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The low frequency part of the gravitational wave spectrum generated by localphysics, such as a phase transition or parametric resonance, is largely fixedby causality, offering a clean window into the early Universe. In this work,this low frequency end of the spectrum is analyzed with an emphasis on aphysical understanding, such as the suppressed production of gravitationalwaves due to the excitation of an over-damped harmonic oscillator and theirenhancement due to being frozen out while outside the horizon. Due to thedifference between sub-horizon and super-horizon physics, it is inevitable thatthere will be a distinct spectral feature that could allow for the directmeasurement of the conformal Hubble rate at which the phase transitionoccurred. As an example, free-streaming particles (such as the gravity wavesthemselves) present during the phase transition affect the production ofsuper-horizon modes. This leads to a steeper decrease in the spectrum at lowfrequencies as compared to the well-known causal $k^3$ super-horizon scaling ofstochastic gravity waves. If a sizable fraction of the energy density is infree-streaming particles, they even lead to the appearance of oscillatoryfeatures in the spectrum. If the universe was not radiation dominated when thewaves were generated, a similar feature also occurs at the transition betweensub-horizon to super-horizon causality. These features are used to showsurprising consequences, such as the fact that a period of matter dominationfollowing the production of gravity waves actually increases their powerspectrum at low frequencies.

Causal gravitational waves as a probe of free streaming particles and the expansion of the Universe