  A first order phase transition at the electroweak scale would lead to the
production of gravitational waves that may be observable at upcoming
space-based gravitational wave (GW) detectors such as LISA (Laser
Interferometer Space Antenna). As the Standard Model has no phase transition,
LISA can be used to search for new physics by searching for a stochastic
gravitational wave background. In this work we investigate LISA's sensitivity
to the thermodynamic parameters encoded in the stochastic background produced
by a phase transition, using the sound shell model to characterise the
gravitational wave power spectrum, and the Fisher matrix to estimate
uncertainties. We explore a parameter space with transition strengths $\alpha <
0.5$ and phase boundary speeds $0.4 < v_\text{w} < 0.9$, for transitions
nucleating at $T_{\text{N}} = 100$ GeV, with mean bubble spacings $0.1$ and
$0.01$ of the Hubble length, and sound speed $c/\sqrt{3}$. We show that the
power spectrum in the sound shell model can be well approximated by a
four-parameter double broken power law, and find that the peak power and
frequency can be measured to approximately 10% accuracy for signal-to-noise
ratios (SNRs) above 20. Determinations of the underlying thermodynamic
parameters are complicated by degeneracies, but in all cases the phase boundary
speed will be the best constrained parameter. Turning to the principal
components of the Fisher matrix, a signal-to-noise ratio above 20 produces a
relative uncertainty less than 3% in the two highest-order principal
components, indicating good prospects for combinations of parameters. The
highest-order principal component is dominated by the wall speed. These
estimates of parameter sensitivity provide a preliminary accuracy target for
theoretical calculations of thermodynamic parameters.
