Wave numbers of Power Spectrum

In summary, the conversation discusses the use of Fourier modes to describe density fluctuations in cosmology and their usefulness in simplifying calculations and predicting the power spectrum of the cosmic microwave background (CMB). Baryonic acoustic oscillations (BAOs) are observed in the power spectrum as a deviation from homogeneity at a preferred scale, and this is also seen in the distribution of dark matter due to its interaction with normal matter through gravity.
  • #1
Madster
22
0
Hi.

I read some basic cosmology where it is always said that density fluctuations, pertubations can be described in modes of waves. In particular if you use linearised theory where δ(x,t) is Fourier transformed δ(k,t).

What exactly is the reason for this? What do the wave modes describe?

Thanks
 
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  • #2
Here, k is the spatial frequency (units of length-1). You're just decomposing the (3D) density field into Fourier modes (sinusoids), in the same way that you are probably familiar with doing for 1D and 2D signals. This is useful because δ(k,t) looks at the amplitude of the perturbations as a function of spatial scale rather than as a function of position in space. Here, large values of k are modes corresponding to small spatial scales, and small values of k are modes corresponding to large spatial scales. The point of a power spectrum is to tell you how much power there is (in the density perturbations) as function of spatial scale.
 
  • #3
Madster said:
Hi.

I read some basic cosmology where it is always said that density fluctuations, pertubations can be described in modes of waves. In particular if you use linearised theory where δ(x,t) is Fourier transformed δ(k,t).

What exactly is the reason for this? What do the wave modes describe?

Thanks
The main benefit here is that in the linear approximation, the waves of different wavelengths behave completely independently. That independence let's us figure out what happens for sub-components of the universe without worrying about the extremely complicated whole, which dramatically simplifies the calculations needed to compute how the physics impacts how these density perturbations change over time.

The linear approximation is valid at long wavelengths and early times. It is especially useful in predicting the power spectrum of the CMB (though with a slight addendum: a spherical harmonic transform is used instead of a Fourier transform, but the concept is the same).
 
  • #4
Thank you very much, another question would be what is observed in the power spectrum.

There is imprinted the baryonic acoustic oscillations. What can I imagine this to be? Is it an amplification of fluctuations at a certain scale of approx. 100 Mpc/h ?
 
  • #5
Madster said:
There is imprinted the baryonic acoustic oscillations. What can I imagine this to be? Is it an amplification of fluctuations at a certain scale [...] ?

That is precisely what BAOs are. A bump or bumps in the power spectrum of the distribution of matter in the universe (as determined by galaxy counts from large-scale surveys). A deviation from smoothness or homogeneity corresponding to a preferred scale: the scale of the sound horizon at the time of recombination.
 
  • #6
Madster said:
Thank you very much, another question would be what is observed in the power spectrum.

There is imprinted the baryonic acoustic oscillations. What can I imagine this to be? Is it an amplification of fluctuations at a certain scale of approx. 100 Mpc/h ?
This is a good website with pretty graphs and images that shows how the BAO signal comes about:
http://astro.berkeley.edu/~mwhite/bao/
 
  • #7
Chalnoth said:
This is a good website with pretty graphs and images that shows how the BAO signal comes about:
http://astro.berkeley.edu/~mwhite/bao/

Thanks the last link hosts really good stuff,
but can you expain to me why there is also a BAO bump in the dark matter distribution? I mean it should not interact with photons by definition, right?
 
  • #8
Madster said:
Thanks the last link hosts really good stuff,
but can you expain to me why there is also a BAO bump in the dark matter distribution? I mean it should not interact with photons by definition, right?
Dark matter interacts with normal matter through gravity. So over time the normal matter bump attracts some dark matter into it (though not nearly as much as the normal matter is attracted to the much larger dark matter bump).
 

1. What are wave numbers in a power spectrum?

Wave numbers in a power spectrum refer to the spatial frequencies of a signal or data set. They are calculated by taking the inverse of the wavelength or distance between two consecutive peaks in the signal.

2. Why are wave numbers important in power spectra analysis?

Wave numbers are important because they allow for the identification of specific frequencies present in a signal or data set. This can provide insight into the underlying patterns or dynamics of the data.

3. How are wave numbers calculated in a power spectrum?

Wave numbers are typically calculated using mathematical formulas or algorithms, such as the Fourier transform. This converts the signal from its original time or space domain into the frequency domain, where wave numbers can be identified and measured.

4. What is the relationship between wave numbers and power in a spectrum?

The power in a spectrum is directly related to the amplitude of the wave numbers present in the signal. This means that higher wave numbers (corresponding to higher frequencies) will have a greater impact on the overall power of the spectrum.

5. Are wave numbers affected by the size or duration of the data set?

Yes, the size and duration of a data set can impact the wave numbers present in the power spectrum. A larger data set may contain more unique frequencies, resulting in a wider range of wave numbers. Similarly, a longer duration of data may capture more cycles of a repeating pattern and result in higher wave numbers.

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