  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.
