  We make use of Friedrich's construction of the cylinder at spatial infinity
to relate the logarithmic terms appearing in asymptotic expansions of
components of the Weyl tensor to the freely specifiable parts of time symmetric
initial data sets for the Einstein field equations. Our analysis is based on
the assumption that a particular type of formal expansions near the cylinder at
spatial infinity corresponds to the leading terms of actual solutions to the
Einstein field equations. In particular, we show that if the Bach tensor of the
initial conformal metric does not vanish at the point at infinity then the most
singular component of the Weyl tensor decays near null infinity as
$O(\tilde{r}^{-3}\ln \tilde{r})$ so that spacetime will not peel. We also
provide necessary conditions on the initial data which should lead to a peeling
spacetime. Finally, we show how to construct global spacetimes which are
candidates for non-peeling polyhomogeneous) asymptotics.
