  We present a novel approach, based entirely on the gravitational potential,
for studying the evolution of non-linear cosmological matter perturbations.
Starting from the perturbed Einstein equations, we integrate out the
non-relativistic degrees of freedom of the cosmic fluid and obtain a single
closed equation for the gravitational potential. We then verify the validity of
the new equation by comparing its approximate solutions to known results in the
theory of non-linear cosmological perturbations. First, we show explicitly that
the perturbative solution of our equation matches the standard perturbative
solutions. Next, using the mean field approximation to the equation, we show
that its solution reproduces in a simple way the exponential suppression of the
non-linear propagator on small scales due to the velocity dispersion. Our
approach can therefore reproduce the main features of the renormalized
perturbation theory and (time)-renormalization group approaches to the study of
non-linear cosmological perturbations. We conclude by a preliminary discussion
of the nature of the full solutions of the equation and their significance.
