Galactic Dark Matter: a Dynamical Consequence of Cosmological Expansion
  This work wants to show how standard General Relativity (GR) is able to
explain galactic rotation curves without the need for dark matter, this
starting from the idea that when Einstein's equations are applied to the
dynamics of a galaxy embedded in an expanding universe they do not reduce to
Poisson's equation but a generalisation of it taking cosmological expansion
into account. A non-linear scheme to perturb Einstein's field equations around
the Robertson-Walker (R-W) metric is devised in order to find their
non-relativistic limit without losing their characteristic non-linearities. The
resulting equation is used to numerically study the gravitational potential of
a cosmological perturbation and applied to a simple galactic model with an
exponentially decreasing baryonic matter distribution. The non-relativistic
limit of GR in a R-W space-time produces a generalised Poisson equation for the
gravitational potential which is non-linear, parabolic and heat-like. It is
shown how its non-linearities generate an effective "dark matter" distribution
caused by both cosmological expansion and the dynamics of the perturbation's
gravitational potential. It is also shown how this dynamical effect gets
completely lost during a linearisation of Einstein's equations. The equation is
then used to successfully fit real galactic rotation curves numerically using a
matter distribution following the shape of a simple S\'ersic luminosity
profile, common to most galaxies, thus without recourse to dark matter. A
relation for the dark to luminous matter ratio is found, explaining the
domination of dark matter in low-mass galaxies. A few rotation curves with a
faster than Newtonian decrease are also presented and successfully fitted,
opening the way to a new possible interpretation of these phenomena in terms of
an effective "anti-gravitational" dark matter distribution, purely geometrical
in origin.
