Phase Transitions and Gravitational Waves
We present a Fisher matrix forecast for the spectral parameter reconstruction of a stochastic gravitational wave background generated by a first-order phase transition in the early universe. We use the LISA and DECIGO instruments for reference and model the source spectrum with a double-broken-power-law template, parameterized by the peak amplitude $\Omega_p$, peak frequency $f_p$, break ratio $r_b$, and intermediate slope $b$. For each detector, we construct a fixed fiducial benchmark with the peak placed in the corresponding sensitivity band, using $f_p=3.0\times10^{-3} \ {\rm Hz}$ for LISA and $f_p=1.0\ {\rm Hz}$ for DECIGO. The LISA baseline includes an unresolved galactic compact-binary foreground, while for DECIGO we compare foreground cases using a full compact binary, a projected post-subtraction compact-binary residual, and an instrument noise only foreground. We find that both detectors constrain the peak parameters $\ln\Omega_p$ and $\ln f_p$ more robustly than the detailed shape parameters $r_b$ and $b$. For the chosen DECIGO benchmark, the projected residual foreground produces negligible parameter degradation relative to the instrument noise only case, while the full foreground lowers the signal-to-noise ratio from $\rho=10$ to $\rho\simeq6.06$ and increases the marginalized uncertainty on $b$ by about $51\%$. These results emphasize that stochastic-background forecasts should distinguish detectability from spectral identifiability.