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