Why is the Primordial Power Spectrum Defined as P(k) = (k^3)/(2π^2)|w_k|^2?

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Discussion Overview

The discussion centers on the definition of the primordial power spectrum in cosmology, specifically the expression P(k) = (k^3)/(2π^2)|w_k|^2, where w_k represents the mode function. Participants explore the theoretical foundations and implications of this definition, seeking clarity on its derivation and significance.

Discussion Character

  • Exploratory, Technical explanation, Conceptual clarification

Main Points Raised

  • One participant questions the rationale behind the definition of the power spectrum, noting that cosmology literature often lacks detailed explanations.
  • Another participant suggests that the power spectrum is the Fourier transform of the correlation function of density perturbations, asking if this insight is helpful.
  • A further contribution reiterates the relationship between the power spectrum and the correlation function, presenting an expression involving the two-point correlation function ξ(r) and a constant C.
  • Another participant explains that the power spectrum indicates how different frequencies contribute to spatial density perturbations, which is a typical outcome of Fourier transforms.

Areas of Agreement / Disagreement

Participants express varying levels of understanding regarding the definition of the power spectrum, with some agreeing on its connection to the Fourier transform of correlation functions, while others seek deeper clarification on its derivation and implications. No consensus is reached on the foundational aspects of the definition.

Contextual Notes

The discussion highlights a lack of detailed derivations in existing literature and the need for clearer explanations regarding the constants and functions involved in the power spectrum's definition.

shinobi20
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Why is the power spectrum defined as
##P(k) = \frac{k^3}{2π^2} |w_k|^2 ##
where ##w_k## is the mode function?

Cosmology books and papers just states that it is defined that way but there are no details on why.
 
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The power spectrum is defined as the Fourier transform of the correlation function of density perturbations. Does this help? Can you write down the expression of the correlation function of density (of curvature) perturbations?
 
bapowell said:
The power spectrum is defined as the Fourier transform of the correlation function of density perturbations. Does this help? Can you write down the expression of the correlation function of density (of curvature) perturbations?
Let ##ξ(r)## be the two point correlation function, then as you said, we define (why?) the power spectrum as

##P(k) = C∫ ξ(r) e^{-ikx} d^3r##

where C is a constant
 
Because the power spectrum tells you how the various frequencies contribute to the spatial density perturbation. That's generally what the Fourier transform tells you.
 

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