A Gravitational Wave Background from Primordial Black Hole Lattices in Matter Dominated Era

We use the wide-used \textsf{Einstein Toolkit} to solve the Einstein constraints and then simulate the expansion of primordial black hole lattices (PBHLs) with different value of $f_{\mathrm{PBH}}$ and $m_{\mathrm{PBH}}$. We find that $f_{\mathrm{PBH}}$ plays an important role during the evolution of PBHLs. Since the motion of primordial black holes (PBHs) caused by the expansion of PBHLs occurs at speeds close to that of light, we expect the emission of gravitational waves (GWs) during the expansion of PBHLs. We use both analytical estimates and numerical simulations to cross check the production of GWs in expanding PBHLs.