Phase Transitions and Gravitational Waves

We present a Fisher-matrix forecast for the detectability of a stochastic gravitational wave background generated by a first-order phase transition in the early universe. We use the DECIGO and LISA missions as reference cases. The source gravitational wave spectrum ${\mathrm{\Omega }}_{\mathrm{G}\mathrm{W}}(f)$ is modeled as the sum of sound wave and turbulence contributions and is parameterized by the transition strength $\alpha$, its inverse duration $\beta \mathrm{/}{H}_{\ast }$, its transition temperature $T_{\ast }$, and the bubble wall velocity $v_w$. For each detector, we construct fiducial models with signal peaking in the sensitivity band of the detector, fixing $T_{\ast }$ and $v_w$, and perform a Fisher analysis on the remaining parameters $\ln \alpha$ and $\ln (\beta \mathrm{/}{H}_{\ast })$. A two-parameter Fisher analysis in $\{\ln \alpha ,\ln (\beta \mathrm{/}{H}_{\ast })\}$, with fixed values of $T_{\ast }$ and $v_w$, yields marginalized $1\sigma$ uncertainties $\sigma (\ln \alpha )\simeq 0.12$ and $\sigma [\ln (\beta \mathrm{/}{H}_{\ast })]\simeq 0.145$. The parameters are strongly correlated, with correlation coefficient $\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{r}\simeq 0.98$. We perform a corresponding analysis for LISA and report marginalized $1\sigma$ uncertainties $\mathrm{\Delta }\alpha \mathrm{/}\alpha \simeq {}_{-0.042}^{+0.044}$ and $\mathrm{\Delta }(\beta \mathrm{/}{H}_{\ast })\mathrm{/}(\beta \mathrm{/}{H}_{\ast })\simeq {}_{-0.107}^{+0.119}$, with correlation coefficient $\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{r}\simeq 0.78$.