We present mid-infrared observations of AGN in the GOODS fields, performed with the Spitzer Space Telescope. These are the deepest infrared and X-ray fields to date and cover a total area of ~0.1 square degrees. AGN are selected on the basis of their hard (2-8 keV) X-ray emission. The median AGN infrared luminosity is at least 10 times larger than the median for normal galaxies with the same redshift distribution, suggesting that the infrared emission is dominated by the central nucleus. The X-ray to infrared luminosity ratios of GOODS AGN, most of which are at 0.5<z<1.5, are similar to the values obtained for AGN in the local Universe. The observed infrared flux distribution has an integral slope of ~1.5 and there are 1000 sources per square degree brighter than ~50 uJy at 3-6 microns. The counts approximately match the predictions of models based on AGN unification, in which the majority of AGN are obscured. This agreement confirms that the faintest X-ray sources, which are dominated by the host galaxy light in the optical, are obscured AGN. Using these Spitzer data, the AGN contribution to the extragalactic infrared background light is calculated by correlating the X-ray and infrared catalogues. This is likely to be a lower limit given that the most obscured AGN are missed in X-rays. We estimate the contribution of AGN missed in X-rays, using a population synthesis model, to be ~45% of the observed AGN contribution, making the AGN contribution to the infrared background at most ~2-10% in the 3-24 micron range, depending on wavelength, lower than most previous estimates. The AGN contribution to the infrared background remains roughly constant with source flux in the IRAC bands but decreases with decreasing flux in the MIPS 24 um band, where the galaxy population becomes more important.
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