Basics of cosmological perturbations

So far, we have been able to build a scenario which solves the horizon, flatness and monopole problems. As we already said, this scenario also explains the existence of an almost scale invariant spectrum in the cosmological perturbations. However, the derivation of this crucial result is significantly more involved than the previous one. First because we need to do a careful study of the cosmological perturbation in the context of general relativity and in an expanding universe. Second because we then need to solve these equations in the specific case of inflation. Third, because perturbation theory only tell of the evolution of cosmological perturbations, so that we need to specify the 

The first step is therefore to understand how to define unambiguously a mapping6. Let us first define two mappings between R4 and the two (unperturbed 

Let us therefore study how the metric components change under the coordinate change i l o i . Let us define 

However, as we already explained, in order to define a perturbation, we want to compare the values of gap at points which have the same coordinates. Here, point P has coordinates x1 with mapping i and coordinates x 1 f1 with mapping i . Let us consider point P which has coordinates x — f11 with mapping i and therefore coordinates x1 with mapping i . What we are interested in is 

We start here from an FRW metric, that is an unperturbed metric such that 

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