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DBI analog of a decaying vacuum cosmology

In this work I discuss the dynamical and thermodynamical equivalence between a general k-essence scalar field cosmology and an arbitrary cosmological model with a decaying vacuum, thus generalizing the approach proposed by Maia and Lima [Phys. Rev. D {\bf 65}, 083513 (2002)]. The formalism obtained is quite general and holds for any non-canonical scalar field model. As a special case I derive a Dirac-Born-Infeld (DBI) model with an exponential potential and constant speed of sound, and show that it is equivalent to a cosmological model with decay law $Λ(H) = 3βH^2$.