  We present simple expressions for the relativistic first and second order
fractional density perturbations for FL cosmologies with dust, in four
different gauges: the Poisson, uniform curvature, total matter and synchronous
gauges. We include a cosmological constant and arbitrary spatial curvature in
the background. A distinctive feature of our approach is our description of the
spatial dependence of the perturbations using a canonical set of quadratic
differential expressions involving an arbitrary spatial function that arises as
a conserved quantity. This enables us to unify, simplify and extend previous
seemingly disparate results. We use the primordial matter and metric
perturbations that emerge at the end of the inflationary epoch to determine the
additional arbitrary spatial function that arises when integrating the second
order perturbation equations. This introduces a non-Gaussianity parameter into
the expressions for the second order density perturbation. In the special case
of zero spatial curvature we show that the time evolution simplifies
significantly, and requires the use of only two non-elementary functions, the
so-called growth supression factor at the linear level, and one new function at
the second order level. We expect that the results will be useful in
applications, for example, studying the effects of primordial non-Gaussianity
on the large scale structure of the universe.
