Early formation of massive, compact, spheroidal galaxies with classical profiles by violent disc instability or mergers

We address the formation of massive stellar spheroids between redshifts z = 4 and 1 using a suite of adaptive mesh refinement hydrocosmological simulations. The spheroids form as bulges, and the spheroid mass growth is partly driven by violent disc instability (VDI) and partly by mergers. A kinematic decomposition to disc and spheroid yields that the mass fraction in the spheroid is between 50 and 90 per cent and is roughly constant in time, consistent with a cosmological steady state of VDI discs that are continuously fed from the cosmic web. The density profile of the spheroid is typically 'classical', with a Sersic index n = 4.5 +/- 1, independent of whether it grew by mergers or VDI and independent of the feedback strength. The disc is characterized by n = 1.5 +/- 0.5, and the whole galaxy by n = 3 +/- 1. The high-redshift spheroids are compact due to the dissipative inflow of gas and the high universal density. The stellar surface density within the effective radius of each galaxy as it evolves remains roughly constant in time after its first growth. For galaxies of a fixed stellar mass, the surface density is higher at higher redshifts.