  Several starburst galaxies have been observed in the GeV and TeV bands. In
these dense environments, gamma-ray emission should be dominated by cosmic-ray
interactions with the interstellar medium ($p_{\rm cr}p_{\rm ism} \to \pi^{0}
\to \gamma\gamma$). Indeed, starbursts may act as proton "calorimeters" where a
substantial fraction of cosmic-ray energy input is emitted in gamma rays. Here
we build a one-zone, "thick-target" model implementing calorimetry and placing
a firm upper bound on gamma-ray emission from cosmic-ray interactions. The
model assumes that cosmic rays are accelerated by supernovae (SNe), and all
suffer nuclear interactions rather than escape. Our model has only two free
parameters: the cosmic-ray proton acceleration energy per supernova
$\epsilon_{\rm cr}$, and the proton injection spectral index $s$. We calculate
the pionic gamma-ray emission from 10 MeV to 10 TeV, and derive thick-target
parameters for six galaxies with Fermi, H.E.S.S., and/or VERITAS data. Our
model provides good fits for the M82 and NGC 253, and yields $\epsilon_{\rm
cr}$ and $s$ values suggesting that supernova cosmic-ray acceleration is
similar in starbursts and in our Galaxy. We find that these starbursts are
indeed nearly if not fully proton calorimeters. For NGC 4945 and NGC 1068, the
models are consistent with calorimetry but are less well-constrained due to the
lack of TeV data. However, the Circinus galaxy and the ultraluminous infrared
galaxy Arp 220 exceed our pionic upper-limit, possible explanations are
discussed.
