## Phase Correlations and Cosmic Structure

I’m indebted to a friend for tipping me off about a nice paper that appeared recently on the arXiv by Franco et al. with the title *First measurement of projected phase correlations and large-scale structure constraints. *The abstract is here:

Phase correlations are an efficient way to extract astrophysical information that is largely independent from the power spectrum. We develop an estimator for the line correlation function (LCF) of projected fields, given by the correlation between the harmonic-space phases at three equidistant points on a great circle. We make a first, 6.5σ measurement of phase correlations on data from the 2MPZ survey. Finally, we show that the LCF can significantly improve constraints on parameters describing the galaxy-halo connection that are typically degenerate using only two-point data.

I’ve worked on phase correlations myself (with a range of collaborators) – you can see a few of the papers here. Indeed I think it is fair to say I was one of the first people to explore ways of quantifying phase information in cosmology. Although I haven’t done anything on this recently (by which I mean in the last decade or so), other people have been developing very promising looking approaches (including the Line Correlation Function (LCF) explored in the above paper. In my view there is a lot of potential in this approach and as we await even more cosmological data and hopefully more people will look at this in future. In my opinion we still haven’t found the optimal way to exploit phase information statistically so there’s a lot of work to be done in this field.

Anyway, I thought I’d try to explain what phase correlations are and why they are important.

One of the challenges we cosmologists face is how to quantify the patterns we see in, for example, 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 realize 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…

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