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A 'galaxy of galaxies' from the Mandelbrot Set
In physical cosmology, fractal cosmology is a set of minority cosmological theories which state that the distribution of matter in theUniverse, or the structure of the universe itself, is a fractal. More generally, it relates to the usage or appearance of fractals in the study of the universe and matter. A central issue in this field is the fractal dimension of the Universe or of matter distribution within it, when measured at very large or very small scales.
The use of fractals to answer questions in cosmology has been employed by a growing number of serious scholars close to the mainstream, but the metaphor has also been adopted by others outside the mainstream of science, so some varieties of fractal cosmology are solidly in the realm of scientific theories and observations, and others are considered fringe science, or perhapsmetaphysical cosmology. Thus, these various formulations enjoy a range of acceptance and/or perceived legitimacy.
A debate still ensues, over whether the universe will become homogeneous and isotropic (or is smoothly distributed) at a large enough scale, as would be expected in a standard Big Bang or FLRWcosmology, and in most interpretations of the Lambda-CDM (expanding Cold Dark Matter) model. Scientific consensus interpretation is that the Sloan Digital Sky Survey suggests that things do indeed seem to smooth out above 100 Megaparsecs. Recent analysis of WMAP, SDSS, and NVSS data by a team from the University of Minnesota shows evidence of a void around 140 Megaparsecs across, however, coinciding with the CMB cold spot, which, if confirmed, calls the assumption of a smooth universe into question. However there are serious hints that the apparent cold spot is a statistical artifact.
In May 2008, another paper was published by a team including Pietronero, that concludes the large scale structure in the universe is fractal out to at least 100 Mpc/h. The paper asserts that the team has demonstrated that the most recent SDSS data shows "large amplitude density fluctuations at all scales" within that range, and that the data is consistent with fractality beyond this point, but inconsistent with a lower scale of homogeneity, or with predictions of large scale structure based solely on gravity. Their analysis shows the fractal dimension of the arrangement of galaxies in the universe (up to the range of 30 Mpc/h) to be about 2.1 (plus or minus 0.1).
However, an analysis of luminous red galaxies in the Sloane survey calculated the fractal dimension of galaxy distribution (on a scales from 70 to 100 Mpc/h) at 3, consistent with homogeneity; they also confirm that the fractal dimension is 2 "out to roughly 20 Mpc/h".
Fractals in theoretical cosmology
In the realm of theory, the first appearance of fractals in cosmology was likely with Andrei Linde’s "Eternally Existing Self-Reproducing Chaotic Inflationary Universe" theory (see Chaotic inflation theory), in 1986. In this theory, the evolution of a scalar field creates peaks that become nucleation points which cause inflating patches of space to develop into "bubble universes," making the universe fractal on the very largest scales. Alan Guth's 2007 paper on "Eternal Inflation and its implications" shows that this variety of Inflationary universe theory is still being seriously considered today. And inflation, in some form or other, is widely considered to be our best available cosmological model.
Since 1986, however, quite a large number of different cosmological theories exhibiting fractal properties have been proposed. And while Linde’s theory shows fractality at scales likely larger than the observable universe, theories like Causal dynamical triangulation and Quantum Einstein gravity are fractal at the opposite extreme, in the realm of the ultra-small near the Planck scale. These recent theories of quantum gravity describe a fractal structure for spacetime itself, and suggest that the dimensionality of space evolves with time. Specifically; they suggest that reality is 2-d at the Planck scale, and that spacetime gradually becomes 4-d at larger scales. French astronomer Laurent Nottale first suggested the fractal nature of spacetime in a paper on Scale Relativity published in 1992, and published a book on the subject of Fractal Space-Time in 1993.
French mathematician Alain Connes has been working for a number of years to reconcile Relativity with Quantum Mechanics, and thereby to unify the laws of Physics, using Noncommutative geometry. Fractality also arises in this approach to Quantum Gravity. An article by Alexander Hellemans in the August 2006 issue of Scientific American quotes Connes as saying that the next important step toward this goal is to "try to understand how space with fractional dimensions couples with gravitation." The work of Connes with physicist Carlo Rovelli suggests that time is an emergent property or arises naturally, in this formulation, whereas in Causal dynamical triangulation, choosing those configurations where adjacent building blocks share the same direction in time is an essential part of the 'recipe.' Both approaches suggest that the fabric of space itself is fractal, however.
