Use of general Relativity to analyse cosmological perturbations in a non-expanding background The theory of cosmological perturbations can be developed in the context of the Newtonian gravitational theory in a non-expanding background. This is all well and good.
However, density fluctuations in the early universe cannot be analysed in the context of General Relativity in a non-expanding background. Why is this so?
 A: You may have misunderstand what Brandenberger means in his paper. He is simply saying that the General Relativity equations for the universe lead to an expanding universe, so of course it would be inconsistent to analyze density fluctuations in the universe, which is expanding, as though it is not. So he simply used Minkowski spacetime to see what it might be like. You can certainly posit density fluctuations in a static general relativistic spacetime. And analyze them perfectly. 
For instance, you can take a perturbation of a spherically symmetric Black Hole, a Schwarzschild solution, which is static, and assume some perturbation, expand it in term of multipoles, and track that over time those multipole moments in the Black Hole radiate away, leaving behind only the monopole moments (i.e., the mass term), and if some of the perturbation was not spherically symmetric, also the dipole moment and some axisymmetric rotation. If the perturbations had been spherically or axially symmetric there would not have been any radiation. No problem doing any of that perturbation analysis in a static spacetime. 
Similarly you could do it In a static spacetime in cosmology, if you had no expansion. The Einstein assumed cosmological constant of a very specific value was put in so it created a static universe (this was found to just not be consistent with the observed expansion of the universe). You get a static solution and you can do density or other perturbations of it. 
