## Inverted Cosmology

Just time for a quick post about a neat little paper by Pontzen et al. that has appeared on the arXiv. Here is the abstract:

The abstract is a model of clarity so there’s no need to add further explanation here. Having A and B simulations in which initial overdensities and underdensities are swapped but everything else is preserved allows a number of interesting things to be studied.

When I read the paper it struck me that it would be fun to use “paired” simulations like this to study statistical properties of the evolved density field that go beyond the usual power spectra discussed in the paper; you can find a nice review of power spectra and their uses here.

Here’s what I mean. 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 shown on the left. 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. These features are manifestations of non-linear dynamics and are not described by linear perturbation theory.

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, i.e. all the information contained in the distribution of phases. However, studying the evolution of Fourier phases in the context of non-linear gravitational evolution is quite tricky for a number of technical reasons. Note that the “paired” simulations of Pontzen et al. are generated in such a way that the A and B simulations also have the same power spectrum, but unlike those shown above, have the same type of morphology, which might allow one to finesse some of these difficulties and separate out the effect of non-linear dynamics from the choice of initial power spectrum in a potentially interesting way.

Just a thought.

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November 18, 2015 at 5:15 am

I am no astrophysicist, but it seems to me the whole point of filamentation and other gravitational clustering processes is that htey spatial-domain clustering, and there’s no reason for their frequency spectrum to contain any more information than their size-scale.

November 18, 2015 at 10:52 am

Not sure I get your point….

November 20, 2015 at 1:38 pm

[…] the little post I did on Tuesday in reaction to a nice paper on the arXiv by Pontzen et al., my attention was drawn today to another […]