We used the Hubble Space Telescope WFC3 near-infrared camera to image the host galaxies of a sample of eleven luminous, dust-reddened quasars at z ~ 2 -- the peak epoch of black hole growth and star formation in the Universe -- to test the merger-driven picture for the co-evolution of galaxies and their nuclear black holes. The red quasars come from the FIRST+2MASS red quasar survey and a newer, deeper, UKIDSS+FIRST sample. These dust-reddened quasars are the most intrinsically luminous quasars in the Universe at all redshifts, and may represent the dust-clearing transitional phase in the merger-driven black hole growth scenario. Probing the host galaxies in rest-frame visible light, the HST images reveal that 8/10 of these quasars have actively merging hosts, while one source is reddened by an intervening lower redshift galaxy along the line-of-sight. We study the morphological properties of the quasar hosts using parametric Sersic fits as well as the non-parametric estimators (Gini coefficient, M_{20} and asymmetry). Their properties are heterogeneous but broadly consistent with the most extreme morphologies of local merging systems such as Ultraluminous Infrared galaxies. The red quasars have a luminosity range of log(L_bol) = 47.8 - 48.3 (erg/s) and the merger fraction of their AGN hosts is consistent with merger-driven models of luminous AGN activity at z=2, which supports the picture in which luminous quasars and galaxies co-evolve through major mergers that trigger both star formation and black hole growth.
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