High-Frequency Solutions to the Constraint Equations

We construct high-frequency initial data for the Einstein vacuum equations in dimension 3+1 by solving constraint on $\mathbb{R}^3$. Our family of solutions $(\bar{g}_\lambda,K_\lambda)_{\lambda\in(0,1]}$ is defined through a expansion similar to geometric optics approach and close particular sense null dust. In order solve equations, we use their conformal formulation main challenge our proof adapt this method context. The application article companion \cite{Touati2022a} where gravitational waves generalised wave gauge.