Secondary Gravitational Wave Signatures from 5D Rotating Primordial Black Holes in the Dark Dimension

We investigate five-dimensional rotating primordial black holes (PBHs) as dark matter candidates within the Dark Dimension (DD) scenario motivated by the Swampland Program. In this framework, a micron-scale extra dimension suppresses Hawking evaporation, allowing PBHs with initial masses $M \gtrsim 10^{10}\,\mathrm{g}$ to survive to the present epoch. Moreover, the memory burden effect, a quantum-gravitational suppression of the evaporation rate by $S^{-p}$, significantly prolongs PBH lifetimes and enlarges the allowed parameter space. We compute the evaporation dynamics for rotating 5D PBHs, derive the enhanced lifetime for $p=2$, and establish the dark matter window $10^{10}\,\mathrm{g} \lesssim M \lesssim 10^{21}\,\mathrm{g}$. The curvature perturbations responsible for PBH formation also generate a stochastic gravitational wave background through second-order scalar-induced effects. Assuming a log-normal primordial power spectrum with $\sigma=1$ and $f_{\mathrm{PBH}}=1$, we calculate the present-day energy density $\Omega_{\mathrm{GW}}h^2$ across the Dark Dimension window. The predicted signals peak at frequencies from nHz to Hz, within the sensitivity ranges of LISA and DECIGO/BBO, while remaining consistent with current CMB spectral distortion bounds. Fisher forecasts show that future observatories can constrain the PBH mass, dark matter fraction, spectral width, and memory burden exponent with percent-level precision. A detection of the predicted gravitational wave background would provide simultaneous evidence for a micron-sized extra dimension, PBH dark matter, and the memory burden effect, offering a decisive test of quantum gravity and extra-dimensional physics.