  We present models of deeply buried ultraluminous infrared galaxy (ULIRG)
spectral energy distributions (SEDs) and use them to construct a
three-dimensional diagram for diagnosing the nature of observed ULIRGs. Our
goal is to construct a suite of SEDs for a very simple model ULIRG structure,
and to explore how well this simple model can (by itself) explain the full
range of observed ULIRG properties. We use our diagnostic to analyze archival
Spitzer Space Telescope IRS spectra of ULIRGs and find that: (1) In general,
our model does provide a comprehensive explanation of the distribution of
mid-IR ULIRG properties; (2) >75% (in some cases 100%) of the bolometric
luminosities of the most deeply buried ULIRGs must be powered by a
dust-enshrouded active galactic nucleus; (3) an unobscured "keyhole" view
through <~10% of the obscuring medium surrounding a deeply buried ULIRG is
sufficient to make it appear nearly unobscured in the mid-IR; and (4) the
observed absence of deeply buried ULIRGs with large PAH equivalent widths is
naturally explained by our models showing that deep absorption features are
"filled-in" by small quantities of foreground unobscured PAH emission (e.g.,
from the host galaxy disk) at the level of ~1% the bolometric nuclear
luminosity. The modeling and analysis we present will also serve as a powerful
tool for interpreting the high angular resolution spectra of high-redshift
sources to be obtained with the James Webb Space Telescope.
