  The tight-coupling approximation (TCA) used to describe the early dynamics of
the baryons-photons system is systematically built to higher orders in the
inverse of the interaction rate. This expansion can be either used to grasp the
physical effects by deriving simple analytic solutions or to obtain a form of
the system which is stable numerically at early times. In linear cosmological
perturbations, we estimate numerically its precision, and we discuss the
implications for the baryons acoustic oscillations. The TCA can be extended to
the second order cosmological perturbations, and in particular we recover that
vorticity is not generated at lowest order of this expansion.
