Archive for Power spectrum

The Power Spectrum and the Cosmic Web

Posted in Bad Statistics, The Universe and Stuff with tags , , , , , , on June 24, 2014 by telescoper

One of the things that makes this conference different from most cosmology meetings is that it is focussing on the large-scale structure of the Universe in itself as a topic rather a source of statistical information about, e.g. cosmological parameters. This means that we’ve been hearing about a set of statistical methods that is somewhat different from those usually used in the field (which are primarily based on second-order quantities).

One of the challenges 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. In effect, the power spectrum is insensitive to the part of the Fourier description of the pattern that is responsible for delineating the cosmic web.

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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SPT and the CMB

Posted in The Universe and Stuff with tags , , , , , , , , , on November 30, 2012 by telescoper

I’ve been remiss in not yet passing on news  from the South Pole Telescope, which has recently produced a number of breakthrough scientific results, including:  improved cosmological constraints from the SPT-SZ cluster survey (preprint here); a new catalogue of 224 SZ-selected cluster candidates from the first 720 square-degrees of the survey (preprint here); the first measurement of galaxy bias from the gravitational lensing of the CMB (preprint here); the first CMB-based constraint on the evolution of the ionized fraction during the epoch of reionization (preprint here); the most-significant detection of non-Gaussianity induced from the gravitational lensing of the CMB (preprint here); and the most precise measurement of the CMB damping tail and improved constraints on models of Inflation (preprint here).

Here’s the graph that drew my eye (from this paper). It shows the (angular) power spectrum of the cosmic microwave for very high (angular) frequency spherical harmonics; the resolution of SPT allows it to probe finer details of the spectrum that WMAP (also shown, at lower l).

Slide1

This is an amazing graph, especially for oldies like me who remember being so impressed by the emergence of the first “acoustic peak” at around l=200 way back in the days of Boomerang and Maxima and gobsmacked by WMAP’s revelation of the second and third. Now there are at least six acoustic peaks, although of progressively lower amplitude. The attenuation of the CMB fluctuations at high frequencies is the result of diffusion damping – similar to the way high-frequency sound waves are attenuated when they pass through a diffusive medium (e.g. a gas).  The phenomenon in this case is usually called Silk Damping, as it was first worked out back in the 1960s by Joe Damping Silk.

Anyway, there’ll be a lot more CMB news early (?) next year from Planck which will demonstrate yet again that cosmic microwave background physics has certainly come a long way from pigeon shit

Power versus Pattern

Posted in Bad Statistics, The Universe and Stuff with tags , , , , , on June 15, 2012 by telescoper

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…