  What is the maximum possible strength of a first-order electroweak phase
transition and the resulting gravitational wave (GW) signal? While naively one
might expect that supercooling could increase the strength of the transition to
very high values, for strong supercooling the Universe is no longer
radiation-dominated and the vacuum energy of the unstable minimum of the
potential dominates the expansion, which can jeopardize the successful
completion of the phase transition. After providing a general treatment for the
nucleation, growth and percolation of broken phase bubbles during a first-order
phase transition that encompasses the case of significant supercooling, we
study the conditions for successful bubble percolation and completion of the
electroweak phase transition in theories beyond the Standard Model featuring
polynominal potentials. For such theories, these conditions set a lower bound
on the temperature of the transition. Since the plasma cannot be significantly
diluted, the resulting GW signal originates mostly from sound waves and
turbulence in the plasma, rather than bubble collisions. We find the peak
frequency of the GW signal from the phase transition to be generically $f
\gtrsim 10^{-4}$ Hz. We also study the condition for GW production by sound
waves to be long-lasting (GW source active for approximately a Hubble time),
showing it is generally not fulfilled in concrete scenarios. Because of this
the sound wave GW signal could be weakened, with turbulence setting in earlier,
resulting in a smaller overall GW signal as compared to current literature
predictions.
