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As a first step towards understanding the angular momentum evolution historyof black holes in merging black-hole/neutron-star binaries, we performpopulation synthesis calculations to track the distribution of accretionhistories of compact objects in such binaries. We find that there are threedistinct processes which can possibly contribute to the black-hole spinmagnitude: a birth spin for the black hole, imparted at either (i) the collapseof a massive progenitor star to a black hole or (ii) the accretion-inducedcollapse of a neutron star to a black hole; and (iii) an accretion spin-up whenthe already formed black hole [via (i) or (ii)] goes through an accretionepisode (through an accretion disk or a common-envelope phase). Our resultsshow that, with regard to accretion-induced spinup in merging BH-NS binaries[method (iii) above], only  {\em accretion episodes associated with common-envelope phases andhypercritical accretion rates} occur in the formation history of merging blackhole/neutron star binaries. Lacking unambiguous experimental information aboutBH birth spins [i.e., regarding the results of processes (i) and (ii)], wechoose two fiducial values for the BH birth angular momentum parameter a=J/M^2,consistent with observations of (i) NS birth spins (a roughly 0) and (ii) X-raybinaries (a=0.5). Using these two fiducial values and a conservative upperbound on the specific angular momentum of accreted matter, we discuss theexpected range of black hole spins in the binaries of interest. We concludewith comments on the significance of these results for ground-basedgravitational-wave searches of inspiral signals from black hole binaries.Comment: Submitted to ApJ. (v1) Uses emulateapj.cls. 5 figures. (v2):  corrected reference list and uses smaller figures (v3): Includes changes in  response to referee comments, including new discussion of XRBs. Figures  merged, so only 3 figures (v4) Minor typo correction, plus updated abstract  posted onlin

Bounds on Expected Black Hole Spins in Inspiraling Binaries