  The low frequency part of the gravitational wave spectrum generated by local
physics, such as a phase transition or parametric resonance, is largely fixed
by 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 a
physical understanding, such as the suppressed production of gravitational
waves due to the excitation of an over-damped harmonic oscillator and their
enhancement due to being frozen out while outside the horizon. 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 (such as the gravity waves
themselves) 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 sizable 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 are 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.
