How to Understand the Primordial Power Spectrum in TASI Lectures on Inflation?

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Discussion Overview

The discussion revolves around understanding the primordial power spectrum as presented in the TASI Lectures on Inflation by William Kinney. Participants explore specific equations and concepts related to the mode function, Bessel functions, and the derivation of integrals within the context of inflationary cosmology.

Discussion Character

  • Technical explanation
  • Mathematical reasoning
  • Homework-related

Main Points Raised

  • Participants inquire about the normalization constant ##\sqrt{-kτ}## in the context of the mode function and its derivation.
  • There is a question regarding how to determine the order of the Bessel function ##ν## from the equations presented.
  • Discussion includes the simplification of the mode function to equation (166) and the relationship of Bessel functions of order 3/2 to trigonometric functions.
  • Some participants suggest comparing equation (162) with the Bessel equation to derive necessary parameters.
  • There is a mention of changing the time variable to eliminate first-order terms for comparison with equation (162).
  • One participant expresses that they have resolved some of their questions but still seeks clarification on the derivation of the power spectrum from integrals in the lecture notes.
  • Suggestions are made for further reading on the two-point correlation function and transformations to Fourier space, indicating a need for supplementary resources.

Areas of Agreement / Disagreement

Participants do not reach a consensus on all questions posed, as some inquiries remain unresolved and others are addressed through various approaches. Multiple viewpoints on the derivation and understanding of equations are present.

Contextual Notes

Participants note the complexity of the latter parts of Kinney's notes, indicating that certain details may be skipped, which contributes to their difficulty in understanding the material.

Who May Find This Useful

This discussion may be useful for students and researchers interested in cosmology, particularly those studying inflationary models and the mathematical frameworks involved in the primordial power spectrum.

Figaro
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I have been reading the TASI Lectures on Inflation by William Kinney, (https://arxiv.org/pdf/0902.1529v2.pdf).
I came across the mode function eq (128) (which obeys a generalization of the Klein-Gordon equation to an expanding spacetime), as I read through until eq (163), I know that it is the Hankel function (though he said it is the bessel function to which it is the solution to the differential equation in eq (162)),

1) How did he get the normalization constant ##\sqrt{-kτ}##?
2) How can I get the order of the Bessel function ##ν##?
3) How did he simplify the mode function to eq (166)?
 
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Figaro said:
I have been reading the TASI Lectures on Inflation by William Kinney, (https://arxiv.org/pdf/0902.1529v2.pdf).
I came across the mode function eq (128) (which obeys a generalization of the Klein-Gordon equation to an expanding spacetime), as I read through until eq (163), I know that it is the Hankel function (though he said it is the bessel function to which it is the solution to the differential equation in eq (162)),

1) How did he get the normalization constant ##\sqrt{-kτ}##?
2) How can I get the order of the Bessel function ##ν##?
3) How did he simplify the mode function to eq (166)?
Compare eq. 162 with the Bessel equation and read off nu and the normalization from there. The mode eq 166 results from the fact that Bessel functions of order 3/2 reduce to sines and cosines.
 
bapowell said:
Compare eq. 162 with the Bessel equation and read off nu and the normalization from there. The mode eq 166 results from the fact that Bessel functions of order 3/2 reduce to sines and cosines.
If I compare it, I'm just getting ##ν = \frac{\sqrt{2-ε}}{1-ε}##.
 
Hrm. Can you write down here the Bessel equation against which you are comparing Eq. 162?
 
bapowell said:
Hrm. Can you write down here the Bessel equation against which you are comparing Eq. 162?
##τ^2 u''_k + τ u'_k + ( τ^2 - p^2 ) u_k = 0##

Are there any other forms? I thought this is the form of the Bessel differential equation?
 
Sure. By changing the "time" variable, you can get rid of the [itex]u'_k[/itex] term, for example. You need to do this in order to compare with Eq. 162, which has no first-order term.
 
bapowell said:
Sure. By changing the "time" variable, you can get rid of the [itex]u'_k[/itex] term, for example. You need to do this in order to compare with Eq. 162, which has no first-order term.
Yes, I compared it with the bessel equation without the first order term. But I have solved my question 2 by using an alternate form of the bessel equation which I found in Boas's book eq 16.1. But I'm yet to solve my other questions.
 
For question 3, look up the Bessel functions of order 3/2. How are they written?
 
bapowell said:
For question 3, look up the Bessel functions of order 3/2. How are they written?
Sorry for the late reply because my laptop was broken. So, I have already worked out my questions but I have another question on equation (173), I can't find out how the power spectrum was derived. I mean, I don't know how he got from the first integral of (172) to the second integral.
 
  • #10
Figaro said:
I don't know how he got from the first integral of (172) to the second integral.

Assume spherical symmetry in k-space, change to spherical coordinates in k-space, and do the angular integration.
 
  • #11
George Jones said:
Assume spherical symmetry in k-space, change to spherical coordinates in k-space, and do the angular integration.
Before that, where can I read more about the two point correlation function? How about how to transform to Fourier space? I'm still new to this since I've just finished Cal I-III , DE and LA. Boas's treatment of Fourier analysis is way too basic so I haven't encountered those things. I'm really having a hard time understanding this lecture notes on Inflation. What do you recommend I should do/read to supplement this lecture notes? The latter part of Kinney's notes are very hard to understand, he skips a lot of details.
 
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  • #12
Figaro said:
Before that, where can I read more about the two point correlation function? How about how to transform to Fourier space?

You might try the second edition of "Fundamentals of Cosmology" by James Rich, pages 248 - 252, 258.

https://www.amazon.com/dp/3642425747/?tag=pfamazon01-20
 
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  • #13
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