The fraction of cluster galaxies that host luminous AGN is an important probe of AGN fueling processes, the cold ISM at the centers of galaxies, and how tightly black holes and galaxies co-evolve. We present a new measurement of the AGN fraction in a sample of 13 clusters of galaxies (M >= 10^{14} Msun) at 1<z<1.5 selected from the Spitzer/IRAC Shallow Cluster Survey, as well as the field fraction in the immediate vicinity of these clusters, and combine these data with measurements from the literature to quantify the relative evolution of cluster and field AGN from the present to z~3. We estimate that the cluster AGN fraction at 1<z<1.5 is f_A = 3.0^{+2.4}_{-1.4}% for AGN with a rest-frame, hard X-ray luminosity greater than L_{X,H} >= 10^{44} erg/s. This fraction is measured relative to all cluster galaxies more luminous than M*_{3.6}(z)+1, where M*_{3.6}(z) is the absolute magnitude of the break in the galaxy luminosity function at the cluster redshift in the IRAC 3.6um bandpass. The cluster AGN fraction is 30 times greater than the 3sigma upper limit on the value for AGN of similar luminosity at z~0.25, as well as more than an order of magnitude greater than the AGN fraction at z~0.75. AGN with L_{X,H} >= 10^{43} erg/s exhibit similarly pronounced evolution with redshift. In contrast with the local universe, where the luminous AGN fraction is higher in the field than in clusters, the X-ray and MIR-selected AGN fractions in the field and clusters are consistent at 1<z<1.5. This is evidence that the cluster AGN population has evolved more rapidly than the field population from z~1.5 to the present. This environment-dependent AGN evolution mimics the more rapid evolution of star-forming galaxies in clusters relative to the field.
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