  In this paper, we analyze the conditions for convergence toward General
Relativity of scalar-tensor gravity theories defined by an arbitrary coupling
function $\alpha$ (in the Einstein frame). We show that, in general, the
evolution of the scalar field $(\phi)$ is governed by two opposite mechanisms:
an attraction mechanism which tends to drive scalar-tensor models toward
Einstein's theory, and a repulsion mechanism which has the contrary effect. The
attraction mechanism dominates the recent epochs of the universe evolution if,
and only if, the scalar field and its derivative satisfy certain boundary
conditions. Since these conditions for convergence toward general relativity
depend on the particular scalar-tensor theory used to describe the universe
evolution, the nucleosynthesis bounds on the present value of the coupling
function, $\alpha_0$, strongly differ from some theories to others. For
example, in theories defined by $\alpha \propto \mid\phi\mid$ analytical
estimates lead to very stringent nucleosynthesis bounds on $\alpha_0$
($\lesssim 10^{-19}$). By contrast, in scalar-tensor theories defined by
$\alpha \propto \phi$ much larger limits on $\alpha_0$ ($\lesssim 10^{-7}$) are
found.
