We report a sensitive X-ray search for the proposed intermediate mass black hole (IMBH) in the massive Galactic cluster, Omega Centauri (NGC 5139). Combining Chandra X-ray Observatory data from Cycles 1 and 13, we obtain a deep (~291 ks) exposure of the central regions of the cluster. We find no evidence for an X-ray point source near any of the clusters proposed dynamical centers, and place an upper limit on the X-ray flux from a central source of f_X(0.5-7.0 keV) <= 5.0x10^-16 erg cm^-2 s^-1, after correcting for absorption. This corresponds to an unabsorbed X-ray luminosity of L_X(0.5-7.0 keV) <= 1.6x10^30 erg s^-1, for a cluster distance of 5.2 kpc, Galactic column density N_H = 1.2x10^21 cm^-2, and powerlaw spectrum with Gamma = 2.3. If a ~10^4 M_sun IMBH resides in the clusters core, as suggested by some stellar dynamical studies, its Eddington luminosity would be L_Edd ~10^42 erg s^-1. The new X-ray limit would then establish an Eddington ratio of L_X/L_Edd <~ 10^-12, a factor of ~10 lower than even the quiescent state of our Galaxys notoriously inefficient supermassive black hole Sgr A*, and imply accretion efficiencies as low as eta <~ 10^-6 - 10^-8. This study leaves open three possibilities: either Omega Cen does not harbor an IMBH or, if an IMBH does exist, it must experience very little or very inefficient accretion.
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