  Different computational techniques for cosmological phase transition
parameters can impact the Gravitational Wave (GW) spectra predicted in a given
particle physics model. To scrutinize the importance of this effect, we perform
large-scale parameter scans of the dynamical real-singlet extended Standard
Model using three perturbative approximations for the effective potential: the
$\overline{\rm MS}$ and on-shell schemes at leading order, and
three-dimensional thermal effective theory (3D EFT) at next-to-leading order.
While predictions of GW amplitudes are typically unreliable in the absence of
higher-order corrections, we show that the reconstructed model parameter spaces
are robust up to a few percent in uncertainty. While 3D EFT is accurate from
one loop order, theoretical uncertainties of reconstructed model parameters,
using four-dimensional standard techniques, remain dominant over the
experimental ones even for signals merely strong enough to claim a detection by
LISA.
