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

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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}##.
 
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.
 
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.
 
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.
 
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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