  This is the first of a sequence of four papers \cite{param1}, \cite{param2},
\cite{param3}, \cite{param4} dedicated to the construction and the control of a
parametrix to the homogeneous wave equation $\square_{\bf g} \phi=0$, where
${\bf g}$ is a rough metric satisfying the Einstein vacuum equations.
Controlling such a parametrix as well as its error term when one only assumes
$L^2$ bounds on the curvature tensor ${\bf R}$ of ${\bf g}$ is a major step of
the proof of the bounded $L^2$ curvature conjecture proposed in \cite{Kl:2000},
and solved jointly with S. Klainerman and I. Rodnianski in \cite{boundedl2}. On
a more general level, this sequence of papers deals with the control of the
eikonal equation on a rough background, and with the derivation of $L^2$ bounds
for Fourier integral operators on manifolds with rough phases and symbols, and
as such is also of independent interest.
