We study the dependence of the observable stochastic gravitational wave background induced by a first-order phase transition on the global properties of the scalar effective potential in particle physics. The scalar potential can be that of the Standard Model Higgs field, or more generally of any scalar field responsible for a spontaneous symmetry breaking in beyond-the-Standard-Model settings thatprovide for a first-order phase transition in the early universe.Characteristics of the effective potential include the relative depth of the true minimum ($E_alpha^4$), the height of the barrier that separates it from the false one ($E_m^4$) and the separation between the two minima in field space ($v$), all at the bubble nucleation temperature. We focus on a simple yet quite general class of single-field polynomial potentials, with parameters being varied over several orders of magnitude. It is then shown that gravitational wave observatories such as aLIGO O5, BBO, DECIGO and LISA are mostly sensitive to values of these parameters in the region $E_alpha sim (0.1-10) times E_m$. Finally, relying on well-defined models and using our framework, we demonstrate how to obtain the gravitational wave spectra for potentials of various shapes without necessarily relying on dedicated software packages.
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