# Wave numbers of Power Spectrum

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

## Answers and Replies

cepheid
Staff Emeritus
Science Advisor
Gold Member
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.

Chalnoth
Science Advisor
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 lets 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).

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 ?

cepheid
Staff Emeritus
Science Advisor
Gold Member
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.

Chalnoth
Science Advisor
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/

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?

Chalnoth
Science Advisor
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).