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

In summary, the conversation discusses the TASI Lectures on Inflation by William Kinney, focusing on the mode function and its simplification to eq (166). Questions are raised about the normalization constant ##\sqrt{-kτ}## and the order of the Bessel function ##ν##, which can be found by comparing eq. 162 with the Bessel equation. The conversation also touches on the power spectrum and the two point correlation function, with a suggested resource for further reading being the book "Fundamentals of Cosmology" by James Rich.
  • #1
Figaro
103
7
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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  • #2
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.
 
  • #3
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}##.
 
  • #4
Hrm. Can you write down here the Bessel equation against which you are comparing Eq. 162?
 
  • #5
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?
 
  • #6
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.
 
  • #7
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.
 
  • #8
For question 3, look up the Bessel functions of order 3/2. How are they written?
 
  • #9
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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Related to How to Understand the Primordial Power Spectrum in TASI Lectures on Inflation?

1. What is a primordial power spectrum?

A primordial power spectrum is a fundamental concept in cosmology that describes the distribution of density fluctuations in the early universe. It is a measure of the fluctuations in the density of matter and radiation that existed shortly after the Big Bang, and it is a key component in understanding the formation of large-scale structures in the universe.

2. How is the primordial power spectrum measured?

The primordial power spectrum is measured using observations of the cosmic microwave background (CMB) radiation, which is the leftover heat from the Big Bang. Scientists use specialized instruments, such as the Planck satellite, to measure the temperature fluctuations in the CMB and extract information about the primordial power spectrum.

3. What is the shape of the primordial power spectrum?

The shape of the primordial power spectrum is typically described as a power law, meaning that the amplitude of fluctuations at different length scales follows a specific mathematical relationship. The exact shape of the power spectrum is determined by the initial conditions of the universe, such as the amount of matter and radiation present at the time of the Big Bang.

4. What can the primordial power spectrum tell us about the early universe?

The primordial power spectrum provides valuable insights into the physical processes that were at play in the early universe. By analyzing its shape and amplitude, scientists can learn about the properties of dark matter and dark energy, the expansion rate of the universe, and the nature of inflation, which is the rapid expansion of the universe in its early stages.

5. How does the primordial power spectrum affect the formation of galaxies and other structures?

The primordial power spectrum is directly related to the formation of large-scale structures in the universe, such as galaxies, clusters of galaxies, and superclusters. The fluctuations in the density of matter and radiation described by the power spectrum serve as the seeds for the formation of these structures, and their distribution and amplitude determine the overall structure of the universe we see today.

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