The low frequency part of the gravitational wave spectrum generated by local physics, such as a phase transition, is largely fixed by causality, offering a clean window into the early Universe. Due to the difference between sub-horizon and super-horizon physics, it is inevitable that there will be a distinct spectral feature that could allow for the direct measurement of the conformal Hubble rate at which the phase transition occurred.
As an example, free-streaming particles present during the phase transition affect the production of super-horizon modes. This leads to a steeper decrease in the spectrum at low frequencies as compared to the well-known causal \(k^3\) super-horizon scaling of stochastic gravity waves. If a sizeable fraction of the energy density is in free-streaming particles, they even lead to the appearance of oscillatory features in the spectrum.
If the universe was not radiation dominated when the waves were generated, a similar feature also occurs at the transition between sub-horizon to super-horizon causality. These features can be used to show surprising consequences, such as the fact that a period of matter domination following the production of gravity waves actually increases their power spectrum at low frequencies.