This is a very interesting and well-written paper based on an impressive dataset. Indeed, I have very few suggestions for improvement.
1) In the Introduction, the authors tend to downplay the extent of paternal care in baboons. However, baboons in the very same population have been shown to engage in true paternal care (Buchan et al 2003).
2) L. 71: This sentence seems to miss a "who".
3) Figs 2B and 2C: There is a mismatch between the legend and the curves.
4) Statistical analyses: It is unclear why the authors adopted a standard null hypothesis testing approach in most analyses (i.e. evaluating the role of independent variables on the basis of p values), but used an information-theoretic approach for the survival analyses (i.e., evaluating the role of independent variables on the basis of AIC). Generally, a consistent approach is recommended (otherwise, one cannot help but feel that maybe the approach with the "best" results was chosen for each analysis).
5) L. 465-467: Among the possible explanations for the observed results, the authors hypothesize that father-infant association and grooming may be correlated with other male care behaviors, which in turn, benefit the infant. It seems data to test this hypothesis should be already available (Buchan et al 2003).
6) L. 476-478: Given the amount of physiological and genetic data available on this population, I also wonder whether it would be possible to test the hypothesis that the strength of father-offspring relationships reflects the infants' phenotypic quality.