Wavelets for studying the large-scale structure (LSS) of the Universe

Post by Jason McEwen

Why wavelets?

It can be insightful to analyse data, or “signals”, in different domains. For example, the frequency spectrum of music is often studied, where contributions to the base and treble are more clearly visible.

Caption

Music in time (lower panel) and frequency (upper panel).  These representations can be computed on your iPhone.

In the Fourier domain, we can probe the frequency content of signals but lose information about the time localisation of signal structure. In the time domain, the reverse is true: we can probe the time content of signals but lose information about the frequency localisation of signal structure.  Wavelets overcome this problem by looking at signal content in time and frequency (scale) simultaneously.  

Often, we may be interested in signals that are defined on domains other than time.  For example, a standard image is a two-dimensional signal defined on a spatial domain.  Nevertheless, for such signals it can also be insightful to view the signal content in the frequency domain, rather than the spatial domain, or in the wavelet domain.

Many physical processes are manifest on particular physical scales, while also spatially localised.  Wavelets are therefore a powerful analysis tool for extracting the fingerprint of a physical process of interest when it is embedded in some background signal.

Wavelet analysis of the CMB

Wavelets have now become a standard analysis technique for studying the anisotropies of the cosmic microwave background (CMB).

In this setting, the signal of interest (the CMB) is defined on the celestial sphere.  We therefore need wavelet transforms defined on the sphere.

CMB

Temperature anisotropies of the CMB defined on the celestial sphere. [Credit: WMAP]

A number of wavelet transforms have been defined on the sphere.  The construction of many of these analysis methods has been motivated directly by the desire to study the CMB but these techniques are of general use for studying signals on the sphere, such as observations made in geophysics and computer graphics.  

Wavelets defined on the sphere are now a prevalent analysis technique for studying the CMB.  In fact, many of the cosmological studies performed in the 2013 analysis and release of Planck data used wavelet methods .

Exact wavelet on the ball for studying LSS

The large-scale structure (LSS) of the Universe, as traced by the distribution of galaxies, is another powerful cosmological probe.  Observations tracing the LSS are made in three-dimensions, with the radial dimension measuring redshift.  These observations therefore also live in spherical space and are made on the ball, i.e. on the sphere augmented with depth information.

LSS

Observations tracing the LSS defined on the ball. [Credit: SDSS]

Recently, we have developed wavelet methods defined on the ball for the purpose of extracting cosmological information from observations tracing the LSS.   More technical details on wavelets on the ball will appear in a future post.

We hope that these types of methods can prove as useful for studying the LSS as they have for studying the CMB.

We’re now busy applying them for various cosmological analyses and will keep you posted!

Related reading

B. Leistedt, J. D. McEwen, Exact Wavelets on the Ball

J. D. McEwen, B. Leistedt, Fourier-Laguerre transform, convolution and wavelets on the ball

F. Lanusse, A. Rassat, J.-L. Starck, Spherical 3D isotropic wavelets

B. Leistedt, H. V. Peiris, J. D. McEwen, Flaglets for studying the large-scale structure of the Universe

Y. Wiaux, J. D. McEwen, P. Vandergheynst, O. Blanc, Exact reconstruction with directional wavelets on the sphere

B. Leistedt, J. D. McEwen, P. Vandergheynst, Y. Wiaux, S2LET: A code to perform fast wavelet analysis on the sphere,

D. Marinucci, D. Pietrobon, A. Balbi, P. Baldi, P. Cabella, G. Kerkyacharian, P. Natoli, D. Picard, N. Vittorio, Spherical needlets for CMB data analysis

J.-L. Starck, Y. Moudden, P. Abrial, M. Nguyen, Wavelets, ridgelets and curvelets on the sphere

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The power of three

Post by Tom Kitching

Today a paper of mine that I have been working on for the last few years was posted to the arXiv.

The subject of the paper is a method called “3D cosmic shear”. It’s a particularly important subject because there are several new telescopes, and surveys that have been designed, or are being built, that intend to use such methods to measure the properties of dark energy. What we found in this paper is that some of the assumptions that have been made about how dark matter and ordinary matter interact in the cosmic web could be incorrect, which means the interpretation of the data from the new experiments will be a bit more complicated than cosmologists previously thought…

What is 3D cosmic shear?

