  It is shown that a first-order cosmological perturbation theory for the open,
flat and closed Friedmann-Lema\^itre-Robertson-Walker universes admits one, and
only one, gauge-invariant variable which describes the perturbation to the
energy density and which becomes equal to the usual Newtonian energy density in
the non-relativistic limit. The same holds true for the perturbation to the
particle number density. Using these two new variables, a new manifestly
gauge-invariant cosmological perturbation theory has been developed. Density
perturbations evolve diabatically. Perturbations in the total energy density
are gravitationally coupled to perturbations in the particle number density,
irrespective of the nature of the particles. There is, in first-order, no
back-reaction of perturbations to the global expansion of the universe.
Small-scale perturbations in the radiation-dominated era oscillate with an
increasing amplitude, whereas in older, less precise treatments, oscillating
perturbations are found with a decreasing amplitude. This is a completely new
and, obviously, important result, since it makes it possible to explain and
understand the formation of massive stars after decoupling of matter and
radiation.
