The gravitational wave (GW) background produced at the cosmological chiral phase transition in a conformal extension of the standard model is studied. To obtain the bounce solution of coupled field equations we implement an iterative method. We find that the corresponding $O(3)$ symmetric Euclidean action $S_3$ divided by the temperature $T$ has a simple behavior near the critical temperature $T_C$: $S_3/T propto (1-T/T_C)^{-gamma}$, which is subsequently used to determine the transitions inverse duration $beta$ normalized to the Hubble parameter $H$. It turns out that $beta/H gtrsim 10^3$, implying that the sound wave period $tau_text{sw}$ as an active GW source, too, can be much shorter than the Hubble time. We therefore compute $tau_text{sw} H$ and use it as the reduction factor for the sound wave contribution. The signal-to-noise ratio (SNR) for Deci-Hertz Interferometer Gravitational Wave Observatory (DECIGO) and Big Bang Observer (BBO) is evaluated, with the result: SNR$^text{DECIGO} lesssim 1.2$ and SNR$^text{BBO} lesssim 12.0$ for five years observation, from which we conclude that the GW signal predicted by the model in the optimistic case could be detected at BBO.
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