Critical behavior and ultraviolet scaling of induced gravitational waves from an early matter-dominated era

Critical behavior and ultraviolet scaling of induced gravitational waves (GWs) from an early matter-dominated (eMD) era are studied in the context of primordial black hole evaporation. The depth of the eMD is characterized by the minimum parameter of the equation of state $\omega_{\min}$ that the Universe can attain during this phase. We identify a critical value $\omega_{c}\sim 7.3\times10^{-3}$ that separates two regimes. For $\omega_{\min}<\omega_{c}$, the GW peak lies at the non-linear cut-off point and requires non-linear dynamics. For $\omega_{\min}>\omega_{c}$, the peak originates from modes that reenter near the matter-radiation equality, and the ultraviolet tail follows a distinct scaling $k^{-3/2}$. This critical behavior provides a clear definition of deep versus shallow eMD and a robust spectral signature for future GW observations.