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We explore the confinement-deconfinement phase transition (PT) of the firstorder (FO) arising in $SU(N)$ pure Yang-Mills theory, based on Polyakov loopmodels (PLMs), in light of the induced gravitational wave (GW) spectra. Wedemonstrate that the PLMs with the Haar measure term, involving modelssuccessful in QCD with $N=3$, are potentially incompatible with the large $N$scaling for the thermodynamical quantities and the latent heat at around thecriticality of the FOPT reported from the lattice simulations. We then proposea couple of models of polynomial form, which we call the 4-6 PLM (with four-and six-point interactions among the basic PL fields which have center charge1) and 4-8 PLM (with four- and eight-point interactions), and discuss how suchmodels can naturally arise in the presence of a heavy PL with charge 2. We showthat those models give the consistent thermodynamical and large $N$ propertiesat around the criticality. The predicted GW spectra are shown to have highenough sensitivity to be probed in the future prospected interferometers suchas LISA, BBO, DICIGO, and TianQin.

Dark Confinement-Deconfinement Phase Transition: A Roadmap from Polyakov Loop Models to Gravitational Waves