  The non-abelian Einstein-Born-Infeld-Dilaton theory, which rules the dynamics
of tensor-scalar gravitation coupled to a $su(2)$-valued gauge field ruled by
Born-Infeld lagrangian, is studied in a cosmological framework. The microscopic
energy exchange between the gauge field and the dilaton which results from a
non-universality of the coupling to gravity modifies the usual behaviour of
tensor-scalar theories coupled to matter fluids. General cosmological
evolutions are derived for different couplings to gravitation and a comparison
to universal coupling is highlighted. Evidences of cosmic acceleration are
presented when the evolution is interpreted in the Jordan physical frame of a
matter respecting the weak equivalence principle. The importance for the
mechanism of cosmic acceleration of the dynamics of the Born-Infeld gauge
field, the attraction role of the matter fluid and the non-universality of the
gravitational couplings is briefly outlined.
