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 $\Omega_{\rm GW}(f)$ is modeled as the sum of sound wave and turbulence contributions and is parameterized by the transition strength $\alpha$, its inverse duration $\beta/H_*$, its transition temperature $T_{*}$, 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 $T_{*}$ and $v_{w}$, and perform a Fisher analysis on the remaining parameters $\ln\alpha$ and $\ln(\beta/H_{*})$. A two-parameter Fisher analysis in $\{\ln\alpha,\ln(\beta/H_{*})\}$, with fixed values of $T_{*}$ and $v_{w}$, yields marginalized $1\sigma$ uncertainties $\sigma(\ln\alpha)\simeq 0.12$ and $\sigma[\ln(\beta/H_{*})]\simeq 0.145$. The parameters are strongly correlated, with correlation coefficient $\mathrm{corr}\simeq 0.98$. We perform a corresponding analysis for LISA and report marginalized $1\sigma$ uncertainties $\Delta\alpha/\alpha \simeq {}^{+0.044}_{-0.042}$ and $\Delta(\beta/H_{*})/(\beta/H_{*}) \simeq {}^{+0.119}_{-0.107}$, with correlation coefficient $\mathrm{corr}\simeq 0.78$.