  We introduce bi-galileon theory, the generalisation of the single galileon
model introduced by Nicolis et al. The theory contains two coupled scalar
fields and is described by a Lagrangian that is invariant under Galilean shifts
in those fields. This paper is the first of two, and focuses on the motivation
and formulation of the theory. We show that the boundary effective theory of
the cascading cosmology model corresponds to a bi-galileon theory in the
decoupling limit, and argue that this is to be expected for co-dimension 2
braneworld models exhibiting infra-red modification of gravity. We then
generalise this, by constructing the most general bi-galileon Lagrangian. By
coupling one of the galileons to the energy-momentum tensor, we pitch this as a
modified gravity theory in which the modifications to General Relativity are
encoded in the dynamics of the two galileons. We initiate a study of
phenomenology by looking at maximally symmetric vacua and their stability,
developing elegant geometric techniques that trivially explain why some of the
vacua have to be unstable in certain cases (eg DGP). A detailed study of
phenomenology appears in our companion paper.
