We present results of Hubble Space Telescope NICMOS H-band imaging of 73 of most luminous (i.e., log[L_IR/L_0]>11.4) Infrared Galaxies (LIRGs) in the Great Observatories All-sky LIRG Survey (GOALS). This dataset combines multi-wavelength imaging and spectroscopic data from space (Spitzer, HST, GALEX, and Chandra) and ground-based telescopes. In this paper we use the high-resolution near-infrared data to recover nuclear structure that is obscured by dust at optical wavelengths and measure the evolution in this structure along the merger sequence. A large fraction of all galaxies in our sample possess double nuclei (~63%) or show evidence for triple nuclei (~6%). Half of these double nuclei are not visible in the HST B-band images due to dust obscuration. The majority of interacting LIRGs have remaining merger timescales of 0.3 to 1.3 Gyrs, based on the projected nuclear separations and the mass ratio of nuclei. We find that the bulge luminosity surface density increases significantly along the merger sequence (primarily due to a decrease of the bulge radius), while the bulge luminosity shows a small increase towards late merger stages. No significant increase of the bulge Sersic index is found. LIRGs that show no interaction features have on average a significantly larger bulge luminosity, suggesting that non merging LIRGs have larger bulge masses than merging LIRGs. This may be related to the flux limited nature of the sample and the fact that mergers can significantly boost the IR luminosity of otherwise low luminosity galaxies. We find that the projected nuclear separation is significantly smaller for ULIRGs (median value of 1.2 kpc) than for LIRGs (mean value of 6.7 kpc), suggesting that the LIRG phase appears earlier in mergers than the ULIRG phase.
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