The book Discovery of Cosmic Fractals by Yurij Baryshev and Pekka Teerikorpi gives an overview of fractal cosmology, and recounts other milestones in the development of this subject. It recapitulates the history of cosmology, reviewing the core concepts of ancient, historical, and modern astrophysical cosmology. The book also documents the appearance of fractal-like and hierarchal views of the universe from ancient times to the present. The authors make it apparent that some of the pertinent ideas of these two streams of thought developed together. They show that the view of the universe as a fractal has a long and varied history, though people haven’t always had the vocabulary necessary to express things in precisely that way.
Beginning with the Sumerian and Babylonian mythologies, they trace the evolution of Cosmology through the ideas of Ancient Greeks like Aristotle, Anaximander, and Anaxagoras, and forward through the Scientific Revolution and beyond. They acknowledge the contributions of people like Emanuel Swedenborg, Edmund Fournier D'Albe, Carl Charlier, and Knut Lundmark to the subject of cosmology and a fractal-like interpretation, or explanation thereof. In addition, they document the work of de Vaucoleurs, Mandelbrot, Pietronero, Nottale and others in modern times, who have theorized, discovered, or demonstrated that the universe has an observable fractal aspect.
On the 10th of March, 2007, the weekly science magazine New Scientist featured an article entitled "Is the Universe a Fractal?" on its cover. The article by Amanda Gefter focused on the contrasting views of Pietronero and his colleagues, who think that the universe appears to be fractal (rough and lumpy) with those of David Hogg of NYU and others who think that the universe will prove to be relatively homogeneous and isotropic (smooth) at a still larger scale, or once we have a large and inclusive enough sample (as is predicted by Lambda-CDM). Gefter gave experts in both camps an opportunity to explain their work and their views on the subject, for her readers.
This was a follow-up of an earlier article in that same publication on August 21 of 1999, by Marcus Chown, entitled "Fractal Universe.". Back in November 1994, Scientific American featured an article on its cover written by physicist Andrei Linde, entitled "The Self-Reproducing Inflationary Universe" whose heading stated that "Recent versions of the inflationary scenario describe the universe as a self-generating fractal that sprouts other inflationary universes," and which described Linde's theory of chaotic eternal inflation in some detail.
In July 2008, Scientific American featured an article on Causal dynamical triangulation, written by the three scientists who propounded the theory, which again suggests that the universe may have the characteristics of a fractal.
Causal dynamical triangulation
Chaotic inflation theory
Large scale structure
Shape of the universe
Science background of fractal light discovery
The original aims of this work were focused on the excess noise properties of microlasers [1-5]. A scaled-up experimental prototype of a microlaser is shown in figure 1. Such small lasers benefit from optimising their light amplification by employing so-called 'unstable cavities'. Within this type of cavity geometry, the circulating light expands to allow high overlap between the amplifying medium and the light itself - see part b) of figure 1. Since the light is repeatedly magnified inside the laser, there will be strong aperturing effects. This is because, as the light beam gets wider, some part of the laser cavity will act as an aperture on the circulating light. The role of the shape of this aperturing part is thus expected to be important for the microlaser output characteristics - see part c) of figure 1.
Fractal laser experiments
Figure 1. a) Experimental configuration of the fractal laser, b) magnification of the circulating light, c)shapes of the aperturing element.
For each shape of cavity aperturing, there was a comparison of theory and experimental results for the laser output. We were rather surprised when a detailed study of the light intensity patterns was undertaken. The columns of figure 2 show transverse laser light profiles when the aperturing element has the following shapes: triangular, rhomboid, pentagonal, hexagonal and octagonal. Looking down each column, one sees progressive development of additional small-scale details as the aspect ratio of each cavity is increased. Some of these output laser modes looked strikingly like 'laser snowflakes'!
Fractal laser modes
Figure 2. Cross-sections of laser beam profiles. Colour-coding is used to distinguish different light intensities.
Of course, real snowflakes are fractals (patterns with proportional levels of more detail when one looks closer and closer). Some further analyses of the laser patterns confirmed that these cavities did indeed result in fractal laser modes [6-19]. In fact, the underlying principle of fractal linear eigenmodes, in systems with magnification greater than one, may have much wider applications. For example, workers at the University of Glasgow were quick to spot that a similar principle could be employed to generate fractal patterns in video feedback systems, and that exact self-similar fractals were possible [20-24].
