs. With a specific mechanicsm for PBHs formation, we forthe first time perform the study of such induced GWs that originate from boththe inflationary era and the radiation-dominated phase. We report that, besidesthe traditional process of generating GWs during the radiation-dominated phase,the contribution of the induced GWs in the sub-Hubble regime during inflationcan become significant at critical frequency band because of a narrow resonanceeffect. All contributions sum together to yield a specific profile of theenergy spectrum of GWs that can be of observable interest in forthcomingastronomical experiments. Our study shed light on the possible joint probe ofPBHs via various observational windows of multi-messenger astronomy, includingthe search for electromagnetic effects with astronomical telescopes and thestochastic background of relic GWs with GW instruments.

When Primordial Black Holes from Sound Speed Resonance Meet a Stochastic Background of Gravitational Waves