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We discuss constraints on which flat directions can have large vacuumexpectation values (VEVs) after inflation. We show that only flat directionswhich are not charged under B-L and develop positive pressure due torenormalization group effects can have large VEVs of order $\Mp$. For example,within the MSSM only the $H_uH_d$ flat direction is found to be viable. Thisstrongly constrains the embedding of a broad class of inflationary models inthe MSSM or some other supersymmetric extension of the SM. For flat directionswith negative pressure, the condensate fragments into very large Q-balls whichwe call Q-giants. We discuss the formation, evolution and reheating of theseQ-giants and show that they decay too late. The analysis requires taking intoaccount new phases of the flat directions, which have been overlooked in theformation and dynamics of the Q-balls. These constraints may be ameliorated byinvoking a short period of thermal inflation. The latter, however, is viable ina very narrow window of parameter space and requires fine tuning.Comment: 40 pages, 3 figure

Constraining modular inflation in the MSSM from giant Q-ball formation