Power versus Pattern

One of the challenges we cosmologists face is how to quantify the patterns we see in galaxy redshift surveys. In the relatively recent past the small size of the available data sets meant that only relatively crude descriptors could be used; anything sophisticated would be rendered useless by noise. For that reason, statistical analysis of galaxy clustering tended to be limited to the measurement of autocorrelation functions, usually constructed in Fourier space in the form of power spectra; you can find a nice review here.

Because it is so robust and contains a great deal of important information, the power spectrum has become ubiquitous in cosmology. But I think it’s important to realise its limitations.

Take a look at these two N-body computer simulations of large-scale structure:

The one on the left is a proper simulation of the “cosmic web” which is at least qualitatively realistic, in that in contains filaments, clusters and voids pretty much like what is observed in galaxy surveys.

To make the picture on the right I first  took the Fourier transform of the original  simulation. This approach follows the best advice I ever got from my thesis supervisor: “if you can’t think of anything else to do, try Fourier-transforming everything.”

Anyway each Fourier mode is complex and can therefore be characterized by an amplitude and a phase (the modulus and argument of the complex quantity). What I did next was to randomly reshuffle all the phases while leaving the amplitudes alone. I then performed the inverse Fourier transform to construct the image shown on the right.

What this procedure does is to produce a new image which has exactly the same power spectrum as the first. You might be surprised by how little the pattern on the right resembles that on the left, given that they share this property; the distribution on the right is much fuzzier. In fact, the sharply delineated features  are produced by mode-mode correlations and are therefore not well described by the power spectrum, which involves only the amplitude of each separate mode.

If you’re confused by this, consider the Fourier transforms of (a) white noise and (b) a Dirac delta-function. Both produce flat power-spectra, but they look very different in real space because in (b) all the Fourier modes are correlated in such away that they are in phase at the one location where the pattern is not zero; everywhere else they interfere destructively. In (a) the phases are distributed randomly.

The moral of this is that there is much more to the pattern of galaxy clustering than meets the power spectrum…

9 Responses to “Power versus Pattern”

  1. Hello Peter – what you say is obviously true in the non-linear regime where modes couple with each other. But in the linear regime, where (assuming no primordial non-Gaussianity), modes evolve completely independently, it isn’t important. (I think you made this point yourself some time ago.)

    By the time we get to non-linear scales, our theoretical prediction for what the auto-correlation function should look like, given an assumed model of inflation, already starts to be a bit suspect does it not? So to use the power spectrum at small scales we’d have to be certain we could actually calculate the thing properly in the first place (and even then, as you say, it doesn’t contain all the information available).

    But so long as we stick to large enough scales that everything is linear, using the power spectrum should be all we need … what is the scale of the simulation in your picture?

    • telescoper Says:

      On large scales, where everything is linear, and if the initial fluctuations are Gaussian then the power-sepctrum is all you need; it’s a complete statistical characterization of the density field.

      The problem is if you want to quantify the small-scale non-linear distribution and/or if the initial fluctuations are non-Gaussian you need to go beyond the power-spectrum. So what you need depends on what questions you want to ask.

  2. I use an illustration like this in talks quite often. I generally use a picture of the person who invited me or some easily recognizable big shot as the input (non-random phase) picture.

    • telescoper Says:

      Indeed. It’s also fun to take two pictures, FT them, swap the phases between the two, and then inverse FT…

      …the phases entirely determine the morphology.

    • I remember seeing them in a talk given by Peter at a conference (with mostly radio astronomers in the audience) in the Netherlands about 12 or 13 years ago. Later that night, I remember seeing other images.

  3. You can’t have randomly shuffled all the phases – you’d end up with a complex galaxy distribution if you did not preserve the symmetries to make the result real.

    Nice illustration tho!

    • telescoper Says:

      Good point. In fact I randomly re-shuffled conjugate pairs of phases to preserve the relations needed to ensure the map is real.

  4. Anton Garrett Says:

    “if you can’t think of anything else to do, try Fourier-transforming everything.”

    Here’s another: If you have a positive-definite quantity about which you are uncertain, normalise it, consider the result a probability density and apply the maximum entropy algorithm to it.

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