Further developments (fractal light research)
In more recent developments, the experimental group in Leiden extended their studies to examine cavity designs that permit a very wide range of fractality - i.e. a greater extent of smaller-scale details [25,26]. At Imperial College London, further investigations have included gaining a much better understanding of the role of the cavity and the mode characteristics in determining the fractal dimension of the resulting laser light [27-30]. A key limitation in our earlier theoretical investigations was that both semi-analytical and full-numerical modelling had been limited to relatively small aspect-ratio cavities (and hence examination of limited ranges of the fractality of the light). The central problem there was in the mathematical description of diffraction from many two-dimensional apertures of widely-varying size.
Magnification of fractal modes
Figure 3. New compact formulations of Fresnel diffraction now allow us to calculate fractal laser modes to an arbitrary level of accuracy. This figure shows: cross-sections of laser beam profiles (top row), and magnified central portions of these mode profiles (bottom row).
At the University of Salford, we subsequently derived two new compact formulations of Fresnel diffraction arising from closed apertures of arbitrary shape. This overcame the earlier modelling limitations, and has allowed us to calculate fractal laser mode patterns with an arbitrary level of detail [31,32]. The detail permitted is only limited by the Fresnel conditions themselves. Figure 3 shows results from some sample calculations, involving the following aperture shapes: triangle, pentagon, hexagon, and decagon. In each case, one can magnify the central details of the laser mode to see even more of the structure present (shown in the frames below).
Further developments at the University of Salford have included proposal and verification of a mechanism for the spontaneous generation of optical fractals in nonlinear systems . We have also been studying various novel configurations and effects in our Video Feedback Laboratory .info
A fractal is a geometric object which basic structure is repeated at different scales. Fractal objects can be natural or artificial (mathematical fractals). Examples of natural fractals are the coastlines, the mountains profiles, the rivers ramifications, the lightning bolts shape, some leaves, trees and vegetables. The main characteristics of a natural fractal is its approximate self-similarity. On the other hand, mathematical fractals are defined by a recursive algorithm and are detailed at any scale of observation; this phenomena is known as exact self-similarity.
THE ANDERSON TRANSITION
Image: The Sierpinski triangle is one of the well-known mathematical fractal by a clear example of exact self-similarity.
The study of the properties of disordered systems has been a field of active research in the last fifty years. One of the most fascinating aspects of these systems, in more than two dimensions, is the appearance of the metal-insulator transition due to the change of the amplitude of the disorder. This transition is known as theAnderson transition. n the metal phase, the internal states of the system are extended, which makes the system to behave like a conductor. In the insulator phase, the internal states are exponentially localised. The metal-insulator transition point, i.e. when the transition from localised to extended states occurs, is characterised by an interesting variety of critical properties. In particular, the internal states show fluctuation on every scale and represent fractal objects. In the transition point, the system is not a metal nor a insulator and the transport properties are considered abnormal.
Image: Example of an internal state for the three-dimensional Anderson model in the metal-insulator transition. In the critical point, the states are fractals but, unlike natural fractals (with approximate self-similarity) or the mathematical fractals (with exact self-similarity), in critical disordered systems the internal states show statistical self-similarity. Note that the critical state is extended in the interior of the cube, which defines the system's volume, but it does not occupy all the permitted volume. For this reason, we say critical states have an effective dimension (fractal dimension) which is smaller than the real dimension of the system, which in this case is 3. Image from L. J. Vasquez, A. Rodriguez, and R. A. Roemer, Phys. Rev. B 78 195106 (2008).
In CRITICALIDAD, Mexican researchers are interested in studying the effect of the fractality of critical states on the properties of electrons transport using systems at the Anderson transition. In particular, the researchers will study measurements of electrons transport like transmission, reflection, conductivity and other related measurements (such as, their distribution, variance, mean and quantum noise) as a function of some parameters of the system (linear size L, number of connected terminals and connection intensity).
Image: Three-dimensional arrangement of points connected to their first neighbours. Also known as three-dimensional Anderson model. In the example, the system linear size is L=3, however the total volume is L^3=27. Each point represents an atom or molecule within a finite crystal network. In this model, the disorder is represented by a random potential assigned to each of the points. Note that the left face of the cube is connected to 9 terminals or ports from where the electrons are beamed. The beamed electrons interact with the points in the network before being dispersed or expelled to the outside of the crystal lattice through its terminals.