Wednesday, September 1, 2010

Cosmologists Love Fourier/Harmonic Space. (Large and Small Scales.)

Cosmologists hardly ever work in real space, meaning, where the variables in the equations are describing real points in the sky.  Instead, cosmologists almost always map their equations to Fourier or Harmonic Space.

But why is this?  There are two answers that both boil down to the universe being homogeneous and isotropic.  First, from the mathematical side, in an isotropic universe Fourier modes decouple making all equations painless to solve.  On the more physical side and intuitive side, since the universe is homogeneous and isotropic, it doesn't make sense to study physics that would affect certain spots in the sky.   Instead it is more meaningful to study physical processes that affect large scale structures on all parts of the sky, or small scale structures on all parts of the sky.

For example, inflation affected the largest scales over the entire sky.  On the other hand, the physics that went on the in primordial photon-baryon plasma effects only the smallest scales.

And this information is exactly what is encoded in the various Fourier/harmonic modes!  This information

Enough Confusing Talk, Give Me Something To Look At!

Okay, even if you got lost in the technical jargon above, at least look at these plots.  Here you see what the CMB looks like if you restrict to specific harmonic modes.  In harmonic space, L becomes your variable instead of x as in real space.  Small L modes refer to large scale information and large L modes refer small scale information.  If you think this is sounds backwards start reading some papers by astronomers. :)

This first plot above shows the information in the CMB for the harmonic modes L between 2 and 10.  As you can see, only information about large scale structure is contained in these modes.  For this reason, the low l modes are very valuable for studying inflation as these structures are not in casual contact today.

The next plot above contains information in the CMB for the harmonic modes L between 2 and 100.   Now we begin to see some small scale structure... on top of that large scale structure we had in the low l modes.

The next plot contains information out to L = 3000.  Now we begin to see how physics affected the very smallest scales. (And if I made this image was in Hi-Res it would be even more impressive!)

Lastly, the plot below shows the CMB dropping the L modes less then 100.  And as one could have guessed, you see no large scale structure!  (Look closely at the plot below and compare to the plot just above and convince yourself this is true.)  These modes tell you nothing about physics that effected the whole sky but everything about physics going on at the smallest scales in the universe.

So, by studying different L harmonic modes, or k Fourier modes, cosmologists can quickly see how physics played out across our universe.  Remember, as there are no preferred positions or directions in the sky, real space information is not very useful. However, physics that exists across the whole sky but only effects large or small scales, like inflation or baryon acoustic oscillations respectively, are brought out quite nicely in harmonic or Fourier space.  And thats why cosmologists love to work in these spaces.


  1. Those of us that work in fluid dynamics also love wavenumber-space. In fact the only real breakthrough in turbulence in the past 75 years, the Kolmogorov cascade, is expressed in wavenumber-space because, like the CMB small-scale turbulence has no preferred direction or position.

  2. Nick that is interesting as I would have never guessed there is no preferred direction in turbulence. (When I think of turbulence I think of wind swirling around an airplane causing it to shake a little. Not the most optimal thing to think about I am sure.)

    Next time I am on a plane I have a new conversation topic. By the time we land I will have people around me taking Fourier transforms so fast people will be calling them FFT.

  3. As the passengers cling on to their seats and grab their children when the turbulence gets bad I will just yell out "Fear not, this can all be better understood in k-space so listen closely to my words."

  4. Interesting Nick this Kolmogorov Cascade, Lewis Fry Richardson (1922)is also interesting for description of multi-scale turbulence.

  5. Cartesian,

    And if I remember right you have an interest in fractals which have interesting scaling properties.


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