This work studies a simplified model of the gravitational instability of an
initially homogeneous infinite medium, represented by $\TT^d$, based on the approximation
that the mean fluid velocity is always proportional to the local acceleration. It is shown
that, mathematically, this assumption leads to the restricted Patlak-Keller-Segel model
considered by J\"ager and Luckhaus or, equivalently, the Smoluchowski equation describing
the motion of self-gravitating Brownian particles, coupled to the modified Newtonian
potential that is appropriate for an infinite mass distribution. We discuss some of the
fundamental properties of a non-local generalization of this model where the effective
pressure force is given by a fractional Laplacian with $0<\alpha<2$, and illustrate
them by means of numerical simulations. Local well-posedness in Sobolev spaces is proven,
and we show the smoothing effect of our equation, as well as a \emph{Beale-Kato-Majda}-type
criterion in terms of $\rhomax$. It is also shown that the problem is ill-posed in Sobolev
spaces when it is considered backward in time. Finally, we prove that, in the critical case
(one conservative and one dissipative derivative), $\rhomax(t)$ is uniformly bounded in
terms of the initial data for sufficiently large pressure forces.