  The chemical enrichments detected in CEMP-$s$ stars are believed to be the
consequence of a mass-transfer episode from a now extinct AGB primary star.
This hypothesis is borne out by the fact that most CEMP-$s$ stars exhibit RV
variations suggesting that they are in binary systems. We use the period
distribution of an observed sample of CEMP-$s$ stars to investigate the
constraints it imposes on our models of binary evolution and on the properties
of the metal-poor binary population in the Galactic halo. We simulate binary
populations using different assumptions about the initial period distribution
and the mass-transfer process, and we compare the predicted period
distributions of our synthetic CEMP-$s$ stars with the observed one. With a set
of default assumptions often made in binary population-synthesis studies, the
percentage of observed CEMP-$s$ systems with periods shorter than 2000 days is
underestimated by almost a factor of 3, and by a factor of 2 between 3000 and
10000 days, while 40% of the simulated systems have periods longer than 10000
days, which is approximately the longest measured period among CEMP-$s$ stars.
To reconcile the results of the models with the orbital properties of observed
CEMP-$s$ stars, one or both of the following conditions are necessary: ($i$)
the specific angular momentum carried away by the material that escapes the
binary system is 2-5 times higher than currently predicted by analytical models
and hydrodynamical simulations of wind mass transfer, and ($ii$) the initial
period distribution of very metal-poor binary stars is significantly different
from that observed in the solar vicinity and weighted towards periods shorter
than about 10000 days. Our simulations show that some, perhaps all, of the
observed CEMP-$s$ stars with apparently constant RV could be undetected
binaries with periods longer than 10000 days, but (...)
