Production of gravitational waves in the early Universe (for example, in the electroweak phase transition) is favoured only if the Universe undergoes a first-order phase transition but not favoured if it suffers a crossover. Why is this so?
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First order phase transitions are characterized by discontinuous first derivatives of the free energy. Specifically, the derivative of Gibbs free energy with respect to pressure determines the density of the gas ($\Delta N/\Delta V$) and this may be discontinuous. The Early Universe was nearly but not quite homogeneous, and near the phase transition temperature the pressure and temperature fluctuations caused certain parts of the Universe to be in different regions of the phase diagram.
In particular, this means that near the temperature of a first-order phase transition, the Universe would become tessellated by discontinuous density jumps. The motion of these randomly places clumps of sharp overdensities due to fluctuations and adiabatic expansion would then be excellent sources of stochastic gravitational waves.
On the other hand, if the phase transition is not associated with such jumps, it may increase the density contrast near phase transition temperature, but this is not enough for very strong gravitational-wave generation. This is the case of crossovers (see both the answers below this question for an excellent discussion what a crossover even is) and generally any higher-order phase transitions.
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