We present Hubble Space Telescope (HST) imaging of 22 ultra-luminous infrared galaxies (ULIRGs) at z~2 with extremely red R-[24] colors (called dust-obscured galaxies, or DOGs) which have a local maximum in their spectral energy distribution (SED) at rest-frame 1.6um associated with stellar emission. These sources, which we call bump DOGs, have star-formation rates of 400-4000 Msun/yr and have redshifts derived from mid-IR spectra which show strong polycyclic aromatic hydrocarbon emission --- a sign of vigorous on-going star-formation. Using a uniform morphological analysis, we look for quantifiable differences between bump DOGs, power-law DOGs (Spitzer-selected ULIRGs with mid-IR SEDs dominated by a power-law and spectral features that are more typical of obscured active galactic nuclei than starbursts), sub-millimeter selected galaxies (SMGs), and other less-reddened ULIRGs from the Spitzer extragalactic First Look Survey (XFLS). Bump DOGs are larger than power-law DOGs (median Petrosian radius of 8.4 +/- 2.7 kpc vs. 5.5 +/- 2.3 kpc) and exhibit more diffuse and irregular morphologies (median M_20 of -1.08 +/- 0.05 vs. -1.48 +/- 0.05). These trends are qualitatively consistent with expectations from simulations of major mergers in which merging systems during the peak star-formation rate period evolve from M_20 = -1.0 to M_20 = -1.7. Less obscured ULIRGs (i.e., non-DOGs) tend to have more regular, centrally peaked, single-object morphologies rather than diffuse and irregular morphologies. This distinction in morphologies may imply that less obscured ULIRGs sample the merger near the end of the peak star-formation rate period. Alternatively, it may indicate that the intense star-formation in these less-obscured ULIRGs is not the result of a recent major merger.
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