We compare the star formation (SF) activity in cluster galaxies to the field from z=0.3-1.5 using $Herschel$ SPIRE 250$mu$m imaging. We utilize 274 clusters from the IRAC Shallow Cluster Survey (ISCS) selected as rest-frame near-infrared overdensities over the 9 square degree Bootes field . This analysis allows us to quantify the evolution of SF in clusters over a long redshift baseline without bias against active cluster systems. Using a stacking analysis, we determine the average star formation rates (SFRs) and specific-SFRs (SSFR=SFR/M$_{star}$) of stellar mass-limited (M>1.3x10$^{10}$ M$_{odot}$), statistical samples of cluster and field galaxies, probing both the star forming and quiescent populations. We find a clear indication that the average SF in cluster galaxies is evolving more rapidly than in the field, with field SF levels at z>1.2 in the cluster cores (r<0.5 Mpc), in good agreement with previous ISCS studies. By quantifying the SF in cluster and field galaxies as an exponential function of cosmic time, we determine that cluster galaxies are evolving ~2 times faster than the field. Additionally, we see enhanced SF above the field level at z~1.4 in the cluster outskirts (r>0.5 Mpc). These general trends in the cluster cores and outskirts are driven by the lower mass galaxies in our sample. Blue cluster galaxies have systematically lower SSFRs than blue field galaxies, but otherwise show no strong differential evolution with respect to the field over our redshift range. This suggests that the cluster environment is both suppressing the star formation in blue galaxies on long time-scales and rapidly transitioning some fraction of blue galaxies to the quiescent galaxy population on short time-scales. We argue that our results are consistent with both strangulation and ram pressure stripping acting in these clusters, with merger activity occurring in the cluster outskirts.
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