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The path integration technique recently developed for the statistical sum ofthe microcanonical ensemble in cosmology is applied to the calculation of theone-loop preexponential factor in the cosmological model generated by aconformal field theory with a large number of quantum species -- the model ofinitial conditions possibly related to the resolution of the cosmologicalconstant and landscape problems. The result is obtained for the family ofbackground cosmological instantons with one oscillation of the FRW scalefactor. The magnitude of the prefactor is analytically and numericallyestimated for fields of various spins conformally coupled to gravity, whichjustifies the validity of semiclassical expansion for this family ofcosmological instantons.

Statistical sum in the CFT driven cosmology