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 ${\ensuremath{\omega}}_{\mathrm{min}}$ that the Universe can attain during this phase. We identify a critical value ${\ensuremath{\omega}}_{c}\ensuremath{\sim}7.3\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}$ that separates two regimes. For ${\ensuremath{\omega}}_{\mathrm{min}}<{\ensuremath{\omega}}_{c}$, the GW peak lies at the nonlinear cutoff point and requires nonlinear dynamics. For ${\ensuremath{\omega}}_{\mathrm{min}}>{\ensuremath{\omega}}_{c}$, the peak originates from modes that reenter near the matter-radiation equality, and the ultraviolet tail follows a distinct scaling ${k}^{\ensuremath{-}3/2}$. This critical behavior provides a clear definition of deep versus shallow eMD and a robust spectral signature for future GW observations.