As the name of the theory indicates, any perturbation theory is developed to study the evolution of perturbations, so is the same what I am trying to tell you here.

If there is no perturbations exist in the Universe, then the Universe must be perfectly homogeneous and isotropic, so it can be described by a FRW metric of the form :{-1,a^2,a^2,a^2}, and a energy momentum tensor of the form :{-\rho,p,p,p}, that's all we need to describe a perfect Universe. (NOTE: here I directly take FRW to describe the real Universe, this is based on the observational fact that the Universe is expanding. )

In such a Universe, each point shares the same properties, so there is no motion, and no energy exchange ( maybe there is, but this can only happen in microscopic world ). Though the real situation is not, it's still very close to a homogeneous and isotropic Universe when averaged over a scale large enough. So introducing small perturbations into the unperturbed metric and energy-momentum tensor small perturbations will help us describing the real universe more accurately.

Also these small perturbations lead us to a possible explanation of the Large Scale Structure we see today. These perturbations were generated at the very early stage of the universe, with time goes on, the universe expands and the cosmic plasma cools, finally the gravitation become dominates in the evolution and over density regions start to attract more and more matter to grow into the structures we see today. This is the main idea of how the large scale structure forms, the details is a little complicated, we will discuss more on this later.

Let's go back ti the definition of the perturbations. As stated above, we know how a homogeneous and isotropic universe looks like, so it's natural to define the perturbations to be derivations of the homogeneous and isotropic quantities. One question arises, are these perturbation uniquely determined?

To answer this question, let's first find out how a coordinate transformation affects the components of physical tensor quantities.

Here the coordinate transformation is not the same as we are familiar with, since we are sitting on earth, and we can make experiments in a very limited space —— on the earth, or at most we can lunch a satellite, which is still located near the earth. So it is an important thing to make the sense of " coordinate transformation " to be clear!

In mathematics on mainflod, we have a topological space, with some properties satisfied ( for more information, find a book and read it by yourself). In GR, we can think that our space-time is such a kind of topological space, and we can define mathematical quantities and each point of this space, but we don't have a specific coordinate. In topological space, there is no concept of distance, until the metric is introduced. Anyway, my task here is to explain the "coordinate transformation" in cosmological perturbation theory, not teach you mainfold.

To describe the perturbations as a function of time and space, we have to choose a useful and simple "new" space-time which is one-to-one with the real SPACE-TIME (background cosmology). Of course we have many choices, and if we choose a good new space-time, our work can be largely simplified.

In simple words, the " coordinate transformation " is just a transition from one coordinate choice to another, we are left unchanged, or with a technical work, passive transformation. (some books on particle physics will discuss this, since symmetries are related to transformations )

After making clear the sense of coordinate transformation, let explain how to choose a gauge.

The total number of perturbations ( metric and matter contents ) exceeds the motion equations and constrain equations, so there are some freedoms left ( just like what you see in Electromagnetic courses ). Generally all the perturbations are non-zero, but according to the transformation laws, we find that in different coordinates, the perturbations also get changed, so there is a hope for us to choose some specific coordinate which can help us to eliminate the some of perturbations, then the problem will get simplified. This is the nature of choosing a gauge.

In simple words, the " coordinate transformation " is just a transition from one coordinate choice to another, we are left unchanged, or with a technical work, passive transformation. (some books on particle physics will discuss this, since symmetries are related to transformations )

After making clear the sense of coordinate transformation, let explain how to choose a gauge.

The total number of perturbations ( metric and matter contents ) exceeds the motion equations and constrain equations, so there are some freedoms left ( just like what you see in Electromagnetic courses ). Generally all the perturbations are non-zero, but according to the transformation laws, we find that in different coordinates, the perturbations also get changed, so there is a hope for us to choose some specific coordinate which can help us to eliminate the some of perturbations, then the problem will get simplified. This is the nature of choosing a gauge.

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