  In this paper we present four simple expressions for the relativistic first
and second order fractional density perturbations for $\Lambda$CDM cosmologies
in different gauges: the Poisson, uniform curvature, total matter and
synchronous gauges. A distinctive feature of our approach is the use of a
canonical set of quadratic differential expressions involving an arbitrary
spatial function, the so-called comoving curvature perturbation, to describe
the spatial dependence, which enables us to unify, simplify and extend previous
seemingly disparate results. The simple structure of the expressions makes the
evolution of the density perturbations completely transparent and clearly
displays the effect of the cosmological constant on the dynamics, namely that
it stabilizes the perturbations. 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.
