Non-Gaussian initial conditions in $\Lambda$CDM: Newtonian, relativistic
  and primordial contributions
  The goal of the present paper is to set initial conditions for structure
formation at non-linear order, consistent with general relativity, while also
allowing for primordial non-Gaussianity. We use the non-linear continuity and
Raychauduri equations, which together with the non-linear energy constraint
determine the evolution of the matter density fluctuation in general
relativity. We solve this equations at first and second order in a perturbative
expansion, recovering and extending previous results derived in the
matter-dominated limit and in the Newtonian regime. We present a second-order
solution for the comoving density contrast in a $\Lambda$CDM universe,
identifying non-linear contributions coming from the Newtonian growing mode,
primordial non-Gaussianity and intrinsic non-Gaussianity, due to the essential
non-linearity of the relativistic constraint equations. We discuss the
application of these results to initial conditions in N-body simulations,
showing that relativistic corrections mimic a non-zero non-linear parameter
$f_{\rm NL}$.
