  The non-abelian generalization of the Born-Infeld non-linear lagrangian is
extended to the non-commutative geometry of matrices on a manifold. In this
case not only the usual SU(n) gauge fields appear, but also a natural
generalization of the multiplet of scalar Higgs fields, with the double-well
potential as a first approximation. The matrix realization of non-commutative
geometry provides a natural framework in which the notion of a determinant can
be easily generalized and used as the lowest-order term in a gravitational
lagrangian of a new kind. As a result, we obtain a Born-Infeld-like lagrangian
as a root of sufficiently high order of a combination of metric, gauge
potentials and the scalar field interactions. We then analyze the behavior of
cosmological models based on this lagrangian. It leads to primordial inflation
with varying speed, with possibility of early deceleration ruled by the
relative strength of the Higgs field.
