Exponentially growing bubbles around early super massive black holes

We addressed the so far unexplored issue of outflows induced by exponentially growing power sources, focusing on early supermassive black holes (BHs). We assumed that these objects grow to 109 M by z=6 by Eddington-limited accretion and convert 5% of their bolometric output into a wind. We first considered the case of energy-driven and momentum-driven outflows expanding in a region where the gas and total mass densities are uniform and equal to the average values in the Universe at z > 6. We derived analytic solutions for the evolution of the outflow, finding that, for an exponentially growing power with e-folding time tS al, the late time expansion of the outflow radius is also exponential, with e-folding time of 5tS al and 4tS al in the energy-driven and momentum-driven limit, respectively. We then considered energy-driven outflows produced by QSOs at the center of early dark matter halos of different masses and powered by BHs growing from different seeds. We followed the evolution of the source power and of the gas and dark matter density profiles in the halos from the beginning of the accretion until z = 6. The final bubble radius and velocity do not depend on the seed BH mass but are instead smaller for larger halo masses. At z=6, bubble radii in the range 50-180 kpc and velocities in the range 400-1000 km s−1 are expected for QSOs hosted by halos in the mass range 3 × 1011 − 1013 M . These radius and velocity scales compare well with those measured for the outflowing gas in the z=6.4 QSO SDSS J1148+5251. By the time the QSO is observed, we found that the total thermal energy injected within the bubble in the case of an energy-driven outflow is Eth ∼ 5 × 1060 erg. This is in excellent agreement with the value of Eth = (6.2 ± 1.7) × 1060 erg measured through the detection of the thermal Sunyaev-Zeldovich effect around a large population of luminous QSOs at lower redshifts. This suggests that QSO outflows are closer to the energy-driven limit than to the momentum-driven limit. We investigated the stability of the expanding gas shell in the case of an energy-driven supersonic outflow propagating within a dark matter halo with Mh = 3 × 1011 M at z=6. We found that the shell is Rayleigh-Taylor unstable already at early times and, by means of a simple model, we investigated the fate of the fragments detaching from the shell. We found that these fragments should rapidly evaporate because of the extremely high temperature of the hot gas bubble if this does not cool. Since the only effective cooling mechanism for such a gas is Inverse Compton by the CMB photons (IC-CMB), which is important only at z ≥ 6, we speculate that such shell fragments may be observed only around high-z QSOs, where IC-CMB cooling of the bubble gas can prevent their evaporation.