Equation decoupling in Cosmological Perturbation theory

In summary, the equations of motion in cosmological perturbation theory are decomposed into scalar, vector, and tensor modes to simplify the solving process. This is based on the different transformation properties of these modes under three-dimensional rotations. This allows for a separate treatment of each mode, making the equations easier to solve.
  • #1
JohnShiu
1
0
Hi,

I am recently reading Weinberg's Cosmology, and getting subtle on Ch5, small fluctuation.

One of the subtle point is on P.225-226 (same as (F.11) -> (F.13) and (F.14) in appendix F). The equations of motion (5.1.24)-(5.1.26) are decomposed into many parts. For example, (5.1.24) is decomposed into (5.1.44), (5.1.45), (5.1.50) and (5.1.53). I found this is very common in cosmological perturbation, I google many times and read the G&C by weinberg, they usually directly decouple the equations into scalar mode and vector mode according to different parts in vector/tensor decomposition.

In my understanding, I don't think (5.1.24) does imply these four equations. I wonder if this is an assumption because these four equations imply (5.1.24), but it seems there should be some meaningful stuff in between (Weinberg wrote: The solutions of Eqs. (F.10)–(F.12) can be classified according to the transformation properties of the dependent variables under three-dimensional rotations). So how do the properties help in decoupling the equation?
 
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  • #2




Hi there,

I understand your confusion with the decomposition of the equations of motion in cosmological perturbation theory. Let me try to explain it in simpler terms.

The reason why the equations of motion (5.1.24)-(5.1.26) are decomposed into different parts is to make it easier to solve the equations. By separating them into scalar, vector, and tensor modes, we can focus on each mode separately and then combine the solutions to get the full solution.

The transformation properties of the dependent variables under three-dimensional rotations play a crucial role in this decoupling process. This is because different modes have different transformation properties, and therefore, they can be treated separately. For example, scalar modes involve only scalar quantities, while vector modes involve vector quantities. This simplifies the equations and makes it easier to solve them.

I hope this helps in understanding why the equations are decomposed in this way. Feel free to ask any other questions you may have. Happy reading!
 

1. What is equation decoupling in Cosmological Perturbation theory?

Equation decoupling in Cosmological Perturbation theory is a mathematical technique used to simplify the equations that describe the evolution of small perturbations in the early universe. It involves separating the equations into two sets, one for the background cosmology and one for the perturbations, which allows for easier analysis and calculation.

2. Why is equation decoupling important in Cosmological Perturbation theory?

Equation decoupling is important because it allows us to study the evolution of small perturbations in the early universe separately from the overall background cosmology. This enables us to better understand the effects of these perturbations on the large-scale structure of the universe and the formation of galaxies and other cosmic structures.

3. How is equation decoupling achieved in Cosmological Perturbation theory?

Equation decoupling is achieved by using a perturbation variable, which is a small quantity that represents the deviation from the background cosmology. This variable is then substituted into the equations of motion and expanded in terms of powers of the perturbation variable. The resulting equations are then separated into background and perturbation equations.

4. What are the limitations of equation decoupling in Cosmological Perturbation theory?

Equation decoupling is a useful technique, but it does have its limitations. It assumes that the perturbations are small and that they evolve slowly compared to the background cosmology. It also neglects any interactions between different types of perturbations, which may be important in some scenarios.

5. How does equation decoupling relate to other theories of cosmological evolution?

Equation decoupling is a specific technique used in Cosmological Perturbation theory, which is a branch of cosmology that studies the evolution of small perturbations in the early universe. It is closely related to other theories of cosmological evolution, such as inflationary cosmology and cosmic inflation, which also study the origins of the universe and the formation of large-scale structures.

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