  We determine the observational signatures of protostellar cores by coupling
two-dimensional radiative transfer calculations with numerical hydrodynamical
simulations that predict accretion rates that both decline with time and
feature short-term variability and episodic bursts caused by disk gravitational
instability and fragmentation. We calculate the radiative transfer of the
collapsing cores throughout the full duration of the collapse, using as inputs
the core, disk, and protostellar masses, radii, and mass accretion rates
predicted by the hydrodynamical simulations. From the resulting spectral energy
distributions, we calculate standard observational signatures (bolometric
luminosity, bolometric temperature, ratio of bolometric to submillimeter
luminosity) to directly compare to observations. We show that the accretion
process predicted by these models reproduces the full spread of observed
protostars in both Lbol - Tbol and Lbol - core mass space, including very low
luminosity objects, provides a reasonable match to the observed protostellar
luminosity distribution, and resolves the long-standing luminosity problem.
These models predict an embedded phase duration shorter than recent
observationally determined estimates (0.12 Myr vs. 0.44 Myr), and a fraction of
total time spent in Stage 0 of 23%, consistent with the range of values
determined by observations. On average, the models spend 1.3% of their total
time in accretion bursts, during which 5.3% of the final stellar mass accretes,
with maximum values being 11.8% and 35.5% for the total time and accreted
stellar mass, respectively. Time-averaged models that filter out the accretion
variability and bursts do not provide as good of a match to the observed
luminosity problem, suggesting that the bursts are required.
