  It is shown that a first-order cosmological perturbation theory for
Friedmann-Lemaitre-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 energy density of the Newtonian theory of
gravity in the limit that all particle velocities are negligible with respect
to the speed of light. The same holds true for the perturbation to the particle
number density. A cosmological perturbation theory based on these particular
gauge-invariant quantities is more precise than any earlier first-order
perturbation theory. In particular, it explains star formation in a
satisfactory way, even in the absence of cold dark matter. In a baryon-only
universe, the earliest stars, the so-called Population III stars, are found to
have masses between 400 and 100,000 solar masses with a peak around 3400 solar
masses. If cold dark matter, with particle mass 10 times heavier than the
proton mass, is present then the star masses are between 16 and 4000 solar
masses with a peak around 140 solar masses. They come into existence between
100 Myr and 1000 Myr. At much later times, star formation is possible only in
high density regions, for example within galaxies. Late time stars may have
much smaller masses than early stars. The smallest stars that can be formed
have masses of 0.2--0.8 solar mass, depending on the initial internal relative
pressure perturbation. It is demonstrated that the Newtonian theory of gravity
cannot be used to study the evolution of cosmological density perturbations.
