Fractal cosmology

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In physical cosmology, fractal cosmology is a set of minority cosmological theories which state that the distribution of matter in the Universe, or the structure of the universe itself, is a fractal across a wide range of scales (see also: multifractal system). 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.

Fractals in observational cosmology[edit]

The first attempt to model the distribution of galaxies with a fractal pattern was made by Luciano Pietronero and his team in 1987,[1] and a more detailed view of the universe’s large-scale structure emerged over the following decade, as the number of cataloged galaxies grew larger. Pietronero argues that the universe shows a definite fractal aspect over a fairly wide range of scale, with a fractal dimension of about 2.[2] The fractal dimension of a homogeneous 3D object would be 3, and 2 for a homogeneous surface, whilst the fractal dimension for a fractal surface is between 2 and 3. The ultimate significance of this result is not immediately apparent, but it seems to indicate that both randomness and hierarchal structuring are at work on the scale of galaxy clusters and larger.

The universe has been observed to be homogeneous and isotropic (i.e. is smoothly distributed) at very large scales, as is expected in a standard Big Bang or FLRW cosmology, and in most interpretations of the Lambda-Cold Dark Matter model. The scientific consensus interpretation is that the Sloan Digital Sky Survey (SDSS) suggests that things do indeed smooth out above 100 Megaparsecs.

One study of the SDSS data in 2004 found "The power spectrum is not well-characterized by a single power law but unambiguously shows curvature...thereby driving yet another nail into the coffin of the fractal universe hypothesis and any other models predicting a power-law power spectrum".[3] Another analysis of luminous red galaxies (LRGs) in the SDSS data calculated the fractal dimension of galaxy distribution (on a scales from 70 to 100 Mpc/h) at 3, consistent with homogeneity; but that the fractal dimension is 2 "out to roughly 20 Mpc/h".[4] In a paper of 2008 entitled "Parabolic drift towards homogeneity in large-scale structures of galaxies" (Physica, A, 387: 3641-3646) D. Queiros-Conde showed that large-scale structure of galaxies are much better described by a scale-dependent fractal dimension drifting from zero to three at a scale around 55 Mpc/h in the context of a "scale-entropy diffusion equation". It gives a way to estimate the number of galaxies in the universe in agreement with Hubble measurements. Moreover, fractal dimension is varying linearly with scale-logarithm. This means that the geometry of galaxy distribution is a "parabolic fractal". Two years later, the same author with M. Feidt showed that the fractal dimension 2 found in numerous studies can be explained as being a problem of measurement. In 2012, Scrimgeour et al. definitively showed that large-scale structure of galaxies was homogeneous beyond a scale around 70 Mpc/h, close to the value found by D. Queiros-Conde.[5]

In 2013, astronomers discovered a large quasar group (LQG) that is 1.6 billion light-years in diameter, far larger than allowed by the cosmological principle, which asserts that the universe should be homogeneous at scales this large.[6]

Fractals in theoretical cosmology[edit]

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"[7] 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"[8] 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[9] and Quantum Einstein gravity[10] 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 2D at the Planck scale, and that spacetime gradually becomes 4D at larger scales. French astronomer Laurent Nottale first suggested the fractal nature of spacetime in a paper on Scale Relativity published in 1992,[11] and published a book on the subject of Fractal Space-Time in 1993.[12]

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[13] 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[14] suggests that time is an emergent property or arises naturally, in this formulation, whereas in Causal dynamical triangulation,[9] 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[15] 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 hierarchical 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?"[16] 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 W. 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.[citation needed]

This was a follow-up of an earlier article in that same publication on August 21 of 1999, by Marcus Chown, entitled "Fractal Universe.".[17] Back in November 1994, Scientific American featured an article on its cover written by physicist Andrei Linde, entitled "The Self-Reproducing Inflationary Universe"[18] 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,[19] written by the three scientists who propounded the theory, which again suggests that the universe may have the characteristics of a fractal.

