Equation decoupling in Cosmological Perturbation theory

[JohnShiu]

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?

 

[AndyW]

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!