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The current LIGO-Virgo observing run has been pushing the sensitivity limitto touch the stochastic gravitational-wave backgrounds (SGWBs). However, nosignificant detection has been reported to date for any single dominated sourceof SGWBs with a single broken-power-law (BPL) spectrum. Nevertheless, it couldequally well escape from existing Bayesian searches from, for example, twocomparable dominated sources with two separate BPL spectra (double-peak case)or a single source with its power-law behavior in the spectrum broken twice(doubly broken case). In this paper, we put constraints on these two cases butspecifically for the model with cosmological first-order phase transitions fromAdvanced LIGO-Virgo's first three observing runs. We found strong negativeevidence for the double-peak case and hence place 95\% C.L. upper limits$\Omega_\mathrm{BPL,1}<5.8\times10^{-8}$ and$\Omega_\mathrm{BPL,2}<4.4\times10^{-8}$ on the two BPL spectra amplitudes withrespect to the unresolved compact binary coalescence (CBC) amplitude$\Omega_\mathrm{CBC}<5.6\times10^{-9}$. We further found weak negative evidencefor the doubly broken case and hence place 95\% C.L. upper limit$\Omega_\mathrm{DB}<1.2\times10^{-7}$ on the overall amplitude of the doublybroken spectrum with respect to $\Omega_\mathrm{CBC}<6.0\times10^{-9}$. Inparticular, the results from the double-peak case have marginally ruled out thestrong super-cooling first-order phase transitions at LIGO-Virgo band.

Searching for double-peak and doubly broken gravitational-wave spectra from Advanced LIGO-Virgo's first three observing runs