We report on the implications of the peak in the cosmic star-formation rate (SFR) at redshift z ≈ 1.5 for the resulting population of low-mass X-ray binaries(LMXB) and for that of their descendants, the millisecond radio pulsars (MRP). Since the evolutionary timescales of LMXBs, their progenitors, and their descendants are thought be significant fractions of the time-interval between the SFR peak and the present epoch, there is a lag in the turn-on of the LMXB population, with the peak activity occurring at z ∼ 0.5 − 1. The peak in the MRP population is delayed further, occurring at z ∼ < 0.5. We show that the discrepancy between the birthrate of LMXBs and MRPs, found under the assumption of a stead-state SFR, can be resolved for the population as a whole when the effects of a time-variable SFR are included. A discrepancy may persist for LMXBs with short orbital periods, although a detailed population synthesis will be required to confirm this. Further, since the integrated X-ray luminosity distribution of normal galaxies is dominated by X-ray binaries, it should show strong luminosity evolution with redshift. In addition to an enhancement near the peak (z ≈ 1.5) of the SFR due to the prompt turn-on of the relatively short-lived massive X-ray binaries and young supernova remnants, we predict a second enhancement by a factor ∼ 10 at a redshift between ∼ 0.5 and ∼ 1 due to the delayed turn-on of the LMXB population. Deep X-ray observations of galaxies out to z ≈ 1 by AXAF will be able to observe this enhancement, and, by determining its shape as a function of redshift, will provide an important new method for constraining evolutionary models of X-ray binaries
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