See also[edit]


  1. ^ Pietronero, L. (1987). "The Fractal Structure of the Universe: Correlations of Galaxies and Clusters". Physica A. 144 (144): 257–284. Bibcode:1987PhyA..144..257P. doi:10.1016/0378-4371(87)90191-9.
  2. ^ Joyce, M.; Labini, F.S.; Gabrielli, A.; Montouri, M.; Pietronero, L. (2005). "Basic Properties of Galaxy Clustering in the light of recent results from the Sloan Digital Sky Survey". Astronomy and Astrophysics. 443 (11): 11–16. arXiv:astro-ph/0501583. Bibcode:2005A&A...443...11J. doi:10.1051/0004-6361:20053658.
  3. ^ Tegmark; et al. (10 May 2004). "The Three-Dimensional Power Spectrum of Galaxies from the Sloan Digital Sky Survey". The Astrophysical Journal. 606 (2): 702–740. arXiv:astro-ph/0310725. Bibcode:2004ApJ...606..702T. doi:10.1086/382125.
  4. ^ Hogg, David W.; Eisenstein, Daniel J.; Blanton, Michael R.; Bahcall, Neta A.; Brinkmann, J.; Gunn, James E.; Schneider, Donald P. (2005). "Cosmic homogeneity demonstrated with luminous red galaxies". The Astrophysical Journal. 624 (1): 54–58. arXiv:astro-ph/0411197. Bibcode:2005ApJ...624...54H. doi:10.1086/429084.
  5. ^ Scrimgeour, M.; et al. (September 2012). "The WiggleZ Dark Energy Survey: the transition to large-scale cosmic homogeneity". Mon. Not. R. Astron. Soc. 425 (1): 116–134. arXiv:1205.6812. Bibcode:2012MNRAS.425..116S. doi:10.1111/j.1365-2966.2012.21402.x.
  6. ^ "Largest Structure in Universe Discovered".
  7. ^ Linde, A.D. (August 1986). "Eternally Existing Self-Reproducing Chaotic Inflationary Universe". Physica Scripta. 15: 169–175. Bibcode:1987PhST...15..169L. doi:10.1088/0031-8949/1987/T15/024.
  8. ^ Guth, Alan (22 June 2007). "Eternal inflation and its implications". J. Phys. A: Math. Theor. 40 (25): 6811–6826. arXiv:hep-th/0702178. Bibcode:2007JPhA...40.6811G. doi:10.1088/1751-8113/40/25/S25.
  9. ^ a b Ambjorn, J.; Jurkiewicz, J.; Loll, R. (2005). "Reconstructing the Universe". Phys. Rev. D. 72 (6): 064014. arXiv:hep-th/0505154. Bibcode:2005PhRvD..72f4014A. doi:10.1103/PhysRevD.72.064014.
  10. ^ Lauscher, O.; Reuter, M. (2005). "Asymptotic Safety in Quantum Einstein Gravity": 11260. arXiv:hep-th/0511260.
  11. ^ Nottale, Laurent (1992). "The theory of Scale Relativity". International Journal of Modern Physics A. 7 (20): 4899–4936. Bibcode:1992IJMPA...7.4899N. CiteSeerX doi:10.1142/S0217751X92002222.
  12. ^ Nottale, Laurent (1993). Fractal Space-time and Microphysics. World Scientific Press.
  13. ^ Hellemans, Alexander — The Geometer of Particle Physics — Scientific American - August, 2006
  14. ^ Connes, A.; Rovelli, C. (1994). "Von Neumann Algebra Automorphisms and Time-Thermodynamics Relation". Class. Quantum Grav. 11 (12): 2899–2918. arXiv:gr-qc/9406019. Bibcode:1994CQGra..11.2899C. doi:10.1088/0264-9381/11/12/007.
  15. ^ Baryshev, Y. and Teerikorpi, P. - Discovery of Cosmic Fractals - World Scientific Press (2002)
  16. ^ Gefter, Amanda - Is the Universe a Fractal? - New Scientist - March 10, 2007: issue 2594
  17. ^ Chown, Marcus - Fractal Universe - New Scientist - August 21, 1999
  18. ^ Linde, Andrei - The Self-Reproducing Inflationary Universe - Scientific American - November 1994 pp. 48-55
  19. ^ Ambjorn, J.; Jurkiewicz, J.; Loll, R. - The Self-Organizing Quantum Universe - Scientific American - July 2008 pp. 42-49


  • Rassem,M. and Ahmed E., "On Fractal Cosmology", Astro. Phys. Lett. Commun. (1996), 35, 311.

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