  This article is the first of two in which we develop a geometric framework
for analysing silent and anisotropic big bang singularities. The results of the
present article concern the asymptotic behaviour of solutions to linear systems
of wave equations on the corresponding backgrounds. The main features are the
following: The assumptions do not involve any symmetry requirements and are
weak enough to be consistent with most big bang singularities for which the
asymptotic geometry is understood. The asymptotic rate of growth/decay of
solutions to linear systems of wave equations along causal curves going into
the singularity is determined by model systems of ODE's (depending on the
causal curve). Moreover, the model systems are essentially obtained by dropping
spatial derivatives and localising the coefficients along the causal curve.
This is in accordance with the BKL proposal. Note, however, that we here prove
this statement, we do not assume it. If the coefficients of the unknown and its
expansion normalised normal derivatives converge sufficiently quickly along a
causal curve, we obtain leading order asymptotics (along the causal curve) and
prove that the localised energy estimate (along the causal curve) is optimal.
In this setting, it is also possible to specify the leading order asymptotics
of solutions along the causal curve. On the other hand, the localised energy
estimate typically entails a substantial loss of derivatives.
  In the companion article, we deduce geometric conclusions by combining the
framework with Einstein's equations. In particular, the combination reproduces
the Kasner map and yields partial bootstrap arguments.
