<p>The power spectrum and the correlation function are related by</p>
<p><span class="math-container">$$\xi(r) = {1 \over 2\pi^2} \int dk\ k^2 P(k) {\sin(kr) \over kr}$$</span></p>
<p>They are (sort of) the Fourier transform of each other. It should not surprise, then, that a peak in one becomes a series of oscillations in the other. Just like the fourier transform of a sinusoidal wave is a dirac delta.</p>
<p>Keep also in mind that the first figure in the question is not exactly the power spectrum. As indicated by the y-axis label, a baseline has been subtracted and it is showing only the oscillations over that baseline.</p>
<p>The full power spectrum looks like this one:</p>
<p><img src="https://upload.wikimedia.org/wikipedia/commons/thumb/e/e9/Planck_2018_Linear_Matter_Power_Spectrum.pdf/page1-800px-Planck_2018_Linear_Matter_Power_Spectrum.pdf.jpg" alt="enter image description here" /></p>

CC BY-SA 4.0
2021-10-11T13:25:00.573Z
I have a one more question. What does the k means in terms of physics?
BAO
CC BY-SA 4.0
2021-10-11T21:45:07.417Z
@BAO $k$ is the comoving wavenumber. We are moving from physical space to fourier space. We describe the density function as a sum (integral) of different oscillating components $\sin(kr)$. The component with wavenumber $k$ corresponds to an oscillation with physical wavelength $\lambda = 2\pi a / k$. The bigger $k$, the smaller the oscillation.
Prallax
CC BY-SA 4.0
2021-10-11T21:50:09.227Z
Please, tell me if you'd like a more thorough explanation of the words I used
Prallax
CC BY-SA 4.0
2021-10-09T06:20:41.193Z
2021-10-09T06:33:08.427Z
2021-10-09T06:33:08.427Z
Prallax
<p><a href="https://i.sstatic.net/Fvo7y.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/Fvo7y.png" alt="enter image description here" /></a></p>
<p>(<a href="https://www.ias.u-psud.fr/Dark_energy/presentations/castanderBAO_081124.pdf" rel="nofollow noreferrer">source</a>)</p>
<p>There is the sentence &quot;the harmonic sequence in the power spectrum correspond to a single peak in the correlation function and that the damping envelope corresponds to the broadening of this peak&quot; in &quot;Eisenstein et al. (2007) APJ_664_675&quot;.</p>
<p>Before, I misunderstood that the BAO peak in the correlation function has many peaks but other peaks are so small that we cannot see these except for  the 150Mpc peak.</p>
<p>But this paper states &quot;a single peak in the correlation function&quot;, so I want to ask: <strong>Why are there many peaks in the power spectrum, but only a single peak in the correlation function?</strong></p>

<cosmology>
What's different of power spectrum and correlation function of BAO?
BAO
# What's different of power spectrum and correlation function of BAO?

(Asked by: username_0 on 2021-10-09)

[![enter image description here](https://i.sstatic.net/Fvo7y.png)](https://i.sstatic.net/Fvo7y.png)

([source](https://www.ias.u-psud.fr/Dark_energy/presentations/castanderBAO_081124.pdf))

There is the sentence "the harmonic sequence in the power spectrum correspond to a single peak in the correlation function and that the damping envelope corresponds to the broadening of this peak" in "Eisenstein et al. (2007) APJ\_664\_675".

Before, I misunderstood that the BAO peak in the correlation function has many peaks but other peaks are so small that we cannot see these except for the 150Mpc peak.

But this paper states "a single peak in the correlation function", so I want to ask: **Why are there many peaks in the power spectrum, but only a single peak in the correlation function?**

**Tags:** cosmology

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## 1 answers

### Answer by username_1 on 2021-10-09. Score: 2 | Accepted answer

The power spectrum and the correlation function are related by

$$\xi(r) = {1 \over 2\pi^2} \int dk\ k^2 P(k) {\sin(kr) \over kr}$$

They are (sort of) the Fourier transform of each other. It should not surprise, then, that a peak in one becomes a series of oscillations in the other. Just like the fourier transform of a sinusoidal wave is a dirac delta.

Keep also in mind that the first figure in the question is not exactly the power spectrum. As indicated by the y-axis label, a baseline has been subtracted and it is showing only the oscillations over that baseline.

The full power spectrum looks like this one:

![enter image description here](https://upload.wikimedia.org/wikipedia/commons/thumb/e/e9/Planck_2018_Linear_Matter_Power_Spectrum.pdf/page1-800px-Planck_2018_Linear_Matter_Power_Spectrum.pdf.jpg)

> **Comments:**
> **username_0**: I have a one more question. What does the k means in terms of physics?  
> **username_1**: @BAO $k$ is the comoving wavenumber. We are moving from physical space to fourier space. We describe the density function as a sum (integral) of different oscillating components $\sin(kr)$. The component with wavenumber $k$ corresponds to an oscillation with physical wavelength $\lambda = 2\pi a / k$. The bigger $k$, the smaller the oscillation.  
> **username_1**: Please, tell me if you'd like a more thorough explanation of the words I used  

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