Let’s start with the 3D part of the name. The Universe is filled with galaxies, which are sprinkled throughout the cosmic web of dark matter (one may say sprinkled like raisins in the infinite hot-cross-bun that is the Universe, but let’s not push the analogy too far).

Plot credit: S. Columbi and Y. Mellier. Matter is distributed as a cosmic web (orange) and galaxies are sprinkled within this web (blue dots). Light from the galaxies (yellow lines) is distorted by the gravitational field of the web and causes a small change in the ellipticity of the galaxy images.

Plot credit: S. Columbi and Y. Mellier. Matter is distributed as a cosmic web (orange) and galaxies are sprinkled within this web (blue dots). Light from the galaxies (yellow lines) is distorted by the gravitational field of the web and causes a small change in the ellipticity of the galaxy images.

We live in one of those galaxies, the Milky Way, so as we observe the Universe what we see are galaxies scattered across the sky (2-dimensions, north-south and east-west, or “right ascension” and declination). We also see galaxies in a 3rd dimension of redshift. This 3rd dimension is a little more complicated to understand: because the speed of light is finite the galaxies we see that are a long way away in the 3rd dimension are not only distant in a space but also distant in time. The 3D distribution of galaxies we see is consists of galaxies that are scattered across space and time.

3D2

But what else happens to light as it propagates towards us? Well, all of the matter from which the cosmic web is made, like all matter everywhere, has a gravitational field, and gravity is a force that pulls things together. You are being pulled by Earths gravity to the ground, the Earth (and you) are being pulled by the Suns gravity, the Sun (and you, and the Earth) are being pulled by all of the other stars in the Milky Way… and so on and so forth. Light doesn’t escape the universal attraction of gravity, and it is also pulled towards objects that have mass. The effect of gravity on a light ray is known as gravitational lensing, what happens (when the gravitational field is weak) is that an additional ellipticity (or “oval-ness”, technically this is the third flattening or third eccentricity of the galaxies observed shape) is imprinted on the image of a distant galaxy. We call this extra ellipticity “shear”.

Plot credit: T. Tyson. The light from distant galaxies is gravitationally lensed by the cosmic web of dark matter, inducing a small change in the ellipticity of the images of galaxies called cosmic shear.

Plot credit: T. Tyson. The light from distant galaxies is gravitationally lensed by the cosmic web of dark matter, inducing a small change in the ellipticity of the images of galaxies called cosmic shear.

The additional ellipticity caused by the cosmic web is known cosmic shear.

3D cosmic shear combines both information from the distances and spatial distribution of galaxies and the gravitational lensing effect of cosmic shear into a single analysis, and in the paper this was applied to a large survey for the first time. Because 3D cosmic shear maps the cosmic web and how it grows with time we can potentially learn about how the Universe evolved, and what its ultimate fate will be.

What did we find?

The paper used the method of 3D cosmic shear to analyse the largest/deepest optical wavelength survey of the sky to date CFHTLenS.

What we found is that matter in the Universe is clumped together less tightly than it should be if dark matter alone was responsible for the structure of the cosmic web.

Previously it has been assumed that the cosmic web of dark matter is not affected by the sprinkling of galaxies on top of it. However what we found is that there is evidence that the cosmic web of dark matter may actually be affected by the galaxies after all, in particular by the supermassive black holes that reside in the centres of the most massive galaxies, so-called Active Galactic Nuclei (AGN).

AGN may be impacting the distribution of dark matter in the Universe. Hubble Space Telescope image of a 5000-light-year-long (1.5-kiloparsec-long) jet being ejected from the active nucleus of the active galaxy M87, a radio galaxy. The blue synchrotron radiation of the jet contrasts with the yellow starlight from the host galaxy.

Alternatively things may be even weirder, another possibility is that what we are seeing the effect of massive neutrinos, or even a combination of effects. But a new mystery, and challenge for cosmologists, now exists. The normal matter and the dark matter seem to have an effect on each other: we may live in a more complicated Universe than we previously imagined. To resolve this mystery more data is now required, and more work to make the data analysis using 3D cosmic shear even more sensitive to the structure of the cosmic web within which we reside.