The Fractal Universe, Part 1

A long time ago I blogged about the Cosmic Web and one of the comments there suggested I write something about the idea that the large-scale structure of the Universe might be some sort of fractal.  There’s a small (but vocal) group of cosmologists who favour fractal cosmological models over the more orthodox cosmology favoured by the majority, so it’s definitely something worth writing about. I have been meaning to post something about it for some time now, but it’s too big and technical a matter to cover in one item. I’ve therefore decided to start by posting a slightly edited version of a short News and Views piece I wrote about the  question in 1998. It’s very out of date on the observational side, but I thought it would be good to set the scene for later developments (mentioned in the last paragraph), which I hope to cover in future posts.

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One of the central tenets of cosmological orthodoxy is the Cosmological Principle, which states that, in a broad-brush sense, the Universe is the same in every place and in every direction. This assumption has enabled cosmologists to obtain relatively simple solutions of Einstein’s General Theory of Relativity that describe the dynamical behaviour of the Universe as a whole. These solutions, called the Friedmann models [1], form the basis of the Big Bang theory. But is the Cosmological Principle true? Not according to Francesco Sylos-Labini et al. [2], who argue, controversially, that the Universe is not uniform at all, but has a never-ending hierarchical structure in which galaxies group together in clusters which, in turn, group together in superclusters, and so on.

These claims are completely at odds with the Cosmological Principle and therefore with the Friedmann models and the entire Big Bang theory. The central thrust of the work of Sylos-Labini et al. is that the statistical methods used by cosmologists to analyse galaxy clustering data are inappropriate because they assume the property of large-scale homogeneity at the outset. If one does not wish to assume this then one must use different methods.

What they do is to assume that the Universe is better described in terms of a fractal set characterized by a fractal dimension D. In a fractal set, the mean number of neighbours of a given galaxy within a volume of radius R is proportional to RD. If galaxies are distributed uniformly then D = 3, as the number of neighbours simply depends on the volume of the sphere, i.e. as R3 and the average number-density of galaxies. A value of D < 3 indicates that the galaxies do not fill space in a homogeneous fashion: D = 1, for example, would indicate that galaxies were distributed in roughly linear structures (filaments); the mass of material distributed along a filament enclosed within a sphere grows linear with the radius of the sphere, i.e. as R1, not as its volume.  Sylos-Labini et al. argue that D = 2, which suggests a roughly planar (sheet-like) distribution of galaxies.

Most cosmologists would accept that the distribution of galaxies on relatively small scales, up to perhaps a few tens of megaparsecs (Mpc), can indeed be described in terms of a fractal model.This small-scale clustering is expected to be dominated by purely gravitational physics, and gravity has no particular length scale associated with it. But standard theory requires that the fractal dimension should approach the homogeneous value D = 3 on large enough scales. According to standard models of cosmological structure formation, this transition should occur on scales of a few hundred Mpc.

The main source of the controversy is that most available three-dimensional maps of galaxy positions are not large enough to encompass the expected transition to homogeneity. Distances must be inferred from redshifts, and it is difficult to construct these maps from redshift surveys, which require spectroscopic studies of large numbers of galaxies.

Sylos-Labini et al. have analysed a number of redshift surveys, including the largest so far available, the Las Campanas Redshift Survey [3]; see below. They find D = 2 for all the data they look at, and argue that there is no transition to homogeneity for scales up to 4,000 Mpc, way beyond the expected turnover. If this were true, it would indeed be bad news for the orthodox among us.

The survey maps the Universe out to recession velocities of 60,000 km s-1, corresponding to distances of a few hundred million parsecs. Although no fractal structure on the largest scales is apparent (there are no clear voids or concentrations on the same scale as the whole map), one statistical analysis [2] finds a fractal dimension of two in this and other surveys, for all scales – conflicting with a basic principle of cosmology.

Their results are, however, at variance with the visual appearance of the Las Campanas survey, for example, which certainly seems to display large-scale homogeneity. Objections to these claims have been lodged by Luigi Guzzo [4], for instance, who has criticized their handling of the data and has presented independent results that appear to be consistent with a transition to homogeneity. It is also true that Sylos-Labini et al. have done their cause no good by basing some conclusions on a heterogeneous compilation of redshifts called the LEDA database [5], which is not a controlled sample and so is completely unsuitable for this kind of study. Finally, it seems clear that they have substantially overestimated the effective depth of the catalogues they are using. But although their claims remain controversial, the consistency of the results obtained by Sylos-Labini et al. is impressive enough to raise doubts about the standard picture.

Mainstream cosmologists are not yet so worried as to abandon the Cosmological Principle. Most are probably quite happy to admit that there is no overwhelming direct evidence in favour of global uniformity from current three-dimensional galaxy catalogues, which are in any case relatively shallow. But this does not mean there is no evidence at all: the near-isotropy of the sky temperature of the cosmic microwave background, the uniformity of the cosmic X-ray background, and the properties of source counts are all difficult to explain unless the Universe is homogeneous on large scales [6]. Moreover, Hubble’s law itself is a consequence of large-scale homogeneity: if the Universe were inhomogeneous one would not expect to see a uniform expansion, but an irregular pattern of velocities resulting from large-scale density fluctuations.

