  What is the influence of cosmology (the expansion law and its acceleration,
the cosmological constant...) on the dynamics and optics of a local system like
the solar system, a galaxy, a cluster, a supercluster...? The answer requires
the solution of Einstein equation with the local source, which tends towards
the cosmological model at large distance. There is, in general, no analytic
expression for the corresponding metric, but we calculate here an expansion in
a small parameter, which allows to answer the question. First, we derive a
static expression for the pure cosmological (Friedmann-Lema\^itre) metric,
whose validity, although local, extends in a very large neighborhood of the
observer. This expression appears as the metric of an osculating de Sitter
model. Then we propose an expansion of the cosmological metric with a local
source, which is valid in a very large neighborhood of the local system. This
allows to calculate exactly the (tiny) influence of cosmology on the dynamics
of the solar system: it results that, contrary to some claims, cosmological
effects fail to account for the unexplained acceleration of the Pioneer probe
by several order of magnitudes. Our expression provide estimations of the
cosmological influence in the calculations of rotation or dispersion velocity
curves in galaxies, clusters, and any type of cosmic structure, necessary for
precise evaluations of dark matter and/or cosmic flows. The same metric can
also be used to estimate the influence of cosmology on gravitational optics in
the vicinity of such systems.
