Decoupling of SVT Metric Perturbations

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In summary, the conversation discusses the concept of perturbation theory in FRW cosmologies and the decoupling of scalar, vector, and tensor perturbations at first order. The reason for this separation is due to the use of a linearized solution, where the linear parts are independent of each other. This approximation is useful for large-scale cosmology, but may break down in smaller systems. The Helmholtz-Hodge decomposition theorem is also mentioned as a contributing factor to the separation of the equations of motion for each perturbation component.
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Zag
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Hello everyone,

I have been studying perturbation theory in the context of FRW cosmologies, and so far have had a really hard time understanding why the SVT (Scalar, Vector, and Tensor) perturbations associated with the metric tensor "decouple" at first order in perturbation theory.

All references avoid explaining this crucial step and simply jump to the final results by mentioning something along the lines: "Because the perturbations decouple, we can write these equations of motion for the scalars, and these equations over here for the vectors, etc."

However, it is not clear to me at all why these 3-scalars, 3-vectors, and 3-tensors which encapsulate the perturbations should evolve independently. In fact, Einstein equations mix them all into the same equation of motion, namely the field equations of general relativity. What is the argument to separate these perturbations into different equations? Where can I find a rigorous mathematical treatment which is not outdated?

Thanks a lot! Any reference and/or comment is appreciated.Zag
 
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I think "at first order" is the clue here. I don't know all of the details, but they're likely using a linearized solution, where the linear parts are independent of one another. I'm pretty sure this is a trivial statement, honestly: first-order expansion in perturbation can have no couplings, because an "##x_1 x_2##" term would be a second-order term.

Taking this approximation is useful as long as the coupling terms are small compared to the non-coupled terms. In General Relativity, this is typically true as long as your density is varying smoothly. Such linearized solutions are useful for large-scale cosmology, such as the cosmic microwave background, but tend to break down in galaxy clusters. Working with such solutions requires recognizing when they break down. So you'd want to have a good treatment of precisely where the terms that aren't first order come in before attempting to apply TeVeS to a real system.
 
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Hello @kimbyd . Thank you for your reply.

You are right. I ended up figuring things out and, indeed, if only first order terms in the perturbations are kept, the very Einstein equations decompose into scalar, vector, and tensor contributions. Since each of these contributions must be unique by the Helmholtz-Hodge decomposition theorem, each of these components end up giving rise to its own equation of motion.Zag
 

Related to Decoupling of SVT Metric Perturbations

1. What is "Decoupling of SVT Metric Perturbations"?

"Decoupling of SVT Metric Perturbations" is a phenomenon in cosmology where the scalar, vector, and tensor perturbations of the metric are treated independently. This means that the perturbations do not interact with each other, allowing for a simpler and more accurate analysis of the evolution of the universe.

2. Why is "Decoupling of SVT Metric Perturbations" important in cosmology?

Understanding the evolution of the universe is a fundamental goal in cosmology. By decoupling the scalar, vector, and tensor perturbations, we can better understand the behavior of each component and how they contribute to the overall evolution of the universe. This allows for more accurate predictions and explanations of cosmological phenomena.

3. How does "Decoupling of SVT Metric Perturbations" affect the study of dark energy and dark matter?

Dark energy and dark matter are two of the most mysterious components of the universe. By decoupling the metric perturbations, scientists can better isolate the effects of dark energy and dark matter on the evolution of the universe. This can lead to a better understanding of their properties and how they influence the expansion of the universe.

4. Is "Decoupling of SVT Metric Perturbations" a well-established concept in cosmology?

Yes, "Decoupling of SVT Metric Perturbations" is a well-established concept in cosmology. It is based on the theory of general relativity and has been extensively studied and validated through observations and experiments. It is a fundamental concept in understanding the evolution of the universe.

5. How does "Decoupling of SVT Metric Perturbations" impact our understanding of the early universe?

The early universe is a crucial period in cosmology, and understanding how it evolved can provide insights into the current state of the universe. By decoupling the metric perturbations, scientists can better analyze the behavior of the early universe and make more accurate predictions about its evolution. This can lead to a deeper understanding of the origins of the universe.

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