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We generalize the cosmological duality between inflation and cycliccontraction under the interchange $a \leftrightarrow H$ to the case ofnon-canonical scalar field theories with varying speed of sound. The singleduality in the canonical case generalizes to a family of three dualitiesconstructed to leave the cosmological acoustic horizon invariant. We find threeclasses of models: (I) DBI inflation, (II) the non-canonical generalization ofcyclic contraction, and (III) a new cosmological solution with rapidlydecreasing speed of sound and relatively slowly growing scale factor, which wedub {\it stalled} cosmology. We construct dual analogs to the inflationary slowroll approximation, and solve for the curvature perturbation in all threecases. Both cyclic contraction and stalled cosmology predict a strongly bluespectrum for the curvature perturbations inconsistent with observations.

Horizon-preserving dualities and perturbations in non-canonical scalar field cosmologies