  The special galileon and Dirac-Born-Infeld (DBI) theories are effective field
theories of a single scalar field that have many interesting properties in flat
space. These theories can be extended to all maximally symmetric spaces, where
their algebras of shift symmetries are simple. We study aspects of the curved
space versions of these theories: for the special galileon, we find a new
compact expression for its Lagrangian in de Sitter space and a field
redefinition that relates it to the previous, more complicated formulation.
This field redefinition reduces to the well-studied galileon duality
redefinition in the flat space limit. For the DBI theory in de Sitter space, we
discuss the brane and dilaton formulations of the theory and present strong
evidence that these are related by a field redefinition. We also give an
interpretation of the symmetries of these theories in terms of broken
diffeomorphisms of de Sitter space.