But above all, it is the principle of Occam’s razor that guides us: in the absence of clear evidence against it, the simplest model compatible with the data is to be preferred. Several observational projects are already under way, including the Sloan Digital Sky Survey and the Anglo-Australian 2DF Galaxy Redshift Survey, that should chart the spatial distribution of galaxies in enough detail to provide an unambiguous answer to the question of large-scale cosmic uniformity. In the meantime, and in the absence of clear evidence against it, the Cosmological Principle remains an essential part of the Big Bang theory.

References

  1. Friedmann, A. Z. Phys. 10, 377–386 ( 1922).
  2. Sylos-Labini, F., Montuori, M. & Pietronero, L. Phys. Rep. 293, 61-226 .
  3. Shectman, S.et al. Astrophys. J. 470, 172–188 (1996).
  4. Guzzo, L. New Astron. 2, 517–532 ( 1997).
  5. Paturel, G. et al. in Information and Online Data in Astronomy (eds Egret, D. & Albrecht, M.) 115 (Kluwer, Dordrecht,1995).
  6. Peebles, P. J. E. Principles of Physical Cosmology (Princeton Univ. Press, NJ, 1993).
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11 Responses to “The Fractal Universe, Part 1”

  1. [...] This post was mentioned on Twitter by Mark Tibbetts, Peter Coles. Peter Coles said: The Fractal Universe, Part 1: http://wp.me/pko9D-1JB [...]

  2. Bryn Jones Says:

    Peter,

    It is usually extremely dangerous to discuss scientific results without reading the original papers, but wouldn’t D =~ 2 be expected from the Las Campanas Redshift Survey as an inevitable consequence of the survey’s strategy? I do recall that the Las Campanas survey used declination strips across the sky: that is to say they measured redshifts of galaxies lying in narrow strips on the sky (a sensible strategy in the face of limited observing time). The volume of the Universe sampled therefore consisted of rather flat wedges.

    Surely D = 2 for such a wedge once the radius R is larger than the typical thickness of the wedge? Perhaps Sylos-Labini et al. considered that in their analysis?

    (Of course, your discussion is now 12 years old and we have incomparably better results from the 2dFGRS and the SDSS.)

    Bryn.

  3. telescoper Says:

    Bryn,

    It’s a bit more complicated that the fractal dimension just reflecting the survey geometry, but it is an issue because it requires you to think about how to deal with boundary effects when trying to estimate D. In other words you have to be very careful in what you say about the fractal dimension on scales larger than the smallest dimension of the survey. This is where there’s a strong divergence between the pro-fractal groups and the more orthodox ones. As I’ll explain in a later post if I’ve got time, the effective dimension you get from newer data clearly varies with scale and tends to 3 on large scales, implying that there is a scale at which homogeneity is reached. In fact when I was at Nottingham me and my student Pan (remember him?) showed that this was true of the PSCz survey (which is all-sky so doesn’t suffer from the flatness problem).

    Peter

  4. Phillip Helbig Says:

    Wow! Another flatness problem in cosmology! :-)

  5. Bryn Jones Says:

    Peter,

    I’ve checked the strategy of the Las Campanas Redshift Survey (Shectman at al., ApJ, 470, 172, 1996), and there were more narrow strips on the sky than I remembered. There were three strips in the South Galactic Cap region, each about 1.5 deg wide and separated by about 1.5 deg. This would give a thicker overall sampled volume than I had thought, and with with a complicated geometry. This illustrates the danger of sounding off without having read the literature that I mentioned above.

    Of course, a proper analysis would require a detailed description of the survey boundaries and probably Monte Carlo simulations. And surely that has been done in the published studies.

    Your choice of the IRAS PSCz catalogue for your work with Pan was very sensible: the all-sky character lessened the issues of survey geometry (but the problems of dust absorption near the Milky Way would have imposed some residual problems). A student doing his M.Sc. dissertation on redshift surveys with me a few years ago showed interest in your work and I had thought he might want to try some simulations of his own in that subject area. However, he ran out of time and his dissertation was mostly a critical review of the recent scientific literature on redshift surveys.

    Bryn.

  6. telescoper Says:

    Bryn,

    PCSz did indeed have the advantage of being nearly all-sky, but it was also extremely sparse. Pan did a really good job on that analysis, I think. He did most of the hard work on it, actually. I just wrote the paper.

    Peter

  7. “Moreover, Hubble’s law itself is a consequence of large-scale
    homogeneity: if the Universe were inhomogeneous one would not expect
    to see a uniform expansion”

    I do not know if universe is homogeneous or fractal. However, I would add to your comment of above that precisely Baryshev and others showed that the field theory of gravity (FTG), worked by Feynman and others, can explain the Hubble law for fractal structures with D=2. Therefore the verification of the law
    cannot be taken as a verification of homogeneity.

  8. [...] that the evidence for cosmological homogeneity isn’t as compelling as most people assume. I blogged some time ago about an alternative idea, that the Universe might have structure on all scales, as would be the [...]

  9. [...] Scientist reminded me that I never completed the story I started with a couple of earlier posts (here and there), so while I wait for the rain to stop I thought I’d make myself useful by posting [...]

  10. […] blogged some time ago about that the idea that the Universe might have structure on all scales, as would be the case if it […]

  11. […] blogged some time ago about that the idea that the Universe might have structure on all scales, as would be the case if it […]

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