In this paper we analyze the spectrum of the primordial gravitational waves (GWs) predicted in the Standard Model*Axion*Seesaw*Higgs portal inflation (SMASH) model, which was proposed as a minimal extension of the Standard Model that addresses five fundamental problems of particle physics and cosmology (inflation, baryon asymmetry, neutrino masses, strong CP problem, and dark matter) in one stroke. The SMASH model has a unique prediction for the critical temperature of the second order Peccei-Quinn (PQ) phase transition $T_c sim 10^8,mathrm{GeV}$ up to the uncertainty in the calculation of the axion dark matter abundance, implying that there is a drastic change in the equation of state of the universe at that temperature. Such an event is imprinted on the spectrum of GWs originating from the primordial tensor fluctuations during inflation and entering the horizon at $T sim T_c$, which corresponds to $f sim 1,mathrm{Hz}$, pointing to a best frequency range covered by future space-borne GW interferometers. We give a precise estimation of the effective relativistic degrees of freedom across the PQ phase transition and use it to evaluate the spectrum of GWs observed today. It is shown that the future high sensitivity GW experiment -- ultimate DECIGO -- can probe the nontrivial feature resulting from the PQ phase transition in this model.
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