Perturbations in FRW metric Considering the FRW metric with perturbations; how can I calculate the Einstein tensor (without a very very disgusting expression which comes from the variation of the difference between the Ricci tensor minus the metric tensor times the curvature)?
 A: The problem concerns standard gravitational perturbation theory. Instead of expanding around say Minkowski space, one expands around the FLRW background.
Using the gauge symmetries and scalar-vector-tensor decomposition, one can reduce the form of the perturbed metric to an incredibly simple form, namely,
$$g_{\mu\nu} = g^{\mathrm{FLRW}}_{\mu\nu} + h_{\mu\nu}= a(t)^2 \begin{pmatrix}
1+2\Psi & 0 & 0 & 0\\ 
 0& 2\Phi -1 & 0 & 0\\ 
 0& 0 & 2\Phi -1 & 0\\ 
 0&  0& 0 & 2\Phi -1
\end{pmatrix}$$
One approach to finding the Einstein equations is noting to first order $\delta R_{\mu\nu} = -\frac12 \Delta_L h_{\mu\nu}$ where $\Delta_L$ is the Lichnerowicz operator. Alternative, one can in this case simply plug in the perturbed metric.
Defining the Hubble parameter, $\mathcal H = \dot a a^{-1}$, we have $G_{00} = 3\mathcal H^2 + 2\nabla^2 \Phi - 6\mathcal H \Phi'$. The spatial part mixed with the time component is,
$$G_{0i} = 2\partial_i(\Phi' + \mathcal H \Psi)$$
and finally the spatial part - the messiest - is,
$$G_{ij} = -(2\mathcal H' +\mathcal H^2)\delta_{ij} + \partial_i \partial_j (\Phi - \Psi)$$
$$ + \left[\nabla^2 (\Psi - \Phi) + 2\Phi'' + 2(2\mathcal H' + \mathcal H^2)(\Phi + \Psi) + 2\mathcal H \Psi' + 4\mathcal H \Phi' \right]\delta_{ij}.$$
Compared to the perturbation equations in full generality, even with gauge fixing, this is a relatively manageable expression. 
It can be further simplified depending on the scenario. Ignoring anisotropic stress, $\Phi = \Psi$ which greatly reduces the equations and in some instances gives us only a Laplace equation to solve.
The situation doesn't get any better than this. In fact, having done perturbation theory of solutions to string theory, I can say the situation can be a lot worse. General relativity is horrendously non-linear, there's no avoiding that.
