  We analyze the link between active galactic nuclei (AGN) and mid-infrared
flux using dust radiative transfer calculations of starbursts realized in
hydrodynamical simulations. Focusing on the effects of galaxy dust, we evaluate
diagnostics commonly used to disentangle AGN and star formation in
ultraluminous infrared galaxies (ULIRGs). We examine these quantities as a
function of time, viewing angle, dust model, AGN spectrum, and AGN strength in
merger simulations representing two possible extremes of the ULIRG population:
one is a typical gas-rich merger at z ~ 0, and the other is characteristic of
extremely obscured starbursts at z ~ 2 to 4. This highly obscured burst begins
star-formation-dominated with significant PAH emission, and ends with a ~10^9
yr period of red near-IR colors. At coalescence, when the AGN is most luminous,
dust obscures the near-infrared AGN signature, reduces the relative emission
from polycyclic aromatic hydrocarbons (PAHs), and enhances the 9.7 micron
absorption by silicate grains. Although generally consistent with previous
interpretations, our results imply none of these indicators can unambiguously
estimate the AGN luminosity fraction in all cases. Motivated by the
simulations, we show that a combination of the extinction feature at 9.7
micron, the PAH strength, and a near-infrared slope can simultaneously
constrain the AGN fraction and dust grain distribution for a wide range of
obscuration. We find that this indicator, accessible to the James Webb Space
Telescope, may estimate the AGN power as tightly as the hard X-ray flux alone,
thereby providing a valuable future cross-check and constraint for large
samples of distant ULIRGs.
