Star-to-star dispersion of r-process elements has been observed in a significant number of old, metal-poor globular clusters. We investigate early-time neutron-star mergers as the mechanism for this enrichment. Through both numerical modeling and analytical arguments, we show that neutron-star mergers cannot be induced through dynamical interactions early in the history of the cluster, even when the most liberal assumptions about neutron-star segregation are assumed. Therefore, if neutron-star mergers are the primary mechanism for r-process dispersion in globular clusters, they likely result from the evolution of isolated, primordial binaries in the clusters. Through population modeling, we find that moderate fractions of GCs with enrichment are only possible when a significant number of double neutron-star progenitors proceed through Case BB mass transfer --- under various assumptions for the initial properties of globular clusters, a neutron-star merger with the potential for enrichment will occur in ~15-60% (~30-90%) of globular clusters if this mass transfer proceeds stably (unstably). The strong anti-correlation between the pre-supernova orbital separation and post-supernova systemic velocity due to mass loss in the supernova leads to efficient ejection of most enrichment candidates from their host clusters. Thus, most enrichment events occur shortly after the double neutron stars are born. This requires star-forming gas that can absorb the r-process ejecta to be present in the globular cluster 30-50 Myr after the initial burst of star formation. If scenarios for redistributing gas in globular clusters cannot act on these timescales, the number of neutron-star merger enrichment candidates drops severely, and it is likely that another mechanism, such as r-process enrichment from collapsars, is at play.
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