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In this work we revisit the growth of small primordial black holes (PBHs)immersed in a quintessential field and/or radiation to the supermassive blackhole (SMBHs) scale. We show the difficulties of scenarios in which such hugegrowth is possible. For that purpose we evaluated analytical solutions of thedifferential equations (describing mass evolution) and point out the strongfine tuning for that conclusions. The timescale for growth in a model with aconstant quintessence flux is calculated and we show that it is much biggerthan the Hubble time.The fractional gain of the mass is further evaluated inother forms, including quintessence and/or radiation. We calculate thecosmological density $\Omega$ due to quintessence necessary to grow BHs to thesupermassive range and show it to be much bigger than one. We also describe theset of complete equations analyzing the evolution of the BH+quintessenceuniverse, showing some interesting effects such the quenching of the BH massgrowth due to the evolution of the background energy. Additional constraintsobtained by using the Holographic Bound are also described. The generalequilibrium conditions for evaporating/accreting black holes evolving in aquintessence/radiation universe are discussed in the Appendix.Comment: 21 pp., 2 Figures, To appear in IJMP

COSMOLOGICAL QUINTESSENCE ACCRETION ONTO PRIMORDIAL BLACK HOLES: CONDITIONS FOR THEIR GROWTH TO THE SUPERMASSIVE SCALE