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We develop a family of deformations of the differential and of thepair-of-pants product on the Hamiltonian Floer complex of a symplectic manifold(M,\omega) which upon passing to homology yields ring isomorphisms with the bigquantum homology of M. By studying the properties of the resulting deformedversion of the Oh-Schwarz spectral invariants, we obtain a Floer-theoreticinterpretation of a result of Lu which bounds the Hofer-Zehnder capacity of Mwhen M has a nonzero Gromov-Witten invariant with two point constraints, and weproduce a new algebraic criterion for (M,\omega) to admit a Calabiquasimorphism and a symplectic quasi-state. This latter criterion is found tohold whenever M has generically semisimple quantum homology in the senseconsidered by Dubrovin and Manin (this includes all compact toric M), and alsowhenever M is a point blowup of an arbitrary closed symplectic manifold.

Deformed Hamiltonian Floer theory, capacity estimates, and Calabi quasimorphisms