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How big is the universe? What is our place in it? These questions have fascinated people since the first observations of the night sky. Possible answers have been fiercely debated, occasionally at the cost of lives.
Here we are interested in a more specific question: is the universe fractal, at least over some range of distances? Because the distances to galaxies is important for this question, we review a bit of the history of how people have attempted to compute these distances.
Swedenborg, Kant, and Lambert proposed a cosmology based on hierarchies. Kant's 1755 essay, "Universal Natural History and Theory of the Heavens, or an Essay on the Constitution and Mechanical Origin of the Entire World-Edifice Treated According to Newtonian Principles," was the first serious scientific cosmological model. Kant imagined the original unorganized mass of the universe evolving into a hierarchical order through the gravitaional attraction of Newton's theory, balanced by an unnamed repulsive force to prevent everything from accreting into a single lump. Evidence of the hierarchical ordering was sparse, to be sure. In the solar system all planets orbit in the same direction, and all satellites orbit the planets in the same direction. Kant thought the Milky Way was a vast disk of stars, orbiting in the same direction. The nebulae were similar disks of stars, galaxies like our own. And so the hierarchy continued, forever in the ideas of Kant and Swedenborg, only to some finite (but large) scale in Lambert's thinking.
By contrast, Newton's cosmology was homogeneous, hence manifestly non-fractal. In the 1930s, the assumption of spatial homogeniety was joined with spatial isotropy to form the Cosmological Principle. This says on very large scales all parts of the universe are more-or-less the same, and every direction looks the same. Perhaps in part this is a reaction to centuries of believing the earth is the center of the universe. While this "we're nothing special" certainly conflicted with Newton's theology, it seems appropriate to modern world views. An additional motivation in relativistic cosmology is that Einstein's equations are much easier to solve under the assumptions of homogenietry and isotropy. Yet stars are organized into galaxies, and galaxies into cluseters of galaxies, so any homogeniety must occur at a very large scale. Einstein's cosmological model takes as hypothesis that the universe is homogeneous and isotropic in space. Homogeneity is generally regarded as essential for Hubble's law, that the velocity of recession of galaxies is proportional to their distance.
Early studies of galaxy distributions revealed clustering hierarchies. Chapters 9 and 33-35 of Benoit Mandelbrot's The Fractal Geometry of Nature present a geometical cartoon that mimicks the appearance of galaxy clusters without the underlying physics (a point evidently missed by some astronomers who wrote early reviews of this book). One cartoon uses Levy flights, the other random tremas. Both exhibit the filaments and walls of the observed distribution of galaxies, but the Levy cartoon is too lacunar, its gaps are larger than observed. The lacunarity of the random trema cartoon can be adjusted to fit observations, and the forgeries thus produced have a convincing appearance.
For over twenty years it has been generally accepted the distribution of galaxies is fractal over relatively small scales. Peebles reports on scales up to 50 million light years, the distribution of galaxies is "remarkably well approximated as a fractal with dimension d = 1.23," but beyond these distances he asserts the distribution becomes homogeneous.
Coleman applies a conditional correlation analysis and deduces a fractal structure over a much larger scale. Current evidence suggests a fractal distribution of galaxies out to 600 million light years, and there is yet no compelling evidence to believe the fractal distribution of galaxies does not extend throughout the universe. Pietronero and coworkers have deduced a dimension of 2 for the galaxy distribution up to 600 million light years.
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In the 1920s, De Vaucouleurs proposed the denisty of matter in the universe obeys a power law relation: denoting by D(r) the density of matter a distance r from the observer, De Vaucouleurs' law is
D(r) = k*r3-d
where d is the dimension of the distribution of matter. A reassuring consequence of this relationship is that there are no preferred observers. "We're nothing special" can be achieved by hierarchical, a well as homogeneous, cosmologies.
Schulman and Seiden proposed a model of galaxy formation in which the birth of
one galaxy stimulates the birth of nearby galaxies. If this process occurs near the
percolation threshold, a fractal distribution with
Summarizing:
| * There is general agreement about the existence of fractal structures out to about 50 million light years. |
| * Recent evidence suggests there is no upper cut-off of the fractal structure, and that the large-scale distribution of mass in the universe has d = 2. Both the absence of an upper cut-off and the value of the dimension are controversial. |
| * Isotropic fractal distributions are compatible with the equivalence of all observers. |
| * The Standard Cosmological Principle, that matter is distributed in a homogeneous and isotropic fashion can be replaced by the Conditional Cosmological Principle, that matter is distributed in a hierarchical and isotropic fashion. |
| * Observationally, hierarchical distribution implies long-range departures from Hubble's Law. |
However, the distance scales to the galaxies is difficult to determine. Twenty-five years ago, concerning the distance scale to the farthest galaxies one of my astronomer teachers remarked,"If you get the right order of magnitude in the exponent, you're doing very well." In the quarter-century that has passed since then, observations have improved significantly. But still, there is much uncertainty. The range of fractality in the universe is not likely to be resolved soon to everybody's satisfaction.
